STUDY OF THE DEVELOPMENT OF SHELL CLOSURES AT

N=32,34 AND APPROACHES TO SUB-SEGMENT

INTERACTION-POINT DETERMINATION IN 32-FOLD
SEGMENTED HIGH-PURITY GERMANIUM DETECTORS

By

Dan-Cristian Dinca

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

2005
 ABSTRACT

STUDY OF THE DEVELOPMENT OF SHELL CLOSURES AT N=32,34 AND

APPROACHES TO SUB-SEGMENT INTERACTION-POINT
DETERMINATION IN 32-FOLD SEGMENTED HIGH-PURITY GERMANIUM
DETECTORS

By

Dan-Cristian Dinca

52−56
The even Ti isotopes have been studied with intermediate-energy Coulomb

excitation and absolute B(E2; 0+ → 2+

1 ) transition rates have been deduced. Our data

confirm the presence of a sub-shell closure at neutron number N = 32 in neutron-

48
rich titanium isotopes above the doubly-magic nucleus Ca and provide no direct

evidence for the predicted N = 34 closure. Large-scale shell model calculations with
the most recent effective interactions are unable to reproduce the magnitude of the

measured strengths in the semi-magic Ti isotopes and their strong variation with
neutron number.

Sub-segment position resolution of the γ-ray interaction points has been demon-
strated for the cylindrically-symmetric 32-fold segmented HPGe detectors of the
NSCL/MSU Segmented Germanium detector Array (SeGA) using digital electronics.

Waveforms of the real charge signals from segments that contain interaction points
and induced charge signals from neighboring segments were digitally recorded by 100

MHz ADCs. Simple integrated quantities (amplitudes, areas, peak times) were ex-
tracted from the waveforms. By analyzing the asymmetry of the induced signals we

could determine the proximity of the interaction point to segments without net-charge
deposition, attaining sub-segment position resolution along the crystals symmetry
axis. The radial position of the interaction point was determined through an analysis

of the rise times of the real charge signals. Although less precise than other methods
involving a complete waveform analysis, the use of integrated quantities simplifies the

problem of sub-segment interaction position estimation.

to my wife, Cornelia

iii
 ACKNOWLEDGMENTS

I am grateful to my advisor, Thomas Glasmacher, for insightful guidance through

all the projects I have been involved in. He gave me the freedom to develop at my
own pace and try my own ideas. I learned a lot from him.
For more than a year Robert Janssens has been my second advisor. Robert made

the titanium experiment an enlightening and enjoyable endeavor. I got more attention
from him than I ever deserved.

I am indebted to Alexandra Gade for her help on the analysis of the titanium
experiment. Her patient answers to my not-so-smart questions made me get a deeper

understanding of what I was studying.

Chris Campbell helped me on the detector development study. Our long discus-
sions were inspiring to me.

I thank the present and former members (or associated members) of the NSCL’s
Gamma Group, Matt Bowen, Jenny Church, Jon Cook, Joachim Enders, Zhiqiang Hu,

Pat Lofy, Wil Mueller, Heather Olliver, Ben Perry (I still remember your parties),
Russ Terry, and Katie Yurkewicz.

Running the NSCL-02002 experiment, ”B(E2) Rates in Even-Even 52−56 Ti”, was a
team effort, involving more than NSCL’s Gamma Group. People that made it possible
and were not mentioned before are Daniel Bazin, Mike Carpenter, Partha Chowdhury,

Alick Deacon, Sean Freeman, Michio Honma, Filip Kondev, Jean-Luc Lecouey, Sean
Liddick, Paul Mantica, Taka Otsuka, Bryan Tomlin, and Ken Yoneda.

I thank Professors B.A. Brown, J. Linneman, S.D. Mahanti, B. Pope and B. Sher-
rill for serving in my PhD Guidance Committee. Dr. Brown developed the theoretical
models and calculations for the energies and reduced transition probabilities for the

neutron-rich titanium isotopes I have measured.

Finally, I thank Cornelia, who deserves my gratitude most. I am blessed to have

iv
met her.

v
Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Reduced transition probabilities to the first 2+ state in 52,54,56 Ti and

development of shell closures at N = 32, 34 . . . . . . . . . . . . . . 7
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Experimental conditions . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Secondary beams . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.2 Particle identification . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.3 Intermediate-energy Coulomb excitation (theory) . . . . . . . 12
2.2.4 Gamma-ray spectroscopic system . . . . . . . . . . . . . . . . 14
2.3 Analysis method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Sub-segment interaction position resolution for the NSCL SeGA de-

tectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1 Physics with fast exotic beams at the NSCL . . . . . . . . . . . . . . 36
3.2 Signal formation in HPGe detectors . . . . . . . . . . . . . . . . . . . 42
3.3 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

A Algorithm description . . . . . . . . . . . . . . . . . . . . . . . . . . 74
A.1 Centered running average . . . . . . . . . . . . . . . . . . . . . . . . 74
A.2 Gaussian smoothing using 9 points . . . . . . . . . . . . . . . . . . . 75
A.3 Signal derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
A.4 Linear fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
A.5 Threshold passing point . . . . . . . . . . . . . . . . . . . . . . . . . 82
A.6 Trapezoidal shaper . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
A.7 A differentiator-integrator shaper . . . . . . . . . . . . . . . . . . . . 88
A.8 Statistics on a waveform segment . . . . . . . . . . . . . . . . . . . . 90
A.9 RC-CR Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

vi
List of Figures

1.1 A section of the nuclear chart showing the region of interest . . . . . 2

1.2 Simplistic view of the orbital structure for 56 Ti above the 1f7/2 shell . 2
1.3 Example of γ-ray interaction with a segmented detector . . . . . . . . 5

2.1 Particle ID, dispersive focal plane vs. time of flight . . . . . . . . . . 10

2.2 Particle ID, time of flight vs. energy loss . . . . . . . . . . . . . . . . 11
2.3 Particle ID, time of flight vs. energy loss. . . . . . . . . . . . . . . . . 11
2.4 Schematics of the projectile-target interaction . . . . . . . . . . . . . 13
2.5 The SeGA rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6 Measurement of the reduced transition probability from the ground
state to the 7/2+ state in the Au target for all beams used in this
experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.7 Laboratory frame coincidence γ-ray spectra for 197 Au . . . . . . . . . 20
2.8 Laboratory frame γ-ray spectra for 197 Au in coincidence with 54 Ti.
Background subtraction applied. . . . . . . . . . . . . . . . . . . . . . 22
2.9 Typical time spectrum for a SeGA detector . . . . . . . . . . . . . . . 22
2.10 Coincidence γ-ray spectra for 76 Ge Doppler-reconstructed event-by-
event in the projectile frame . . . . . . . . . . . . . . . . . . . . . . . 24
2.11 Coincidence γ-ray spectra for 52 Ti Doppler-reconstructed event-by-
event in the projectile frame . . . . . . . . . . . . . . . . . . . . . . . 24
2.12 Coincidence γ-ray spectra for 54 Ti Doppler-reconstructed event-by-
event in the projectile frame . . . . . . . . . . . . . . . . . . . . . . . 25
2.13 Coincidence γ-ray spectra for 56 Ti Doppler-reconstructed event-by-
event in the projectile frame . . . . . . . . . . . . . . . . . . . . . . . 25
2.14 Projectile frame γ-ray energy in coincidence with 76 Ge. Background
subtraction applied . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.15 Projectile frame γ-ray energy in coincidence with 52 Ti. Background
subtraction applied . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.16 Projectile frame γ-ray energy in coincidence with 54 Ti. Background
subtraction applied . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.17 Projectile frame γ-ray energy in coincidence with 56 Ti. Background
subtraction applied . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.18 Comparison of the measured 2+ 1 excitation energies with the results
of large-scale shell model calculations using the GXPF1 (dashed lines)
and GXPF1A (solid lines) effective interactions. . . . . . . . . . . . . 30

vii
2.19 Comparison of the measured absolute B(E2; 0+ → 2+ 1 ) transition strengths
with the results of large-scale shell model calculations using the GXPF1
and GXPF1A effective interactions . . . . . . . . . . . . . . . . . . . 31
2.20 Comparison of the measured absolute B(E2; 0+ → 2+ 1 ) transition strengths
with the results of large-scale shell model calculations using the GXPF1
and GXPF1A effective interactions using the effective charges calcu-
lated from Ref. [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.1 The energy resolution dependence versus the detector angle for a typ-
ical beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 Laboratory frame (upper panel) and projectile frame (lower panel) γ-
ray energy spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Segment labeling scheme for SeGA crystals . . . . . . . . . . . . . . . 46
3.4 Weighting potential for segment E, transversal cut . . . . . . . . . . . 47
3.5 Weighting potential along the trajectories “1” and “2” . . . . . . . . 48
3.6 Weighting potential for segment E, longitudinal cut . . . . . . . . . . 49
3.7 Example of real and transient charge signals . . . . . . . . . . . . . . 50
3.8 Experimental setup of the SeGA scanning stand . . . . . . . . . . . . 52
3.9 Segment signals for an interaction closer to the E4 side of the D4
segment (Quadrant 3) . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.10 Segment signals for an interaction closer to the E4 side of the D4
segment (Quadrant 4) . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.11 Segment signals for an interaction closer to the C4 side of the D4
segment (Quadrant 3) . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.12 Segment signals for an interaction closer to the C4 side of the D4
segment (Quadrant 4) . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.13 The average miss of the algorithm in estimating the interaction position
for several positions of the collimator . . . . . . . . . . . . . . . . . . 58
3.14 Histograms of the algorithm misses for each event in the data set when
pulse shape analysis is involved compared to the case with no PSA . 59
3.15 Amplitudes of the transient signals for segments F4 and D4. z = 0 mm. 60
3.16 Amplitudes of the transient signals for segments F4 and D4. z = 2 mm. 61
3.17 Amplitudes of the transient signals for segments F4 and D4. z = 4 mm. 62
3.18 Amplitudes of the transient signals for segments F4 and D4. z = 6 mm. 63
3.19 Amplitudes of the transient signals for segments F4 and D4. z = 8 mm. 64
3.20 Central contact waveforms corresponding to a photopeak event at
1332 keV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.21 Segment waveforms corresponding to a photopeak event at 662 keV . 66
3.22 Time to reach 30% of the full amplitude (t30) plotted against the
position of the collimator in the radial direction . . . . . . . . . . . . 67
3.23 Time to reach 60% of the full amplitude (t60) plotted against the
position of the collimator in the radial direction. . . . . . . . . . . . . 68
3.24 Time to reach 90% of the full amplitude (t90) plotted against the
position of the collimator in the radial direction . . . . . . . . . . . . 69

viii
3.25 Time to reach 30% of the full amplitude plotted against the time to
reach 90% of the full amplitude. The radius is color coded (arbitrary
units). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.26 The central contact signal corresponding to a 1332 keV event and a
trapezoidal shaper with integration time of 1 µs and flat-top duration
of 0.5 µs applied on it . . . . . . . . . . . . . . . . . . . . . . . . . . 71

