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LIBRARY
Michigan State
University

 

 

 

This is to certify that the
dissertation entitled

TRANSITION RATES BEYOND “Ca:
EFFECTIVE CHARGE IN THE pf SHELL

presented by

JONATHAN MICHAEL COOK

has been accepted towards fulﬁllment
of the requirements for the

PhD. degree in Physics and Astronomy

 

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Date

 

MSU is an Aﬁinnative Action/Equal Opportunity Employer

 

PLACE IN RETURN BOX to remove this checkout from your record.
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TRANSITION RATES BEYOND 480a:
EFFECTIVE CHARGE IN THE pf SHELL

By
Jonathan Michael Cook

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Physics and Astronomy

2009

ABSTRACT

TRANSITION RATES BEYOND 480a:
EFFECTIVE CHARGE IN THE pf SHELL

By
J Onathan Michael Cook

The well-known magic numbers Of stable nuclei can vanish and new ones appear
in the neutron-rich nuclei far from stability. The GXPFI effective interaction
predicts the appearance of an N = 32 subshell gap and an N = 34 Shell closure in
Ti and Ca. The investigation Of these Shell gaps developed into a question of the
appropriate effective Charge in the pf shell. The trend Of the measured transition
matrix elements in the neutron-rich Ti isotopes is reproduced with effective charges
derived from an analysis of isospin analogue states in 51 Fe and 51Mn, cap a: 1.15 and
en 2 0.8, that differ from the standard effective charges, ep = 1.5 and en = 0.5.
The proton and neutron effective Charges are mixed in the case of Ti, and to
separate the two components the neutron effective charge is determined via the
intermediate-energy Coulomb excitation of 500a. To reduce the uncertainty in
the measurement a model is developed to simulate in-beam response functions for
a position sensitive NaI(Tl) 7-ray detector. The effective Charge is found to be
en = 0.77(13), indicating that the enhanced neutron effective Charge derived in

the upper pf shell is applicable in the lower pf shell.

ACKNOWLEDGMENT

I would like to express my thanks to my advisor, Thomas Glasmacher, and to
Alexandra Gade for the large amount of time she Spent guiding me during the
creation of this dissertation. I entered graduate school with a great group of
guys in my class: Matthew Amthor, Ivan Brida, Andy Rogers, Terrence Strother,
and Wes Hitt. I have gained much from knowing my research group and others:
Ania Kwiatkowski, Przemek Adrich, Matt Bowen, Chris Campbell, Christian Di-
get, Cristian Dinca, Andrew Ratkiewicz, Kyle Siwek, Russ Terry, Dirk Weisshaar,
Heather Zwahlen. Finally, this dissertation was made possible by the outstanding

staff of the National Superconducting Cyclotron Laboratory.

iii

TABLE OF CONTENTS

List of Tables .................................................. vi
List of Figures ................................................. vii
Introduction ................................................... 1
The nuclear shell model and transition strength measurements . 5
2.1 The beginnings ............................. 5
2.2 The nuclear landscape ......................... 6
2.3 The nuclear shell model ........................ 8
2.3.1 Form Of the Shell model .................... 8
2.3.2 Effective interactions ...................... 10
2.3.3 Effective charge ......................... 12
2.3.4 The pf shell ........................... 17
2.4 'D'ansition-rate measurement techniques ............... 21
2.4.1 Transition rates ......................... 21
2.4.2 Intermediate-energy Coulomb excitation ........... 26
2.4.3 The accuracy of intermediate—energy Coulomb excitation ex-
periments ............................ 28
Experimental apparatus ........................................ 32
3.1 Isotope production ........................... 32
3.2 The 8800 Particle Spectrograph and the APEX NaI(Tl) scintillator
Array .................................. 34
3.2.1 The 8800 Particle Spectrograph ................ 35
3.3 The APEX NaI(Tl) Scintillator Array ................. 37
3.3.1 The detector ........................... 37
3.3.2 Electronics ............................ 42
3.3.3 Calibrations ........................... 45
3.3.4 Efficiency ............................ 46
Simulation ..................................................... 51
4.1 Detector Geometry ........................... 52
4.2 7-ray generator ............................. 52
4.3 Interaction physics ........................... 56
4.4 Detector and electronics modeling ................... 61

iv

 

5 Data analysis and experimental results ......................... 69

5.1 52Ti B(E2;0f’ ——> 2?) measurement .................. 70
5.1.1 Particle identiﬁcation ...................... 71
5.1.2 'y-ray spectrum ......................... 75
5.1.3 Cross section .......................... 79
5.1.4 In-beam efficiency ........................ 82

5.2 50Ca B(E2; 01f —» 2?) measurement .................. 84
5.2.1 Particle identiﬁcation ...................... 84
5.2.2 Cross section and B(E2;01+ —+ 21*) .............. 85

5.3 Effective charge in the pf shell ..................... 89

5.4 Conclusion ................................ 91

A Simulated response functions of APEX ......................... 93

A.1 GEANT4 ................................. 93
A.1.1 Input ﬁles ............................ 93
A.1.2 Detector construction ...................... 99
A.1.3 'y—ray generator ......................... 122
A14 Angular distribution ...................... 131
A15 Event action ........................... 135
A16 Double-Sided scintillator model ................ 141

Bibliography ................................................... 155

 

 

 

 

2.1

2.2

3.1

5.1

5.2

LIST OF TABLES

The properties of 50Ca ..........................

The dominant ’y-ray multipolarities for a U, — J fl transition with
parity change Arr .............................

The resolution of APEX compared to that typical of a 3”x3” NaI(Tl)
crystal ...................................

The cross section and B(E2;01+ —> 2?) of 52Ti used to determine
the in-beam efficiency of APEX .....................

The measured 50Ca cross section leading to the ﬁnal transition rate
value. ..................................

vi

20

23

43

84

1.1

1.2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3.1

LIST OF FIGURES

The measured energy levels of neutron-rich Ti isotopes compared tO
shell-model calculations. ........................ 3

The measured transition strengths of neutron—rich Ti isotopes com—
pared to Shell-model calculations with the standard and suggested

effective charges. ............................ 4
The Chart of the N uclideS ........................ 7
Diagram of the energy levels of the Shell model. ........... 11

The spin arrangement of the T2 = 0 and T2 = 1 two-nucleon states. 15

The systematics of the ﬁrst excited states in even-even, neutron-rich
pf-Shell nuclei. ............................. 18

Experimental B(E2;0'1f —> 2?) transition rates for neutron-rich Ti
isotopes compared to shell-model calculations with the standard and
modiﬁed effective Charges. ....................... 19

The neutron effective charges derived from the Ca isotopes ...... 21

Schematic of Coulomb excitation of a nucleus from an initial state
Ii) to a ﬁnal bound state If) and the ensuing 'y decay ......... 25

Schematic of a projectile nucleus scattering in the electromagnetic
ﬁeld of an inﬁnitely heavy target nucleus ................ 27

The percent differences between adopted and measured B (E2; 0;" —-> 2?)
transition rates for published test cases in intermediate-energy Coulomb

excitation measurements ......................... 31

The Coupled Cyclotron Facility ..................... 33

vii

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

3.10

3.11

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

The S800 Particle Spectrograph ..................... 37

The laboratory-frame energy of a 1 MeV 7 ray produced in a frame

moving at B = 0.3. ........................... 38
A photo of the APEX Array ....................... 39
The scintillation process ......................... 40

The energy spectrum of an 88Y 7-ray source produced by APEX. . 42
A schematic Of the APEX electronics .................. 44

The ﬁt of the 898 keV y—ray energy peak in one Slice Of detector 2
for the energy calibration. ....................... 46

The position response of APEX for an 88Y 'y-ray source. ...... 48

A simulated position-energy matrix for a double—Sided scintillator
bar with a high-resolution Signal input into the discriminator. . . . 49

A Simulated position-energy matrix of a double—Sided scintillator bar
with a low-resolution signal input into the discriminator. ...... 50

The APEX detector geometry constructed in GEANT4. ....... 53

An isotropic angular distribution (solid line) is compared to the E2
distribution produced by the Coulomb excitation of 52Ti ....... 55

The simulated *y-ray emission position along the beam line ...... 56
A kinematic diagram of a ”y ray Compton scattering on an electron. 59
The idealized detector response to 2 MeV ’y rays. .......... 60

The relationship between the angle of incidence «p to the crystal

surface and the angle Of emission 0 from the scintillation point. . . . 62
The position response of APEX for an 88Y y-ray source. ...... 65
The energy response of APEX for an 8BY ”y-ray source. ....... 66

The Coulomb excitation energy spectrum of 76Ge ........... 67

viii

4.10

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9

5.10

5.11

5.12

5.13

5.14

The position response of the 76Ge photopeak. ............ 68
The particle identiﬁcation Spectrum with 52Ti indicated. ...... 72
The correlation between the position at CRDCI in the S800 focal

plane and the time of ﬂight. ...................... 73

The correlation between the angle in the S800 focal plane and the

time of flight. .............................. 74
A histogram showing the dependence Of the ion chamber on the

dispersive position. ........................... 75
The corrected particle identiﬁcation spectrum with 52Ti indicated . 76
The time Spectrum of a Single APEX PMT. ............. 78
The scattering angle of 52Ti nuclei with the maximum and minimum

scattering angles indicated ........................ 79
The y-ray energy Spectrum for 52Ti. ................. 80

Fit Of the response function plus continuum background to the 52Ti
’y-ray spectrum. ............................. 83

The particle identiﬁcation spectrum with 50Ca indicated ....... 85
The corrected particle identiﬁcation spectrum with 50Ca indicated. 86
The lower-lying states of 50Ca ...................... 88
The difference in the angular distribution of 50Ca from that of 52Ti. 89

The ﬁt of the response function plus continuum to the 50Ca 'y—ray
energy spectrum. ............................ 90

Images in this thesis/ dissertation are presented in color.

ix

Chapter 1

Introduction

The well-investigated energy-level spacing of stable nuclei may be altered in exotic
nuclei with the magic numbers Of the stable isotopes disappearing and new magic
numbers arising. The CXPFI interaction was developed as a uniﬁed effective
interaction for the pf shell, and a N = 32 subshell gap and a N = 34 shell closure
were predicted to appear for neutron-rich Ti and Ca isotopes. The measurement
of the energy levels of the neutron-rich Ti isotopes conﬁrmed the N = 32 subshell
Closure but did not ﬁnd evidence for the N = 34 shell closure. The energy-level
measurement was used to reﬁne the GXPFl into the GXPFlA effective interaction.
The measurement of the transition rates of the neutron-rich Ti isotopes provides
additional evidence for the N = 32 subshell closure but again a N = 34 shell
Closure is not apparent. The GXPFIA interaction fails to reproduce the transition
rate systematics with the standard effective charges. New effective Charges for the
pf Shel] were suggested from a measurement of transition rates in isospin analogue
states in 51Fe and 51Mn. The suggested effective charges reproduce the trend of
the transition rates but not the amplitudes. A measure of the effective charge for

nucleons in similar orbits to those of the neutron-rich Ti isotopes is proposed.

The GXPFI interaction was developed for shell-model calculations in the full

pf shell, which permits the investigation Of excitations across the N and Z = 28
shell gap[1]. By ﬁtting the model parameters to emperical binding energies and
energy levels, calculations with the GXPFl interaction reproduces experimental
excitation energies of the ﬁrst excitated states E (2?) in the Ca and Ti and pre-
dicted an N = 32 subshell gap and an N = 34 shell closure. The energy gap arises
from an attractive proton-neutron interaction, with the uf5/2 orbit increasing in
energy as the occupation of the 7r f7 /2 orbit decreases to create a gap between the
Vp1/2 and the uf5/2 orbits. The experimental energy levels of the neutron-rich Ti
isotopas measured via ,6 decay and deep inelastic reactionsl2, 3, 4, 5] are compared
to shell-model calculations in Figure 1.1. The high E (2?) at N = 32, near that
of the well-known N = 28 magic number, indicates a subshell closure. The ex-
perimental E (2?) for N = 34 lies 300 keV lower than the energy calculated with
the GXPFl interaction. The GXPFl interaction was developed using data from
mostly stable nuclei, and the energy levels of the neutron-rich Ti isotopes were
included to develop a modiﬁed effective interaction refered to as GXPF1A[6| that
places the E (2?) within 23 keV Of the measured value. The difference between the
two interactions lies primarily in the [21/2 and f5/2 matrix elements. The E(2if)
of 56Ti Sits near those of the non-magic N = 28, 30 nuclei, and no evidence is pro-
vided for a shell closure at N = 34. Additionally, a two—proton knockout reaction
was used to ﬁnd E(2fL) = 2562 keV for 52Ca; the high excitation energy relative

to the nearby non-magic nuclei, z 1 MeV, suggests an N = 32 subshell Closure[7].

To further examine the predicted shell closures, the transition strengths Of the
neutron-rich Ti isotopes were measured (Figure 1.2a). The transition strength at
N = 32 approaches that of the the magic N = 28, conﬁrming a subshell gap,
and the transition strength at N = 34 is sufﬁciently increased that no evidence
is provided for a shell closure. The shell-model calculation using the standard

effective charges, ep = 1.5 and en = 0.5, to parameterize the averaged effects

26 28 3'0 3'2 34N

 

 

 

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$1600 1 ,_ __

> - .l.

2’ f -- :

a - '

m 800

f; 0 Experiment

% " --- GXPF1
— GXPF1A

 

 

 

 

 

048T. 50%. 52TI “TI 56Ti

Figure 1.1: The measured energy levels of neutron-rich Ti isotopes compared to
shell-model calculations with the GXPFI and GXPF1A interactions.

of states outside Of the truncated shell-model space fails to reproduce the data.
Using the effective charges ep z 1.15 and en z 0.8 derived in a study of T; 2
ii A = 51 states, the trend is reproduced although the amplitudes are not
(Figure 1.2b). The structure in the calculated transition strengths is due to the
enhanced neutron effective charge, suggesting that a measurement Of the neutron
effective charge would help to illuminate the underlying nuclear structure. The
GXPF1A interaction utilizes a 40Ca core, and 50Ca is conﬁgured with 10 valence
neutrons, two of which lie above the N = 28 Shell gap. The goal of this thesis
is the measurement of the transition strength of 50Ca to determine the neutron

effective charge near the proposed N = 32,34 shell gaps.

 

 

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N = 0,5 ..... N = 0.8 “.-

5 e" — GXPF1A a e" — GXPF1A

m 0 . L m 0 .

 

 

 

 

 

48'“ 5011 52'” 54'“ 56-” 48'“ 50'” 52“ 54'” 5611
Figure 1.2: The measured transition strengths of neutron-rich Ti isotopes com-
pared to shell-model calculations with (a) the standard effective charges, 6,, = 1.0
and en = 0.5, and (b) the suggested effective charges, (31) z 1.15 and en z 0.8.

Chapter 2

The nuclear Shell model and

transition strength measurements

2. 1 The beginnings

The ﬁeld Of nuclear physics was born with Henri Becquerel’s discovery of radioac-
tivity. Following work by Ernest Rutherford and Pierre and Marie Curie, James
Chadwick discovered the neutron in 1932|8]. Now certain that the nucleus con—
tained positively charged protons and chargeless neutrons, physicists began to
probe the internal structure of the nucleus. Subsequent discoveries showed that
the chemical properties of the nucleus depended on the number of protons, and
the nucleus can contain varying number of neutrons to form isotopes. The protons
and neutrons, collectively referred to as nucleons, were found to interact through
three forces: the weak nuclear force, the strong nuclear force, and the Coulomb
force. The weak nuclear force is the means by which neutrons become protons
and vice versa. The strong nuclear force is produced by the short-range attractive
potential of each nucleon. The Coulomb force causes a repulsion between protons.

In contrast to the easily-studied and well-known Coulomb force, the strong nuclear

force, taking place on the femtometer scale of the nucleus, is not well understood.
Theoretical models constructed to explain and to predict the interactions within
the nucleus are broadly classiﬁed into two groups: collective and Single-particle
models.

Collective models picture the nucleus as a group Of nucleons in concerted action.
Vibrations in the shape of the nuclear matter effect energy levels characteristic of
the harmonic oscillator, and rotation of a static deformation produces the energy
levels of a rotor. Collective models have had great success examining the structure
of the nucleus as a whole at the cost of losing Sight of individual nucleons.

In contrast, Single-particle models approach the nucleus as a group of individual
nucleons. The interactions between the nucleons determines the attributes of the
nucleus; for example, an excitation Of the nucleus involves the movement Of one
or more nucleons from one speciﬁc conﬁguration to another. The single-particle
models elucidate the fundamental interactions governing the nucleus but become
increasingly cumbersome as the number Of nucleons increases. The advancement of
single-particle models into heavier nuclei remains at the forefront of nuclear physics
research as we continue to better understand the more exotic nuclei produced by

continually improving facilities.

2.2 The nuclear landscape

In analogy to the chemical table of elements, the isotopes can be arranged to form
the Chart Of the Nuclides, shown in Figure 2.1. The nuclear neutron number N
increases moving to the right of the chart, and the proton number Z increases
moving upward. Each block on the chart is an isotope Of mass number A = N + Z.
The isotopes in black are stable—they do not spontaneously decay into another

isotope. The stable isotopes form a “valley of stability,” which follows N = Z for

126

82

 

Proton Number 2
8

 

 

 

28
20
8 .
2 .
2 8 20 28 so 82 126 184
Neutron Number N

Figure 2.1: The Chart of the N uclides with neutron number increasing to the right
and proton number increasing towards the top. The stable nuclei are drawn in
black, the known unstable nuclei in dark grey, and predicted nuclei in light grey.
The magic numbers are indicated on the axes.

lighter isotopes and veers toward a neutron excess for higher mass nuclei. The
known unstable nuclei are shown in dark grey and predicted nuclei in light grey.
The unstable isotopes undergo spontaneous B decay or a emission, transforming
them into another isotope. The right edge of the nuclei shown in the chart is the
neutron drip line—the point beyond which a neutron added to the nucleus will not

form a bound system. Similarly, the proton drip line is found on the left edge.

Many systematic features of nuclei can be seen on this chart. The proton
and neutron numbers labeled on the axes are called magic numbers. These magic
numbers, 2, 8, 20, 28, 50, 82, and 126, correspond to particularly stable nuclear
conﬁgurations as indicated by a high ﬁrst excited state and a low probability of
transition to that state. Several other physical observables indicate the existence

of the magic numbers. Nuclear binding energies, for example, display major dis-

 

continuities at the magic numbers, as do neutron and proton separation energies.

The nuclear magic numbers are analogous to features found in the atom. In
particular, the neutron and proton separation energies corresponds to the atomic
ionization energy. The ionization energies have a discontinuity at the noble gases,
Z = 2, 10, 18, 36, 54, and 86 elements that are particularly stable and that for
a long time were thought inert. The electron conﬁgurations of the noble gases
have their electrons placed in closed shells, i.e. one conﬁguration of electrons
has been ﬁlled and the next electron must be added to a conﬁguration with a
different angular momentum or radius. Between the major shell Closures with
correspondingly major changes in properties there may be smaller subshells with
smaller changes in properties, such as may be seen between the Group 2 and Group
3 elements. The recognition of the similarity between the nuclear and the atomic
properties led to the development of a theoretical model of the nucleus inspired by

shell structure.

2.3 The nuclear shell model

2.3.1 Form of the Shell model

The investigation of the Shell-like effects of the nucleus culminated in the develop—
ment of the nuclear shell model by Maria Goeppert-Mayer and J. Hans D. Jensen,
for which they were awarded the 1963 Nobel Prize in Physics[9]. The shell model
is a Single-particle model consisting of a central potential Vc and a spin-orbit po-
tential V1.3. The natural appearance of shell structure can be seen even in the
simple case Of a square well potential with N, Z >> 1 non-interacting nucleons,
where ﬁlling the square well according to Fermi statistics results in nucleons lying
in successively higher energy levels. The last level ﬁlled in this Fermi gas model

corresponds to an energy and momentum referred to as the Fermi surface. The

Fermi gas model, however, does not present a means of calculating wavefunctions

that a more sophisticated model is capable of providing.

The shell model is based on the Simpliﬁcation that each nucleon moves in a
potential that represents the sum effect of the interactions with the remaining
A — 1 nucleons. The strong nuclear force is known to have a short range, and the
central potential is correspondingly expressed over a ﬁnite distance. Nonetheless,
for computational Simplicity the central potential is commonly represented as a
harmonic oscillator potential, which despite extending to inﬁnity is often sufficient.

The harmonic oscillator potential is
Vc(r) = —mw2r2 (2.1)

where m is mass and w is chosen to reproduce the nuclear mean square radius,
hw z 40A—1/3 MeV. The harmonic oscillator potential has quantum numbers nm,

ny, nz, and ms with energy levels lying at

E = (n+%)hw. (2.3)

The m3 quantum number is the spin orientation of the nucleons, which are spin-1/2
fermions and can therefore take the values Of 21:23. Employing the Pauli Exclusion
Principle, which states that two identical fermions may not occupy the same quan-
tum state simultaneously, it can be seen that the magic numbers of the harmonic
oscillator occur at "tot = 1 with occupancy 2, "tot = 2 with total occupancy 8,
and so on with shell closures at 20, 40, 70, and 112. While the initial nuclear magic

numbers Of the nucleus are reproduced, the higher lying ones are not.

Mayer’s and Jensens’ breakthrough was the realization that a strong spin-orbit

potential leads to the correct magic numbers. The spin-orbit potential V1.3 rep-
resents the splitting of energy levels of total angular momentum j' = f+ 8' and
originates in the exchange of vector mesonsllO]. The observed Shell closings are
obtained when the Sign of the potential lowers the j = l + 1/2 levels. Altogether,

the central potential takes the form
V(r) = Vc(r) — W3(r)l - s. (2.4)

This potential reproduces the nuclear magic numbers Of the single-particle Shell
structure diagramed in Figure 2.2. The states are labeled using nlj notation, with
n as the radial quantum number that increments for each repetition of l. The
spin-orbit potential can be seen in the Splitting of the j degeneracy so that 1p3/2
lies lower than the 1121/2 and in the formation of large energy gaps as occupancy
reaches the magic numbers. The Pauli Exclusion Principle gives rise to a 23' + 1
occupancy for each Shell. The Shells created between the magic number occupancies
are termed the p shell for the 1133/2 and 1121/2 states, the sd Shell for the 1d5/2
through the 1d3/2 states, and pf Shell for the 1 f7 /2 through the 1f5/2 states.

2.3.2 Effective interactions

The actual interaction between nucleons is considerably more complex than repre-
sented thus far. To approximate the true nuclear potential for a single nucleon, a
Hamiltonian is formed from a mean potential and the two-body interaction. The
mean potential is chosen such that the Off—diagonal matrix elements of the Hamil-
tonian are minimized and provides a means of calculating the single-particle states.
Diagonalizing the full Hamiltonian determines a basis in which the Hamiltonian is
solved. The resulting basis states consist of mixtures of the Single-particle states,

a Situation referred to as conﬁguration mixing. In this manner, a large part of the

10

 

 

“ "111/2 12
, 2d3/2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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2d <’__'_,”;g7/2 8
19 (I ’ - $2 6
\
\
‘
\
\
~
\
\
‘199/2 10
2 - _ _ I ’
p “;...f‘~291/2 2
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‘117/2 8
,-1d
28 _ _ ”K , 3/2 4
1d ”’ f f ' 251/2 2
fffff 1d5/2 6
______ 191/2 2
1p ____
‘ ‘ ‘193/2 4
1s ------- 131/2 2
nl nlj 2j+1

Figure 2.2: Diagram of the energy levels of the shell model. Shown on the left are
the energy levels Of an isotropic harmonic oscillator potential. On the right are the
single—particle levels of the harmonic oscillator with a spin—orbit interaction. It is
the radial quantum number that increases with the orbital angular momentum l.
j = l :l: s is the total angular momentum, and each orbital can be occupied by a
maximum of 23' -l- 1 neutrons or protons. Large shell gaps lead to the occurrence
Of magic numbers as indicated.

11

nucleon-nucleon interaction has been replaced by a mean ﬁeld, and the residual
interaction remaining may be reduced to the point that the inﬁnite-dimensional
Hamiltonian can be solved in a small subset of the original space. This procedure
of determining the effective interaction in a shell-model subspace is referred to as
renormalization.

There are natural subsets of states over which the Hamiltonian is renormalized.
Since low-energy states are Of interest, the excitation of only nucleons just below
the Fermi surface is considered. These valence nucleons lie above a large number of
states with very low likelihood of excitation, which can be considered an inert core.
The states lying far above the Fermi surface have similarly very little contribution
to the excitation and may be ignored. The full shell model has then been reduced
to no more than a few states containing the valence nucleons and their low-lying
excited states. A nearby doubly-magic nucleus is typically used for the core. With
a chosen core and valence space, an effective interaction can be developed from
ﬁts to empirical data to include the averaged effects of the interaction between va-
lence nucleons and nucleons outside of the model space. Two effective interactions
accepted as standard are the Cohen-Kurath in the valence p shell and the USD in
the valence sd Shell[11, 12].

2.3.3 Effective charge

The Cohen-Kurath and USD interactions have had considerable success in model-
ing the features of relevant nuclei. Theory agrees satisfactorily with experiment on
energy levels, beta-decay, and magnetic moments. In addition, the off-diagonal ma-
trix elements reproduce the magnetic dipole transition rates, providing additional
conﬁrmation Of the wavefunctions. However, the wavefunctions derived from the
effective interactions fail to produce correct electric quadrupole transition rates.

The failure can be seen clearly in the case of 17O, which has a 160 core with a

12

valence neutron excited into the 231/2 state. The electric quadrupole operator is
not expected to act strongly on a chargeless neutron, and the predicted lifetime
for decay to the 1d5/2 orbit is long. Empirically, the lifetime is quite short, as if

the excited nucleon were a proton.

This discrepancy may be resolved by considering that when the valence neutron
moves through the core it must disturb the inert core nucleons. The disturbance
leads to a polarization of the core protons, which is the underlying mechanism
causing the electric quadrupole transition of a neutron. The polarization of the
core protons may be parameterized in the original scheme of a core plus a valence
nucleon by adding an effective Charge to the neutron, en = (In, in units Of elemen-
tary charge. Equivalently, valence protons carry an effective charge 6,, = (1 + 61,).
The factors (in and 5,, are referred to as the polarization charges and, at ﬁrst glance,
are expected to be equal due to the charge independence of the strong force. In the
effective charge scenario, level schemes, beta-decay, and magnetic moments (heavy
nuclei may require an analogous effective 9 factor) are computed with the derived
wavefunctions; electric quadrupole matrix elements are calculated with the addi—
tion of polarization charges. Since accurate wavefunctions are prerequisite to the
derivation Of the effective charge, the polarization charges were ﬁrst determined
from simple conﬁgurations, such as one or two nucleons outside of the 160 core.
Experimentally, it has been found that 6p 2 6n = 0.5 is adequate to reproduce elec-
tric quadrupole transition rates for these nuclei[13], and this value of polarization

charge has come to be accepted as the standard value.

The electric quadrupole, or E2, transition rates are typically expressed in terms
of the reduced transition rate B(E2). The B(E2;OfL —+ 2?) for an even-even

nucleus is related to theory and effective charge by

13

 

where An and Ap are the neutron and proton transition matrix elements. The
scaling Of the effective charge on the transition rate is apparent. The B(E2) is
discussed further in Section 2.4.1.

In examining the mechanism underlying the initial assumption Of the equiva—
lence of the proton and neutron polarization charge, it is useful to introduce isospin.
The formulation of isospin neglects the Coulomb force and treats the proton and
neutron as a single type Of particle, the nucleon. Isospin algebra is that of a spinor,

and the nucleon has two states,

tm = +1 /2 (2.6)

where tz is the 2 component Of the isospin f: A nucleus is then characterized by a

total isospin and 2 component of

T’ = (2.7)

EME
.srl

2
I
N

T2 =

 

For the simplest case Of a two—nucleon interaction, the 72-12 system has the prop-
erties Of T; = +% + % = +1 and T = 1. Similarly, for the p—p system T; = —1
and T = 1. The p—n system has the properties of T2 = 0 and T = 1 or 0. The sig-
niﬁcance of the result can be visualized by considering the two-nucleon state with
l = 0. As shown in Figure 2.3, the T7, = +1 and T2 = —1 states have antiparallel
spin as required by the Pauli principle, and the T = 1 p—n state is Similarly anti-
symmetric. Since the strong force is Charge independent, all T: = 1 conﬁgurations
have the same energy. The p—p and n-n states, and hence the antisymmetric p—n

state, are unbound. In contrast, the symmetric T2 = 0 state is the bound deuteron.

14

 

S=1, T=0

n-n

Lg,

Tz=+1 TZ=0 Tzz-l

Figure 2.3: The spin arrangement Of the T2 = 0 and T2 = 1 two—nucleon states.
The T2 = 1 isospin triplet states on the top are unbound. The antisymmetrized
T2 = 0 pm singlet state is bound, Signaling that a stronger interaction is possible
for p—n than for 72-72 or p—p.

The conclusion is that a divergence in the polarization charges Of the proton and
neutron would lead to an enhancement of the neutron polarization charge relative

to that of the proton.

The meaning Of effective charge is illustrated by models where the effective
charge is not required. In the case of 6Li, a nO—core shell model, in which all nu-
cleons are active in a 6M model Space, was developed by Navratil et al.[14]. In
this model, the E2 transition matrix elements can be calculated directly without
resorting to effective charges. Since the full-space eigenvectors are known, a two—
body E2 Operator renormalized over a Oliw subspace can be calculated explicitly.
Operating on the renormalized wavefunctions reproduces the full-space transition
matrix elements. On the other hand, introducing effective charges of 6,, = 1.527
and en = 0.364 to the full-space E2 operator acting on the truncated model space
also replicates the full-space matrix elements. Therefore, the model-space trunca-
tion is capable of causing operator renormalization that is characterized by effective
charges similar to the standard effective charges, ep = 1.5 and en = 0.5. The po-

larization of the core is due to a coupling of the valence nucleons with the giant

15

quadrupole resonance of the core protons. For the no-core shell model, the calcu-
lation of the full—space eigenvectors is unfortunately intractable for all but small A
systems due to the rapidly increasing number of possible conﬁgurations in higher
mass nuclei, a region where the parameterization of core polarization offered by

effective charge allows calculations to be performed.

The giant quadrupole resonance arises in the collective model Of the nucleus.
TIeating the nucleus as a harmonic oscillator, the quadrupole resonances take the
form a An = 2 excitation, that is, excitations to E = 2hw states. The isoscalar
component of the effective charge, 815 = %(ep + en), is coupled to the isoscalar
component of the resonance, a quadrupole-shaped oscillation Of the nuclear matter.
The isovector component Of the effective charge, 81V = gen — ep), instead couples
to a resonance of neutrons and protons moving with a phase shift Of 7r. In the no-
core shell model of 6Li, the difference in the renormalization of the isoscalar and
isovector parts of the operator results in the formation of neutron effective charge.
The signiﬁcance of the giant quadrupole resonance to the effective charge was
studied by comparing a microscopic model of the excitations to the macroscopic,
collective modelll5]. The microscopic model is formed by explicitly adding An =
2 excited states to the Shell-model space with a delta-function for the effective
interaction. For 42Ca, the microscopic model calculates 6;, = 0.06(07) and (in =
0.57(03). In contrast, the macroscopic model is found to be much more sensitive
to the isovector component of the resonance, and polarization charges of 8,, = 0.19

and en = 0.90 are calculated.

As shown above, each core has a unique internal conﬁguration and because
each isotope has a unique arrangement of valence nucleons, and the polarization
charges are not ﬁxed across the entire nuclear landscape. Polarization charges of
6,, = 0.2 and (in = 0.5 were derived for the full so! shell[16] and e.g. were recently

applied with success to transition rates in 36’38Si[17]. As mentioned in the following

16

section, polarization charges of ep 2 1.15 and 6,, z 0.80 have been suggested for

the pf shell.

2.3.4 The pf Shell

The pursuit of uniﬁed effective interactions beyond the p and 5d Shells led to
the development of the GXPF1A effective interaction for the pf shell[6]. The
GXPF1A effective interaction was developed from the G matrix and was ﬁt to 700
experimental energy levels in 87 mostly stable nuclei. The GXPF1A interaction
was then used to predict energy levels and transition rates in exotic nuclei. Nuclei
further away from the valley of stability are of particular interest since the well-
known magic numbers of stable nuclei may grow less prominent or vanish and
new Shell gaps may appear[18]. The existence of Shell gaps is of great importance
because the placement and ordering of the orbitals is the root of the shell model.

The GXPF1A interaction successfully reproduces the energy levels of nuclei
in the lower pf Shell. The systematics Of even-even nuclei are interesting due to
their regular structure: a 0+ ground state with a 2+ ﬁrst excited state using J W
notation. The systematics Of the 2: state in neutron-rich Ca and Ti isotopes are
plotted in Figure 2.4 and compared to the excitation energies calculated with three
effective interactions, the GXPF1A, the Older GXPFl, and the KB3G[19]. The
high E(2:f) characteristic Of a Shell gap is clear at N = 28, and a subshell closure
is indicated at N = 32.

In the investigation of the shell structure Of pf -shell nuclei, the B (E2; 0? ——> 2?)
of neutron-rich Ti isotopes were measured by Dinca et al. [20]. The B(E 2; Oil" —+ 2'11“)
is sensitive to the overlap Of the initial and ﬁnal state wavefunctions rather than
being an eigenvalue of the system. The characteristic low transition probability
is seen at the magic N = 28 in Figure 2.5a, and a similar rate is seen for the

subshell closure at N = 32 while the other isotopes are found to have transition

17

 

Ex (MeV)

 

 

 

 

 

’>‘
d)
.5,
Lil oExp.
--- GXPF1
---- KB3G
0 . . _ g—g QXPFIA‘ A
20 24 28 32 36 40
Neutron Number

Figure 2.4: The systematics Of the ﬁrst excited states in even-even, neutron-rich
pf-shell nuclei. The experimental results in dots is compared to the results Of
calculation with three effective interactions, the GXPF1 (dot-dashed), the KB3
(dashed), and the GXPF1A (solid). Figure from Reference [6].

18

 

2.6 2.8 3.0 3.2 3.4 N 2.6 29 3.0 3.2 3.4 N

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

gene-(a) ] g 800*(b)
N I (3" I
3 8’,
+3 I r:
__ N
l 400 l 400-
”f ' . 05 I
’23“ . ep =15 0 Experiment - ‘25” _ ep = 1.15 0 Experiment .
.. ..... GXPF1 - --- GXPF1
N e = 0.5 N e = 0.8
E " — GXPF1A y, " -— GXPF1A
co 0 CD 0

 

48-h 561—, 525“ 54‘1"I 565“ 48-i—I 50’1-I 523” 5411 561-,

Figure 2.5: Experimental B(E2;0'1f —> 2:”) transition rates for neutron-rich Ti
isotopes compared to Shell-model calculations using the GXPF1A interaction with
(a) standard effective charges and (b) modiﬁed effective charges suggested by du
Rietz et al.[21]. Figure from Dinca et al.[20].

rates roughly twice as high. Intriguingly, the transition rates calculated with the
GXPF1A interaction do not reproduce the data. However, it is instructive to
separate the contribution to the transition rate of active states, the theoretical
transition amplitudes, and the effects from states outside of the active subspace,
effective charge.

Effective charge has previously been studied in the upper pf shell by du Rjetz
et al.[21]. By measuring the lifetime of analogue states in the T; = :l:1/2 mir-
ror nuclei 51Fe and 51Mn, it was possible to isolate the effective charge. When
compared with shell-model calculations, the ideal effective charges were found to
be 6,, z 0.80 and (2p m 1.15. Recall that an enhanced neutron effective charge
is suggested by antisymmetry. The result of using these polarization charges to
calculate transition rates in neutron-rich Ti isotopes is shown in Figure 2.5b. The
trend of the transition rates is reproduced if the amplitudes are not.

Because of the infeasibility Of calculating the interaction in the entire conﬁg-
uration space Of a pf shell nucleus, effective charge forms a key measure of the

interaction of the valence nucleons with the core. The standard effective charges

19

 

T1/2 13.9 (6) 8

Sn 6353(9) MeV
3,, 1.228x104 (4) MeV
QB- 4966 (17) MeV
E(2f) 1026(1) keV

 

Table 2.1: The half-life, neutron separation energy, proton separation energy, Q
value of 8 decay, and energy of the ﬁrst 2+ level of 50Ca. Data taken from Refer-
ence [22].

have been shown to be insufﬁcient for both A = 51, T; = :l:1/2 mirror nuclei
and the neutron-rich Ti isotopes. The effective charges deduced in the upper pf
shell reproduce the trend but not the amplitude of transition rates measured in the
lower pf shell. The question remains: what effective charges properly parameterize

the core polarization in the pf shell?

The Ti isotopes discussed here have both proton and neutron valence nucleons;
thus the individual contribution to the core polarization iS disguised. An ideal
measurement of effective charge would isolate the effective charge of the proton
and neutron individually. Using a 40Ca core, the Ca isotopes are modeled with
valence neutrons alone. The E2 transition is solely due to the polarization Of the
core protons by the valence neutrons and therefore provides a means of measuring
the neutron effective Charge alone. The neutron effective charge of N > 40 Ca
isotopes derived using experimental B(E2; 0]" -—> 2?) values and GXPF1A shell-
model calculation are shown in Figure 2.6. The core polarization is enhanced near
40Ca as described in Reference [15] and decreases as the neutron number increases.
However, all of the experiment data in the ﬁgure is from isotopes below the N = 28
shellgap. 50Ca, with two neutrons in the 2193/2 orbit, is more Similarly conﬁgured
to the N = 32,34 Ti isotopes where the Shell closures have been predicted. It is

for this reason that the E2 transition rate of 50Ca is presented here.

20

 

9 2.0
f, 1.8
3 1.6

1.4
° 1.2
a 1.0
ﬁes
0.6
0.4

0.2
o o l 1 l l l l l l l_
42Ca 4403 4603 48Ca 5003

on
'l'l'lﬁl‘l'l'l'l'ITl

 

 

 

Figure 2.6: The neutron effective charge derived from the Ca isotopes’ experimental
transition strengths and amplitudes calculated with the GXPF1A interaction. For
50Ca the standard effective charge and that suggested by du Rjetz et al. are
indicated with dashes. Data from References [23, 24]

2.4 Transition-rate measurement techniques

2.4. 1 Transition rates

Portions of the following section have been adopted from Reference [25], which was
written by the author of this dissertation. Transition rates are typically expressed
in terms of the reduced transition rate matrix element. The B(aA) Of a general

transition from state J,- to state J f is deﬁned as

 

B(UA;J,- —-> Jf) 2 2J1

.-+1""f”"""*""i>'2 (2.8)

where a signiﬁes either an electric or magnetic transition, A is the orbital angular
momentum of the transition, and J” is the total angular momentum of the initial
and ﬁnal states. The matrix element on the right is a reduced matrix element, i.e.

the Wigner-Eckart Theorem has been used to separate the M 3 dependence, leaving

21

the matrix element dependent on J only. The magnetic substates are normally not
observed separately, and the initial states have been averaged and the ﬁnal states
summed. The multipolarity of the emitted 'y ray is determined by selection rules

that follow from the conservation of angular momentum,
[Ji—JfISASJi+Jf, (2.9)
and from the conservation of parity,

A (—1))‘ EA radiation (2 10)
71' = .
(-——1))‘+1 MA radiation

For photons with wavelengths much greater than the nucleus (the nucleus is char-
acterized by a length scale of ~ 10 fm and the wavelength of a 1 MeV photon is
~ 103 fm), the probability Of a transition decreases rapidly with higher multipo—
larity. The lowest allowed multipolarities for AJ = |J,- — J f] transitions, with
the next highest allowed multipole in parenthesis to indicate that it is usually not
Signiﬁcant, are listed in Table 2.2[26]. In this investigation of effective charge, the
E2 excitation from the 0? ground state to the 2]" ﬁrst excited state in even-even

nuclei is analyzed, and the quantity Of interest is
B(E2;0T —> 2f) = I<2TII02II0T>I2 (2.11)

with Q2 as the electric quadrupole operator. The lifetime Of the state 7' is related

to the reduced transition excitation rate by

4.081 x 103
'r = ,
E3 B(E2; of —+ 21*)

 

(2.12)

22

 

 

 

Parity change Change of angular momentum [Ji — J f]
Arr 0 or 1 2 3 4 5
No M1(E2) E2(M3) M3(E4) E4(M5) M5(E6)
Yes E1(M2) M2(E3) E3(M4) M4(E5) E5(M6)

 

Table 2.2: The dominant y-ray multipolarities for a |J,- — J f] transition with parity
change Arr. The lowest multipolarity is indicated, and the second is Shown in
parenthesis to indicate that it is usually not signiﬁcant. The condition A S J,- + J f
is assumed to be satisﬁed, and the J,- = J f = 0 transition is not allowed[26].

with T in ps, the transition energy Ex in MeV, and B(E2) in e2fm4.

Transition matrix elements are experimentally found by measuring the lifetimes
of excited states or the electromagnetic cross sections to the excited states pro—
vided the excitation mechanism is understood. Lifetime measurements such as the
Doppler-Shift Attenuation Method (DSAM) and the Recoil Distance Doppler Shift
(RDDS) method are based on the analysis of Doppler-Shifted 'y-ray peak shapes.
In DSAM, nuclei are excited following fusion-evaporation reactions, Coulomb ex-
citation, or inelastic scattering and, when the stopping power Of the nuclei in the
material is known, the Doppler shifts of the 7 rays emitted by the recoiling reaction
residues determine the points in time at which decay occurred and hence the life-
time Of the excited state (suitable for 'r < 1 ps) [27]. The RDDS method similarly
uses the Doppler Shift of *y rays emitted by an excited, recoiling nucleus to deter-
mine the lifetime Of the state. A stopper is placed downstream of the target and the
intensity ratio of ’y rays emitted in—flight and stopped for different target-stopper
distances provides a measure of the lifetime in the range 10—12 s < 'r < 10—9 s[28].
The lifetime measurement methods have been recently extended to in-flight mea—
surements of fast beams. In one method, the ”y decay of a 7' z 1 nS state in a
reaction fragment moving at 5 z 0.40 produced a photopeak with low-energy tail
due to a Doppler reconstruction that assumes 'y-ray emission from the target loca-
tion. The lifetime was extracted from the Doppler-shifted peak shape[29]. Another

method similar to RDDS has the stopper replaced by a degrader to Slow rather

23

than stop the beam. The ’7 rays emitted before and after the degrader are Doppler
shifted into separate photopeaks; the intensity ratio between the photopeaks for

different target-degrader distances provides a measure of lifetimes over the range

of 5—500 ps[30].

Nuclear resonance fluorescence (N RF ), electron scattering, and Coulomb exci-
tation determine transition rates through the measurement of cross sections. In
.a typical NRF experiment, a continuous photon spectrum (bremsstrahlung) irra-
diates a target of stable nuclei. The target nuclei are excited by the radiation
and de—excitation '7 rays are subsequently emitted with an angular distribution in-
dicative of the transition multipolarity. The energy-integrated cross section of the
scattered 7 rays is inversely proportional to the lifetime of the excited state[31].
Electron scattering utilizes a simpliﬁed form of low-energy Coulomb excitation
where the form factor in the Born approximation is related to the multipolarity
of the transition. The transition rate can be extracted from the value of the form
factor]32]. Due to the well-understood nature of the interaction and the ease of
producing a large projectile ﬂux, electron scattering is one of the most accurate

methods of determining transition probabilities.

In Coulomb excitation, the interaction of the electromagnetic ﬁelds of the target
nuclei and projectile nuclei leads to excitations with subsequent ’y-ray emissions.
The number of photons N7,f__,,- Observed in an inverse-kinematics Coulomb ex—

citation experiment with y—ray tagging is related to the excitation cross section

by
Nmf—n'

0i—+f = NTNBE (2.13)

where NT is the number of target nuclei (in units of cm—2), N B is the number
Of beam nuclei, and e is the efﬁciency of the experimental setup. N B can be

determined prior to interaction with the target, and NT is given by the target

24

I

possible (‘3

feeding , I
J
If >
Coulomb
excitation 7 decay
| i)

 

Figure 2.7: Schematic Of Coulomb excitation of a nucleus from an initial state It) to
a ﬁnal bound state | f) and the ensuing 7 decay with a possible feeding transition
from a higher state Shown.

thickness. The efﬁciency accounts for the intrinsic and geometric efﬁciencies of all
detector systems involved. Equation 2.13 assumes only one excited state; if more
states than one are excited, possible feeding from higher excited states must be

considered (see Figure 2.7).

The excitation cross section can be related to the reduced transition probability
through various approaches. Coulomb excitation has long been employed at ener-
gies below the Coulomb barrier of the pro jectile-target system, where a Rutherford
trajectory is assumed[33], and was proposed 30 years ago for higher energies[34].
Measurements of projectile Coulomb-excitation cross sections at beam energies well
above the Coulomb barrier[35, 36] are ideal for rare-isotope experiments with low
beam rates, which can be Offset by reaction targets that are about 100—1000 times
thicker than for below-barrier energies. Post-target particle identiﬁcation permits
inverse-kinematic reconstruction of each pro jectile-target interaction. Experiments
at intermediate beam energies also allow for the unambiguous isotopic identiﬁca-
tion of incoming beam particles on an event-by—event basis, which is not generally
possible at contemporary low-energy ISOL facilities. For this intermediate-energy
Coulomb excitation work, the relativistic theory developed by Winther and Alder,

which involves a semiclassical approach with ﬁrst-order perturbation theory, has

25

been utilized]34]. Distorted—wave Born approximation calculations have also been
used tO determine transition rates from cross sections[35] and are in agreement

with the excitation theory developed by Winther and Alder.