A.1 Gaussian filter coefficients . . . . . . . . . . . . . . . . . . . . . . . . 76

ix
List of Tables

2.1 Summary of secondary beam properties. . . . . . . . . . . . . . . . . 9

2.2 Summary of scattering angle cuts. The first entry for 52 Ti corresponds
to the 256 mg/cm2 target and the second to the 518 mg/cm2 target. . 23
2.3 Summary of measured transition energies (2+ +
1 → 0g.s. ), number of par-
ticles detected and Coulomb excitation cross sections. . . . . . . . . . 23
2.4 Comparison of measured B(E2; 0+ → 2+ 1 ) values (labeled B(E2; ↑) in
the table) with shell model calculations using the GXPF1 interaction
as well as the recently proposed GXPF1A interaction. The two 52 Ti
entries correspond to separate measurements with Au targets of differ-
ent thicknesses; (a) 256 mg/cm2 , (b) 518 mg/cm2 and (c) the weighted
average of the two. Data on the excitation of the Au target by the
various Ti isotopes are given as well. . . . . . . . . . . . . . . . . . . 28

x
Chapter 1

Introduction

Beyond the valley of β stability, the large asymmetry between proton and neutron

numbers is expected to modify the shell structure. Over the last few years, the appear-
ance and disappearance of magic numbers [2–6] and the formation of new regions of

deformation [7] have been observed. Phenomena such as the “island of inversion”,
where the shell inversion occurs near the ground state, illustrate that significant

changes can occur in the nuclear structure of neutron-rich nuclei. Modifications in

the oribital energy spacings affect properties such as nuclear excitation modes and
nuclear shapes.

Specialized experimental methods have been developed to measure quantum-

mechanical observables with low exotic beam rates in order to elucidate the structure

of neutron-rich nuclei. Using thick secondary targets to increase luminosity, the meth-
ods involved include γ-ray spectroscopy following deep-inelastic scattering, Coulomb
excitation of fast projectiles or reaccelerated beams, β-decay studies, or transfer re-

actions.
56
Beta-decay studies of Cr indicated a possible sub-shell gap at neutron number

N=32 [8]. The nearest doubly-magic nucleus is 48 Ca (see Figure 1.1) and if the magic
numbers established for stable nuclear species are considered, the next sub-shell gap

would be expected at N=40. A possible cause of the orbital reordering has been

1
 32 Ge
31 Ga
30 Zn
29 Cu
28 Ni
27 Co
26 Fe 44
25 Mn
24 Cr 42
23 V
22 Ti 40
21 Sc
Z 20 Ca 38
20 28 30 32 34

N
Figure 1.1: A section of the nuclear chart showing the region of interest. The black
squares are the β-stable nuclei. The three titanium isotopes (52,54,56 Ti) are circled.

1g9/2 10
1f5/2 6
2p1/2 2
2p3/2 4
28 28
1f7/2 8

π ν
56
Figure 1.2: Simplistic view of the orbital structure for Ti above the 1f7/2 shell

identified as the proton-neutron monopole interaction [9]. As the number of protons

occupying the πf7/2 shell decreases, the πf7/2 − νf5/2 monopole pairing interaction

strength weakens. As a consequence, the neutron orbital νf5/2 is shifted up in energy.

Coupled with the rather significant spin-orbit splitting between the νp1/2 and νp3/2

orbitals, this may lead to new sub-shell gaps (Figure 1.2). Shell-model calculations
using the effective interaction GXPF1 [10] revealed the possibility of sub-shell gaps
at neutron numbers N=32 and N=34.

To investigate the development of these shell gaps further, β-decay measurements

at the NSCL [11, 12] in conjunction with prompt γ-ray spectroscopy following deep-
54 56
inelastic reactions at Gammasphere [13, 14] were performed for Ti and Ti. The
energies of the first excited states in these nuclei supported the notion of a shell gap at

2
N=32, but not at N=34. It has been shown that knowledge of the level scheme alone

is not sufficient to identify shell closures [15, 16]. To search for additional evidence
of the newly proposed shell gaps, we have measured an additional observable, which

provides a direct measure of the degree of collectivity in a nucleus. The reduced tran-
sition matrix elements to the first excited 2+ state in 52,54,56
Ti have been determined
via intermediate-energy projectile Coulomb excitation [17] at the Coupled Cyclotron

Facility of the National Superconducting Cyclotron Laboratory. Coulomb excitation is

a sensitive probe to study the characteristics of quadrupole collectivity in nuclei with

even number of protons and neutrons [18–20]. Simultaneous with the measurement
of the de-excitation γ-ray energy, giving the energy spacing between the two bound

states, the Coulomb excitation cross section is related to the electromagnetic matrix
elements B(Eλ) and B(Mλ) of multipolarity λ. The B(E2) strength relates to the de-
gree of collectivity and provides insight into details of the many-body wave functions

for nuclei near closed shells. The first experiment involving Coulomb excitation of
radioactive beams was performed by inelastically scattering a 8 Li radioactive beam
nat
at 14.6 MeV on a Na target [21–23]. Gamma-rays emitted by exotic beams under-
going Coulomb excitation were measured first in an experiment involving scattering
76 208
of Kr on a Pb target at 237 MeV [24].
The first part of the thesis describes the measurement of the reduced transition
probabilities to the first 2+ state in the exotic nuclei 52
Ti, 54
Ti and 56
Ti via the

intermediate-energy Coulomb-excitation method and discusses the development of

shell closures for neutron numbers N=32 and N=34. The experimental method and

the analysis procedure have been extensively described in numerous publications [17,
25–28]. Emphasis will be placed on the characteristics of this experiment and the

nuclei investigated.
Experiments aimed at studying very exotic nuclei are characterized by low beam
rates. Gamma-rays emitted from fast moving nuclei (v/c=0.3) are detected at Doppler

shifted energies in the laboratory frame. Under current experimental conditions mea-

3
surements in the regime of limited statistics would gain from an enhanced peak-to-

background ratio. Even for relatively large bin sizes of the γ-ray energy spectra in
the projectile frame (Figures 2.10, 2.11, 2.12,2.13) the peak widths are large. An im-

portant factor in the energy resolution of a photopeak is the Doppler reconstruction

mechanism. A better correction of the Doppler broadening would reduce the number
of background counts under the peak, thus reducing the uncertainty in the peak area

and any cross sections derived from it. Energy resolution is also a crucial when two
γ-ray transitions close in energy have to be resolved.

The concept of γ-ray tracking gained a lot of attention in the past years. Collabo-
rations like GRETA/GRETINA [29, 30] and AGATA [31] were formed to address the

use of γ-ray tracking for nuclear physics. The method involves the determination of
the γ-ray interaction points inside a semiconductor-based detector with high spacial
accuracy, down to the size of the charge cloud or better (1-2 mm). Based on the inter-

action positions, the path of the γ-ray is reconstructed inside the detector. The path
reconstruction process involves the digitization of the signals from highly-segmented

semiconductor detectors (usually high-purity germanium). The recorded signals are

fitted with a base of precalculated signals to derive the interaction positions. Once

the positions are known, all possible scattering histories are computed and a figure of
merit is built. The path with the highest figure of merit is chosen to be the photon
path. If the γ-ray scattering path is known, also the first point of interaction is de-

termined. Knowing the first interaction point is crucial for Doppler correction. The
relation between the photon energy in the projectile frame and the laboratory frame

is given by:

Eproj = γElab (1 − βcosθ) (1.1)

where Elab and Eproj are the γ-ray energies in the laboratory frame and projectile

4
 Detector

γ
θ
Beam
Target
Figure 1.3: Example of γ-ray interaction with a segmented detector

frame, respectively. The beam velocity is β = v/c and

1
γ=
(1.2)
1 − β2

is the relativistic factor. The angle θ is taken between the scattered beam and the
γ-ray direction. This angle can be calculated by knowing the position where the
nucleus emitted the photon and the point where the γ-ray first interacted with the

detector. Figure 1.3 depicts an example of a γ-ray interacting three times within a
segmented detector. Only the first interaction point defines the angle necessary for

Doppler reconstruction.
The second part of the thesis deals with a study targeted at improving the inter-
action position resolution for the segmented germanium detectors used at NSCL for

γ-ray spectroscopy. A sub-segment interaction position resolution offers the possibil-

ity of a better Doppler reconstruction for in-beam γ-ray spectroscopy, leading to an

enhanced peak-to-background ratio. It will be demonstrated that a simpler approach

5
can yield good performance, without involving a full γ-ray tracking which requires

expensive hardware and software solutions.