2.4.2 Intermediate-energy Coulomb excitation

The most important difference between low- and intermediate-energy Coulomb ex-
citation is possibility of nuclear interactions occurring above the Coulomb barrier.
However, the inclusion of nuclear contributions to the measurement of electromag-
netic transition rate can be prevented in heavy-ion reactions by considering only
those projectiles scattered within a maximum scattering angle representing a “safe”
minimum impact parameter bmin (see Figure 2.8). The radius Rim beyond which
the Coulomb interaction dominates deﬁnes the minimum impact parameter to be
allowed in the experiment. Wilcke et (21. use elastic scattering data to predict
Rint for interactions between various nuclei[37]. For 46Ar it has been shown that
varying bmin where bmz-n 2 Hint has little effect on the measured transition rate
value[38]. The minimum impact parameter is related to the maximum scattering

angle in the center-of-mass system 031,,“ by

em = geetwgzgwm) (2.14)
Zprontare2
0' 2 2
mac 8

where ,8 = v/c and 7 = 1/1/1 — ,62 are the velocity and Lorentz factor of the
beam, and m0 is the reduced mass of the two nuclei.
The adiabatic cutoff of the Coulomb excitation process leads to reduced exci-

tation probability beyond a maximum excitation energy
he
Egm z 1733 (2.15)

26

p projectile

/ 0

target

 

Figure 2.8: Schematic of a projectile nucleus scattering in the electromagnetic
ﬁeld of an inﬁnitely heavy target nucleus. For a ﬁxed beam velocity )6 = v/c, the
scattering angle 0 depends on the impact parameter b. A maximum scattering
angle is chosen in the experiment to restrict the minimum impact parameter.

where b is the impact parameter, and intermediate-energy beams can excite states
at higher excitation energies compared to low-energy beams. For example, 26Mg
impinging on a 209Bi target with a beam velocity of ,6 = 0.36 has an adiabatic
cutoff of E?“ z 6 MeV[39]. However, the possibility of feeding from excitations
to states above the ﬁrst 2+ state must be considered when calculating the excita-
tion cross section[36]. Photons are used to identify the inelastic scattering process
to bound excited states and hence target thickness is not constrained by the need
to preserve momentum resolution to differentiate elastic and inelastic scattering.
Higher energy beams allow for the use of thicker targets, and the number of scat—
tering centers can be increased by as much as a factor of 1000 over low-energy
experiments, permitting an equivalent decrease in the number of required pro-
jectile nuclei. In typical intermediate-energy Coulomb excitation experiments, 1
beam particle in 103-104 interacts with the target nuclei and multiple excitations
are Signiﬁcant only to this small factor]36]. The wide range of scattering angles in-
herent in low-energy Coulomb scattering require large solid-angle detectors; a few

degreae of acceptance sufﬁces for intermediate—energy Coulomb excitation. At the

27

NSCL, SeGA[40], an array Of 18, 32-fold segmented, high-purity Ge 7-ray detec-
tors, and APEX[41], 24 position-sensitive NaI(Tl) crystals, are used for Coulomb
excitation measurements in conjunction with a phoswitch detector or the 8800

spectrographl42] for event-by-event particle identiﬁcation. Similar setups are em-

ployed at GANIL[43], GSI[44], and RIKEN]35].

The angular distribution of the 7 rays emitted depends on the multipolarity
of the transition, electric quadrupole in this case, and on the minimum impact
parameter. The magnetic substates in the ﬁnal state are populated depending on
the impact parameter of the interaction, which varies from bmin to inﬁnity. The
angular distribution can be calculated as described in Reference [45]. Although the
angular distribution is symmetric with respect to the beam axis, the projectile is
traveling at E z 0.3, and the Lorentz boost produces a forward-focused distribution

in the laboratory frame.

2.4.3 The accuracy of intermediate-energy Coulomb excita-

tion experiments

Several reports on initial results from low-energy Coulomb excitation measure-
ments on 30Mg[46, 47, 48] questioned the accuracy of the intermediate-energy
approach and speculated that nuclear excitations are mixed with the Coulomb
interaction regardless of the restriction to small scattering angles. The particular
case of 30Mg now seems resolved in that the previously reported discrepancyl49]
has disappeared in the published low-energy result and is now in agreement with
one intermediate—energy result]50], but not with another[51]. Responding to the
general question raised, an examination of the accuracy of the intermediate-energy
Coulomb excitation method was undertaken by comparing the “test cases” mea-

sured at intermediate beam energies at the National Superconducting Cyclotron

28

Laboratory at Michigan State University to adopted reduced transition matrix ele—
ment values based on four or more independent measurements with complementary
techniques that are available in the literature[25]. These test cases were measured
over the past decade with the identical setups and during the same experiments
used for measurements of unknown transition matrix elements. While these test
cases have been individually reported previously in peer-reviewed journals together
with the respective new measurements, their collective comparison to adopted val-
ues here reafﬁrms intermediate-energy Coulomb excitation as an accurate method

relative to other transition rate measurement techniques.

The advantages of intermediate-energy Coulomb excitation are most pronounced
when the method is applied to exotic nuclei with low production rates. Un-
der these circumstances, the statistical uncertainty dominates. This difﬁculty is
present irrespective of the method applied, and, therefore, only high statistics
intermediate-energy Coulomb excitation measurements will be considered in de—
termining the method’s accuracy. A summary of intermediate-energy Coulomb
excitation measurements of previously published transition rates along with their
respective adopted values can be found in Figure 2.9(a). For thaee Coulomb exci-
tation test cases, no feeding was Observed. The adopted values are those compiled
by Raman[52] where four or more independent transition rate measurements using
any of the above techniques have been made for each nucleus. In the calculation
of the adopted transition rate for 40Ar, one Of eight experimental values was mea
sured using intermediate-energy Coulomb excitation, and for 36Ar, two Of eight.
The error bars on the adopted values represent the relative uncertainties. The
average difference from the adopted value is 6% and only one data point exceeds
10%. Note that all measurements are in agreement with their respective adopted

values.

Figure 2.9(b) shows the relative differences between measured B ( E2; 0]“ —> 2?)

29

transition rates and the adopted value[52] for 26Mg. The shaded area represents
the uncertainty of the adopted value. The measurements were made using low—
energy (2:, 2'7) Coulomb excitation, NRF, DSAM, RDDS, and electron scatter-
ing. These traditional transition rate measurements have an average difference
of 23% from the adopted value for 26Mg. The right-most data point, which de-
viates from the adopted value by 3%, was measured by Church et al.[39] using
intermediate-energy Coulomb excitation at a beam energy of 66.8 MeV / nucleon.
This speciﬁc measurement illustrates the more general point made in Figure 2.9(a)
that intermediate-energy Coulomb excitation measurements that are not limited
by statistics can readily measure transition rates with an accuracy of about 5% to

a precision Of about 10%.

30

 

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31

Chapter 3

Experimental apparatus

The experiment was performed using the Coupled Cyclotron Facility (CCF) at
the National Superconducting Cyclotron Laboratory (NSCL) at Michigan State
University. The three steps of radioactive ion beam production are primary beam
production, fragmentation, and isotopic separation. At the CCF, a beam of sta-
ble ions is accelerated to approximately 0.5a and impinge upon a beryllium tar-
get. The fragmentation reaction produces a large number of different isotopes,
and the isotopes of interest, 50Ca and 52Ti, are isolated by the A1900 Fragment
Separator and transported to the experimental vault where intermediate-energy,
inverse-kinematics Coulomb excitation with 7—ray tagging is performed using the

S800 Particle Spectrograph and the APEX NaI(Tl) 7—ray detector.

3.1 Isotope production

The Coupled Cyclotron Facility at the NSCL consists of an electron cyclotron
resonance ion source, two coupled cyclotron accelerators, and a fragment separator.
In this experiment, a stable beam of 76Ge was produced with a range of charge

statas by the ARTEMIS ion source. Since a cyclotron can only accelerate a single

32

 

RT—ECR 20 ft
0‘13... 7 - ;
ion sources c " ~‘ WW
SC-ECR 10 m
, 9
K1200 , " ' é l
7' A1 900 if focal plane

production :- . - :1 _ ’1

stripping target { - ‘.
bu \

wedge

Figure 3.1: The Coupled Cyclotron Facility. A beam of 76Ge is produced by the
ion source, accelerated by the cyclotrons, and impinged on the production target.
The fragments of interest are selected in-ﬂight by the A1900 Fragment Separator
and directed into the experimental vault.

charge state, one of the high intensity charge states, 76Ge+12, was accelerated by
the K500 cyclotron to 0.156c. The 76Ge nuclei are then stripped of electrons by
passing through a 600 ,ug/cmz carbon foil, and the K1200 cyclotron accelerates
the 76Ge+30 ions to 0.480c. The cyclotron operates at a frequency of 22.5 MHz
(Figure 3.1).

The beam impinged on a 517 mg/cm2 98c production target, where a portion
of the beam underwent fragmentation. In the ﬁrst part of the two-step fragmen-
tation process is a quick peripheral collision that produces a highly excited pre-
fragment. The prefragment subsequently decays over a longer time scale through
statistical nucleon emission, resulting in a broad distribution of stable and exotic
nuclei[53]. The fragmentation reaction produces fast beams, allowing the study
of short-lifetime isotopes, and it is independent of the chemical properties of the
beam. The reaction does, however, generate a broad momentum distribution.

The beam fragments pass into the A1900 Fragment Separator[54|, which uses
a Bp—AE-Bp selection to isolate the isotope of interest. TWO dipole magnet bend

the beam, selecting a magnetic rigidity Bp = p/Z, where p = mv is the mo-

33

mentum of the fragments with relativistic mass m. The ions passed through an
369(1) mg/cm2 achromatic aluminum wedge, introducing a Z-dependent energy
loss described by the Bethe—Bloch equation. A second magnetic rigidity restriction
on the wedge-induced isotopic momentum spread completes the isolation of the
isotope of interest. Typically, the A1900 produces a cocktail beam consisting of
the desired isotope and several other isotopes of similar A and Z. The 500a beam
produced by the A1900 for this experiment consists primarily of ggCa, £80, ggTi,
and 3%“ and the ggTi beam contained signiﬁcant proportions of ggSc, 33’54V, and
gSCr. In this experiment, the momentum aperture at the wedge position limits
the momentum spread to Ap/p = 3% for 50Ca, and to Ap/p = 0.5% for 52Ti. The
difference in momentum spread is due to the production rate; higher rates permit
greater selectivity.

At the exit of the A1900 is the extended focal plane, or XF P, where a scintillator
is placed for time-of—ﬂight measurements, which will be discussed in the following
section. Horn the XFP, the ions are transported to the experimental vault. The
beam reaches the S800 Particle Spectrograph with an efﬁciency of approximately
70%. For the case of 50Ca, 1010 particles/second (pps) of 76Ge directed onto the

production target produce 102 pps of 50Ca at the focal plane of the S800.

3.2 The SSOO Particle Spectrograph and the APEX
NaI(Tl) Scintillator Array

The intermediate—energy Coulomb excitation measurement is performed using the
8800’s particle identiﬁcation and tracking capabilities coupled with the APEX
N aI(Tl) array to identify inelastically scattered beam projectiles. The 7 rays pro-
duced by nuclei moving in intermediate-energy beams requires position-sensitive

detectors for Doppler correction. One possible solution is diffusively—reﬂective scin-

34

tillator bars with photomultiplier tubes (PMTs) on both ends, and this method is
utilized by the APEX NaI(Tl) scintillator array.

3.2.1 The 8800 Particle Spectrograph

The S800 Spectrograph is a high-resolution, high-acceptance particle spectrograph
used for particle identiﬁcation and tracking [42]. As can be seen in Figure 3.2, the
beam from the A1900 is provided at the 8800 Object, from which the analysis beam
line directs the beam to the target position where a variety of radiation detectors
may be stationed. In this experiment, the APEX NaI(Tl) array is stationed at
the target position to tag inelastic scattering events. The beam continues into
the spectrograph where dipole magnets spread the beam according to magnetic

rigidity and ﬁnally into a detector package in the focal plane.

The S800 Focal Plane consists of an ion chamber for AE measurements, two
position-sensitive Cathode Readout Drift Chambers (CRDCs), and three scintil-
lators for time—of-ﬂight measurements (tof). The ion chamber and scintillators
coupled with a scintillator at the exit of the A1900 allow for particle identiﬁcation
by plotting tof cc 1) versus AE' cc Z2. The CRDCS are separated by a meter; the
two position measurements determine the angle at which a particle passes through
the focal plane. The transport of ions from one focal plane to another can be repre-
sented in general by a transport matrix T that describes the relationship between

the positions x and y and angles from the beam axis a z tana and b m tan ,8.

{A (M

y 31

a

\bjtarget ijFP

35

 

 

 

 

The ion paths measured in the 8800 Focal Plane can be mapped to the target focal
plane by inverting T, which was determined by mapping the S800 dipole magnets.
The scattering angle reconstruction has an uncertainty of 005°. The three focal
plane scintillators are of 5, 10, and 15 cm thickness; only the ﬁrst is required for
heavy ions. The ﬁrst scintillator is the source of the S800 particle trigger and
can be used for time-of-flight measurements between the focal plane and the S800

Object or the A1900 XFP.

The S800 Spectrograph can run in two modes, dispersion matched and focused.
In both modes, the beam provided by the A1900 is focused in space and dispersive
in momentum at the S800 Object, where a timing scintillator is located. Since
passing through the scintillator increases the angular dispersion of the beam, a
focused position minimizes the total phase space of the beam. In focus mode,
the analysis line is achromatic, and the beam is focused to a spot 1 cm FWHM
at the at the target while remaining dispersed in momentum. The beam is then
chromatic at the S800 Focal Plane, with the beam momentum width convoluted
with the momentum loss in the target. The momentum acceptance of the S800 in
focus mode is 4%. The focal plane image is located at CRDCl, where the beam
is focused in angle, and the dispersive position of the beam ion is dependent on
momentum. In dispersion-matched mode, the entire system is achromatic, with
the momentum spread of the beam at the object canceled at the focal plane.
with the cost of a wide dispersion in momentum and space at the target. The
momentum acceptance of the S800 in this mode is 1%, which corresponds to a
beam width of 11 cm at the target. Since the intrinsic momentum of the beam is
focused at the focal plane, this mode is provides a high resolution measurement
of the momentum loss in the target. In this experiment, the higher acceptance of
focus mode is needed to offset the low production rate of exotic nuclei. Moreover,

identifying the Coulomb-excitation event with '7-ray tagging removed the need for

36

focal plane

 

 

8800 analysis line 8800 spectrograph

 

4 N D

 

Figure 3.2: The S800 Particle Spectrograph. The beam enters from the left, passes
through the target area where the 'y—ray detector is located, and is transported into
the focal plane for particle identiﬁcation and tracking.

high momentum resolution.

3.3 The APEX N aI(Tl) Scintillator Array

3.3.1 The detector

The identiﬁcation of intermediate-energy Coulomb excitation events through the
detection of emitted '7 rays typically occurs with beam velocities of [3 = 0.3 at the
NSCL. The transformation from the projectile to laboratory frame introduces a
large Doppler shift as shown in Figure 3.3. The angular resolution may be achieved
through segmentation of the detector, such as is done in the 18, 32-fold segmented,
high-purity Ge detectors in the Segmented Ge Array (SeGA)[40] and the soon-to-
be-completed CsI(Na) array CAESAR, or with a position-sensitive detector such
as the APEX detector. The APEX detector was chosen for this experiment for
its higher efﬁciency, z 7% at 1 MeV compared to the z 2.5% efﬁciency of SeGA.

The detector was built as trigger detector for position annihilation radiation as

37

 

 

 

 

 

%‘ I
5 I
g 1200 _—-
1 100E—
1000 :-
900 r
i-
800 —
1 I I I I I I I I I I I I I l I I I I I I I I I I I I
-200 -100 0 100 200
position (mm)

Figure 3.3: The laboratory-frame energy of a 1 MeV 7 ray produced in a frame
moving at ﬂ = 0.3 as seen by the APEX detector. The energy is spread from
800 keV to 1.3 MeV.
part of the APEX experiment at the ATLAS accelerator at Argonne National
Laboratory[55] and will be referred to by the shortened name “APEX” in the
following discussion.

The APEX Array [56, 41] consists of 24 trapezoidal cylinders of NaI(Tl) ar-
ranged in a barrel conﬁguration. Each detector bar is jacketed in 0.4 mm of steel
with a 1.1 cm thick quartz window leading to a photomultiplier tube at both ends.
The bars are 55.0 cm long and 6.0 cm thick, and they have inner and outer trape-
zoidal faces of 5.5 and 7.0 cm respectively. The barrel of detectors is surrounded
by a 1.9 cm thick lead tube for background shielding, and the entire array is sur-
rounded by and mounted on a 1.0 cm thick steel tube. The steel tube has wheels
that allow the array to be mounted on rails. The shielding is a key component of
the array; the large volume of NaI(Tl) is highly efﬁcient, and without the shield-
ing 'y-ray source calibration measurements would have a large deadtime due to

background radiation. The array is shown in Figure 3.4.

38

 

Figure 3.4: A photo of the APEX Array with twenty-four N aI(Tl) bars with PMTs
at each end and a lead and steel shield.

Inorganic scintillators with activators such as NaI(Tl) function by producing
light when radiation falls on the crystal. The material must be transparent to the
emitted light, and the luminescence decay time must be short. As is typical of the
structure of insulators and semiconductors, the N aI crystal lattice permits electrons
to lie in discrete energy bands, a lower valence band and a higher conduction
band (Figure 3.5). Incident radiation deposits energy that boosts electrons from
the valence band to the conduction band. Photon decay from the conduction
band to the valence band is an inefﬁcient process and produces light of the same
energy as the band gap. This light may then boost another electron into the
valence band, reducing the light output of the crystal. The introduction of an
activator atom, thallium in this case, produces local energy levels in the forbidden
band. The conduction-band electron quickly encounters an activator site, where it
can recombine with a hole, leading to light emission. The reduced energy of the
recombination transition leads to light in the visible region where the crystal is
transparent.

During the experiment, APEX sits at the target position of the S800 ﬂush
against the beamline gate valve to the S800’s entrance quadrupole magnet. APEX

must sit close to the magnet because the projectile '7 rays are forward focused but

39

conduction band

 

activator
excited states

excitation from
band gap (
Y-ray interaction

activator
ground state

 

 

 

valence band

Figure 3.5: The scintillation process. The interaction of 7 rays with the scintilla-
tor crystal excites electrons from the valence band to the conduction band. The
electron drifts to an activator site where it decays to the valence band through a
multi-step process including photon emission.

the array is large enough that center of the array is approximately 60 mm upstream
from the target position of the S800. The large magnetic ﬁelds produced by the
quadrupole magnet far exceed those in which a standard PMT can function; APEX
uses the Hamamatsu H2611 PMT, whose closely-spaced ﬁne mesh dynodes function
in ﬁelds beyond 1 T. However, the trade-off for PMTs that function near the

beamline magnet is a reduced pulse-height resolution relative to standard PMTs.

Position resolution is achieved by diffusively grinding the sides of the NaI(Tl)
crystal bar, causing the incident light to reﬂect in a random direction. The cu-
mulative effect is an exponential attenuation of scintillation photons Np}, as they
move away from the scintillation position z as measured from the center of the bar
of length L. In this case, the number of photoelectrons produced in the PM'Ils of

quantum efﬁciency a, is then

N

NPE,1 = 61-39-1363—“(L/2H) (3-2)
N h ._ _
NPE,2 = 62—3-6 “(L/2 2’)

40

where u is the light attenuation coefﬁcient per unit length. A value of p = 0.047/ cm
was chosen to achieve a position resolution of 3.0 cm FWHM. With the PMT and
ampliﬁer gains and analog-to-digital conversion collectively symbolized by a factor

9;, the recorded detector output is

A1 = 91NPE,1 (33)

A2 = 92NPE,2-

Solving this system of equations for 2: leads to

1 A2 9262
= — — —1 — . .4
Zm 2.” (n A1 n 9161 (3 )

The energy E of the 7 ray is proportional to Nph,

N h _
Erec = V A1142 = V€1€29192—§_e #L- (35)

The end face of the crystal has an area of 37.5 cm2. It is attached to a quartz
window of area 15.2 cm2 that connects to a 10.2 cm2 active PMT face. The
geometry of the detector reduces the number of scintillation photons reaching the
photocathode by 73%, and optical transmission losses lead to a further reduction
of around 25%. The exponential attenuation of the scintillation photons, their
low transmission to the PMT, and the magnetic-ﬁeld resistant PMTs, each one
vital to the feasibility of the experiment, altogether lead to an energy resolution
signiﬁcantly degraded from the typical resolution of an N aI(Tl) detector as shown
in Table 3.1. An energy spectrum from an 88Y 7-ray source detected by APEX is

shown in Figure 3.6.

41

 

E

counts/4 keV

3 3
iiHIH'TTHIHr

 

Illll

 

I I I I I I I I I LII I I I I LIII I I I I I I I I -
0 800 1000 1200 1400 1600 1800 2000 2200

energy (keV)

Figure 3.6: The energy spectrum of an 88Y 7-ray source produced by APEX.
3.3.2 Electronics

The electronics for recording data from APEX were assembled speciﬁcally for this
experiment and consist two parts: one that digitizes the electrical pulse and another
to select those events that are coincident with a beam particle. The large mass
of the array very efﬁciently absorbs background radiation, and the coincidence
timing provides a means of rejecting all 7—ray data that is not time-correlated with
a beam particle, reducing the room background to a very small part of the in-beam

background.

A high-voltage power supply provides approximately 2 keV to each of the 48
PM'I‘s. Signal cables carry the output of those 48 PMTs to CAEN N568B Spec-
troscopic Ampliﬁers. The N 5688 provides three outputs, a shaped and ampliﬁed

signal, that same signal with a further 10x ampliﬁcation (not used in this exper-

42

 

detector energy resolution
(keV) (% FWHM)

 

APEX 662 19
898 16

1836 11

typical NaI 662 7

 

Table 3.1: The resolution of APEX compared to that typical of a 3”x3” NaI(Tl)
'crystal.

iment), and a fast, ﬁxed-ampliﬁcation signal for timing. The PMT voltage and
ampliﬁcation were chosen as described in the following section. The shaping time
of 1 us was chosen to maximize resolution while minimizing deadtime. The shaped
and ampliﬁed signals are then digitized by CAEN V785 ADCs.

The timing signal travels from the ampliﬁer fast output into a LeCroy 3420
Constant Fraction Discriminator. Signals that exceed the threshold for a. given
channel are passed through to a CAEN C469 Gate and Delay Generator and
are multiplexed into a single signal, the APEX trigger. The multiplexed signal
is passed to the S800 trigger logic, where it is combined with the S800 trigger.
Two types of triggers are accepted, a downscaled 8800 particle trigger to measure
the number of beam projectiles and an APEX-8800 coincidence trigger to record
projectile-correlated 7 rays. The master trigger is then returned to the APEX
electronics, where it is split by a gate generator. The trigger gate generator pro-
duces a short logic pulse that starts a CAEN V775 TDC that is later stopped by
the now-delayed signal for each channel coming from the gate and delay generator.
The trigger gate generator also produces a long gate that opens the CAEN V785
ADC to accept the shaped signal from the ampliﬁer. In this manner, the energy
of particle-coincident 7 rays and the time difference between particle and 7-ray

detection is recorded. A schematic of the electronics is shown in Figure 3.7.

43

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44

3.3.3 Calibrations

APEX must be calibrated for energy and position, and the calibration consists of
an approximate hardware calibration and a precise software calibration. A planar-
collimated 60Co 7-ray source is used for the calibrations. This source consists of
a 60C0 source sandwiched between two Heavimet disks so that 7 rays are emitted
radially in a plane. The disks are placed concentric with APEX so that e.g. with
the source at the center of the array all detectors are illuminated at only the center
of the bars. The hardware calibration was performed by placing the source at the
center of the bar, adjusting the PMT voltage until the signals from both sides each

bar were roughly equal and then ﬁnely adjusting with the ampliﬁer.

For the position calibrations, data were collected with the collimated source
position positioned at 1.5” increments along the length of the detector. The peaks
in the reconstructed spectra were ﬁtted, and a third order polynomial was used
to map the ﬁt centroids onto the actual position of the source. The range of the
position calibration is limited by the breakdown of the exponential attenuation

model at the ends of the bars.

The energy calibration utilized 137Cs and 88Y 7—ray sources, providing calibra-
tion points below, near, and above the energy of interest at 1 MeV. 60Co is not
used for the energy calibration because the two 7—ray peaks at 1173 and 1332 keV
are not well separated in the energy spectrum. Software gates were placed on the
uncalibrated position so that ﬁfteen energy calibration slices were created for each
bar. For each source 7—ray energy, the centroid of the reconstructed energy was ﬁt
with a Gaussian and linear background as shown in Figure 3.8. A linear function
was used to map the centroids onto the 7-ray source energies. In this manner, the

energy spectrum was calibrated.

45

 

counts/bin
.5
8

   

ITIIIIfIIIlIIIIII

 

 

 

 

’0 .— .
00 E .............
40 '_
— I I I I I I I I I I i I I I L I I I I I I I I I I
500 600 700 800 900
energy (ch)

Figure 3.8: The ﬁt of the 898 keV 7-ray energy peak in one slice of detector 2 using
a Gaussian with a linear background as the ﬁtting function. The centroid channel,
along with those of the 662 and 1836 keV photopeaks of 7—ray calibration sources,
was mapped to the source energy to complete an energy calibration.

3.3.4 Efficiency

The total efﬁciency of the array was calculated using calibrated 7 sources at the
target position and summing the energy spectra from each bar. The 7-ray peak was
ﬁt using a Gaussian plus a linear background, and was compared to the number of
7 rays produced by the source while accounting for deadtime. Three of the twenty-
four detectors were not functioning for this experiment, and are not included in

the analysis. The measured efﬁciency is 11.6% at 898 keV and 7.4% at 1836 keV.

An intriguing effect comes to light upon examining the position spectra of the
energy peaks. As shown in Figure 3.9, the 1836 keV 7 ray of 88Y produces an
isotropic distribution of a point source; however, the 898 keV 7 ray does not. To
see how this can happen, consider the ideal case, where the detector response
would resemble what is simulated in Figure 3.10. The source, 88Y in this case,

produces an isotropic distribution of 7 rays in the position response of the detector.

46

Recalling that the energy and timing signals are separated by the ampliﬁer, let
it be assumed that without altering the energy signal a high-resolution signal is
fed into the discriminator, which cuts off energies below a ﬁxed minimum. The
output signal from each PMT has a position dependence, which leads to the ﬁxed-
value threshold being realized as a position-dependent threshold. The exponential
attenuation of the scintillation photons towards the end of the bar coupled with
the acceptance of all events that exceed either threshold results in the appearance
of an exponential increase in the threshold moving away from the end of the bar.
In this idealized case, a 600 keV 7 ray that interacts at 150 mm would exceed the
threshold on the positive side of the detector but not on the negative, and vice
versa if the 7 ray had interacted at -150 mm. Had the 7 ray fallen on the center of
the bar, the threshold would not be exceeded on either side, and the event would

not be recorded.

Figure 3.11 illustrates this effect in APEX. The low resolution of the array
coupled with the short shaping time of the fast output of the ampliﬁer leads to
the discriminator triggering on widely varying signals for 7 rays of a single energy.
Rather than a high-resolution signal being fed into the discriminator, the discrim-
inator acts on a signal of lower resolution than the energy signal, resulting in an
indistinct threshold. This threshold effect distorts the 898 keV position response
but not that of the 1836 keV 7 ray in Figure 3.9. It is probable that this effect was
present in previous work with APEX [57, 41], with the extent of the efﬁciency loss
depending on the amplitude of the threshold. Reducing the position-dependent
effect of the threshold can be accomplished by reducing the light attenuation fac-
tor u, but doing so would reduce the position resolution of the detector—a vital
component of Doppler reconstruction. Alternatively, the threshold may be re-
duced, increasing deadtime due to the many low-energy events at the extremes of

the detector. The position-dependent threshold, an inherent characteristic of the

47

 

898 keV

counts/4 mm

IIIITIIIIIIIITIIIII

IIII

 

 

 

'11

 

 

position (mm)
Figure 3.9: The position response of APEX for an 88Y 7—ray source placed at

60 mm. The 1836 keV 7 ray produces an isotropic distribution (with a slight effect
from the maximum range of the ADC) while the 898 keV 7 ray does not.

detector, forces a trade-off between efﬁciency, position resolution, and deadtime.

Fortunately, the experiment is possible with this effect.

48

 

 

-200 -150 -100 -50 50 100 150 200
position (mm)

Figure 3.10: A simulated position-energy matrix for a doublesided scintillator
bar with a high-resolution signal input into the discriminator. The exponential
attenuation of the scintillation photons is realized as exponential increase in the
threshold towards the center of the bar.

49

 

 

position (mm)

Figure 3.11: A simulated position-energy matrix of a double-sided scintillator bar
such as those of APEX with a low—resolution signal input into the discriminator.
This ﬁgure and the ideal case ﬁgure differ only in the threshold. Although the
efficiency has decreased, the low resolution leads to an indistinct threshold energy
cut off.

50

Chapter 4

Simulation

The key quantity in this investigation of nature is a cross section, the ratio of the
number of 7-ray emitted to the number of possible Coulomb excitation reactions.
In the previous chapter it was explained that measuring the AE and time of ﬂight
of a particle is sufﬁcient to identify the isotope, and the detection of 7 rays will
now be considered in detail. The difﬁculty of 7-ray spectroscopy lies in the fact
that a monoenergetic 7 ray will not produce a monoenergetic response. Instead,
7 rays interact with materials via three major processes, the photoelectric effect,
the Compton effect, and pair production, to produce a detector response function
extending from zero energy to somewhat above the 7-ray energy. Coulomb-excited
beam projectiles adds the complexity of 7-ray emission in an electric quadrupole
angular distribution folded with a B m 0.3 Doppler boost. A model of the detector
response that includes the angular distribution of 7—ray emission and the kinemat-
ics of the projectile can produce response functions for ﬁtting to the data. The
response function can translate the shape of the 7-ray energy spectrum into the
number of 7 rays detected. With a known efﬁciency the number of 7 rays emitted

is determined.

In this chapter, the focus is the determination of response functions to extract

51

the number of 7 rays detected from the energy spectrum. A simulation of the
detector response of APEX was created using GEANT4 [58, 59], a C++ toolkit
developed at CERN for simulating the interactions of radiation with matter. There
are four parts to the simulation: the detector geometry, the 7-ray generator, the
interaction physics, and the model of the detector and electronics. The model is
then compared to the detector response of laboratory-frame 7—ray sources and in-
beam Coulomb excitation reactions. Efﬁciency will be discussed in the following

chapter.

4.1 Detector Geometry

The basic building block of the APEX Array is a single detector bar. The sim-
ulation bar consists of the Na] crystal, steel jacket, and quartz window with the
dimensions described in Section 3.3. The 24 bars are arranged in a barrel and
surrounded by the lead and steel shields as shown in Figure 4.1. Finally, the 6”
Al beam pipe was added, and target foils are inserted when needed for Coulomb
excitation simulations. The 66”x22”x0.5” aluminum table on which APEX sits was
included to determine if backseattering from large objects outside the array affected
the detector response; a negligible difference was noted, and no other objects were

included.

4.2 7-ray generator

7-ray calibration sources used in the laboratory emit 7 rays of ﬁxed energies isotrop-
ically from a point. The sources contain an unstable isotope that B decays into
excited states of the daughter isotope, which then transition to the ground state by

7-ray emission. The proportion of B decays that produce a given 7 ray is termed

52

 

Figure 4.1: The APEX detector geometry constructed in GEANT4. On the left is
a view looking down the beam pipe with the target at the center and the array
surrounding. On the right is view of APEX from the outside.

the intensity ratio of that 7 ray. The characteristics of the source are modeled in
the simulation for a single event through the following process: the intensity ratio
is used to determine if each of the 7 rays emitted by the source will be emitted in
that decay. The 7 ray that is selected to be emitted has its energy and a direction
chosen randomly from an isotropic distribution passed to the primary event gen-
erator queue. Once all 7 rays are in the queue, the event generator produces all of
them at once from a single point. For example, the 898 keV line 88Y has an inten-
sity of 94.0%, and the 1836 keV line has an intensity of 99.4%. Each primary event
has a 94.0% probability of containing an 898 keV 7 ray computed by sampling a
random number from a ﬂat distribution. On the occasion that the 898 keV 7 ray
is to be emitted, a random, isotropically—distributed vector is selected, and the 7
ray is added in the primary event generator queue. The 1836 keV 7 ray is similarly
treated, and in this example will also be emitted. A second random direction is
chosen, and subsequently the two 7 rays in queue are emitted. The simulation

then tracks those 7 rays until they have deposited all energy or they have left the

53

simulation’s world.

The in-beam 7 ray generator includes a number of additional features. In the
projectile frame, the 7 rays are not distributed isotropically but rather with an

electric quadrupole distribution along the beam axis as discussed in Section 2.4.2.

P(0pmj) = {a0L0(sin ﬁmoj) + a2L2(sin Oproj) + a4L4(sin apr0jll sin Oproj (4.1)

and an equiprobable distribution in (ppm. While the ¢proj component, lying
perpendicular to the beam direction, is not altered in the transition from the
projectile frame to the laboratory frame; the 7—ray energy and the E2-distributed
9 component are relativistically boosted. The relation between the projectile and

laboratory frame is given by

cos Oproj + 6
1 + ,BCOS oproj,

 

01.11; = (43)

where ,6 ~ 0.3. In Figure 4.2 the isotropic angular distribution is compared to
the E2 angular distribution emitted by 52Ti after Coulomb excitation on a “atAu
target with a midtarget velocity 6 = 0.363. In the experiment, the [3 of emission

forms a distribution according to the beam momentum width 619 with

p (4.4)

5P _ 255
_ ’7 —

[3
Experimentally, the incoming beam has a momentum selected by the magnetic
rigidity of beam-line magnets and a width chosen by the slits in the A1900. Passing
through the target broadens the momentum, which is then measured in the S800.

Ultimately, the position resolution of APEX is insufficient for the array to be

sensitive to momentum widths of a few percent typical at the NSCL.

54

 

P (6)

 

 

 

 

9 (deg)

Figure 4.2: An isotropic angular distribution (solid line) is compared to the E2
distribution produced by the Coulomb excitation of 52Ti on a m"tAu target at
[3 = 0.363.

In the laboratory frame, the projectile source emission position is spread over
a large volume relative to the point-like calibration source. The beam impinges
on the target with a normal distribution of approximately 1 cm FWHM in both
the vertical and horizontal directions. The position of the emission in the beam
direction depends on the lifetime of the excited state. The B (E2) is related to the
lifetime T by the relation given in Equation 2.12. Since the de—excitation transition
is subject to exponential decay, the position of 7 ray emission is exponentially
distributed. Thus, with the excitation occurring on average in the middle of the

target, the probability of emission at a distance d from the target center is governed

by
_ d
PemissionId) = 8 ﬁg (4'5)

A plot of this distribution is given for 52Ti in Figure 4.3. The majority of the 7

are emitted outside of the target.

55

a

 

§

countdOJ mm
3 8

d

IIIII—IWIIIIIIIIIITTIIIIIIIIITTIIﬁ

8338

 

 

 

IIII [ILLIIIII I Plilllgigl

0 1 2 3 4 5
2 position (mm)

 

0

Figure 4.3: The simulated 7-ray emission position along the beam line with z =
0 at midtarget. The exponential decay in time of the excited states leads to
an exponential decay in position of 7-ray emission. The target is approximately
0.1 mm thick (one bin), and most 7 rays are emitted outside of the target.

4.3 Interaction physics

7 rays deposit energy in materials via three major processes, the photoelectric
effect, pair production, and Compton scattering [60]. All three processes involve
an abrupt transfer of photon energy to electron energy with the photon either
vanishing or scattering. The photoelectric effect is dominant at lower energies,
Compton scattering predominates at the 1 MeV energies discussed here, and pair
production grows more important at higher energies. These three interactions
produce the characteristic form of the detector response, a Gaussian photopeak

atop a Compton continuum extending toward low energies.

In the photoelectric effect, the photon interacts with an atom and vanishes. An
electron is ejected from the atom, mostly probably from the most tightly bound,

or K, shell if the photon energy 17.11 is sufﬁcient. The photoelectron carries with it

56

a kinetic energy

Ee_. : hV _ Eb (4.6)

where Eb is the original binding energy of the ejected electron. The ionized atom
quickly absorbs a free electron or rearranges its shells, either releasing x rays or an
Auger electron. The energy of the x rays and Auger electrons is re—absorbed after
traveling typically less than 1 mm. The result of the photoelectric effect is then a
photoelectron that carries most of the 7-ray energy and local effects of lower energy.
If nothing escapes from the detector, the full energy of the 7 is deposited in the
detector, and the result of many monoenergetic 7 rays undergoing photoelectric
effect interactions is a delta function in the energy spectrum at the incident 7-ray

energy, E7.

The absorption of the full photon energy means that the photoelectric effect
is the ideal interaction for determining the energy of the incident 7 ray. The

probability of photoelectric absorption per atom 'T is approximately

7' oc —. (4.7)

The strong dependence on Z is the reason lead is used for APEX’s shield and part

of the reason NaI (Z (I) = 53) is an excellent scintillator.

Compton scattering is the most likely interaction at energies around 1 MeV
that are of interest for this experiment. The process occurs when a 7-ray photon
interacts with an electron and is deﬂected by an angle 6 with respect to the original
trajectory as shown in Figure 4.4. In doing so, the photon transfers some energy

to the recoiling electron. The conservation of energy and momentum leads to an

57

expression for the energy hV' of the deﬂected photon,

hu

hu' = h
1 + Inn—:20 — cos 0)

 

(4.8)

where moc2 is the energy of an electron at rest. To illustrate the results of Compton
scattering on the energy response of the detector, consider two extreme cases. For
a very small scattering angle 0 z 0 the recoil electron absorbs very little energy.
In the case of a very large scattering angle where 0 z 1r the kinetic energy E e_ of

the recoil electron is

hz/
hu’ [[92,r = (4-9)

1 + 2hu/m0c2
ZhV/m062 )
1 + 2hu/m0c2

 

 

E8- [9:7r 2 fax — hu’ = hl/ ( (4.10)

Thus, Compton scattering deposits anywhere from zero to ABC = hu — E e- [9:7r
energy in the interaction material. For monoenergetic 7 rays the result is a peak
beginning AEC below the incident photon energy with a continuum extending to

zero.

The third signiﬁcant process by which 7 rays interact with materials is pair
production. Pair production occurs when a 7 ray of at least 1.02 MeV vanishes in
the presence of an atom to produce an electron-positron pair. The energy of the
photon above the rest mass of the electron-positron pair is carried away as kinetic
energy.

_ 2
E6- + Ee+ — hu — 2m0c (4.11)

The kinetic energy of the electron and positron is lost within a few millimeters
of the interaction point. The thermalized positron will annihilate with another

electron and produce two photons of energy m0c2. If the detector is large enough,

58

Recoil electron

 
    

Incident photon
E=hv

 

Scattered photon
E=hv’

Figure 4.4: A kinematic diagram of a 7 ray of energy hu Compton scattering on
an electron at an angle 0 with energy hu’.

these photons will be reabsorbed. For a monoenergetic 7 ray the result is a delta

function at E = hu —— m0c2.

Altogether, pair production, Compton scattering, and the photoelectric effect
produce the characteristic form of the detector response to 7 rays. As shown in the
idealized detector response to a 2 MeV 7 ray in Figure 4.5, there is a photopeak
at 2 MeV consisting of the 7 rays that deposit all of their energy into the detector
through any of these interactions with the rest of the spectrum formed by 7 rays
that deposit only a portion of their energy. The most likely process of partial
energy deposition is for a 7 to Compton scatter out of the detector. Therefore, on
the low-energy side of the photopeak is a gap of width AEC produced by a lack of
Compton-scattered 7 rays followed by the Compton edge. The gap contains some
events due to single photons undergoing multiple Compton-scattering interactions.
The Compton continuum continues from the Compton edge down to low energies
where the backseatter peak is formed. The backseatter peak is created by 7 rays
that have undergone a head-on collision in another material and scattered back

into the detector, producing a peak at E z EC. Finally, there is the possibility of

59

 

 

3 C .. 5 as
r "5 a.
5 - 3 a i ..
E104? x .5 10 10 to
3 — r_0 O in O
O :- 8. E E o H
0 - ‘- '- ﬂ 2 cu 2
F a C 3 0‘ m a.
35 S O .E 1:
3 8 > .0 at q; %
10 E— .Q 1» E E S 2
U 2
: 3 2 E a q
: : «,2 m e N
“‘- 01 31 8
102.

 

 

 

 

 

 

 

 

10
1
IIIIIIIIIIIIIIIIIIIIIIIILLLIIIIIIIILIII I
0 200 400 000 800 10001200 40 100018002000
energy(keV)

Figure 4.5: The idealized detector response to 2 MeV 7 rays showing the photo-
peak, Compton edge, single- and double-escape peaks, annihilation peak, and the
backscatter peak. In the idealized case, all interactions except Compton scattering
result in delta functions.

pair production. Pair production that occurs inside the detector most likely leads
to annihilation and absorption of the energy by the detector. However, a portion
of the annihilation photons escape the detector, causing a single-escape peak to
form at E = hu’ - m002 and double—escape peak at E = hu’ — 2m0c2. Finally,
pair production may occur in surrounding materials, and annihilation photons will
produce a delta function at E = moc2. While the principles discussed here for the
ideal case form the basis of the actual output of the detector, there are signiﬁcant

differences as described in the following section.

60

4.4 Detector and electronics modeling

Once energy has been deposited, the simulation treats the detectors as described
in Section 3.3. The 'y ray deposits energy in a few locations, which are individually
converted into scintillation photons that are attenuated towards the ends of the
bar. The number of scintillation photons Np], produced depends on the energy
deposited Edep, the efﬁciency of the scintillation process (em-m = 12% for NaI),

and the energy of the scintillation photons (the average is Eph = 3 eV).

E e '
Nph = m (4.12)
huph

While the attenuation accounts for the gross process moving away from the location
of the interaction, there is a local effect that must be included: some of the photons
will never move along the length of the bar. Due to the complexity of determining
what portion of light would be transmitted for every location in the bar, a simpliﬁed
model is implemented. The angle of incidence wait for total internal reﬂection for
N aI(Tl) is

wait = arcsin (M) = 32.7° (4,13)
nNaI

where nvac = 1 and n Na] 2 1.85 are the indices of refraction. By examining
Figure 4.6, one notices that the range of total internal reflection, w = (1120,“, 90°),
is equivalent to the emission angle range 0 = (0°, 90° — t/Jcn't). Thus, the portion
of scintillation photons traveling in each direction that survive the ﬁrst interaction

with the scintillator wall is approximately
f = 1 — sin «pm, z 46%. (4.14)

Finally, the scintillation photons are attenuated to the ends of the bar. Due to

the geometry of the detector, only T = 27% of the photons pass through the

61

\\\\\\\\\\\\\\\ \\\\\\\\

Crystal surface

    

Scintillation point

 

\\\\\\\\\\\\\\\\\\\\\\\\

Figure 4.6: The relationship between the angle of incidence z/J to the crystal surface
and the angle of emission 0 from the scintillation point.

quartz window and into the PMT, which has a quantum efﬁciency of 20% at the
wavelength emitted by NaI(Tl). At this point, an additional factor of k = 0.88
is applied to the number of photoelectrons to account for the simplicity of the
model, e.g. the angular range cutoff and the transmission through the window,
and the use of ﬁxed values for integrated quantities, e.g the quantum efﬁciency.

The method of determining the value of k is described below.

As an example, a 1 MeV 7 ray may deposit 700 keV at z = 10 cm and 300 keV
at z = 8 cm. The numbers of scintillation photons N,- that reach the end of each

bar are

 

 

N1 = Edepfsdntf (”W 2+ Z1) = 1668 (4.15)
2,21! h
P

2727p}!

where the index 3' has been summed over the two interactions. After passing

through the windows and into the PMTs, the numbers of photoelectrons produced

62

in the PMTs are

NPE,1 = T€quantkN1 = 79 (4.17)

The number of photoelectrons in the ﬁrst stage of the PMT is the point of mini-
mum statistics for the system. Statistical ﬂuctuations are factored in by sampling
Poisson distributions with mean N p3, For this example, let N pEJ = 90 and
N p33 = 171.

The N p13,,- are now used to reconstruct the position and energy of the 7 ray.
The reconstructed position is given by

1 NPE,1
Zrec = — 1n

2;! NPE,1

 

= 9.4 cm. (4.19)

with the coefﬁcient of linear attenuation u = 0.047/ cm. The energy is

 

Em = “/1?ng NP” = 976 keV. (4.20)

The scaling factor g is determined by ﬁtting the centroid of the simulated pho-
topeak and mapping it to the 7 ray energy. The reduction factor k given above
was determined by matching the photopeak width of the simulation to that of the
data. In this manner, the energy and position spectra of the 1836 keV 7 ray in

88Y are reproduced.

The energy threshold has an effect on the detector response at lower energies as
discussed in Section 3.3.4. The threshold is modeled by reconstructing the discrim-
inator voltages for each PMT signal. These voltages are scaled by a gain factor gt),
that converts the physical voltage to the proper value when applied to the simula-

tion’s PMT output. Because the ampliﬁer’s fast timing output has a short shaping

63

time, the signal entering the discriminator has a large variation in amplitude. This
variation in the amplitude of the input to the discriminator is represented in the
simulation by a variation in the amplitude of the threshold. Each detector has a
ﬁxed mean threshold; each simulated event samples a normal distribution about
the mean to determine if an effective threshold has been exceeded and the event
accepted. The gain factor and threshold width were determined by the process of
x2 minimization of the simulated response function to the position spectrum of the
898 keV 7 ray. Because the spectrum is an energy-gated position spectrum, both

the energy and position response of APEX are reproduced by the x2 minimization.