6
Chapter 2

Reduced transition probabilities to

the first 2+ state in 52,54,56Ti and
development of shell closures at
N = 32, 34

2.1 Overview

Shell structure is the foundation for much of our present understanding of atomic

nuclei, although most of our knowledge about the ordering and location in energy
of the single-particle states remains empirical. In this context, neutron-rich nuclei

have become the focus of recent theoretical and experimental efforts.1 The on-going
investigations are motivated to a large extent by expectations of substantial modifi-

cations of shell structure in nuclei with a sizable neutron excess [33–38]. Such alter-
ations can have a considerable impact on global nuclear properties such as the nuclear
1
Reprinted excerpts and figures with permission from: D.-C. Dinca, R.V.F. Janssens, A. Gade,
D. Bazin, R. Broda, B.A. Brown, C. M. Campbell, M. P. Carpenter, P. Chowdhury, J. M. Cook,
A.N. Deacon, B. Fornal, S. J. Freeman, T. Glasmacher, M.Honma, F.G. Kondev, J.-L. Lecouey, S.
N. Liddick, P. F. Mantica, W. F. Mueller, H. Olliver, T. Otsuka, J. R. Terry, B. A. Tomlin, K.
Yoneda, Physical Review C, Volume 71, Number 4, 041302, 2005. Copyright 2005 by the American
Physical Society. [32]

7
shape or the type of excitations characterizing the low-energy level schemes. One of

the proposed causes for the reordering of single-particle states is the proton-neutron
monopole interaction [9]. This interaction has recently been invoked to account for

the presence of a sub-shell gap at N = 32 in neutron-rich nuclei located in the vicinity

48
of doubly-magic 20 Ca28 . At present, experimental evidence for the existence of this
N = 32 gap rests solely on the low and medium spin (I ≤ 12) level sequences of the
52 52−56 52−58
Ca [39], Ti [11, 13, 14] and Cr [8] even-even isotopes.
It is the purpose of our work to track the evolution of this sub-shell gap further,

through the measurement of the electromagnetic transition rates to the first excited
52,54,56
states of the Ti isotopes with the technique of intermediate-energy Coulomb

excitation [17]. Such rates provide one of the most sensitive probes of nuclear struc-
ture. In deformed nuclei, transition strengths are related to the magnitude of the
deformation, while in nuclei in the vicinity of closed shells, they are of great value in

probing the details of the many-body wavefunctions. In fact, these rates have often
highlighted properties that were unexpected on the basis of level energies alone. For

example, the B(E2; 0+ → 2+

1 ) value in doubly-magic
56
Ni was found to be larger than
anticipated [16], while that measured for 136 Te, with 2 protons and 2 neutrons outside
132
the doubly-magic Sn nucleus, is surprisingly small [40].
In the particular case discussed here, the transition rates represent a sensitive test
of the most modern effective interactions that have been developed to describe pf -

shell nuclei [10]. It is shown that the data support the view of a sizable shell gap at
N = 32, but that there is no experimental evidence for an additional sub-shell closure

predicted to occur at N = 34. Moreover, detailed comparisons between the data and
the calculations also indicate shortcomings of the proposed effective interactions in

reproducing the observed trend of the B(E2) values with neutron number.

8
 Table 2.1: Summary of secondary beam properties.
Beam Energy Velocity Velocity Target thickness
(before target) (before target) (mid target)
(MeV/nucl.) (v/c) (v/c) (mg/cm2 )
76
Ge 81 0.392 0.375 256
52
Ti 89 0.408 0.399 256
52
Ti 81 0.408 0.388 518
54
Ti 88 0.406 0.398 256
56
Ti 88 0.406 0.387 518

2.2 Experimental conditions

2.2.1 Secondary beams

The measurements were carried out at the Coupled Cyclotron Facility of the Na-

tional Superconducting Cyclotron Laboratory using secondary beams produced in

76
fragmentation of Ge primary beam at an energy of 130 MeV/nucleon. Following

the 380 mg/cm2 9 Be production target, the species of interest were selected with
the A1900 fragment separator [41] and directed to the target position of the high-
resolution S800 magnetic spectrograph [42].

Four settings of the A1900 separator were used in the experiment. First, the
76
Ge primary beam was degraded to 81 MeV/nucleon and sent onto a 256 mg/cm2
197
Au target as a check of the technique and the setup. Following this measurement,
secondary beams of the three even Ti isotopes of interest, all with an energy of
197
89 MeV/nucleon, were then selected in succession and directed onto Au targets of
256 mg/cm2 and 518 mg/cm2 thickness. The thinner Au target was used with the
52
Ti and 54 Ti fragments, the thicker with 52 Ti and 56 Ti. For a primary beam intensity

of 10 pnA, the three Ti settings resulted in average rates on target of 9000 Hz (52 Ti),
600 Hz (54 Ti) and 40 Hz(56 Ti). Each incoming beam particle was identified on an

event-by-event basis, and the isotopes of interest represented respectively 58, 28 and
10% of the flux of incoming particles. A summary of the secondary beam properties
is given in Table 2.1.

9
 57
Cr
Time of flight (au)
55
V

54
Ti

Dispersive focal plane position (au)

Figure 2.1: Particle identification spectrum for the 54 Ti setting. The dispersive focal
plane is on the horizontal axis and the time of flight on the vertical axis. The number
of counts is color coded and linear scale and saturated at 10% of the full scale.

2.2.2 Particle identification

As an example, the particle-identification spectra are presented for the 54 Ti isotope in

Figures 2.1, 2.2, and 2.3. A software gate requiring the coincidence of events in both
regions of interest in the dispersive focal plane versus time-of-flight spectrum and the

time of flight versus energy loss spectrum define the particle identification conditions.
The particle time of flight was measured between a scintillator at the object position
of the spectrograph and the E1 scintillator at the back of the S800 focal plane. E1

also acted as a particle trigger detector. The focal plane ion chamber measured the
energy loss and the Cathode Readout Drift Chambers (CRDCs) provided the beam

position and angle on an event-by-event basis. For a detailed description of the S800
spectrograph see References [42] and [43].

10
 57
Cr
Energy loss (au)

55
54
V
Ti

Time of flight (au)

Figure 2.2: Particle identification spectrum for the 54 Ti setting. The number of counts
is color coded and saturated at 10% of the full scale.

Counts

57 104
Cr
Energy loss (au)

54
Ti
55
V 103

102

Time of flight (au)

Figure 2.3: Particle identification spectrum for the 54 Ti setting. The number of counts
is color coded and log scale.

11
2.2.3 Intermediate-energy Coulomb excitation (theory)

Interactions between the target and projectile nuclei occur above the Coulomb barrier.
The preponderence of Coulomb excitations over nuclear reactions can be ensured
by selecting only impact parameters larger than the sum of the nuclei radii plus

several femtometers. This is equivalent to the selection of the most forward scattering
angles [17, 26, 27]. The impact parameter b depends on the scattering angle in the

center-of-mass frame θcm in the following way (Figure 2.4, Equation 2.1).

 
a θcm
b = cot , (2.1)
γ 2

where γ is the relativistic factor (1.2) and a is dependent on the projectile and target
atomic numbers (Zproj and Ztar respectively) and the reduced mass of the target-

projectile system (m0 ) via:

Zproj Ztar e2
a= . (2.2)
m0 c2 β 2

1 1 1
= + . (2.3)
m0 mproj mtar

Based on the kinematics of the process a relationship can be established between

the scattering angles in the center-of-mass frame (θcm ) and laboratory frame (θlab ).

The theory of relativistic Coulomb excitation was pioneered by A. Winther and

K. Alder [25]. Their work assumes that, to first order, the straight-line trajectory of

the projectile is perturbed by the recoil, rescaling the impact parameter (Equation
2.4).

πa
b→b+ . (2.4)
2γ

Winther and Alder’s theory link the reduced transition probabilities in the pro-

jectile to the cross section via:

12
 γ

β
θlab
Projectile
b

Target
Figure 2.4: Schematics of the projectile-target interaction. The target nucleus is con-
sidered at rest after the interaction (infinite mass).

13
 
  
Ztar e2 Bpro (πλ, 0 → λ) 2(1−λ)  (λ − 1)
−1
f orλ ≥ 2
σπλ ≈ π b (2.5)

c e2 min

 2 ln(bmax /bmin ) f orλ = 1

where π is the parity of transition and λ its multipolarity. The limits of the impact

parameters considered are denoted bmin and bmax .

2.2.4 Gamma-ray spectroscopic system

The Au target was surrounded by SeGA, an array of fifteen, 32-fold segmented, ger-

manium detectors [44] arranged in two rings with central angles of 90◦ and 37◦ relative
to the beam axis (See Figure 2.5). The forward ring contained 7 detectors while the

other 8 were located at 90◦ . The high degree of segmentation is necessary to correct
for the Doppler shift of the γ rays emitted in flight (on an event-by-event basis).

2.3 Analysis method

The reduced transition matrix elements B(E2; 0+ → 2+

1 ) can be deduced from the

Coulomb excitation cross section using the exact relationship of the Equation 2.5

(Reference [25]). Simulations with the code GEANT3 [45] reproduced the efficiency
of SeGA measured with the standard calibration source 152 Eu source and provided the

detector response for the in-beam data by taking into account the Lorentz boost (see
Reference [27] for further details). This reference also describes the particle identifi-

cation and the determination of the scattering angle carried out on an event-by-event
basis with the focal plane detector system [43] of the S800 spectrograph.
The Coulomb excitation cross section is given by the relationship

Nγ
σ= (2.6)
tot Ntarget Nbeam

14
 Backward ring Forward ring
o o
90 37

Incoming
beam

Target

Figure 2.5: The segmented germanium detectors distributed in two rings around the
secondary target position.

15
Nγ is the number of photons de-exciting for the state of interest. If the state is

not only populated via Coulomb excitation, feeding corrections have to be applied.
This was not the case for the present work. tot is the total efficiency of experimental

setup, including the gamma-ray detector effiency, particle detector efficiency and data
acquisition system dead time corrections. Ntarget is the number of nuclei in the target,
and Nbeam is the number of incoming projectiles. The number of nuclei in the target

(density per unit area) is calculated as

NA × ρ
Ntarget = , (2.7)
A

where NA is Avagadro’s number (NA = 6.022 × 1023 particles/mol), ρ is the target

density (thickness) in g/cm2 , and A is the atomic mass of the target nuclei in g/mol.
Nbeam is calculated by multiplying the number of particles integrated in the down
2
scaled (D/S) particle identification spectrum NpartD/S times the D/S factor set in
the trigger electronics fD/S .

Nbeam = NpartD/S × fD/S . (2.8)

The total efficiency is the sum of the efficiencies of the two rings (Figure 2.5).

tot = 37o + 90o (2.9)

For one ring of SeGA, the efficiency using the method of GEANT simulations, is

ring = (Eγ )GEANT

lab × δGEANT × Af . (2.10)
2
To reduce the dead time of the data acquisition system the trigger condition is set to a logic “OR”
between the particle detector (S800) trigger and the γ-ray detector (SeGA) trigger in coincidence
with S800. The particle trigger is down scaled, meaning that only one every number of events in
recorded. This is equivalent to a random sample of the particles not detected in coincidence with
γ-rays.

16
where (Eγ )GEANT
lab is the efficiency simulated with GEANT in the projectile frame

assuming an isotropic emission, δGEANT is the GEANT scaling coefficient, and Af is

an angular factor that accounts for the anisotropic angular distribution of the emitted

γ-rays.
A discrepancy on the order of 1 to 3.5 percent is found between the GEANT-
simulation photopeak efficiency for the calibration sources and the measured effi-

ciency. To account for this discrepancy when the efficiency in the projectile frame is
calculated, a scaling coefficient δGEANT is defined for the energy range of the γ-rays of

interest. The uncertainty for this coefficient also includes the uncertainty in the value
of the calibration source activity.