Continuing the example from above, the simulated mean thresholds in this bar
may be Eth,1 = 159 and Eth,2 = 123. Although N pE,2 > Eth,2a the discriminator
may or may not trigger due to the statistical nature of the process. Following the
conﬁguration of the APEX electronics, if either threshold on a bar is exceeded,
the data are recorded from both PMT channels. The ﬁnal result of the APEX
simulation is shown in Figure 4.7. The isotropic position response of the 1836 keV
7 ray is reproduced, as well as the threshold-inﬂuenced response of the 898 keV
7 ray. The energy response is shown in Figure 4.8. The disparity in the position
response between the simulation and the data is due to the breakdown of the
exponential attenuation at the ends of the detector. By rejecting the ends of
the bars, the difference in the 898 keV peak area in the simulation and data are
reduced to 1% compared to the 7% difference shown here. However, the 7-rays
from excited beam projectiles are forward focused, and the efﬁciency in the center
of the array is reduced due to the threshold. The signiﬁcance of this discrepancy
may be gauged by plotting the positions of those events that fall within a gate
on the 76Ge photopeak shown in Figure 4.9 and those events within an equal-
width gate on the high-energy side of the photopeak. The difference between the

two histograms is representative of the interaction position of photopeak 7 rays.

64

 

g mung/4 mm

IlllllllIlllllllllllllllllll

 

 

 

 

 

l l l l l l l l l l l l l l l l l l l l l l l 11 l I
300 -200 400 o 100 200 300

position (mm)

Figure 4.7: The position raeponse of APEX for an 88Y 7-ray source placed at 6 cm.
The 1836 keV 7 ray (lower) and 898 keV 7 ray (upper) data are shown in black
and the simulation in grey.

Figure 4.10 indicates that the photopeak events mostly lie in the region where the
simulation and data diverge. It is for this reason that the simulation is used to
determine a peak shape and the efﬁciency is established relative to the previously
measured transition rate of 52Ti.

The maximum laboratory-frame scattering angle encountered in this experi-
ment is 306° for 52Ti. The acceptance of the S800 is 20 msr formed in an approxi-
mately ellipsoid shape of 7° in the dispersive direction and 10° in the nondispersive
direction. Moving the target upstream from the position of optimal acceptance to
the S800 would increase the proportion of 7-rays emitted within the angular range
subtended by APEX. However, the beamline magnets were known to be misaligned
during this experiment, resulting in a reduction of the nondispersive acceptance.
A measurement of the acceptance of the S800 is prerequisite to determining an

upstream target position, and the measurement in turn requires that a scattering

65

 

§

3

TTllllllllllllll]

counts/4 keV

  

EEEE

   

)—-
r.—
_
—
_

 

 

LJJllllnglllllllLLlllll[L411

800 1000 1200 1400 1600 1800 2000
energy (keV)

C

Figure 4.8: The energy response of APEX for an 88Y 7—ray source placed at 6 cm.
The data is shown in black and the simulation in grey.

chamber be installed at the target position rather than APEX. Ultimately, the
misalignment was not quantiﬁed but merely ﬁxed soon after the experiment. It
is possible to have positioned the target with trial and error, a time-consuming
process subject to the limitation of determining an acceptance cutoff in the low

statistics of the affected larger scattering angles.

The goal of the simulation is to produce response functions for ﬁtting to a
7-ray spectrum from a Coulomb excitation reaction. The primary beam of an
experiment can be directed into the experimental area with high intensity and
little cost in time. The primary beam for this experiment, 76Ge, was Coulomb
excited on a 209Bi target with a speed of ,6 = 0.396 at midtarget to furnish a test
of the simulated 7-ray energy peak shape of projectile Coulomb excitation. This
test measurement was made in six hours in contrast with the two days required for

the 50Ca measurement. The results of this response function ﬁt to the data with

66

 

‘é‘

   

counts/20 keV

.0
e
.0
.0
e..
0..
a...
0..

150

100

IIIIIIIIIIIIIITIIIIIT

 

 

l l l 1 l I l l I l l l I l l l l l l l

400 600 000 1000 1200 1400
energy (keV)

 

Figure 4.9: The Coulomb excitation energy spectrum of 76Ge traveling at B =
0.396 with a ﬁt of the response function (thick, solid line) with a continuum back-
ground (thick, dashed line).

an exponential plus a constant background is shown in Figure 4.9. Fitting the
same data with a Gaussian function rather than the response function produces a
width 6% larger and an area 10% larger; the minimization, however, results in a X2
two orders of magnitude larger. Apparently, the constrained width and Compton

continuum of the response function are necessary facets of the ﬁtting function.

67

 

counts/10 mm

     
     

I '.
-

 

   

a.
I .J

I ' a I
o ' e
a.
I
- I I I I I I I I I I I L4 I I I I I I I I I I I

-200 -100 o 100 200 ‘300
position (mm)

IT]ITTIIIIIIIIIITTSIIIIIITIllllIlllllllllllll

 

 

D
_

 

go

Figure 4.10: The position response of the events contained in a gate on the 76Ge
photopeak of Figure 4.9 (solid) compared to those of an equal-width gate placed
on the high energy side of the photopeak (dashed). The difference in the two
histograms is representative of the position response of photopeak events, which
lie mostly in the region where the simulation diverges from the data.

68

Chapter 5

Data analysis and experimental

results

With the experimental apparatus described and the response of APEX simulated,
the process of extracting the neutron effective charge is now demonstrated. Due
to the divergence between the simulated position response of APEX and the data
near the ends of the bars, the B(E2;0if —+ 2?) of 52Ti is used to determine the
efﬁciency of APEX. The B(E2;0'1Jr —> 21") of 50Ca is then found, and a neutron
effective charge is calculated. First, the 52Ti nuclei are distinguished from the
beam contaminants in the focal plane of the S800 Spectrograph. The particle
identiﬁcation includes corrections for measurement effects such as variations in the
ﬂight paths of the beam nuclei to better separate the isotopes. By selecting the
7 rays that are time-correlated with the scattered 52Ti nuclei, the random 7-ray
background is minimized. To avoid the possibility of nuclear excitations, only 7
rays emitted by nuclei that are scattered by less than a maximum angle related to
a minimum impact parameter will be accepted, and, to reduce the beam-correlated
background, the 7 rays from nuclei that are scattered at very small angles, i.e. the

interactions with large impact parameters, will be rejected. With this selection

69

of events, the 7 rays are used to tag the Coulomb excited nuclei and thereby
measure the Coulomb excitation cross section of 52Ti. Using the intermediate-
energy Coulomb excitation theory of Alder and Winther[34], the B(E2; 0? —> 2?)
of 52Ti is deduced from the angle-integrated Coulomb excitation cross section.
The transition rate is compared to a previous measurement to ﬁnd the efﬁciency
of APEX. Next, using the same method described for 52Ti, the 7—ray cross section
of 50Ca is extracted from the data. Feeding of the 2?” state from excitations to
higher-lying states is inferred to be small from shell-model considerations. The
B(E2; 0? —+ 2?) is deduced, and the similarity in the reaction kinematics and 7-
ray distribution and energy of 52Ti and 50Ca permit the efﬁciency determined from
the 52Ti reaction to be applied to the 50Ca case. Comparing the transition rate
of 50Ca to that predicted by the GXPF1A effective interaction leads to a value of

the neutron effective charge.

5.1 52Ti B(E2;0{L —+ 2?) measurement

The efﬁciency for the 50Ca cross section measurement will be determined in this
section through the measurement of the B(E2; 0? —> 2?) of 52Ti. The particle
identiﬁcation procedure will be examined, and corrections to the particle identiﬁ—
cation spectrum will be detailed. The software gates on the APEX times and the
projectile scattering angle will be demonstrated to produce a reduced-background
7-ray energy spectrum of 52Ti. A simulated response function will be ﬁt to the en-
ergy spectrum, and a cross section determined. The observed B(E2; 0? -—> 21*) of
52Ti will be deduced. From comparison to a previous measurement, the efﬁciency

will be found.

70

5 . 1 . 1 Particle identiﬁcation

The procedure for identifying isotopes in the A1900 is described in Section 3.1.
From the A1900, the beam is directed onto the secondary target at the target
position of the S800. From the magnetic rigidity of the S800 Analysis Line and a
calculation of the energy loss of the beam in the target using the program LISE[61],
the magnetic rigidity of the S800 Spectrograph magnets is selected to guide the
elastically-scattered and Coulomb-excited nuclei to the focal plane. In this ex-
periment, the 52Ti arrived on target as a member of a beam cocktail and was
identiﬁed in the focal plane of the S800 Spectrograph. The AE-tof spectrum of
the 52Ti beam is shown in Figure 5.1. On the horizontal axis is the time of ﬂight
between the 8800 Object scintillator and the 8800 Focal Plane scintillator, and on
the vertical axis is the energy loss in the focal plane ion chamber. The relative
intensities and positions of the loci are compared to the particle identiﬁcation his-
togram from the focal plane of the A1900 to identify the nuclei. The 52Ti locus is
indicated in the ﬁgure along with the primary contaminant, 53V. The 52Ti beam
reached the focal plane with 66% purity, and 5.5 hours of data were collected at

an average total beam rate of 3.6x 103 pps.

In this experiment, the limited beam rate requires that the focus mode of the
S800 be used. The momentum spread at the focal plane causes a broadening of
the loci in the particle identiﬁcation spectrum, which can be reduced by the intro-
duction of corrections for beam-parameter correlations. For example, by sweeping
out a larger arc in the S800 dipoles, a high-momentum particle will enter the focal
plane at a different position than a low-momentum particle of the same species.
Since the time of ﬂight between the S800 Object and Focal Plane depends on the
path taken by the projectile, a correlation is formed between the time of ﬂight

and the dispersive position in the focal plane. Similarly, a particle entering the

71

 

    

  

,Ig'll,l‘l.ll,l|IIll

   

  
 

4-

g,
400:—

no

, . ,'..
. , - i
. .1.‘
- . - i
‘4. .3.
.~ ' . ,-
. , .
. -x_ . - ., ;. .
. .- .. -
‘.' ..‘ 2,
‘ ' I " "1.4 ‘u I
. " ‘ v .' '.

 

 

300 ‘ 0 i
7 -650 -600 -550 '400
101 (ch)

Figure 5.1: The particle identiﬁcation spectrum with 52Ti and the other primary
constituent 53V indicated. Corrections for beam parameter dependencies have not
been implemented. Compare with Figure 5.5, which does have the dependencies
removed.

72

 

   
        

 

 

50,] F -
g : 140
31,3, _
a 50—. 120
g _- , .
'7, ‘5; 1 00
h 1'
§~ 7: " so
a 0 ~.

I . . so

-50 —- 40
: .; ' :3. . 20
'100; l I I l 1 ;.-I..|J—L:l 2'..- L -".r“l q-l - . I. . " :1- - . . o
-750 -700 -650 -600 ~550 -500 450 -400 -350
tof (ch)

Figure 5.2: The correlation between the position at CRDCI in the S800 focal plane
and the time of ﬂight. A linear correction is applied to the time of ﬂight to improve
the particle-identiﬁcation spectrum.

3800 at a larger angle from the beam direction will travel further than a particle
entering at a smaller angle, also leading to a. correlation between the time of ﬂight
and the angle in the focal plane. The correlation between the time of ﬂight and
the dispersive position in the ﬁrst CRDC is illustrated in Figure 5.2, and the cor-
relation between the time of ﬂight and the angle in the focal plane can be seen in
Figure 5.3. The dependency of time of ﬂight on the dispersive position and angle
in the focal place is removed by introducing linear corrections. The corrected time

of ﬂight is

tcorr = tfp—obj — ammCRDCl — agﬂfp (5.1)

where $031301 is the dispersive position on CRDCl and t fpnobj is the time of
ﬂight between the S800 Object and Focal Plane. The factors ax and a9 are their

respective corrections.

In these data and in data collected during previous experiments using the $800,

73

 

 

 

’6
8
0
a 100
C
n .—
2 I L;-
g 80
a _
g E; so
~ I 40
-0.05—
_ 20
'0. _ . ' h i f" o
-700 650 6% -550 -500 450

400
tof (ch)

Figure 5.3: The correlation between the angle in the S800 focal plane and the
time of ﬂight. A linear correction is applied to the time of ﬂight to improve the
particle-identiﬁcation spectrum.

the energy loss in the focal place ion chamber is dependent on the dispersive
position as shown in Figure 5.4. The origin of this dependency is not known, and
a phenomenological correction is applied to remove the dependency. The correction

takes the form

AEebc‘O‘x) :c < :20
AEcgrr = (5.2)
AE :L‘ > (1:0

with AE the measured energy loss in the ion chamber and $0 and b chosen to

make AEcorr constant in the dispersive direction a: at CRDCI.

The AE—tof spectrum of the 52Ti beam with beam parameter corrections is
shown in Figure 5.5. The reduced width of the loci relative to the uncorrected
particle identiﬁcation is noticeable for this Ap/ p = 0.5% beam and will be much
more signiﬁcant in the case of 50Ca due to the larger 3% momentum width. A

software gate is placed on the 52Ti locus in the corrected spectrum to select the

74

 

 

  

E _ 30
1.5. :‘ -
c 200 —.
o _. 25
E H
1 l I
3 0° 2°
é 0 _—j- 15
'° I
.100 :—- 10
-200 :— 5
300 300 °

 

900
AE (ch)

Figure 5.4: A histogram showing the dependence of the ion chamber, AE, on
the dispersive position, 3:. A phenomenological correction is applied to make AE
constant in z. The data shown here is from the 50Ca beam, where the effect is
more pronounced.

particle-correlated 7—ray events.

5.1.2 7-ray spectrum

The immediate goal is to extract the spectrum of the 52Ti 21'" —> Oil- transition at
1050 keV from the background radiation. Due to the equiprobable emission of 7
rays axially from the beam direction, the 7-ray energy spectra of the individual
APEX bars can be summed into one histogram, and all APEX energy histograms
shown in this work are summed. Three detectors were not functioning during the
course of this experiment and are not included in the analysis. The photopeak
is expected to contain a few hundred counts spread over approximately 180 keV
FWHM. Since APEX counts at over 4 kHz on room background, the background
radiation must be minimized so as not to overwhelm the photopeak.

With the 52Ti particle identiﬁcation gate applied, the 7—ray energy spectrum

75

 

 

 

g 966
§ 000 50°
Ig- .
700 ‘ 400
600 300
500 200
400 1 00
am 1 1 1 1 L14 1 - ‘11 l 2 l 1 '1 :1: 1L11 1 j_L 1...: . r. .1 _]_1 I 1.:I'1--1 o
-700 -550 -500 -550 -500 450 400
tot, corrected (ch)

Figure 5.5: The corrected particle identiﬁcation spectrum with 52Ti and the other
primary constituent 53V indicated.

contains only those event that included a 7—ray trigger within the 200 ns width of
the particle-7 coincidence gate (see Section 3.3 for timing details). With 4.4x 107
52Ti nuclei in 5.5 hours, the coincidence gate was open for 8.8 s, or 0.044% of the
total data collection time. Further, a 7 ray was not detected in coincidence with
all projectiles, and multiple 7 rays up to the number of active detector bars can be
recorded for each particle event during the longer ADC gate; however, the 4.5x 105
particle-7 coincidence triggers with a 1.7 as ADC gate leads to an open gate on
each of the 21 ADCs 3.8 x 10‘3% of the total collection time. Random background
is signiﬁcant only to this small factor.

In addition to the room background, the energy spectrum contains beam—
correlated background that can be distinguished from the promptly emitted 7
rays in the time spectrum of each PMT channel. This time spectrum is the dif-

ference between the detection of a projectile in the 8800 and the detection of a 7

76

ray in APEX. Since the rise time of NaI(Tl) is less than 5 ns, the 7 rays emitted
promptly upon the occasion of a 52Ti nucleus passing through the target will pro-
duce a timing peak within the 200 ns coincidence gate. The time spectrum for a
single PMT channel is shown in Figure 5.6. On the left side of the spectrum are
the random background events, and in the center is a Gaussian peak of prompt
7 rays. Extending to the right are beam-correlated background events, such as
from target breakup and the creation of short-lived isotopes in the beam pipe.
A software gate is placed on the prompt 7—ray events so that off-prompt 7 rays
are omitted from the energy spectrum. Since the thresholds effect discussed in
Section 3.3 prevent a large portion of 7-rays from simultaneously surpassing the
threshold in both channels of a detector bar, the gates on the two time spectra
are combined with a logic OR to form the time gate of the detector. The detector

time gate is applied to each detector bar individually.

The ﬁnal gate is placed on the scattering angle of the projectile with two pur-
poses, to minimize background like the previous gates and to avoid nuclear con-
tributions to the excitation. The Rutherford-like cross section of the laboratory
scattering angle of 52Ti is histogrammed in Figure 5.7. A gate is placed to reject
nuclei scattering at large angles to avoid those reactions with small impact param-
eters where nuclear excitations are possible. The maximum scattering angle shown
in Figure 5.7 corresponds to the minimum impact parameter bmin = 13.5 fm as
related by Equation 2.14. The rejection of small angles is due to the fact that
Coulomb excitation is more probable for smaller impact parameters (larger scat-
tering angles) while elastic scattering favors small scattering angles; therefore,
removing the very forward scattered nuclei reduces the background by a large
amount while removing a smaller proportion of the angle-integrated Coulomb ex-
citation cross section. The effect of this minimum angle gate is demonstrated in

Figure 5.8. The shaded histogram is the energy spectrum of 52Ti with time gates

77

prompt peak
I

I

 

   
  

 

beam-correlated
background

      

random
background

 

 

 

4o

20

300 400 -200 0 200 400
time (oh)

Figure 5.6: The time spectrum of a single APEX PMT. This spectrum shows the
difference in time between the detection of a beam projectile within the 52Ti gate
by the 8800 and the detection of a 7 ray by APEX. The low random background
can seen on the left side of the spectrum. The 7 rays promptly emitted after the
nuclei pass through the target form a Gaussian peak in the center of the spectrum,
and the beam-correlated background continues to the right. One channel is 227 ps.

and a scattering angle range of [0, 0min): and the solid line histogram is the energy
spectrum of 52Ti with time gates and an angle range of [Om-n, 0mm) as shown in
Figure 5.7. The 2? —+ Oil" 7-ray photopeak at 1050 keV is visible in the latter
case and is not in the former. The analysis of the 50Ca data shares a selection
of this same impact parameter range; therefore, the minimum impact parameter
is selected to not exceed the nuclear interaction radius of both nuclei, and the
maximum, bm = 40.0 fm, is chosen to optimize the peak-to-background ratio of

the two 7-ray energy spectra.

78

6min 9max

 

 

.5 _
350000 :-
40000 —
: small
I 10% of particles 'mrrﬁteters
30000 — 85% of Coulomb-excitation pa
I cross section
20000
90% of particles
1 0000 15%.of goulomb-
excrtatlon
cross section
L.

 

 

 

 

 

I I I I I I I I I I I I I I I I I l r ‘ ‘ ‘ l A I ‘ 1 l l l I I I L
°o 1o 20 so 40 so so 70

scattering angle (mred)

Figure 5.7: The scattering angle of 52Ti nuclei with the maximum and minimum
scattering angles indicated. The relative cross section for each section demonstrates
that the signal-to—noise ratio is improved within the angle range [6mm 0mm).

5.1.3 Cross section

The 7 rays tag the Coulomb excited nuclei to determine the excitation cross sec-
tion. The number of 7 rays observed is determined by ﬁtting the energy spectrum
generated by the gates discussed in the previous section with a simulated response
function. Lacking the knowledge of the efﬁciency of APEX due to the divergence
between the simulated position response and the data, the observed cross section

and the associated uncertainty will be calculated.

The simulated response function is created as described in Chapter 4. The
threshold parameters are determined though the x2 minimization to an 88Y source
measurement taken just after the in—beam measurement. The decay lifetime is
known from a previous intermediate-energy Coulomb excitation measurement by
Dinca et al. (see the next section for details). The position of emission along the

beamline is shown in Figure 4.3. The midtarget velocity ﬁmid and the after-target

79

 

>500

counts/90 ke
§

 

IIIIIIIIIITTTIIITTIII IIIIIIIII IIIIIIIIIIIIIIIII

 

 

 

0 LJ I I I I I I I I I I I I L I I I I I I I I
600 800 1000 1200 1400 1600
energy(keV)

Figure 5.8: The 7-ray energy spectrum for 52Ti with time gates applied. The
shaded histogram includes the scattering angle range [0, 6min)1 and the black line
histogram includes the range [0mm 19mm). Although the shaded histogram in-
cluded 90% of the scattered 52Ti nuclei, no photopeak presents itself clearly above
the background. The peak-to—background ratio can be improved by removing these
very forward scattered nuclei, leaving the unshaded histogram.

80

velocity .Bpost are calculated from the magnetic rigidity of the beamline magnets
and the thickness of the target, 184 mg/cm2 n“"tAu. The majority of the 7 rays

are emitted outside of the target, and ﬁpost is used for Doppler reconstruction.

The response function generated by the GEANT4 simulation for the 2? —> 0?—
transition and a continuum background is ﬁt to the 7—ray energy spectrum of 52Ti.
The continuum is an exponential plus a constant and is allowed to vary with the
ﬁt. Figure 5.9 displays the 7-ray spectrum with the continuum as a thick, dashed
line and the ﬁt drawn as a thick, black line. The number of 7 rays observed is

Afit

lcoinc

N’Yobs : (5'3)

with A fit as the amplitude of the ﬁt and lcomc as the livetime for the coinci-
dence trigger. The number of 52Ti nuclei N B is similarly scaled by the livetime
of the particle trigger. With the number density NT of the target known, the
efficiency of APEX is the remaining factor required to calculate the cross section

from Equation 2.13. Instead, the observed cross section will be deﬁned as

_ N7,f-*i

Ui-rfobs _ NT NB (5'4)

and the efﬁciency will be addressed in the following section. Since the 2?" state
decays through 7 emission, the number of emitted 7 rays is equivalent to the
number of excitations. No feeding was observed in Dinca’s measurement of the
B(E2;0’1*' —+ 21+) of 52Ti, and none is assumed here. The resulting cross section
is 98(17) mb. Poisson statistics for N B and livetimes provide a small uncertainty
contribution. The major contributions to the uncertainty are the ﬁt (13%) and
the simulated response function (10%). The ﬁt uncertainty is taken from the

covariance matrix for the ﬁt parameters and the data. The uncertainty in the

81

response function originates in the simulation parameters. The placement of the
target was varied by 0.5 cm, a sufﬁciently large distance to be just noticeable
in the position response of APEX, to ﬁnd an uncertainty of 3%. The simulated
threshold parameters are estimated to contribute 8% to the uncertainty. The
threshold settings for the 50Ca beam are different from those of the 52Ti beam to
reduce deadtime, and the uncertainty is the difference in efﬁciency between the two
threshold settings coupled with the difference between the the optimal simulated
energy response and the amplitude of the 898 keV peak in 88Y. The simulated
threshold amplitude strongly affects the response function, with the diffusiveness

contributing to a lesser degree.

5.1.4 In—beam efﬁciency

The observed transition rate is calculated using the theory of Alder and Wintherl34],
and the efﬁciency of APEX is determined by comparing the observed transi-
tion rate of 52Ti to a previously measured value[20]. The Alder and Winther
formulation of the theory of intermediate-energy Coulomb excitation is intro-
duced in Section 2.4.2. The observed transition rate of 52Ti is deduced to be
B(E2; 0? —> 2?) = 382.6(65.9) e2fm4. The conversion of the angle-integrated
cross section to the transition rate depends on the the angular range over which
the integration occurs, and the uncertainty in the scattering angle (66 = 05°) is
added in quadrature to the cross-section uncertainty to form the total transition
rate uncertainty.

The B(E2;0'f —+ 2:) of 52Ti has previously been measured by Dinca et al.
using intermediate-energy Coulomb excitation with the Segmented Germanium
Array at the NSCLIZO]. SeGA has been used many times for transition rate
studies with success as demonstrated not only by Dinca’s measurement of the

B (E2; 0?" —> 2i") of the high-intensity, stable 76Ge beam shown in Figure 2.9a but

82

 

countsno keV
f ' Us

8
O
I I I

250
200 '

150 _

 

 

 

 

100_

50;

E
oLLIJllIIJdIllIIlIlIlIII
600 800 1000 1200 1400 1600
energy(keV)

Figure 5.9: Fit of the response function plus continuum background to the 52Ti
7-ray spectrum. The ﬁt is the thick, black line and the continuum is dashed.

83

 

natAu target 184 mg / cm2

6 0.364

N, 2279 (303)

E(2f) 1033 (78) keV
gobs 98 (17) mb
B(E2;0i: —4 290,3 383 (69) eifm:
B(E2;0l —> 21 )Dinca 567 (51) e fm
e 0.67 (14)

 

Table 5.1: The cross section and B(E2; 01'" —> 2?) of 52Ti is compared to to the
B(E2; 01F —> 2?") measured by Dinca et al.[20] to derive the efﬁciency.

also by the other published test cases shown in the ﬁgure. Additionally, the mea-
surement of 52Ti utilized natAu targets of two different thicknesses, permitting a
further veriﬁcation of the method by measuring the B(E2;0'1" -—> 2:") of the tar-
gets. Finally, the transition rate of 52Ti was found to be in agreement with an

earlier measurement by Brown et al.[62].

Dinca determined B(E2; 01+ —> 2?) = 567(51) e2fm4, and the observed transi-

tion rate is scaled to the Dinca value by the factor

B E2;0+ _. 2+
5 = B [(9 . +1 +1)0b3 = 0.67(13). (5.5)
( 2.01 ->21)

 

Dinca

This efficiency coupled with the efﬁciency of the simulation is the true efﬁciency
of the APEX Array for this reaction. The photopeak efﬁciency is approximately

8.5%. The results are summarized in Table 5.1.

5.2 50Ca B(E2; 01* —+ 2;“) measurement

5.2.1 Particle identiﬁcation

The gating and calculation for the 50Ca transition rate follows the same method

that was demonstrated for the 52Ti transition rate. The particle-identiﬁcation

84

 

  

 

 

20W:
£750:- 45

E 40
700:

E 35
650:

E 30
600.—

: 25
5505— 20
500; 15
450E-

: 10
4005— . . .. _. . 5
350:. I I |' I 1 L 1. L I 1'- ‘1: 1 1_L-r11'-'I1'J__1 1 1" o

-800 -700 -600 -500 -400 -300 -200
tof(ch)

Figure 5.10: The particle identiﬁcation spectrum with 50Ca and the other primary
constituents 53Ti and 53V indicated. Corrections for beam correlations have not
been implemented.

spectrum without corrections for the 500a cocktail beam is shown in Figure 5.10.
The loci widths are signiﬁcantly decreased by the application of corrections to the
time of ﬂight and energy loss in the ion chamber, the results of which are shown in
Figure 5.11. 5OCa was delivered with 7% purity, with 53Ti and 54V as additional

beam components, and two days of data were recorded.

5.2.2 Cross section and B(E2;0f ——> 21+)

The 7-ray spectrum background is reduced for the 50Ca measurement following
the method prescribed for 52Ti, timing gates on the prompt 7—ray time peak of
each PMT channel and a selection on the scattering angle. The contribution from
feeding is found to be small from shell-model considerations, and the observed

de-excitation cross section is calculated. The 500a Coulomb excitation reaction is

85

 

 

35° 1 1 L 1‘_L 1'1 1 1 I1.:17'-.1:.:1_L111111-1";~ 4 1'1 I 1 1 1 '1 0
-800 -700 -600 -500 400 -300 -200
tot (ch)

Figure 5.11: The corrected particle identiﬁcation spectrum with 50Ca and the
other primary constituents 53Ti and 54V indicated.

shown to be similar to that of 52Ti, demonstrating the propriety of applying the

previously determined efﬁciency.

The scattering angle of 50Ca is restricted to avoid nuclear excitation and to min-
imize background. The minimum and maximum impact parameters are those used
for 52Ti although the corresponding scattering angles are slightly shifted accord-
ing to the A and Z of 50Ca (Equation 2.14). The simulated threshold parameters
were determined through the X2 minimization to an 88Y source spectrum before
the 50Ca data were collected. The response function with an exponential plus con-
stant continuum ﬁtted to the 7-ray energy spectrum is displayed in Figure 5.14.
The ﬁt is the thick, black line, and the continuum is dashed. The resulting ob-
served cross section is 18.4(4.9) mb with the error largely due to uncertainty in the
ﬁt (24%). The other sources of uncertainty are the same as those given for the 52Ti

case, with uncertainty of the simulated response function dominating (10%) after

86

the ﬁt. Additions to the cross section due to feeding from excitation to higher-lying

states will be examined with nuclear structure considerations.

A level diagram of the lower-lying states of 50Ca is shown in Figure 5.12. The
21" state lies at 1027 keV, and feeding is possible from the 23' level at 2999 keV,
the I?” level at 3519 keV, and the 3; level at 3993 keV. The 3519 keV level has
been identiﬁed as a If conﬁguration in a recent deep inelastic transfer reaction
measurementl63]. Table 2.2 shows that 0? —> 11+ is an M1 transition, which
is suppressed at intermediate beam energiesl64]. In Reference [7], the 3; state
in 52Ca is found to be due to cross-shell excitations of protons, a situation un-
likely to change signiﬁcantly with a small reduction in the number of valence
neutrons. The excitation to the 31" in 50Ca may therefore be estimated from the
B(E3; 0?" —> 31—) = 6.5x 103 e2fm6 transition rate in 48Ca[22]. A ﬂ-decay studyl65]
determined a 38(5)% branch to the 2iF state, resulting in a contribution of 0.49 mb
to the 7-ray cross section. The feeding contribution from the 2; state may be esti-
mated from the GXFPIA predicted excitation rate, B (E2; 0? -—r 23’ ) = 41.3 e2fm4
(en = 0.8). The ,B-decay measurement and a deep inelastic transfer reaction study
did not reveal feeding from the 23’ to the Of. Assuming a branching ratio of at
most 1% leads to an estimated contribution of 0.10 mb to the cross section. To—
gether, feeding from the 2; and 31— may contribute up to @059 mb to the cross
section. However, this value is the feeding contribution to the cross section, not
the observed cross section. Since the feeding contribution is small, the transition
rate will initially be calculated assuming no feeding. A similar treatment of feeding

can be found in Reference [66].

The efﬁciency determined with the 52Ti measurement can be used for the 50Ca
measurement due to the similar kinematics of the two reactions and the nearly
equivalent 7-ray angular distribution. Both reactions occur on m“Au targets, and

the )6 at midtarget, the average location of the Coulomb excitation, differs by

87

 

 

 

 

 

 

 

'3‘; 8
3' N g: 3223
8 9.’
1+ 3’. 8 3519
#2
2+ °’
'- M
3
3
>
9
0
C
LU
8
2+ 4- 1026
__ 0+ 0

 

 

 

 

 

Figure 5.12: The lower-lying states of 50Ca from the NNDC[22] with J 7' determined
through ,6 decayl65] and deep inelastic transfer reactions[63].

 

natAu target 245 mg/cm2
6 0.363

N, 973 (238)

E(2;*') 1085 (80) keV

a 27.3 (9.1) mb
B(E2;0f—.2;t) 120 (41) e2fm4
An 14.14 efm2
en 0.77 (13)

 

Table 5.2: The measured 50Ca cross section leading to the ﬁnal transition rate
value, B(E2;0{r —> 2f) = 120(41) e2fm4.

0.1%. The energy of the 50Ca 2? state lies 34 keV lower in energy, less than the
FWHM of the photopeak. The calculated angular distributions of 7-ray emission
are shown in Figure 5.13; the 50Ca and 52Ti distributions very nearly overlap.
Using the theory of Alder and Winther, the 50Ca transition rate is found to be
B(E2;0i" ——+ 21") = 120(41) e2fm4. The results are summarized in Table 5.2.
Returning to the feeding contribution, the 0.59 mb contribution to the cross section

is ~2% of the cross section, an insigniﬁcant amount relative to the 33% uncertainty.

88

 

 

 

 

 

 

 

0.00, 4 -m,
0.04:
0002’ ’ ‘‘‘‘‘‘‘ ‘
’ s ’ s
A * ’ \ I’ ‘s
Q L I I s .
V L ‘ l ‘
a, 0.00r ,4 A
4 ‘ I ‘
i 1 I .
-0.02: g ,' f
. 1 ”
004' ‘ ' ‘
— . P \ I 1
t ‘
_0.w’. - 9 #1 1m . . . 1 1 n n 4 .
0 50 100 150
0(deg)

Figure 5.13: The difference in the angular distribution of 50Ca from that of 52Ti
(solid line) is small. The difference in the angular distributions of 76Ge and 52Ti
(dashed) is shown for comparison.

5.3 Effective charge in the pf shell

In Section 2.3.4 the question of effective charge in the pf shell is proposed. The neu-
tron transition amplitude of 50Ca is predicted by the GXF PIA to be 14.14 efm2[24].

Following from Equation 2.5, the neutron effective charge is then

BE2;0+ 2+
en=\/ ( 2112—, 1)=0.77(13) (5.6)
n

 

 

The larger effective charge suggested by du Reitz’s study of 51Fe and 51Mn mirror
nuclei, en z 0.8, is indicated by the value derived in this work. According to
empirical results[l6], the difference in effective charge ranges from the Coulomb
value, 6,, - en = 1.0, to 0.35. The proton effective charge is then expected to lie

between 1.01 and 1.55, a range that includes the du Rietz value ep z 1.15.

89

 

counts/90 keV
8

 

 

 

250

200

 

 

 

 

150—11J111l111l1L1IL111111l
600 800 1000 1200 1400 1600
energy(keV)

Figure 5.14: The ﬁt of the response function plus continuum to the 50Ca 7-ray
energy spectrum. The ﬁt is drawn as a thick, black line, and the continuum is
dashed.

90

5.4 Conclusion

In Section 2.3.3, the core polarization was shown to arise from E = 2m excitation
’ of the core protons. The operator derived in a microscopic model that included
2p-2h excitations in the shell-model space resulted in a neutron effective charge
en = 0.57(03). The operator in the macroscopic model, where the polarization
charge is the result of excitations to An = 2 harmonic oscillator vibrations in
the nuclear core, produced a neutron effective charge en = 0.90. Because the
matrix element calculated with the GXPF1A effective interaction agrees with two
older interactions, the GXPF1[1] and the KB3[19], the derived effective charge
suggests that the operator derived with the macroscopic, vibrational model better
reproduces the true operator in this shell-model subspace than the microscopic
shell model operator. The enhanced neutron effective charge additionally indicates

a strengthened isovector quadrupole resonance.

The effective charge of 50Ca inferred in this thesis conﬁrms the use of an en-
hanced neutron effective charge to reproduce the trend of the neutron-rich Ti tran-
sition strengths measured by Dinca et al.[20] (Figure 2.5). The E(2'1f) 2562 keV
of 52Ca measured by Gade et al. [7] suggests an N = 32 subshell closure, in line
with the conclusion drawn from the investigation of the Ti isotopes. However,
there remains no evidence for a shell closure at N = 34; a deﬁnitive statement is
prevented by the large uncertainty in the B(E2; 0?" ——> 21*) of 56Ti. The measure-
ment of the E(2f) of 54Ca and the B(E2;0i" ——> 2?) of 52’54Ca would decisively
resolve the question of the N = 34 shell closure. The measurement of the transition
strength of these two nuclei awaits increased beam production rates and improved

experimental equipment.

In the coming months the new CAESAR (CAEsium Iodide ARray) 7—ray de-

tector will be deployed at the NSCL. To avoid the loss in resolution inherent in

91

the double-sided PMT conﬁguration of APEX, CAESAR will use segmentation for
position sensitivity. The 192 square cylinders of CsI(Na) will have one face of each
crystal nearly entirely covered by a PMT rather than a small fraction. Additionally,
the threshold cutoff will occur at a ﬁxed energy for each detector segment, lead-
ing to an array efﬁciency that will not vary signiﬁcantly with position. Magnetic
shielding will permit high-resolution PMTs to be used near the beamline magnets.
The expected in—beam resolution of CAESAR is < 10% FWHM at 1 MeV with an
efﬁciency of 40%, a great improvement over the 17% FWHM resolution and < 10%
efﬁciency of APEX. The high efﬁciency of CAESAR will make possible the study
of nuclei with currently impractically low production rates, leading to an exciting

expansion in our knowledge of nuclear structure.

92

Appendix A

Simulated response functions of

APEX

A.1 GEANT4

The GEANT4 model of the APEX Array played an essential role in the measurement
described in this paper. Understanding the effect of the thresholds on the efﬁciency
would have been a considerably more difficult task without the ability to test the
hypothesis with a Monte Carlo model. In consideration of the important role of
the simulation, abbreviated portions of key classes are reproduced to permit the

replication of the results.

A. 1.1 Input ﬁles

The input ﬁles contain the following information:

#Mult: multiplicity of primary gamma rays
#Energy: comma separated list of gamma-ray energies (keV) of
# length MULT

93

#Intensity: comma separated list of gamma-ray intensities (Z) of
# length MULT

#DecayDist: comma separated list of gamma-ray transition decay

# constants in distance (mm) traveled during the

# excited states’ lifetimes of length MULT

#BetaMid: midtarget beta

#BeatPre: pretarget beta

#BetaPost: posttarget beta

#BeamWidthFWHM: horizontal,vertical spacial beam FWHM in mm

#BeamMomWidth: beam momentum width (Z); must be defined after

# BetaPre
#AngDist: "uni" unidirectional, "iso" isotropic, or "E2" E2
# angular distribution

#E2Coeff: comma-separated list of E2 angular distribution
# coefficients

#TargetPos: x,y,z target position in centimeters

#Target: 0 for no target, "Au" or "Bi" to insert target
#ReconDist: assumed distance in mm of decay after center of
# target for Doppler reconstruction in mm; state
# after TargetPos

#Collimator: O for no collimator, anything else to insert

# the collimator

#Visualization: O for no visualization, anything else for

visualization
#Verbosity: 0 for none, 1 for some, 2 for all
#Alpha: energy scaling value in keV per channel
#Mu: linear attenuation coefficient in inverse centimeters

94

# Amplitude_{up/dn} = Alpha * exp(-Mu*(L/2+-X))

# where L is the crystal length and X is interaction point

The 76Ge input ﬁle:

Mult: 1

Energy: 562.93
Intensity: 100.0
DecayDist: 3.1

BetaMid: 0.3960
BetaPre: 0.4092
BetaPost: 0.3814
BeamWidthFWHM: 10.0,10.0
BeamMomWidth: 3.0
AngDist: E2

E2Coeff: 1.0,-0.665148,-O.241809
TargetPos: 0,0,6.2
Target: Bi

ReconDist: 3.1
Collimator: 0
Visualization: 0
Verbosity: 0

Alpha: 7.87160

Mu: 0.047
The 52Ti input ﬁle:

Mult: 1

Energy: 1049.73

95

Intensity: 100.0
DecayDist: 0.6

BetaPre: 0.3726
BetaMid: 0.3633
BetaPost: 0.3533
BeamWidthFWHM: 10.0,10.0
BeamMomWidth: 3.0
AngDist: E2

E2Coeff: 1.0,—0.593296.-0.188789
TargetPos: 0,0,6.2
Target: AuThin
ReconDist: 0.6
Collimator: 0
Visualization: 0
Verbosity: 0

Alpha: 7.87160

Mu: 0.047
The 50Ca input ﬁle:

Mult: 1

Energy: 1026
Intensity: 100.0
DecayDist: 3.1
BetaPre: 0.3850
BetaMid: 0.3637
BetaPost: 0.3378

BeamWidthFWHM: 10.0,10.0

96

BeamMomWidth: 3.0

AngDist: E2

E2Coeff: 1.0,-0.587135,-0.184617
TargetPos: 0,0,6.2

Target: Au

ReconDist: 3.0

Collimator: 0

Visualization: 0

Verbosity: 0

Alpha: 7.87160

Mu: 0.047

Compilation constants are deﬁned in Constantsh, and data structures are de-

ﬁned in Data.h.

/*

--Compilation Constants—-

General:
G4int NUM_PRIMARY: maximum number of primary gammas possible

G4int NUM_DETECTOR: number of detectors

ApexDetectorConstruction:
G4bool REMOVE_NON_DETECTORS: remove all objects except the detectors
(note: does not affect the heavimet collimator)

G4bool CHECK_0VERLAPS: Check for geometric overlaps

97

#ifndef CONSTANT8_H

#define CONSTANTS_H

#define NUM_PRIMARY 2

#define NUM_DETECTOR 24

#define NUM_ANGDISTCOEFF 3

#define REMOVE_NON_DETECTORS false

#define CHECK_0VERLAPS false

#endif

#ifndef DATA_H

#define DATA_H 1

#include <globals.hh>

#include <G4ThreeVector.hh>

typedef struct PrimaryShot_t {
// Laboratory frame data
G4double Beta;
G4ThreeVector EmisPos; // location of gamma emission
G4double Energy;
G4double Theta;
G4double Phi;

// Particle frame data

98

G4double PFEnergy;
G4double PFTheta;
G4double PFPhi;

};

typedef struct DetectorOutput_t {
G4int NumHits;
/* position of first hit on detector */
G4ThreeVector FirstHitPos;
/* total energy deposited in the crystal */
G4double EnergyDep;
G4double EnergyUp; /* energy out of up PMT */
G4double EnergyDn; /* energy out of dn PMT */
G4double EnergyRec; /* reconstructed energy */
/* reconstructed energy, Doppler corrected */
G4double EnergyRecDop;

G4double PositionRec; /* reconstructed position */

#endif

A.1.2 Detector construction

The 24-bar NaI(Tl) 7—ray detector APEX is constructed in the class ApexDe-
tectorConstruction along with the lead background shielding and the aluminum

table.

99

#ifndef ApexDetectorConstruction_h

#define ApexDetectorConstruction_h 1

#include <globals.hh>

#include <G4LogicalVolume.hh>

#include <G4VUserDetectorConstruction.hh>
#include <G4Region.hh>

#include <sstream>

using stdzzstringstream;

#include "CApexInitialization.hh"

class ApexDetectorConstruction : public G4VUserDetectorConstruction {

public:
ApexDetectorConstruction();
~ApexDetectorConstruction();

G4VPhysicalVolume* Construct();

private:
G4LogicalVolume* vault_log; // world volume
G4Logica1Volume* beampipe_log;
G4LogicalVolume* shieldLead_log; // Pb cylindrical shield
G4LogicalVolume* shieldSteel_log; // steel cylindrical shield
G4LogicalVolume* table_log;
G4LogicalVolume* heavimet_log;

G4LogicalVolume* target_log;

100

G4LogicalVolume* apexBar_log; // Mother; creates steel for jacket

G4LogicalVolume* apexBarVac_log; // Vacuum placed inside mother

G4LogicalVolume* apexBarCrysta1_log;// Crystal placed inside
// vacuum
G4Logica1Volume*

window_log; // quartz windows place in vac

G4VPhysicalVolume*
G4VPhysicalVolume*
G4VPhysicalVolume*
G4VPhysicalVolume*
G4VPhysicalVolume*
G4VPhysica1Volume*
G4VPhysicalVolume*

G4VPhysicalVolume*

G4VPhysicalVolume*
G4VPhysicalVolume*
G4VPhysica1Volume*
G4VPhysicalVolume*

G4VPhysicalVolume*

vault_phys;
beampipe-phys;
shieldLead_phys;
shieldSteel_phys;
table_phys;
heavimetUp_phys;
heavimetDn_phys;

target_phys;

apexBar_phys[23];
apexBarVac_phys;
apexBarCrystal_phys;
windowUp_phys;

windoan_phys;

G4Region* aCrystalRegion; // crystal cut region

//Measurements

static const G4double m_HeavimetRadius;

static const G4double m_HeavimetCylHeight;

101

static const G4double m_HeavimetGap;

G4ThreeVector m_HeavimetUpPos;

G4ThreeVector m_HeavimetDnPos;

G4bool

m_Target;

G4String m_TargetMaterial;

CApexInitialization* m_pApexInit;

stringstream m_InfoSS;

G4int

static const G4bool m_RemoveNonDetectors;

static const G4bool m_CheckOverlaps;

m_NumberOfDetectors;

// log to store with output file

// number of detectors

void CheckOverlaps(G4VPhysicalVolume* volume) const;

#endif

#include
#include
#include
#include
#include
#include

#include

"ApexDetectorConstruction.hh"

"ApexCrystalSD.hh"
"Constants.h"
<G4Materia1.hh>
<G4MaterialTable.hh>
<G4Element.hh>

<G4ProductionCuts.hh>

102

#include
#include
#include
#include
#include
#include
#include
#include
#include
#include

#include

<G4ElementTable.hh>
<G4Box.hh>
<G4Tubs.hh>
<G4Trd.hh>
<G4LogicalVolume.hh>
<G4ThreeVector.hh>
<G4PVPlacement.hh>
<G4SDManager.hh>
<G4VisAttributes.hh>
<G4Color.hh>

<G4NistManager.hh>

//measured

 

const G4double ApexDetectorConstruction::m_HeavimetRadius =7.0*cm;
const G4double ApexDetectorConstruction::m_HeavimetCy1Height=7.6*cm;
const G4double ApexDetectorConstruction::m_HeavimetGap =4.7*mm;
// Remove all objects except the detectors?