The magnetic substates are not evenly populated by the Coulomb excitation.
This process should be accounted for in the analysis. Discussions regarding the γ-ray
angular distribution (W (θproj )) can be found in References [20,25,46,47]. Nθproj is the

number of γ-rays detected by the detector array at a given angle θ. The index “proj”
indicates that the quantity is considered in the reference frame of the projectile.


Nθproj W (θproj ) sin θproj
θproj
Af = (2.11)
1 
Nθproj sin θproj
4π θ
proj

The uncertainty of the experimental cross section is

 2  2  2  2
∆Nγ ∆Nbeam ∆Ntarget ∆tot
∆σ = σ + + + (2.12)
Nγ Nbeam Ntarget tot

where


∆Nγ = Nγ (2.13)

 2  2
∆ρ ∆NA
∆Ntarget = Ntarget + (2.14)
ρ NA

17
 Af and (Eγ )GEANT
lab are based on GEANT simulations and analytical calculations

and the uncertainties induced by them are considered negligible compared to the other
sources. The main source of uncertainty are the peak area (Nγ ) and the simulation

scaling coefficient δGEANT (5 - 6%).

2.4 Results
76 197
Inelastic scattering of the primary Ge beam on a Au target was used to validate
76
the experimental technique. In Ge, the reduced transition probability has previ-
ously been determined through Coulomb excitation at energies below the Coulomb

barrier [48]. The relevant spectrum measured in the projectile frame, for scattering an-
gles restricting the impact parameter of the reaction to values larger than the sum of

the two nuclear radii plus 5 fm, is given in Figure 2.10. Using the Winther-Alder the-
ory of relativistic Coulomb excitation [25], the angle-integrated cross section measured

under these conditions translates into a value of B(E2; 0+ → 2+ 2

1 ) = 2923(346) e fm
4

that compares well with the adopted one of 2780(30) e2 fm4 [48]. From the same
measurement, a similar comparison can be made for the excitation of the Au target

and good agreement is again found between the present data and the literature [49]:
B(E2; 3/2+ → 7/2+ ) = 4472(951) versus 4494(409) e2 fm4 .

For all settings the reduced transition probability to the 7/2+ state in the Au
target was also calculated. Figure 2.6 shows the B(E2; 3/2+ → 7/2+ ) for all settings.
52
The first value labeled Ti is for the 256 mg/cm2 197
Au target thickness and the
second is for the 518 mg/cm2 . The value labeled “adopted” is the one from [49]. The
197
velocity of the excited Au target nuclei is very small and no Doppler correction

is needed. The de-excitation gamma-ray peaks have higher resolution (Figure 2.7)
than the ones for gamma-rays emitted by fast moving nuclei because the Doppler

broadening has no significant contribution at that velocity. The 197 Au nuclei from the
target can be considered at rest.

18
Figure 2.6: Measurement of the reduced transition probability from the ground state
to the 7/2+ state in the Au target for all beams used in this experiment.

19
 50

40
Counts / keV

30
511

20

10

0
450 500 550 600 650 700 750
Energy (keV)

Figure 2.7: Laboratory frame coincidence γ-ray spectra for 197 Au. The nucleus is
Coulomb excited by the electromagnetic field of the incoming 54 Ti beam.

20
 Background subtraction is employed in this type of experiment when the hardware

time gates of the data acquisition triggering system have to be wide and permit a large
number of events that are uncorrelated with the reaction of interest to be recorded.

Another situation in which the background subtraction is used is when the counts
40 60
from background lines (e.g. K from the environment, Co from activated pieces of
beam pipe) are repositioned in the reconstructed projectile frame energy spectrum

by Doppler correction in the same region as the in-beam peaks of interest. A software
“prompt time” gate is defined around the prompt time peak of each germanium

detector time spectrum (see Figure 2.9). A logic “OR” gate is defined for the prompt
time peaks of all the SeGA detectors. The same procedure is applied for a gate with

the same time duration (width), placed after the prompt time peak. When using the
background subtraction method, the data are scanned first requiring a logic “AND”
of the software particle identification gate and the prompt time of SeGA. Then data

are scanned with a logic “AND” condition for the particle identification gate and the
off-prompt SeGA time. The resulting γ-ray spectrum is subtracted from the spectrum

obtained for the prompt time. The result is a background subtracted spectrum. As an
197
illustration of the results, a background subtracted spectrum for Au is displayed
197
in Figure 2.8. Compared to the Figure 2.7 the background around the Au peak is
more flat (e.g. the 511 keV line disappeared).
Figures 2.10, 2.11, 2.12, and 2.13 show representative coincidence γ-ray spectra
76 52−56
for Ge and the even Ti isotopes Doppler reconstructed event-by-event in the
76
projectile frame. The energy at mid-target for Ge was 73.5 MeV/nucleon, and the

minimum impact parameter 17.6 fm as deduced from the maximum scattering an-
gle in the center-of-mass frame of projectile and target, θcm < 3.1◦ . For 52
Ti the

corresponding values for the 256 mg/cm2 and 518 mg/cm2 Au targets were 82.4 and
79.1 MeV/nucleon, respectively, with θcm < 3.1◦ and < 3.3◦ and similar minimum im-
pact parameter of 13.9 fm. The spectrum measured with the thinner target is shown
54 56
in the Figure 2.11. For Ti and Ti, the respective projectile energies were 83.3

21
 40

Counts / keV 30

20

10

-10
450 500 550 600 650 700 750
Energy (keV)
197 54
Figure 2.8: Laboratory frame γ-ray spectra for Au in coincidence with Ti. Back-
ground subtraction applied.

800

600
Counts / (3.125 ns)

Prompt time gate Off-prompt time gate

400

200

0
100 200 300 400 500 600 700
Time (ns)

Figure 2.9: Typical time spectrum for a SeGA detector. Prompt time gate is depicted
in dashed line. The off-prompt time gate, equal in width with the prompt time gate
is in dotted line.

22
Table 2.2: Summary of scattering angle cuts. The first entry for 52 Ti corresponds to
the 256 mg/cm2 target and the second to the 518 mg/cm2 target.
Beam Minimum impact Center-of-mass Laboratory-frame
parameter angle angle
◦
(fm) () (◦ )
76
Ge 17.6 3.06 2.18
52
Ti 13.9 3.10 2.42
52
Ti 13.9 3.29 2.57
54
Ti 14.0 3.20 2.48
56
Ti 14.1 3.58 2.75

Table 2.3: Summary of measured transition energies (2+ +

1 → 0g.s. ), number of particles
detected and Coulomb excitation cross sections.
Beam Energy Particles detected Cross section Target thickness
+ +
21 → 0g.s.
(keV) (mb) (mg/cm2 )
76
Ge 562.6(6) 26 × 106 394(47) 256
52
Ti 1050(2) 130 × 106 119(16) 256
52 6
Ti 1049(2) 67 × 10 125(16) 518
54 6
Ti 1497(4) 92 × 10 83(15) 256
56
Ti 1123(7) 6 × 106 155(51) 518

and 78.6 MeV/nucleon, with minimum impact parameters of 14.0 fm and 14.1 fm

computed from θcm < 3.2◦ and < 3.6◦ . The arrows indicate the expected location
of transitions deexciting the 2+
2 levels. A summary of minimum impact parameters

and the center-of-mass and projectile frame maximum scattering angles is presented
in the Table 2.2. Table 2.3 shows the measured transition energies (2+ +
1 → 0g.s. ) the

number of particles detected (with down scaler correction) and Coulomb excitation
cross sections measured for the angular ranges in Table 2.2.
For all settings in this experiment, projectile and laboratory frame, the areas of

the peaks of interest were determined with and without background subtraction. In
all situations no difference between the results with statistical significance was found.

Figures 2.14, 2.15, 2.16, 2.17 show particle-γ coincidence spectra for Doppler corrected
γ-ray energy with background subtraction applied.

With the reliability of the technique demonstrated, attention can now turn to the

23
 500
76Ge
400 v/c = 0.356
2+1 0+g.s.
Counts / 8 keV

300

200

100

0
200 400 600 800 1000 1200
Energy (keV)
Figure 2.10: Coincidence γ-ray spectra for 76 Ge Doppler-reconstructed event-by-event
in the projectile frame.

52Ti
v/c = 0.385
+ +
100 21 0g.s.
Counts / 8 keV

+ +
2 2 21
50

0
600 800 1000 1200 1400
Energy (keV)
Figure 2.11: Coincidence γ-ray spectra for 52 Ti Doppler-reconstructed event-by-event
in the projectile frame. The possible location of the 2+ +
2 → 21 transition is shown by
an arrow

24
 50
+ +
54Ti
22 21
40 v/c = 0.397
Counts / 8 keV

30 2+1 0+g.s.

20

10

1000 1200 1400 1600 1800 2000

Energy (keV)
Figure 2.12: Coincidence γ-ray spectra for 54 Ti Doppler-reconstructed event-by-event
in the projectile frame. The possible location of the 2+ +
2 → 21 transition is shown by
an arrow.

56Ti
v/c = 0.362
10 + +
21 0g.s.
Counts / 12 keV

0
800 1000 1200 1400 1600
Energy (keV)
Figure 2.13: Coincidence γ-ray spectra for 56 Ti Doppler-reconstructed event-by-event
in the projectile frame.

25
 250
76Ge
v/c = 0.356
Counts / 8 keV 200

150 2+1 0+g.s.

100

50

0
200 400 600 800 1000 1200
Energy (keV)
76
Figure 2.14: Projectile frame γ-ray energy in coincidence with Ge. Background
subtraction applied.

70 52Ti
60 +
21 0g.s.
+ v/c = 0.385
Counts/ 8 keV

50
40
30
20
10
0
600 800 1000 1200 1400
Energy (keV)
52
Figure 2.15: Projectile frame γ-ray energy in coincidence with Ti. Background sub-
traction applied.

26
 30
54Ti
25 v/c = 0.397
Counts / 8 keV
20 2+1 0+g.s.

15

10

0
1000 1200 1400 1600 1800 2000
Energy (keV)
54
Figure 2.16: Projectile frame γ-ray energy in coincidence with Ti. Background sub-
traction applied.

56Ti
10
v/c = 0.362
+ +
8 21 0g.s.
Counts / 8 keV

0
800 1000 1200 1400 1600
Energy (keV)
56
Figure 2.17: Projectile frame γ-ray energy in coincidence with Ti. Background sub-
traction applied.