// (note: does not remove the heavimet collimator)
const G4bool
ApexDetectorConstruction::m_RemoveNonDetectors=REMOVE_NON_DETECTORS;
// Debug for overlapping volumes?

const G4bool

ApexDetectorConstruction::m_Check0verlaps = CHECK_0VERLAPS;

103

ApexDetectorConstruction::ApexDetectorConstruction()

:aCrystalRegion(0)

m_InfoSS<<"ApexDetectorConstruction Info:"<<G4endl;

// Get pointer to data initialization object
m_pApexInit = CApexInitialization::Instance();

m_NumbeerDetectors = m_pApexInit->Num0fDetectors(l;

// Check if valid target entered.
m_TargetMaterial = m_pApexInit->Target();
if (m_TargetMaterial=="Au"ll
m_TargetMaterial=="Bi"l|
m_TargetMaterial=="AuThin")
m_Target=true;
else if (m_TargetMaterial=="0")
m_Target=false;
else {
m_Target=false;
G4cerr<<"ERROR> Invalid target material. "
<<"No target included."<<G4endl;

}

m_HeavimetUpPos = m_HeavimetDnPos = m_pApexInit->TargetPosition();
m_HeavimetUpPos +=

G4ThreeVector(0.0,0.0,(m_HeavimetCylHeight+m_HeavimetGap)/2);

104

 

m_HeavimetDnPos +=

G4ThreeVector(0.0,0.0,-(m_HeavimetCylHeight+m_HeavimetGap)/2);

G4VPhysicalVolume* ApexDetectorConstruction::Construct() {

G4cout<<"Constructing Detectors....";

if (m_Targetllm_Check0verlapsllm_RemoveNonDetectors)
G4cout<<G4end1;

if (m_RemoveNonDetectors) {
G4cout<<" Only detector bars created.\n";

m_InfoSS<<"0nly detector bars created."<<G4end1;

// -------- Material Definitions --------

//

G4NistManager* man = G4NistManager::Instance();

G4Material* Al man->Find0rBuildMaterial("G4_A1");

G4Material* Pb man->Find0rBuildMaterial("G4_Pb");

G4Materia1* W = man->Find0rBuildMaterial("G4_W");

G4Material* Fe man—>Find0rBui1dMaterial("G4_Fe");
G4Material* Au = man->Find0rBuildMaterial("G4_Au");

G4Material* Bi = man->Find0rBuildMaterial("G4_Bi");

105

 

G4Material* air man->Find0rBuildMaterial("G4_AIR");

G4Material* vacuum man->Find0rBuildMaterial("G4_Galactic");

G4Material* NaI

man—>Find0rBuildMateria1("G4_SODIUM_IODIDE");

G4Material* quartz man->Find0rBuildMaterial("G4_SILICON_DIOXIDE");

// -------- Volumes -------

// From Kaloskomis, et al.

const G4double kalInnerDetectorRadius = 42.8*cm/2;

//~——- Vault (world volume) ——-—
// Arbitrarily chosen world half size
// 4*m x 4*m x 4*m

//x axis: up

//y axis: south

//z axis: east, downstream along beam line

const G4double vault_x = 2*m;
const G4double vault_y = 2*m;
const G4double vault_z = 2*m;

G4Box *vau1t_box =

new G4Box("vault_box",vault_x,vault_y,vault_z);
vault_log = new G4LogicalVolume(vault_box, air,
"vault_log",0,0,0);
// GEANT doesn’t allow world volume to rotate.

// You’ll have to convert to SeGA/S800 coordinates

106

 

 

// by yourself. Z is the beam axis.
vault_phys = new G4PVP1acement(0,G4ThreeVector(0,0,0),

vault_log,"vault",0,false,0);

//-——- Beampipe —---
//

const G4double innerRadiusOfPipe 15.24*cm/2; // 6" pipe

const G4double outerRadiusOfPipe innerRadiusOfPipe + 2.*mm;

const G4double halfLengthOfPipe = 1.5*m;
const G4double startingAngleOfPipe = 0.*deg;
const G4double spanningAngleOfPipe = 360.*deg;

if (!m_RemoveNonDetectors) {
G4Tubs* beampipe_tub =
new G4Tubs("beampipe_tub", innerRadiusOfPipe,
outerRadiustPipe, halfLengthOfPipe,
startingAngleOfPipe, spanningAngleOfPipe);
beampipe_log =

new G4LogicalVolume(beampipe_tub,Al,"beampipe_log",0,0,0);

// centered in world concentric with z axis
const G4double beampipePos_x = 0*m;

const G4double beampipePos_y = 0*m;

const G4double beampipePos_z = 0*m;
beampipe_phys =

new G4PVPlacement(0,

107

 

G4ThreeVector(beampipePos_x,
beampipePos_y,
beampipePos_z),

beampipe_log, "beampipe",

vault_log, false, 0);

Checvaerlaps(beampipe_phys);

//—--- Heavimet Collimator --—-

//

if (m_pApexInit—>Collimator()) {

const G4double innerRadiustHeavimet = 0.0*cm;
const G4double startingAngleDfHeavimet = 0.0*deg;
const G4double spanningAngleDfHeavimet = 360.0*deg;

G4Tubs* heavimet_tub =
new G4Tubs("heavimet_tub",
innerRadiusOfHeavimet,m_HeavimetRadius,
m_HeavimetCylHeight/2,
startingAngleOfHeavimet,spanningAngleDfHeavimet);
heavimet-log =

new G4LogicalVolume(heavimet_tub,w,"heavimet_log",0,0,0);

heavimetUp_phys =

new G4PVPlacement(0,m_HeavimetUpPos,

108

heavimet_log,"HeavimetUp",
vault_log, false, 0);
heavimetDn_phys =
new G4PVPlacement(0,m_HeavimetDnPos,
heavimet_log,"HeavimetDn",
vault_log, false, 0);
CheckOverlaps(heavimetUp_phys);

CheckOverlaps(heavimetDn_phys);

//——-— Target ----
//

if (m_Target) {

/*

* The targets were measured to be the following thicknesses:

*

Au: 0.100 mm

*

Bi: 0.258 mm

*

Both of the measurements are a about 0.005 mm larger than

i

the values computed below. I assume that my micrometer

*-

skills are only as good as good as this difference.

*

Jon Cook — 20060315

109

G4double massThicknesstTarget;

G4Material* targetMaterial;

if (m_TargetMaterial == "Au") {

G4cout<<" Target: 519 mg/cm2 Au"<<G4endl;
m_InfoSS<<"Target: 519 mg/cm2 Au"<<G4endl;
massThicknessOfTarget = 518.84*mg/cm2;
targetMaterial = Au;

} else if (m_TargetMaterial == "AuThin") {
G4cout<<" Target: 184 mg/cm2 Au"<<G4endl;
m_InfoSS<<"Target: 184 mg/cm2 Au"<<G4end1;
massThicknesstTarget = 184*mg/cm2;
targetMaterial = Au;

} else if (m_TargetMaterial == "Bi") {

G4cout<<" Target: 245 mg/cm2 Bi"<<G4endl;
m_InfoSS<<"Target: 245 mg/cm2 Bi"<<G4end1;
massThicknesstTarget = 245*mg/cm2;
targetMaterial = Bi;

} else {

G4cerr<<"ERROR> Unknown target specified"<<G4endl;
m_InfoSS<<"ERROR> Unknown target specified"<<G4endl;
massThicknesstTarget = 0.0000001*mg/cm2;

targetMaterial = vacuum;

const G4double halfSideLengthOfTarget = 5.0*cm/2;

110

const G4double halfThicknessOfTarget =
(massThicknessOfTarget/targetMaterial->GetDensity())/2;
G4cout<<" Thickness of target:"<<halfThicknessOfTarget/mm*2

<<" mm"<<G4endl;

G4Box* target_box =
new G4Box("target_box", halfSideLengthOfTarget,
halfSideLengthOfTarget, halfThicknessOfTarget);
target_log =
new G4LogicalVolume(target_box,targetMaterial,
"target_log",0,0,0);
target_phys =
new G4PVP1acement(0,m_pApexInit->TargetPosition(),
target_log,"target_phys",
vault_log, false, 0);
CheckOverlaps(target_phys);
} else {

G4cout<<" No target included."<<G4end1;

//---- Detector Bar ----

//
// 55.0 x 6.0 x 5.5(7.0) cm (L x H x W) according to Kaloskamus

const G4double halfInnerWidthBar = 5.5*cm/2;
const G4double halfDuterWidthBar = 7.0*cm/2;
const G4double halfHeightBar = 6.0*cm/2;

111

const G4double halfLengthBar 55.0*cm/2;

const G4double jacketThickness 0.4*mm; // Kaloskamis

const G4double halfInnerWidthBarVac

halfInnerWidthBar - jacketThickness;

const G4double halfOuterWidthBarVac

half0uterWidthBar - jacketThickness;

const G4double halfHeightBarVac

halfHeightBar - jacketThickness;

const G4double halfLengthBarVac

halfLengthBar;
// No jacket on the ends of the bar due to difficulties with window

// extending beyond mother volume (apexBarVac).

//-- Steel Jacket --
G4Trd* apexBarJacketSolid_trd =
new G4Trd("apexBarJacketSolid_trd",
halfDuterWidthBar, halfInnerWidthBar,
halfLengthBar, halfLengthBar,
halfHeightBar);
apexBar_log = new G4LogicalVolume(apexBarJacketSolid_trd, Fe,

"apexBar_log", 0, 0, 0);

//-- Vacuum inside Jacket --
G4Trd* apexBarVac_trd =

new G4Trd("apexBarVac_trd",

112

 

halfOuterWidthBarVac, halfInnerWidthBarVac,
halfLengthBarVac, halfLengthBarVac,
halfHeightBarVac);
apexBarVac_log =
new G4Logica1Volume(apexBarVac_trd, vacuum,
"apexBarVac_log",0,0,0);
apexBarVac_phys =
new G4PVPlacement(0, G4ThreeVector(0,0,0), //unrotated, centered
apexBarVac_log,
"apexBarVac_phys",
apexBar_log, // in an individual detector
false,0);

CheckOverlaps(apexBarVac_phys);

//—— NaI Crystal and PMT windows --

const G4double halfThicknessWindow = 1.1*cm/2;
const G4double innerRadiusWindow = 0.0*cm;
const G4double outerRadiusWindow = 4.4*cm/2;
const G4double startingAngleWindow = 0.0*deg;
const G4double spanningAngleWindow = 360.0*deg;
const G4double crystalWidthReduction = 1.22*mm;
const G4double crystalHeightReduction = 1.22*mm;

const G4double halfInnerWidthBarCrystal
halfInnerWidthBarVac - crystalWidthReduction;

const G4double halfDuterWidthBarCrystal =
halfOuterWidthBarVac - crystalWidthReduction;

const G4double halfHeightBarCrystal =

113

halfHeightBarVac - crystalHeightReduction;
const G4double halfLengthBarCrystal =
halfLengthBarVac - (2*halfThicknessWindow);
m_pApexInit—>SetHa1fLength0fCrystal(halfLengthBarCrystal);
G4cout<<"\n APEX crystals reduced by "
<<2*crystalHeightReduction/mm<<" mm (height) and "
<<2*crystalWidthReduction/mm<<" mm (width)."<<G4endl;
m_InfoSS<<"APEX crystals reduced by "
<<2*crystalHeightReduction/mm<<" mm (height) and "

<<2*crystalWidthReduction/mm<<" mm (width)."<<G4endl;

G4Trd* apexBarCrystal_trd =
new G4Trd("apexBarCrystal_trd",
halfOuterWidthBarCrystal, halfInnerWidthBarCrystal,
halfLengthBarCrystal, halfLengthBarCrystal,

halfHeightBarCrystal);

apexBarCrystal_log =
new G4Logica1Volume(apexBarCrystal_trd, NaI,

"apexCrystalBar_log",0,0,0);

apexBarCrystal_phys -
new G4PVPlacement(0, G4ThreeVector(0,0,0),
apexBarCrystal_log,
"apexBarCrystal_phys",
apexBarVac_log,
false,0);

CheckOverlaps(apexBarCrystal_phys);

114

G4Tubs* window_tub = new G4Tubs("window_tub",
innerRadiusWindow,outerRadiusWindow,
halfThicknessWindow,

startingAngleWindow,

spanningAngleWindow);

window_log =

new G4Logica1Volume(window_tub,quartz,"window_log",0,0,0);

const G4double windowUpPos =
halfLengthBarCrystal+halfThicknessWindow;
const G4double windoanPos = —windowUpPos;
G4RotationMatrix windowRM;
G4double theta = 90*deg;
windowRM.rotateX(theta);
windowUp_phys =
new G4PVPlacement(G4Transform30(windowRM,
G4ThreeVector(0,windowUpPos,0)),
window_log,"WindowUp", apexBarVac_log, false, 0);
windoan_phys =
new G4PVPlacement(G4Transform30(windowRM,
G4ThreeVector(0,windoanPos,0)),
window_log,"Windoan", apexBarVac_log, false, 0);
CheckOverlaps(windowUp_phys);

CheckOverlaps(windoan_phys);

// physical implementation of APEX bars

115

//
const G4double detRadius = kalInnerDetectorRadius + halfHeightBar;
const G4double startingAngle = 0.*deg;

const G4double incrementAngle = 360.*deg / m_NumberOfDetectors;

for (G4int detectorNumber=0;

detectorNumber<=(m_NumberOfDetectors-1); detectorNumber++){

G4double phi = startingAngle + incrementAngle*detectorNumber;

G4double detPos_x

detRadius*cos(phi);

G4double detPos_y detRadius*sin(phi);

G4double detPos_z

0; // position along beamline fixed
G4RotationMatrix rm;

G4double theta = 90*deg;

rm.rotateX(theta);

rm.rotateZ(phi-90*deg);

// Store rotation so that internal position can

// be reconstructed later.

m_pApexInit->SetDetRotMatrix(detectorNumber,rm);

char physName[1024];
sprintf(physName,"apexBar_phys:%d",detectorNumber);
apexBar_phys[detectorNumber] =
new G4PVPlacement(G4Transform3D(rm,G4ThreeVector(detPos_x,
detPos_y,

detPos_z)),

116

 

 

 

apexBar_log, physName,
vault_log, false, detectorNumber);

CheckOverlaps(apexBar_phys[detectorNumber]);

}

//--—- Array Shielding --—-
//

// Lead part

//

 

G4double innerRadiusOfTube =

kalInnerDetectorRadius + 2*halfHeightBar + 0.75*cm;
G4double outerRadiusOfTube = innerRadiusOfTube + 2.1*cm;
const G4double halfLengthOfTube = 65*cm/2;

const G4double startingAngleOfTube

0.*deg;

const G4double spanningAngleOfTube 360.*deg;

// centered in world concentric with z axis

const G4double ShieldPos_x = 0*m;
const G4double ShieldPos_y = 0*m;
const G4double ShieldPos_z = 0*m;

if (!m_RemoveNonDetectors) {
G4Tubs* shieldLead_tub =
new G4Tubs("shieldLead_tub", innerRadiusOfTube,
outerRadiusOfTube, halfLengthOfTube,

startingAngleDfTube, spanningAngleDfTube);

117

shieldLead_log = new G4Logica1Volume(shieldLead_tub, Pb,
"shieldLead_log", 0, 0, O);
shieldLead_phys =
new G4PVPlacement(0,
G4ThreeVector(ShieldPos_x,
ShieldPos_y,
ShieldPos_z),
shieldLead_log, "shieldLead",
vault_log, false, 0);

CheckOverlaps(shieldLead_phys);

// Steel part
//

// uses dimensions given in Lead section above
//
innerRadiustTube

outerRadiusOfTube;

outerRadiusOfTube

innerRadiustTube + 1.*cm;

if (!m_RemoveNonDetectors) {
G4Tubs* shieldSteel_tub =
new G4Tubs("shieldSteel_tub", innerRadiustTube,
outerRadiusOfTube, halfLengthOfTube,
startingAngleDfTube, spanningAngleOfTube);
shieldSteel_log = new G4Logica1Volume(shieldSteel_tub, Fe,
"shieldSteel_log", 0, 0, 0);

shieldSteel_phys =

118

new G4PVP1acement(0,
G4ThreeVector(ShieldPos_x,

ShieldPos_y,ShieldPos_z),
shieldSteel_log, "shieldSteel",
vault_log, false, 0);

CheckOverlaps(shieldSteel_phys);

//---- Table ----
//
const G4double halfLength_x

1.27*cm/2; // equivalent to 1/2 inch

const G4double halfLength_y 56.3*cm/2;

167.5*Cm/2;

const G4double halfLength_z
// approximate value!!!

—1.*(innerRadiusOfTube + 7.*cm);

const G4double tablePos_x

const G4double tablePos_y 0.; // centered on beamline
// Table edge aligned with end of PMT
const G4double tablePos_z = -(halfLength_z — halfLengthOfTube)

+ 10*cm;

if (!m_RemoveNonDetectors) {
G4Box* table_box =
new G4Box("table_box", halfLength_x,
halfLength_y, halfLength_z);
table_log =

new G4Logica1Volume(table_box, Al, "table_log", 0,0,0);

119

table_phys =
new G4PVPlacement(0,
G4ThreeVector(tab1ePos_x,
tablePos_y, tablePos_z),
table_log, "table", vault_log, false, 0);

CheckOverlaps(table_phys);

// ———————— Make Sensitive Detectors --------
// Make the crystals the active volumes
G4SDManager* SDman = G4SDManager::GetSDMpointer();
G4String CrystalSDname = "Apex/Crystal”;
ApexCrystalSD* CrystalSD =

new ApexCrystalSD( CrystalSDname, "CrystalCollection" );
SDman->AddNewDetector(CrystalSD);

apexBarCrystal_log->SetSensitiveDetector(CrystalSD);

// -------- Visualization Options --------
//
vault_log->SetVisAttributes(G4VisAttributes::Invisible);

apexBarVac_log—>SetVisAttributes(G4VisAttributes::Invisible);

G4Color windowBlue (0.0, 0.75, 1.0, 0.75);
G4Color targetMetallic (0.537, 0.439, 0.302);

G4Color lead (0.5, 0.5, 0.5);

120

 

G4int shieldLineSegments = 50;

G4VisAttributes* windowVisAtt = new G4VisAttributes(windowBlue);
window_log->SetVisAttributes(windowVisAtt);
if (m_Target) {
G4VisAttributes* targetVisAtt =
new G4VisAttributes(targetMetallic);
targetVisAtt->SetForceSolid(true);
target_log—>SetVisAttributes(targetVisAtt);
}
G4VisAttributes* leadVisAtt = new G4VisAttributes(lead);
leadVisAtt->SetForceLineSegmentsPerCircle(shieldLineSegments);
shieldLead_log—>SetVisAttributes(leadVisAtt);
G4VisAttributes* steelShieldVisAtt = new G4VisAttributes();
steelShieldVisAtt—>
SetForceLineSegmentsPerCircle(shieldLineSegments);

shieldSteel_log->SetVisAttributes(steelShieldVisAtt);

m_pApexInit->SetInfoDetConstruct(m_InfoSS.str());

G4cout<<" done.\n\n";

// Returns the pointer to the physical world:

return vault_phys;

121

 

void
ApexDetectorConstruction::

CheckOverlaps(G4VPhysicalVolume* volume) const

if (m_Checvaerlaps)

volume->Check0verlaps();

A. 1.3 7-ray generator

The ApexPrimaryGeneratorAction class manages the emission of 7 rays.

#ifndef ApexPrimaryGeneratorAction_h
#define ApexPrimaryGeneratorAction_h 1
#include "Data.hh"

#include "CApexInitialization.hh"
#include "CRootManager.hh"

#include "CAngularDistribution.hh"

#include <G4VUserPrimaryGeneratorAction.hh>

class G4ParticleGun;

class G4Event;

class

ApexPrimaryGeneratorAction : public G4VUserPrimaryGeneratorAction {

public:

122

 

 

ApexPrimaryGeneratorAction();
~ApexPrimaryGeneratorAction();

void GeneratePrimaries(G4Event* anEvent);

private:
G4ParticleGun* m_pParticleGun; // gamma—ray gun
CApexInitialization* m_pApexInit; // input manager
CRootManager* m_pRootManager; // output manager
CAngularDistribution* m_pAngDist; // angular distribution src
G4double m_BetaEmission; // beta (v/c) of emitted gamma ray

G4ThreeVector m_EmissionPos; // position of gamma source
G4LorentzVector m_GammaVector; // gamma-ray 4 vector
G4ThreeVector m_ProjMomentum; // unit vector of proj. momentum

PrimaryShot_t m_Shot; // data about this primary event

// ptr to function producing a random, unitary G4ThreeVector

// in a given distribution

AngDistFn m_AngDistRandomVector;

void DopplerShiftGamma(); // Doppler shift gamma to lab frame

void Print() const; // print contents of m_Shot

};

#endif // #ifndef ApexPrimaryGeneratorAction_h

#include "ApexPrimaryGeneratorAction.hh"

123

 

#include "Constants.h"

#include <globals.hh>

#include <G4Event.hh>

#include <G4ParticleGun.hh>
#include <G4ParticleTable.hh>
#include <G4Partic1eDefinition.hh>

#include <Randomize.hh>

#define CALL_PTR_MEMBER_FN(ptrTonject,ptrToMember)

((ptrTonject)->*(ptrToMember))

ApexPrimaryGeneratorAction::ApexPrimaryGeneratorAction () {

G4cout << "Creating primary event generator...";

// Get places to find and put information
m_pApexInit = CApexInitialization::Instance();
m_pRootManager = CRootManager::Instance();

m_pAngDist = new CAngularDistribution();

// Number of particles to be shot in one invocation
const G4int n_partic1e = 1;

m_pParticleGun = new G4ParticleGun(n_partic1e);

// Tell gun to fire gammas
G4ParticleTable* particleTable =

G4Partic1eTable::GetParticleTable();

124

G4String particleName = "gamma";
G4ParticleDefinition* particle =
particleTable->FindParticle(particleName);

m_pParticleGun->SetParticleDefinition(particle);

// Projectile moves along 2 axis

m_ProjMomentum = G4ThreeVector(0.0,0.0,1.0);

// Select angular distribution

G4String angularDistribution = m_pApexInit->AngularDistribution();

if (angularDistribution=="uni") {
m_AngDistRandomVector =
&CAngu1arDistribution::UnidirectionalDirection;
G4cout<<"\n Unidirectional angular distribution selected.\n";
} else if(angularDistribution=="iso") {
m_AngDistRandomVector =
&CAngularDistribution::IsotropicRandomDirection;
G4cout<<"\n Isotropic angular distribution selected."<<G4endl;
} else if(angularDistribution=="E2") {
m_AngDistRandomVector = &CAngularDistribution::E2RandomDirection;
G4cout<<"\n E2 angular distribution selected."<<G4end1;
} else {
m_AngDistRandomVector =
kCAngularDistribution::UnidirectionalDirection;
G4cerr<<"\n No angular distribution selected; "

<<"using unidirectional."<<G4endl;

125

G4cout << " done.\n";

ApexPrimaryGeneratorAction::”ApexPrimaryGeneratorAction() {
delete m_pParticleGun;

delete m_pAngDist;

void ApexPrimaryGeneratorAction::GeneratePrimaries(G4Event* anEvent)
{
if (m_pApexInit—>VerbosityLevel()>0)

G4cout <<"\n\n\n ----- GeneratePrimaries(G4Event*) ----- \n";

/* Clear all primary and detector info in the event buffer.
* Clearing the buffer can’t be done in
* ApexEventAction::BeginOvaentAction because GeneratePrimaries
* is called before BegianEventAction.

* Of course.

*/

m_pRootManager—>C1earEvent();

/* For each gamma, check if it’s going to be generated this

126

 

* instance,sample an emission position, sample a random
* direction from the chosen distribution, boost it if

* necessary, fire away, and record data.

*/

for(unsigned i=0; i<m_pApexInit->PrimaryMultiplicity(); i++ ) {

if (CLHEP::RandFlat::shoot()<=m_pApexInit->GammaIntensity(i)){

// within branching ratio probability, shoot a gamma

// Determine position of emission
m_EmissionPos =
G4ThreeVector(

CLHEP::RandGauss::shoot(m_pApexInit->TargetPosition().x(),
m_pApexInit->BeamSigmaX()),
CLHEP::RandGauss::shoot(m_pApexInit->TargetPosition().y(),
m_pApexInit->BeamSigmaY()),
m_pApexInit->TargetPosition().z() +
CLHEPzzRandExponential::shoot(

m_pApexInit—>GammaLengthDecayConst(i))
);

m_pParticleGun->SetParticlePosition(m_EmissionPos);

// Create Lorentz Vector of gamma ray
m_GammaVector =

G4LorentzVector(m_pApexInit—>GammaEnergy(i) *

127

 

 

CALL_PTR_MEMBER_FN(m_pAngDist,
m_AngDistRandomVector)(),

m_pApexInit->GammaEnergy(i));

// Record what is happening in the particle frame

m_Shot.PFEnergy m_GammaVector.getT();

m_Shot.PFTheta m_GammaVector.theta();

m_Shot.PFPhi

m_GammaVector.phi();

 

// Boost!

if (m_pApexInit~>BetaMidtarget()!=0.0)
D0pplerShiftGamma();

else

m_BetaEmission=0.0;

// Record what is happening in the lab frame

m_Shot.Beta m_BetaEmission;

m_Shot.BmisPos m_EmissionPos;

 

m_Shot.Bnergy m_GammaVector.getT();

m_GammaVector.theta();

m_Shot.Theta

m_Shot.Phi m_GammaVector.phi();

m_pRootManager->AddPrimaryData(&m_Shot,i);

// Fire the gamma ray!

128

m_pParticleGun->SetParticleMomentumDirection

(m_GammaVector.getV());

m_pParticleGun—>SetParticleEnergy(m_GammaVector.getT());

m_pParticleGun->GeneratePrimaryVertex(anEvent);

} else {

// not
m_Shot
m_Shot
m_Shot
m_Shot
m_Shot
m_Shot

m_Shot

m_EmissionPos

within branching ratio probability; record null event

.PFEnergy = 1.0*keV;
.PFTheta = ~5.0*deg;
.PFPhi = -185.0*deg;
.Energy = 1.0*keV;
.Beta = 0.0;

.Theta = -5.0*deg;
.Phi = ~185.0*deg;

G4ThreeVector(0.*cm,0.*cm,—10.*cm);

m_pRootManager->AddPrimaryData(&m_Shot,i);

if (m_pApexInit->VerbosityLeve1()>0)

Print();

} // end loop over gamma list

if (m_pApexInit->VerbosityLeve1()>0)

G4COUt <<" _____________________________________ \nu ;

129

 

 

/* Takes m_GammaVector, which is already set in the
* particle frame for the i’th gamma ray, and Doppler boosts
* into the lab frame along the z axis using the betas given
* in the input file and provided here by m_pApexInit.
* In my case, the momentum width is much greater than the
* momentum acceptance, and a block function is assumed.
* The position resolution of APEX is so poor that the
* momentum distribution doesn’t matter.
* Also, for 76Ge betaPostTarget*c*lifetime = 3 mm
* while the target is 0.258 mm thick. The gamma is emitted

* after the target.

void ApexPrimaryGeneratorAction::DopplerShiftGamma() {

G4double scalinngBetaDueToBeamWidth =
1 + m_pApexInit->BeamBetaWidth()*CLHEP::RandFlat::shoot(-0.5,0.5);
m_BetaEmission =
m_pApexInit->BetaPosttarget()*scalingOfBetaDueToBeamWidth;

m_GammaVector.boostZ(m_BetaEmission);

return;

130

 

 

 

A.1.4 Angular distribution

The CAngularDistribution class returns a random vector from a unidirectional,

isotropic, or E2 distribution.

/*
* Class for creating isotropic, unidirectional, and

* E2 angular distributions.

:1:
* Written by Jon Cook
*

*/

#ifndef CANGULARDISTRIBUTION_H

#define CANGULARDISTRIBUTIUN_H

#include <globals.hh>

#include <Rtypes.h>

#include <TF1.h>

#include "CApexInitialization.hh"

class CAngularDistribution {

public:

131

 

 

 

CAngularDistribution();

~CAngularDistributionO;

void Coefficients(G4double a0, G4double a2, G4double a4);
G4ThreeVector UnidirectionalDirection();

G4ThreeVector IsotropicRandomDirection();

G4ThreeVector E2RandomDirection();

 

private:

 

static const UInt_t kaumbeerAngDistCoeff =
(UInt_t)NUM_ANGDISTCOEFF;

CApexInitialization* prexInit;

// 0,2,4 Legrange coefficients

Double_t fAngularCoeff[kaumberOfAngDistCoeff];

TF1* fAngularDistribution;

};

typedef G4ThreeVector (CAngularDistribution::*AngDistFn)();

#endif // #ifndef CANGULARDISTRIBUTION_H

#include "CAngularDistribution.hh"

#include <G4RandomDirection.hh>

// Function can’t be member of the class. See TF1 documentation.

Double_t CAngularDistributionAngularFunction(Double_t* theta,

132

Double_t* param) {

Double_t cosThetaSq = cos(*theta);
cosThetaSq *= cosThetaSq;
Double_t value = 1/(4*TMath::Pi()) *
(
param[0] +
param[l] * 1./2.*(3.*cosThetaSq-1.) +
param[2] * 1./8.*(35.*cosThetaSq*cosThetaSq-30.*cosThetaSq+3.)
) * sin(*theta);

return value;

CAngularDistribution::CAngularDistribution() {

for (UInt_t i=0; i<kaumber0fAngDistCoeff; i++)

fAngularCoefffi]=0.0;

prexInit = CApexInitialization::Instance();

Coefficients(prexInit->AngDistCoeff(0),
prexInit—>AngDistCoeff(1),

prexInit—>AngDistCoeff(2));

/*
* Create angular distribution on [0,pi) of E2 transition using

* even Legendre Polynomials up to order 4.

133

*

Using 180*8 = 1440 binning for 1/8 degree resolution

* corresponding to approximately 1 mm resolution at APEX.

*

Since the bin is approximated by a parabola and APEX has
* a position resolution of some centimeters, 1 mm resolution

* is quite sufficient.

fAngularDistribution = new TF1("AngDist“,
CAngularDistributionAngularFunction,
0.0, TMathzzPi(),
kaumbeerAngDistCoeff);
Int_t npx = 1440;
fAngularDistribution->Sethx(npx);
fAngularDistribution->SetParName(0,"a0");
fAngularDistribution->SetParName(1,"a2");
fAngularDistribution->SetParName(2,"a4");
fAngularDistribution->SetParameter(0,fAngularCoeff[0]);
fAngularDistribution->SetParameter(1,fAngularCoeff[1]);

fAngularDistribution->SetParameter(2,fAngularCoeff[2]);

CAngularDistribution::"CAngularDistribution() {

delete fAngularDistribution;

134

 

void CAngularDistribution::Coefficients(G4double a0, G4double a2,

G4double a4) {

fAngularCoeff[0] = (Double_t)aO;
fAngularCoefffl] = (Double_t)a2;
fAngularCoeff[2] = (Double_t)a4;

G4ThreeVector CAngularDistribution::UnidirectionalDirection() {
G4ThreeVector randomVector = G4ThreeVector(3.,0.,1.);

return randomVector.unit();

G4ThreeVector CAngularDistribution::IsotropicRandomDirection() {

return G4RandomDirection();

G4ThreeVector CAngularDistribution::E2RandomDirection() {
G4ThreeVector randomVector = G4RandomDirection();
randomVector.setTheta((G4double)fAngularDistribution->GetRandom());

return randomVector;

135

A. 1.5 Event action

The processing of the event is handled by the ApexEventAction class.

#ifndef ApexEventAction_h
#define ApexEventAction_h 1
#include <G4UserEventAction.hh>
#include <globals.hh>

#include "CApexInitialization.hh"
#include "CRootManager.hh"

#include "CApexCalculator.hh"

class G4Event;

class ApexEventAction : public G4UserEventAction
{
public:
ApexEventAction();

virtual ~ApexEventActionO;

virtual void BeginOvaentAction(const G4Event*);
virtual void EndeEventAction(const G4Event*);

inline void SetDrawFlag(G4bool val) { drawFlag = val; };

private:

CApexInitialization* m_pApexInit;

136

 

CRootManager*

vCApexCalculator

G4int

m_pRootManager;
m_vCalculator;

m_CrystalCollID;

G4bool drawFlag;

};

#endif

#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include

#include

"ApexEventAction.hh"
"ApexCrystalHit.hh"
"ApexCrystalSD.hh"
<G4Event.hh>
<G4EventManager.hh>
<G4HCofThisEvent.hh>
<G4VHitsCollection.hh>
<G4TrajectoryContainer.hh>
<G4Trajectory.hh>
<G4VVisManager.hh>
<G4SDManager.hh>
<G4UImanager.hh>
<G4ios.hh>

<vector>

using stdzzvector;

137

ApexEventAction::ApexEventAction()

:drawFlag(false)

m_pApexInit = CApexInitialization::Instance();
m_pRootManager = CRootManager::Instance();

m_vCalculator = vCApexCalculator(m_pApexInit->Num0fDetectors());

ApexEventAction::"ApexEventAction()
{s}

void ApexEventAction::BegianEventAction(const G4Event*) {
if(drawF1ag)
{
G4VVisManager* pVVisManager =
G4VVisManager::GetConcreteInstance();

if(pVVisManager)

G4UImanager::GetUIpointer()->ApplyCommand("/vis"/draw/current");

138

void ApexEventAction::EndOvaentAction(const G4Event* evt ) {

G4SDManager * SDman = G4SDManager::GetSDMpointer();
G4String colNam;
m_CrystalCollID =

SDman—>GetCollectionID(colNam="CrystalCollection");

//G4int eventNum = evt->GetEventID();

G4HCofThisEvent * HCE = evt->GetHCofThisEvent();

ApexCrystalHitsCollection* crystalHC = 0;

if(HCE)
crystalHC =

(ApexCrystalHitsCollection*)(HCE->GetHC(m_CrystalCollID));

if(crysta1HC) {

G4int n_hit = crystalHC->entries();

/*
From HC get detector number, energy, and position
Pass info to CApexCalculator and calculate

Pass calculated values to CRootWriter

*/

for(size_t j=0;j<m_pApexInit—>Num0fDetectors();j++)

139

m_vCalculator[j].Clear();

// create array of hit detectors

vector<G4int> vHitDetector;

G4int detNum = —1;

for(G4int i=0;i<n_hit;i++) {

detNum=(*crystalHC)[i]->DetNum();
m_vCalculator[detNum].AddHit((*crystalHC)[i]—>Edep(),

(*crystalHC)[i]—>Position());

// If detector is newly hit in this event, add it to the list
//of hit detectors
G4bool NewHitDet = true;
for (size_t i=0; i<vHitDetector.size(); i++) {

if (vHitDetector[i]==detNum)

NewHitDet = false;

};
if (NewHitDet) {

vHitDetector.push_back(detNum);

};

// Calculate PMT effects for each hit detector and

140

// add the results to the event

for (size_t i=0; i<vHitDetector.size(); i++) {

DetectorOutput_t* output =
m_vCalculator[ vHitDetector[i] ].Calculate();

m_pRootManager->AddDetectorData(output, vHitDetector[i]);

if (m_pApexInit~>VerbosityLevel()>0)
if (output->EnergyRec>0.0)

m_vCalculator[ vHitDetector[i] ].Print(vHitDetector[i]);

// Write this event to file

m_pRootManager->WriteEvent();

} // endif(crystalHC)

if(drawFlag)
{
G4VVisManager* pVVisManager =
G4VVisManager::GetConcreteInstanceC);

if(pVVisManager)

if(crysta1HC) crystalHC->DrawAllHits();

G4UImanager::GetUIpointer()~>ApplyCommand("/vis"/show/view");

141

A.1.6 Double-sided scintillator model

The deposited energy is treated according to the model described in Section 3.3 in

the class CApexCalculator.

#ifndef CApexCalculator_h
#define CApexCalculator_h
#include <vector>

#include <G4Types.hh>

#include <G4ios.hh>

#include <G4String.hh>

#include <G4ThreeVector.hh>
#include <G4RotationMatrix.hh>
#include <Randomize.hh>

#include "CApexInitialization.hh"

#include "Data.hh"

class CApexCalculator {

public:

CApexCalculator();

~CApexCalculatorO;

142

void AddHit(G4double energy, G4ThreeVector pos);
DetectorDutput_t* Calculate();
void Clear();

void Print(G4int detNum) const;

G4int NumHits() const;

G4double DepositedEnergy() const;
G4ThreeVector FirstHitPosition() const;
G4double AmplitudeUp() const;

G4double AmplitudeDn() const;

G4double ReconEnergy() const;

G4double D0ppReconEnergy() const;

G4double ReconPosition() const;

private:

CApexInitialization *m_pApexInit;

// fixed values

// scintillation efficiency

static const G4double m_ScintEfficiency;

// average scintillation photon energy
static const G4double m_AveragePhotonEnergy;
// cumulative NaI scintillation effect
static const G4double m_ScintEffects;

// index of refraction of air

static const G4double m_IndexRefractAir;

143

 

// index of refraction of NaI

static const G4double m_IndexRefractNaI;

// critical angle of the NaI-air joint

static const G4double m_CriticalAngle;

// proportion of scintillation photons surviving TIR
static const G4double m_ProportionTransmitted;

// cumulative effects at the scintillation point
static const G4double m_ScintPointEffects;

// ratio of window to crystal area

static const G4double m_WindowSizeFactor;

// transmission efficiency through the window
static const G4double m_WindowLossFactor;

// cumulative effect of the window

static const G4double m_WindowTransmission;

// PMT quantum efficiency

static const G4double m_PMTquantEff;

// factor my which the statics are reduced to match data
static const G4double m_StaticsScaleFactor;

// cumulative PMT and window effect

static const G4double m_WindowPMTeffects;

// inner radius of the APEX array

static const G4double m_ApexRadius;

// depth of gamma interaction with the crystal
static const G4double m_InteractionDepth;

// radius of gamma interaction with APEX

static const G4double m_InteractionRadius;

144

// raw values

// number of interactions

G4int m_NumHits;

// energy deposited by one hit

G4double m_Energy;

// internal point of current interaction
G4ThreeVector m_InternalPosition;

// internal point of first interaction

G4ThreeVector m_FirstHitPosition;

// calculated values

// sum of energies deposited for one primary event
G4double m_RawSumEnergy;

// number of scintillation photons that reach the up
// end of the crystal

G4double m_PhotonsUp;

// number of scintillation photons that reach the dn
// end of the crystal

G4double m_PhotonsDn;

// amplitude of up PMT output

G4double m_AmplitudeUp;

// amplitude of dn PMT output

G4double m_AmplitudeDn;

// reconstructed energy

G4double m_ReconEnergy;

// reconstructed position

145

G4double m_ReconPosition;
// time component is Doppler-corrected reconstructed
// energy; x and y are fixed values

G4LorentzVector m_GammaVector;

// data structure for output of interaction

DetectorOutput_t m_Dutput;

void TransportHitToPMT();
void CalculatePMT();

void Reconstruct();

void DopplerCorrect();

void CollectConfigInfo();

typedef std::vector<CApexCalcu1ator> vCApexCalculator;

#endif // #ifndef CApexCalculator_h

#include "CApexCalculator.hh"

#include <sstream>

//(see Knoll, page 233)
//

// Efficiency of conversion of deposited energy to photons

146

const G4double CApexCalculator::m_ScintEfficiency = 0.12;
// Average energy of a scintillation photon
const G4double CApexCalculator::m_AveragePhotonEnergy = 3*eV;

//Number used at runtime
const G4double CApexCalculator::m_ScintEffects =

m_ScintEfficiency/m_AveragePhotonEnergy;

//Tota1 Internal Reflection

//

// Hecht page 94

const G4double CApexCalculator::m_IndexRefractAir = 1.00029;

// http://www.detectors.saint-gobain.com/Media/Documents/

// 30000000000000001004/

// SGC_Scintillation_Properties_Chart_52206.pdf

const G4double CApexCalculator::m_IndexRefractNaI = 1.85;

// Hecht page 121

const G4double CApexCalculator::m_CriticalAngle =
asin(m_IndexRefractAir/m_IndexRefractNaI);

// 05115 Analysis logbook page 43

const G4double CApexCalculator::m_ProportionTransmitted =

( 1 - sin(m_CriticalAngle) );

// Cumulative effects at the scintillation point

// Nph(to be attentuated) = Edep*fScintPointEffects

//

// Division by two is to account for the fact that on average

// half of the photons go to one PMT and half to the other.

147

//
const G4double CApexCalculator::m_ScintPointEffects =

m_ScintEffects*m_ProportionTransmitted/2.0;

/* Photon transmission from crystal to PMT */

/* Ratio of PMT active area to crystal end area.

* Kaloskamis pg 449 gives 0.27

* Apmt/Acrystal = pi*(3.6/2)“2 / ((5.5+7.0)/2*6.0) = 0.2714
*/
const G4double CApexCalculator::m_WindowSizeFactor = 0.27;

// Efficiency of transmission from crystal to window. Knoll page 330.

0.75;

const G4double CApexCalculator::m_WindowLossFactor
// Combined Effects

// Nearly equal to fWindowSizeFactor*fWindowLossFactor, but here

// using Przemek’s value from Guide 7.

const G4double CApexCalculator::m_WindowTransmission = 0.27;

// quantum efficiency of PMT

//Knoll page 330, R2490-05 data sheet on page 30 of analysis logbook
0.20;

const G4double CApexCalculator::m_PMTquantEff

const G4double CApexCalculator::m_StaticsScaleFactor 0.88;

//Number used at runtime

const G4double CApexCalculator::m_WindowPMTeffects

m_WindowTransmission*m_PMTquantEff*m_StaticsSca1eFactor;

148

 

/* Interaction radius for Doppler correction.
* The interaction depth was selected by testing for the ideal
* value according to the simulation. The radius comes from

* Kaloskamis.

*/
const G4double CApexCalculator::m_ApexRadius = 42.8*cm/2;
const G4double CApexCalculator::m_InteractionDepth = 2.0*cm;

const G4double CApexCalculator::m_InteractionRadius

m_ApexRadius + m_InteractionDepth;

CApexCalculator::CApexCalculator() {
srand((unsigned)time(0));
m_pApexInit = CApexInitialization::Instance();

CollectConfigInfo();

void

CApexCalculator::AddHit(G4double energy, G4ThreeVector pos) {

m_Energy = energy;

m_RawSumEnergy += energy;

/*
* A trapezoid is defined with inconvenient coordinates.

* I adjust coordinate system here so that z is the world 2

149

* (i.e. along the length of the bar), x is toward the smaller
* parallel face, and y is a measure of closeness to the slanted
* face. I assume a right-handed system to determine the

* orientation of y. The origin remains at the center of the bar.

*/

m_InternalPosition = G4ThreeVector(pos.z(),pos.x(),pos.y());

if (m_NumHits==0)

m_FirstHitPosition = m_InternalPosition;
m_NumHits++;
TransportHitToPMTC);

return;

// Changes to this function may affect the energy calibration.

void CApexCalculator::TransportHitToPMT() {

// Convert energy to photons (see Knoll, page 233)

G4double photons = m_Energy * m_ScintPointEffects;

// Attenuate photons as they travel towards the PMT’s
//photons at PMT = photons*exp[-mu*(DetLength/2 — hitPos)
// To be correct, the Up has + and On has -; the original is

//the opposite

150

 

m_PhotonsUp +=

photons*exp( -1*m_pApexInit->Mu()*
(m_pApexInit->HalfLength0fCrysta1()+
m_InternalPOsition.getZ())
);
m_PhotonsDn +=

photons*exp( —1*m_pApexInit->Mu()*
(m_pApexInit—>Ha1fLength0fCrysta1()-
m_InternalPosition.getZ())
);
m_Energy = sqrt(-1.0);
m_InternalPosition = G4ThreeVector();

return;

// To be called after all hits have been transported to the PMT

DetectorOutput_t* CApexCalculator::Calculate() {

CalculatePMT();
Reconstruct();

DopplerCorrect();

m_Output.NumHits m_NumHits;

m_0utput.FirstHitPos m_FirstHitPosition;

m_Dutput.EnergyDep m_RawSumEnergy;

151

 

m_Dutput.EnergyUp m_AmplitudeUp;

m_Output.EnergyDn m_AmplitudeDn;

m_0utput.EnergyRec m_ReconEnergy;

m_Output.EnergyRecDop m_GammaVector.getT();

m_Output.PositionRec m_ReconPosition;

return &m_0utput;

/* Reduces photon number by window and PMT effects to find
* the minimum number of photons (first stage of PMT). Smears
* minimum photon number into Poisson distribution to find
* m_Amplitude{Up,Dn}.
* Changes to this function may affect the energy calibration.
*/

void CApexCalculator::CalculatePMT() {

// calculate number of photoelectrons produced in PMT’s

G4double photoElectronsUp m_WindowPMTeffects*m_PhotonsUp;

G4double photoElectronsDn m_WindowPMTeffects*m_PhotonsDn;

// Smear into Poisson distributions

photoElectronsUp =
CLHEP::RandPoisson::shoot((double)photoElectronsUp);

photoElectronsDn =

152

CLHEPzzRandPoisson::shoot((double)photoElectronsDn);

// Poisson returns an int. Smear int througout the range of
// the bin to avoid funny binning prOperties, e.g., at Up=2*Dn.
m_AmplitudeUp =

photoElectronsUp +

( (G4double)rand()/((G4double)RAND_MAX + (G4double)1.0) );
m_AmplitudeDn =

photoElectronsDn +

( (G4double)rand()/((G4double)RAND_MAX + (G4double)1.0) );

return;

// Uses m_Amplitude{Up,Dn} to determine the unscaled reconstructed

//energy and position of the event in this bar.

void CApexCalculator::Reconstruct() {

if ( (m_AmplitudeDn<=0.0) ll (m_AmplitudeUp<=0.0) ) {

—30.0*cm;

m_ReconPosition

m_ReconEnergy 1.0*keV;

} else {

m_ReconPosition (1/(2*m_pApexInit->Mu()))

* log(m_AmplitudeDn/m_AmplitudeUp);

m_ReconEnergy m_pApexInit->A1pha()

153

 

* sqrt(m_AmplitudeUp*m_AmplitudeDn);
};

return;

/*

* Doppler shift reconstructed energy from lab frame to particle
* frame using the midtarget beta and the reconstructed position.
* A massless Lorentz vector for the gamma ray is constructed

* then boosted to the lab frame.

* Since x and y are not changed in a boost, they are given

* incorrect fixed values that don’t matter. The important point
* is that r“2 = sqrt(x‘2 + y‘2) is the interaction radius so

* that unit() returns a unit vector with the the correct theta

* of the reconstructed gamma emission direction.

void CApexCalculator::DopplerCorrect() {

G4ThreeVector gammaDirection =
G4ThreeVector(m_InteractionRadius, 0.0,

m_ReconPosition - m_pApexInit->D0ppReconZPos());

gammaDirection = gammaDirection.unit();

m_GammaVector =

154

G4LorentzVector(m_ReconEnergy*gammaDirection, m_ReconEnergy);

m_GammaVector.boostZ(—1*m_pApexInit->BetaPosttarget());

return;

155

Ill

[2]

[3|

[4]

[5]

[6]

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