27
Table 2.4: Comparison of measured B(E2; 0+ → 2+ 1 ) values (labeled B(E2; ↑) in
the table) with shell model calculations using the GXPF1 interaction as well as the
recently proposed GXPF1A interaction. The two 52 Ti entries correspond to sepa-
rate measurements with Au targets of different thicknesses; (a) 256 mg/cm2 , (b) 518
mg/cm2 and (c) the weighted average of the two. Data on the excitation of the Au
target by the various Ti isotopes are given as well.

Nucleus B(E2; ↑) B(E2; ↑) B(E2; ↑) B(E2; ↑)

(e2 fm4 ) (e2 fm4 ) (e2 fm4 ) (e2 fm4 )
expt. GXPF1 GXPF1A 547 keV Au
52
Ti (a) 593(80) 427 435 4114(627)
52
Ti (b) 548(70) 427 435 4063(455)
52
Ti (c) 567(51) 427 435
54
Ti 357(63) 453 446 4279(672)
56
Ti 599(197) 483 448 6356(2227)

three even-mass Ti isotopes of interest. The analysis was carried out following the
76
prescription given above for Ge. In each case, the cross section for the excitation of
the first 2+ level was extracted from the γ-ray yields measured in spectra corrected

for the Doppler shift and the response of the SeGA detectors (representative spectra
are shown in Figures 2.10, 2.11, 2.12, 2.13), with appropriate restrictions on the

scattering angle of the Ti fragments (see discussion above and Ref. [27]). Table 2.4
presents the derived B(E2; 0+ → 2+
1 ) values. In the case of
52
Ti, measurements were
carried out with two targets of different thickness in order to ensure the validity of

the experimental approach when thicker targets are required to compensate for lower
fragment yields, as is the case here for 56 Ti. The two 52 Ti data points are in excellent

agreement (Table 2.4). Furthermore, they also agree with an earlier measurement [50],
though the errors are large: B(E2; 0+ → 2+ +515 2 4
1 ) = 665−415 e fm . Additional confidence

in the transition rates of Table 2.4 comes from the data gathered simultaneously for

Coulomb excitation of the target: the B(E2; 3/2+ → 7/2+ ) values for the excitation
197
of Au agree with each other and with the adopted value [49] (see Table 2.4).

The values in Table 2.4 assume that the excitation of the 2+

1 levels of interest

occurs in a one-step, direct process without significant contribution(s) from higher-

28
lying 2+ states to the measured 2+ +
1 → 0 γ-ray yields. In
52
Ti, a number such higher

2+ states are known [51], and a 2+ 54

2 level has also been proposed tentatively in Ti [13].

In the data for both nuclei, none of the γ rays associated with decays from these levels

towards the ground state and the yrast 2+

1 level was observed, as illustrated in Figures

2.11 and 2.12 where the location of the 2+ +

2 → 21 transitions in
52,54
Ti is given with
arrows. The absence of peaks indicates that any feeding correction must be small.

Furthermore, as discussed below, these excited levels are understood in the context of
the shell model and the associated reduced transition probabilities are calculated to

be smaller by an order of magnitude or more than the B(E2; 0+ → 2+

1 ) values under

discussion here. The largest such strengths is predicted to occur for the 2+
2 level in

52
Ti. In this case, the upper limit for the 2+ +
2 → 21 intensity obtained from the data

translates into a maximum correction to the B(E2; 0+ → 2+ 2 4

1 ) value of 34 e fm , i .e.,

well within the error bars of Table 2.4. In all other cases the contributions from higher

2+ levels would be even smaller and it is concluded that possible feeding corrections
do not affect the values of Table 2.4 significantly.

2.5 Discussion

Experimental evidence for a shell closure is usually inferred from at least two observ-

ables derived from nuclear spectra: the energy of the first excited state and the reduced
transition probability to the same level. The former is expected to be rather large,

reflecting the sizable energy gap associated with a shell or sub-shell closure, and the
latter is anticipated to be small and comparable to single-particle estimates. Figures
2.18 and 2.19 present the two physical quantities of interest for all even Ti isotopes

with mass A = 48 to A = 56. From the figure, a clear anti-correlation between the
two observables can be readily seen: while the E(2+
1 ) energies increase significantly

at N = 28 and N = 32 (Figure 2.18), the B(E2; ↑) strengths are lowest for these
two neutron numbers (Figure 2.19). Furthermore, both these physical quantities also

29
 26 28 30 32 34N
E(2+1) Energy (keV)
1600

800
Experiment
GXPF1
GXPF1A
0 48 50 52 54 56Ti
Ti Ti Ti Ti
Figure 2.18: Comparison of the measured 2+ 1 excitation energies with the results of
large-scale shell model calculations using the GXPF1 (dashed lines) and GXPF1A
(solid lines) effective interactions.

differ markedly from the corresponding values at neutron numbers N = 26, 30 and
34. For 50 Ti, the well known shell closure at N = 28 translates into a small transition

probability: with the B(E2) value of Figure 2.19, the deexcitation from the 2+
1 level

to the ground state has a strength of only 5.6 single-particle units.

The fact that the excitation energy and the reduced transition probability observed
54 50
in Ti are comparable to those in Ti (see Figure 2.19 and Table 2.4) then suggests

that the Ti isotope with N = 32 is as good a semi-magic nucleus as its N = 28

counterpart and, hence, that a substantial sub-shell gap must occur at N = 32.
Conversely, the fact that the three other Ti isotopes have 2+
1 excitation energies lower

by several hundreds of keV and B(E2; 0+ → 2+

1 ) values higher by a factor of ∼ 2 can

be interpreted as an experimental indication for the absence of sub-shell gaps in the

30
 26 28 30 32 34 N
2 fm 4)

800
2+
1 )(e

400
B(E2, 0g.s.
+

Experiment
ep = 1.5e
GXPF1
en = 0.5e
GXPF1A
0 48
Ti 50
Ti 52Ti 54
Ti 56Ti

Figure 2.19: Comparison of the measured absolute B(E2; 0+ → 2+ 1 ) transition

strengths with the results of large-scale shell model calculations using the GXPF1
(dashed lines) and GXPF1A (solid lines) effective interactions. The B(E2; 0+ → 2+ 1)
value for 52 Ti is the weighted average of the two measurements given in Table 2.4.

31
neutron single-particle spectrum at N = 26, 30 and 34.

Diagonalization of Hamiltonian matrices when the full valence-nucleon space is

considered is difficult due to limitation in computing power. For the pf -shell, an

effective interaction, GXPF1 [10], was proposed, based on microscopic effective inter-
action [52] based on the Bonn-C potential.
Large-scale shell model calculations with the GXPF1 effective interaction, opti-

mized for the description of pf -shell nuclei [10], attribute the onset of a N = 32 gap
in neutron-rich Ca, Ti and Cr nuclei to the combined actions of the 2p1/2 − 2p3/2

spin-orbit splitting and the weakening of the monopole interaction strength between
f7/2 protons and f5/2 neutrons. The dashed lines in Figure 2.18 represent the results

of calculations with this interaction: while the N = 32 gap in the Ti isotopes is ac-
counted for, the calculations also predict an additional gap at N = 34 that is not
borne out by experiment. As pointed out in References [11, 14], the data suggest in-

stead that the energy spacings between the p3/2 , p1/2 and f5/2 neutron orbitals, as well
as the degree of admixture between these states in the wavefunctions of the 56 Ti yrast

excitations, require further theoretical investigation. This has been done recently by
Honma et al. [53] with the introduction of a modified version of the interaction, la-

beled GXPF1A, in which the matrix elements of the interaction involving mostly the
p1/2 orbital have been readjusted. It is worth pointing out that the evaluation of the
properties of this orbital from experimental data is particularly challenging since it

contributes little angular momentum to any given state. Traces of its impact are often
obscured as a result. The solid lines in Figure 2.18 indicate that the GXPF1A calcu-

lations reproduce the E(2+

1 ) energies. In fact, they provide a satisfactory description

of all the known levels in the even Ti nuclei, including those above the 6+ level in
54
Ti which involve neutron excitations across the N = 32 shell gap [53]. They also
describe the odd Ti nuclei satisfactorily [54].
Shell-model predictions of the reduced transition matrix elements were calculated

32
using
(ep Ap + en An )2
B(E2 ↑) = , (2.15)
2Ji + 1

where Ap and An are the E2 matrix elements calculated in the model space matrix
of the proton and neutron. ep and en are the total effective charges of the proton

and neutron [55]. The choice of effective charge values for protons and neutrons is
related to the core polarization models [56]. The effective charges account for average

effects of the renormalization from wavefunction admixtures outside the model space
and center-of-mass corrections. Ji is the spin of the initial state, 0 in the case of
52,54,56
the ground states of Ti. The MSHELL code [57] was used to carry out the
computations.
For all the even Ti isotopes, the wavefunctions of the 2+ levels are dominated by

(f7/2 )2 proton configurations coupled to ground and excited states of the neutron con-
figurations. This is reflected in the proton and neutron amplitudes Ap and An from

which the E2 matrix elements are computed (see below). For the GXPF1 interac-
tion, these (Ap , An ) amplitudes, in units of efm2 , have respective values of (8.8,15.4),
48−56
(10.7,9.5), (9.0,14.4), (10.7,10.6), and (11.8,8.7) for the even Ti. The theoretical

shell gaps for neutrons at N = 28, 32 and 34 result in reduced An amplitudes and
50,54,56
in excitation spectra for Ti that most closely reflect the (f7/2 )2 proton struc-

ture. The deviation of the experimental 56 Ti spectrum from theory indicates a weaker
shell gap at N = 34. As stated above, the new GXPF1A interaction [53] improves

the agreement, and the new amplitudes (Ap , An ) = (10.3,11.4) reflect a larger neu-
tron admixture. With this interaction, the calculated p1/2 − f5/2 shell gap at N = 34
is still significant, i.e., 2.5 MeV. Furthermore, this gap is calculated to increase to
54
3.5 MeV for Ca [53], so that a neutron sub-shell closure is still predicted in this
case. The B(E2, ↑) rates computed from the (Ap , An ) values (Equation 2.15) with

conventional effective charges of ep = 1.5e and en = 0.5e overestimate the measured

transition rates for the N = 28 and 32 nuclei (Table 2.4, Figure 2.19). Moreover, they

33
 26 28 30 32 34 N
2 fm 4)
800
2+
1 )(e

400
B(E2, 0g.s.
+

ep ~ 1.15e
en ~ 0.8 e
0 48
Ti 50
Ti 52Ti 54Ti 56Ti

Figure 2.20: Comparison of the measured absolute B(E2; 0+ → 2+ 1 ) transition

strengths with the results of large-scale shell model calculations using the GXPF1 and
GXPF1A effective interactions using the effective charges calculated from Ref. [1].

are rather constant as a function of neutron number, in contrast with the oscillating
behavior observed in the experiment. An oscillation related to the neutron shell gaps

is present in the An amplitudes. It is possible that, for neutron-rich nuclei, the neutron
en effective charge needs to be increased, while keeping the isoscalar effective charge

ep + en constant. Such a modification in the ep and en charges could result in a better

agreement with experiment. Recent data [1] on analogue states in A = 51, Tz = ±1/2
mirror nuclei suggest that this may well be the case and values of ep ∼ 1.15e and

en ∼ 0.8e were proposed. While these values would induce a small staggering in the
calculated B(E2) values (see Figure 2.20), they are not sufficient to bring experiment

and theory in agreement. Additional data on pf -shell nuclei are needed to investigate
this issue further.

34
 In summary, the present data on absolute E2 transition rates, together with earlier

work on excitation energies, confirm the presence of a sub-shell closure at neutron

48
number N = 32 in neutron-rich Ti nuclei above Ca, an observation in agreement

with the results of shell model calculations with the most recent effective interactions.
However, the data do not provide any direct indication of the presence of an additional
N = 34 sub-shell gap in the Ti isotopes. Moreover, the measured B(E2; 0+ → 2+
1)

probabilities highlight the limitations of the present large-scale calculations as they

are unable to reproduce in detail the magnitude of the transition rates in the semi-

magic nuclei and their strong variation across the chain of neutron-rich Ti isotopes.

35
Chapter 3

Sub-segment interaction position

resolution for the NSCL SeGA
detectors

3.1 Physics with fast exotic beams at the NSCL

The National Superconducting Cyclotron Laboratory (NSCL) at Michigan State Uni-

versity has designed and purchased an array of eighteen 32-fold segmented high-purity
germanium (HPGe) detectors [44]. The Segmented Germanium Array (SeGA) was

optimized for nuclear spectroscopy of fast exotic beams produced by projectile frag-
mentation. Since its commissioning in 2001 the types of experiments in which SeGA

played a central role ranged fromγ-ray spectroscopy following β-decay [58, 59] and
transfer reactions to Coulomb excitation [27, 32], inelastic proton scattering [60, 61],
one- or two-nucleon knockout or fragmentation [62, 63]. Despite the broad range of

reaction types studied, all experiments except β-decay measurements share common
characteristics. The nuclei that compose the secondary beams delivered to the ex-

perimental stations have velocities in the 0.15c - 0.65c range at the γ-ray emission
time. For most experiments the detector multiplicity is one with the highest proba-

36
bility, meaning that the array will have on average about one detector triggering for

a given event. Gamma-ray multiplicity per detector is also mostly 1 (a detector is hit
by a single γ-ray), and there is no need to disentangle the signals from two or more

photons. The triggered count rate is usually lower than 1 kHz. This affects the cost
of the digital data acquisition system because some of the digital signal processing
(DSP) features can be implemented in software in commercial off-the-shelf comput-

ers and components. SeGA is used in conjunction with an ancillary particle detector
that triggers the data acquisition (e.g. NSCL’s S800 spectrograph). In addition to

reducing the trigger rate in the γ spectroscopic system, ancillary detectors also pro-
vide information about the incoming and outgoing beam (direction, velocity, isotope

identification). SeGA is not designed to be used as a Compton camera. Photons come

mostly from the target or from room background. Logic “AND” trigger conditions
with the particle detectors reduce the room background random coincidences to a

manageable level. Also, various particle tracking detectors can be used to estimate
the γ-ray emission point. Due to the relativistic velocities at which the nuclei travel,

the emitted γ-rays are Doppler shifted in the laboratory frame where the measure-
ment is performed. Gamma-rays detected in the laboratory frame at forward angles

have energies higher than in the center-of-mass, while those detected at backward an-
gles have energies lower than in the center of mass. The energy in the center-of-mass
frame is the one characteristic of the nuclear transition, and it must be Doppler recon-

structed from the energy measured in the laboratory frame. The Doppler corrected
energy depends on the measured γ-ray energy in the laboratory frame, the velocity of

the beam, and emission angle with respect to the velocity of the beam (see Equation
1.1). The angle between the projectile velocity and the γ-ray emission direction is

determined if the emission point, the velocity vector of the nucleus, and the first in-
teraction point with the detector are known. The source of γ-rays of interest is known
as the target is usually placed in the center of the array. The emission point can be

estimated using the particle detectors placed before and after the secondary target.

37
The lifetimes of the excited states are usually very small. Typically the γ-ray emis-

sion occurs while the nucleus is still inside the target. The determination of the first
interaction point with the detector depends on the characteristics of the given γ-ray

spectroscopy system. For a given energy and orientation of the crystals the efficiency
of the system is proportional to the inverse square of the average distance between
detectors and target. The energy resolution describes how well two energy peaks can

be separated and determines how large the peak-to-background ratio is. It depends
on the intrinsic resolution of the detector (crystal plus accompanying electronics), the

uncertainty in the beam velocity and the opening angle of the detector. It is given by
the relationship:

 2  2  2  2
∆Eγ β sin θ 2 −β + cos θ 2 ∆Eint
= (∆θ) + (∆β) + .
Eγ 1 − β cos θ (1 − β 2 )(1 − β cos θ) Eγ
(3.1)

where β is the velocity of the nucleus that emitted the γ-ray at the time when the
nuclear transition occurred, θ is the emission angle with respect to the velocity of

the beam and Eγ is the γ-ray energy. In a first approximation, the uncertainty in the
determination of the target position is neglected. The intrinsic contribution ∆Eint is

not strongly dependent on energy. The uncertainty in the beam velocity is due to the
fact that, if the de-excitation takes place inside the target, the velocity of the beam
at that point is not known. The ∆β term depends on the lifetime of the excited state

and on the target thickness. The uncertainty in the opening angle of the detector is
the angle subtended by the active volume of the detector that measured the γ-ray.

By segmenting the outer contact of the detector in a number of slices and quadrants
the interaction position can be determined with higher accuracy. The opening angle

is reduced approximately linearly with the number of transverse segment groups.

Each of SeGAs 18 coaxial detectors is 32-fold electronically segmented in 8 longi-
tudinal slices and 4 axial quadrants [44]. In the experiment setups the SeGA detectors

are placed with their symmetry axes parallel to the beam axis to take advantage of

38
the 8 longitudinal segment groups for Doppler reconstruction. In the case of SeGA

detectors, if the segment where the first interaction occurred could be established, the
uncertainty in the opening angle would be the angle subtended by a single segment.

Figure 3.1 shows the energy resolution dependency on the angle of the center of the
detector with respect to the beam axis for a set of typical parameters, considering,
for simplicity, the lifetime of the excited state to be zero. For the angles at which

detectors can be physically placed, even if the uncertainty induced by the opening
angle is reduced to zero, there will still be the uncertainty due the beam velocity

that is dependent on the target characteristics and the lifetime of the excited state
(considered zero in derivation of the Equation 3.1)

In the determination of the first interaction position, presently the segment with
maximum energy deposited is selected. This simple algorithm gives satisfactory results
because low-energy photons tend to deposit most of their energy in the first interaction

and the photons with higher energy tend to scatter forward. The forward scattering
does not change the primary γ-ray emission angle by a significant amount. Figure 3.2
52
shows an example of a γ-ray energy spectrum from the Coulomb excitation of Ti
on a 197 Au target, measured in the laboratory frame for the upper panel and Doppler

reconstructed with the “hit on maximum” algorithm in the lower panel.

In the lab frame only the background lines are clearly visible. After Doppler cor-
52
rection the Ti appears and the background lines are smeared out. Only the first

interaction of the γ-ray with the detector is of importance for Doppler reconstruc-
tion. Neglecting the loss in energy resolution due to the finite lifetime of an excited

state, the effective energy resolution obtained in experiments is lower than the the-
oretical prediction also because the segment with the highest energy deposited is

not always where the γ-ray first interacted with the detector. These mispredictions
lead to a degradation of the energy resolution for Doppler corrected spectra. There
are several approaches to address this problem. The γ-ray tracking methods [29, 30]

involve a very precise (down to 1-2 mm) determination of the interaction position.

39
 3

2
∆E / E (%)

0
0 60 120 180
θ (deg)
Figure 3.1: The energy resolution dependence with the detector angle for a 52 Ti beam
with β = 0.385 and uncertainty ∆β = 0.024. The energy of the γ-ray is considered
ECM = 1.049 MeV (projectile frame). The opening angle of the detector is considered
∆θ = 3 deg, corresponding to a SeGA detector places at 20 cm from target. The
intrinsic constant is ∆Eint = 0.002. The dotted line indicates the contribution from
the beam velocity uncertainty. The dash-dot line corresponds to the contribution
from the opening angle of the detector. The contribution of the intrinsic resolution
is plotted in dashed line. The total energy resolution is plotted the continuous line.
The conditions are typical for fast exotic beams physics at the NSCL.

40
 60
β = 0.
Counts/ 2 keV

40 1173 1332 1460

20

100 β = 0.385
Counts/ 8 keV

1049

50

600 800 1000 1200 1400 1600

Energy (keV)

Figure 3.2: Laboratory frame (upper panel) and projectile frame (lower panel) γ-ray
energy spectra. In the laboratory frame, the 1173 keV and 1332 keV lines correspond
to the 60 Co decay and the 1460 keV line to the 40 K decay. In the projectile frame, the
1049 keV line corresponds to the 52 Ti 2+ +
1 → 0gs transition.

41
With the interaction points and the energies deposited in each of them determined,

the probability of each type of scattering history is calculated for every permutation
of interaction points using the scattering angles from segment to segment and ener-

gies and a figure of merit is built. Based on the value of the figure of merit the most
probable interaction path is determined. For Doppler reconstruction purposes the first
interaction point in that path is considered. Another approach is to use an energy-

weighted position. Instead of choosing one of the segment or sub-segment positions to

be the first interaction, an average position based on physics considerations is chosen.

In any of the above mentioned approaches, finer segmentation is needed to reduce

the uncertainty in the scattering angles between the interaction points. One way to

accomplish this is by using the properties of the signals formed in the semiconductor.

3.2 Signal formation in HPGe detectors

Semiconductor radiation detectors are the unquestionable favorites of the high-resolution

γ-ray spectroscopy scientific community. Their energy resolution is unmatched by
scintillators. In addition, semiconductor-based detectors have high efficiency, due to

their large atomic number. For the energy range of interest, the photoelectric effect
cross section is proportional to Z 3.5 , the Compton effect to Z, and the pair produc-

tion with Z 2 . The high-purity germanium (HPGe), with the atomic number equal to
32 is the material of choice for high-resolution spectroscopy. Semiconductor materials

investigated thus far that have a Z higher germanium (HgI2 , CdTe, GaAs) suffer from
charge trapping problems that lead to incomplete charge collection, depending on the
interaction position inside the crystal. Such problems make these materials unsuit-

able for most high-resolution applications. A further advantage of Ge is the maturity

of the technology to grow, cut, implant and diffuse impurities in crystals. To some

extent the crystals can be machined in different shapes and the impurity profile can
be controlled. The NSCL SeGA detectors are close-ended, coaxial cylindrically sym-

42
metric. The impurity profile is of the n-type, meaning that the outer contact is a thin

region implanted with boron atoms (50 µm) and the central contact region is lithium
diffused (500 µm). The Boron is a substitution acceptor in germanium forming a p+

region. On the central contact side, the lithium acts as a an interstitial donor atom in
germanium forming an n+ impurity region. The central contact is biased to a voltage
varying from 4000 V to 5000 V depending on the detector (positive polarity). The sig-

nal from the inner contact is collected by an AC-coupled preamplifier and the signals
from the outer segments are collected by a DC-coupled preamplifier, also connected

to the ground. For more details see Reference [44].

A γ-ray photon with the energy in the range of interest for the experiments per-

formed at the NSCL can interact with the germanium detector via four major pro-
cesses: photoelectric effect, Compton effect, Rayleigh (coherent) scattering, and pair
production. From several keV up to around 3 MeV, the photoelectric effect and the

Compton scattering are the dominant interaction modes. The Rayleigh scattering is
of importance for low energy photons and although the threshold is at 1.022 MeV, the

pair production mechanism starts to play a role only for photon energies higher than
5 MeV. Viewed from the crystals point of view, an interaction with a photon pro-

duces an electron (for Compton scattering and photoelectric effect) or an electron and
a positron in the case of a pair production event. For the photon energies of interest,
the electron has an energy much larger than the other electrons bound in the crystal

lattice. The primary electron interacts with the lattice via ionization (electron direct
and indirect ionization) and non-ionization processes (phonon excitations). During

the stopping process, the primary electron creates a large number of electron-hole
pairs. The number of electron-hole pairs can be estimated from the average energy

spent to move an electron from the valence band into the conduction band. This
energy is 2.96 eV for Ge at 77 K. As an example, a 1 MeV photon that deposits all
its energy in the crystal produces around 3.4 ×105 electron-hole pairs. The 2.96 eV

energy is larger than the Ge band gap at 77 K because some of the primary electron

43
energy is lost in the production of crystal lattice vibrations (phonons).

The ensemble of created electron-hole pairs is called the charge cloud. In normal
conditions, the pairs recombine locally. In the case of the detector, the crystal is

biased, and the charges start moving in opposite directions, depending on the sign of
their charge. Electrons promoted into the conduction band and their vacancies left in
the valence band (holes) move as independent charge carriers.

Signal formation at the electrodes is described by the Ramo-Shockley theorem

via the weighting potential method [64, 65]. The weighting potential is not a real

potential. It is a measure of the coupling between the charge carriers at the specified
position in space and the sensing electrode. It can be calculated by solving Laplace’s

equation in the crystal volume for the sensing electrode placed at a potential of one
volt and all the other electrodes grounded. After an interaction, the electrons and
holes move in the real electric field of the detector, but the signals they produce at

the collecting electrodes are determined by the weighting potential.

Briefly, this is the recipe on how the signal shapes can be calculated. If the prop-

erties of the crystal (shape, impurity profile, dielectric constant) and the voltage bias
are known, the net space charge density (ρ(r)) can be calculated from the distribution

of donor atoms (Nd (r)) via the relationship:

ρ(r) = eNd (r) (3.2)

where e is the electron charge. The electric potential (Φ(r)) throughout the crystal

can be obtained by solving the Poisson equation:

ρ(r)
∇2 Φ(r) = − (3.3)


where  is the dielectric constant of germanium. There are only a few trivial cases

for which this equation can be solved analytically (see for example [66]). Most of the
time a numerical solver is needed. The electric field inside the crystal is the gradient

44
of the electric potential.

E(r) = −∇Φ(r) (3.4)

The charge carrier drift velocity is then:

vdrif t,c (rc ) = µc E(rc ) (3.5)

The subscript c stands for charge carrier, since the electron and hole mobilities (µe
and µh ) are different, depending on the effective mass of the carrier which in turn is

a tensor whose components vary depending upon the crystal axes orientation. For a
study concerning the anisotropy of the electron drift velocity in germanium crystals
at high electric fields and low temperature and its influence on the charge collection

process see Reference [67].

Net charges inside the crystal create image charges in the inner and outer elec-

trodes. These image charges create the signals that are picked up by the preamplifiers.
It is a common missconception that the charges collected at the electrodes create the
signals. Instead, their image charges are the ones responsible for generating signals.

The real charge collected is the integral of the image charge induced. According to
the Ramo-Shockley theorem the current induced in the electrodes is:

ic = qc .vdrif t,c Ew (3.6)

where Ew is the weighting field calculated from the weighting potential.

For a SeGA detector, the labeling scheme of the segments is the one provided in

Figure 3.3.
Two cross sections of the weighting potential for segment E are plotted in Figures

3.4 and 3.6. Figure 3.4 shows a cross section of the weighting potential for segment E,
2 mm above the separation plane between the segments E and D. Let’s assume that

two charge carriers are moving on the paths “1” and “2”. The charge moving on the

45
 H ... D ... A

Figure 3.3: Segment labeling scheme for SeGA crystals.

red path (labeled “1”) is sensed differently by the electrode E than a charge moving

on the blue path (labeled “2”). The weighting potential along paths “1” and “2” are
plotted in Figure 3.5. If the charge carrier is of the same type (electron or hole), the

electric field being axially symmetric, the signal produced by the charge moving on
path “1” will have a larger amplitude than the charge on path “2”. Let’s consider

a longitudinal cross section of the weighting potential for segment E at 45 degrees

(midway between the quadrants) (Figure 3.6). A charge moving in segment D 2 mm
above segment E will be sensed differently in segment E than a charge moving 4 mm

above segment D. The signal amplitude of the trajectory at 2 mm will be different

(larger) from the amplitude at 4 mm or more.

To qualitatively illustrate the possibility of sub-segment radial resolution, an ex-

ample of three interactions is presented in Figure 3.7. It is important to keep in mind
that the sensing/collecting electrodes for the segments are on the outer border of

46
 Weighting potential for segment E
30 V

20 0.2
1
10
2 0.15
y (mm)

0.1
-10

-20 0.05

-30
0
-30 -20 -10 0 10 20 30
x (mm)
Figure 3.4: Weighting potential for segment E, transversal cut 2 mm above the border
between segments E and D.

47
 0.25
Weighting potential (V)

0.2
1
0.15
2
0.1

0.05

0
0 5 10 15 20 25 30
Radius (mm)
Figure 3.5: Weighting potential along the trajectories “1” and “2” in Figure 3.4.

48
 80 V

0.8
70

0.7
60
0.6
50
H (mm)

D 0.5
40
E 0.4
30
F
0.3
20
0.2

10 0.1

0 0
0 10 20 30
R (mm)

Figure 3.6: Weighting potential for segment E, longitudinal cut at 45 degrees.

49
 1
2
3 3
2
1

3 2 3
1
1 2
Figure 3.7: Example of real and transient charge signals.

the detector. After the electron-hole pairs are created, the electrons move toward the

central contact and the holes toward the outer contacts (segments).
For the interaction point labeled “1”, both electrons and holes are close to the
sensing electrode. At first electrons dominate, so the shape of the signal that peaks

rapidly. But since they move away from the electrode their contribution becomes
smaller and smaller. The holes give a strong signal at first, but they are fully collected

sooner than the electrons and their contribution soon dies out. For the interaction
point labeled “3”, the electrons are rapidly collected by the central contact and are

far from the sensing electrode. Their contribution is small. The holes travel almost
the entire radius of the detector and produce a small signal at first that gets amplified
as the charge carriers get closer to the collecting electrode. The interaction point “2”

is an intermediate situation. For the neighboring segments, the transient signal is

a variation of positive and negative signals that integrate to zero. Signals from the

SeGA detectors cannot be read directly from the crystal in a technically feasible way.
Each channel has a charge-sensitive preamplifier to condition the signal. The charge-

50
sensitive preamplifier works in the first approximation as an integrator. The Figure

3.7 also shows the signals from the interacting and neighboring segments after the
preamplifier. They have different rise times depending on the interaction position. In

the interaction segments, because real charge is collected, the preamplifier integrates
to a given value proportional to the charge created in the segment, which in turn is
proportional to the energy deposited in the interaction.

3.3 Experimental setup

A γ-γ coincidence setup was assembled. Figure 3.8 shows the SeGA scanning stand

where the experiment was performed. The purpose of the setup was to investigate
the segments and the central contact signals for a single interaction Compton process.

The 662 keV γ-rays from a collimated 137 Cs source were directed at the SeGA detector
perpendicular to the detector axis. The heavimet collimator had a length of 100 mm

with a cylindrical hole with a diameter of 2 mm. The distance between the radioactive
source and the outer surface of the detector was about 150 mm, limiting the maximum
diameter of the beam spot at the interaction region to around 5mm. A 3”x3” Bicron

NaI(Tl) scintillation detector was placed underneath the SeGA detector to allow
detection of γ-rays scattered from the germanium crystal. Two 50 mm thick lead

bricks were used to create a collimating slit between the two detectors of about 2 mm
in width. Two Struck 100 MHz sampling rate SIS3300 12-bit and two SIS3301 14-bit

flash analog-to-digital converters were used to digitize the waveforms from the central
contact and 31 segments out of a total of 32. For the coincidence measurements one
channel was used for the pre-amplified NaI detector signal, reducing the number of

digitized signals from segments to 30. To include all the shape information contained
in the pulse, the length of the trace was set to include the rise time part of a segment

signal with energy deposited (comparable with the drift time of 200 ns - 500 ns) and
parts of the signal before and after the rising/falling edge. A number of 128 samples

51
 Figure 3.8: Experimental setup of the SeGA scanning stand.

per waveform is sufficient to extract timing and position information. Various settings

have been tried, from 128 (∼1.3 µs) to 1024 samples per trace(∼10µs).
All 4 digitizing modules were placed on a VME backplane communicating with

the data acquisition PC via an SBS/Bit3 interface. The experimental setup used
the standard NSCL data acquisition software, NSCLDAQ [68] for reading out the

modules and NSCLSpecTcl [69] for data analysis. The DSP filters and the software
gates particular to the analysis of this experiment were implemented as C++ classes
and Tcl/Tk scripts.

3.4 Results

The collimated 662 keV photon beam has been swept across the D4 segment, parallel

to the quadrant segmentation plane between segment groups 1,4 and 2,3. Due to
the geometry of the setup, beside random coincidences from the room background

52
 137
and cosmic rays, only the 622 keV γ-rays from the Cs source that interact once

in the HPGe detector, scatter at about 90 deg (deposited energy 370 keV) and then
interact with the NaI(Tl) detector are expected to trigger the data acquisition. After

the calibration of both detectors, software gates can be imposed, such that the event
for which the energies deposited in the germanium and NaI have values close to the
single Ge scattering at 90 deg scenario. In Figures 3.9, 3.10 and 3.11, 3.12 two relevant

events are presented. In Figures 3.9 and 3.10, signals are for an interaction when the
collimator is placed closer to the E side, and Figures 3.11 and 3.12 for an interaction

closer to the C side. The D4 segment signal shows the deposition of 370 keV electron
energy on which the γ-ray initially scattered. The amplitude of the induced signal in

segment E4 is larger compared to the amplitude in segment C4 in the case showed in

Figures 3.9 and 3.10. In Figures 3.11 and 3.12 the transient signal amplitudes in C4
are larger than the ones in E4.

Waveforms and simple quantities based on waveforms often have arbitrary units
assigned in the plots presented. The flash ADCs have a full range of 1 V, from -0.5 V

to +0.5 V. Each segment has its own DC offset. Depending on the ADC type, the
full 1 V range is digitized into 4096 bins (12-bit ADC) or 16384 bins (14-bit ADC).

The average sensitivity of the segment preamplifier is ∼ 125 mV/MeV.

For each linear position of the collimator, the differences in the induced signal
amplitudes for segments E4 and C4 were histogrammed and fit with a Gaussian. The

linear position of each run was plotted versus the corresponding Gaussian centroid.
A linear fit to the data was performed. To diminish the effect of incorrect position

determination, any position calculated to be outside a +/-3.7 mm window centered

on the segment center is forced into the window’s edge.

Using this algorithm, the same data set was then used to determine the algo-
rithm’s success. The average miss was calculated as the average over a given data
set of the absolute value of the difference between the expected interaction point and

the calculated value. The average miss for a SeGA detector using only the contact

53
 2025

2020

2015

2010

2005 E3
Signal amplitude (channel)

2025

2020

2015

2010

2005
D3
2015

2010

2005

2000
C3
1995
0. 0.5 1.0 1.5 2.0 2.5
Time (µs)
Figure 3.9: Segment signals for an interaction closer to the E4 side of the D4 segment
(Quadrant 3).

54
 2025

2020

2015

2010

2005 E4
Signal amplitude (channel)

2010

1990

1970
D4
1950
2025

2020

2015

2010
C4
2005
0. 0.5 1.0 1.5 2.0 2.5
Time (µs)
Figure 3.10: Segment signals for an interaction closer to the E4 side of the D4 segment
(Quadrant 4).

55
 2025

2020

2015

2010
E3
Signal amplitude (channel)

2005
2025

2020

2015

2010

2005
D3
2015

2010

2005

2000

1995
C3
0. 0.5 1.0 1.5 2.0 2.5
Time (µs)
Figure 3.11: Segment signals for an interaction closer to the C4 side of the D4 segment
(Quadrant 3).

56
 2025

2020

2015

2010
E4
Signal amplitude (channel)

2005

2010

1990

1970
D4
1950
2020

2015

2010

2005

2000
C4
0. 0.5 1.0 1.5 2.0 2.5
Time (µs)
Figure 3.12: Segment signals for an interaction closer to the C4 side of the D4 segment
(Quadrant 4).

57
 3
Average miss (mm)

0
0 2 4 6 8
Linear position (mm)
Figure 3.13: The average miss of the algorithm in estimating the interaction position
for several positions of the collimator. In dashed line is plotted the theoretical average
miss when no sub-segment position resolution is assumed.

58
 0.25

0.2
Fraction of events

0.15

0.1

0.05

0
-8 -6 -4 -2 0 2 4 6 8 10
Position (mm)

Figure 3.14: Histograms of the algorithm misses for each event in the data set when
pulse shape analysis is involved (darker shade) compared to the case with no PSA
(lighter color shade).

segmentation is 2.5 mm (Figure 3.13). This can be calculated by considering a uni-

form distribution of the events (see also Figure 3.14) and the fact that the miss in
this case in the absolute value of the γ-ray interaction position. An average over the

entire distribution gives a value which is a quarter of the length of the segment. The
algorithm shows an improvement and works well across the length of the segment. A
histogram of the misses for each event in the data set (all positions) is presented in

Figure 3.14. The region colored with a lighter shade shows the expected distribution
of misses when using only the contact segmentation and no pulse shape analysis.

For multiple interactions in a single segment, a gate on the 662 keV energy de-
posited in segment E4 was placed and the collimated beam was swept from the F

side to the D side. Only the information from photons depositing their full energy
in the targeted segment is recorded. With this requirement, the photon’s interaction
history must include either a photoelectric effect, of a succession of Compton inter-

actions in the segment followed by a photoelectric effect. At that energy, the cross

59
 80 1

0.9
70
Transient signal amplitude in D4 (ch)

0.8
60
0.7
50 0.6

40 0.5

0.4
30
0.3
20
0.2
10
0.1
0 0
0 10 20 30 40 50 60 70 80
Transient signal amplitude in F4 (ch)

Figure 3.15: Amplitudes of the transient signals for segments F4 (horizontal) and D4
(vertical). The collimator is moved by 2 mm for each graph. First position.

section for the Compton scattering is significantly larger than the cross section for the
photoelectric effect. Most likely the photon scatters several times before it is finally
absorbed. Induced signal amplitudes in F4 (horizontal axis) and D4 (vertical axis) are

plotted against each other in Figures 3.15, 3.16, 3.17, 3.18, 3.19. The case in Figure
3.15 corresponds to a position closer to the F side and the case in Figure 3.19 to a

position of the collimator closer to the D side. The intermediary positions in Figure
3.16 through Figure 3.18 are for the collimator moved 2 mm each time. Plotted are

only the positive amplitudes, for which the induced signal amplitude calculation was
most reliable. Statistically it still can be estimated on which side of the segment the
interactions occurred.

60
 80 1

0.9
70
Transient signal amplitude in D4 (ch)

0.8
60
0.7
50 0.6

40 0.5

0.4
30
0.3
20
0.2
10
0.1
0 0
0 10 20 30 40 50 60 70 80
Transient signal amplitude in F4 (ch)

Figure 3.16: Amplitudes of the transient signals for segments F4 (horizontal) and D4
(vertical). The collimator is moved by 2 mm for each graph. Second position.

61
 80 1

0.9
70
Transient signal amplitude in D4 (ch)

0.8
60
0.7
50 0.6

40 0.5

0.4
30
0.3
20
0.2
10
0.1
0 0
0 10 20 30 40 50 60 70 80
Transient signal amplitude in F4 (ch)

Figure 3.17: Amplitudes of the transient signals for segments F4 (horizontal) and D4
(vertical). The collimator is moved by 2 mm for each graph. Third position.

62
 80 1

0.9
70
Transient signal amplitude in D4 (ch)

0.8
60
0.7
50 0.6

40 0.5

0.4
30
0.3
20
0.2
10
0.1
0 0
0 10 20 30 40 50 60 70 80
Transient signal amplitude in F4 (ch)

Figure 3.18: Amplitudes of the transient signals for segments F4 (horizontal) and D4
(vertical). The collimator is moved by 2 mm for each graph. Fourth position.

63
 80 1

0.9
70
Transient signal amplitude in D4 (ch)

0.8
60
0.7
50 0.6

40 0.5

0.4
30
0.3
20
0.2
10
0.1
0 0
0 10 20 30 40 50 60 70 80
Transient signal amplitude in F4 (ch)

Figure 3.19: Amplitudes of the transient signals for segments F4 (horizontal) and D4
(vertical). The collimator is moved by 2 mm for each graph. Fifth position.

64
 90%

30%

Figure 3.20: Central contact waveforms corresponding to a photopeak event at

1332 keV.

To investigate the radial position resolution, the focus was placed on the rise time
analysis because it was expected that there would be a dependence of the rise times

with the radial position of the interaction [70]. Figures 3.20 and 3.21 show waveforms
taken from the central contact for events corresponding to a photopeak event from a

1332 keV γ-ray and from a segment in which the energy from a 662 keV γ-ray was
fully deposited inside the segment, respectively.
Using the coincidence setup with the collimated source from the front of the

detector and the scattered γ-rays detected at around 90 degree measurements were
made for a number of radial positions. The rise time of the waveform was calculated

as the time it takes to go from 10% of its total height to a given fraction of the total
amplitude. Rise times for 30%, 60%, and 90% fractional amplitudes, denoted t30,

t60 and t90 respectively, were measured. In Figures 3.22, 3.23, and 3.24, the three
rise times are plotted function of the measured radii. The gaps on the radius axis

65
 30%

90%

Figure 3.21: Segment waveforms corresponding to a photopeak event at 662 keV.

Only the interactions in which the γ-ray energy was fully deposited in the segment
are selected.

66
 50

45 140

40
120
35
100
Radius (mm)

30

25 80

20 60
15
40
10
20
5

0 0
0 50 100 150 200 250 300 350 400
t30 (ns)
Figure 3.22: Time to reach 30% of the full amplitude (t30) plotted against the position
of the collimator in the radial direction.

correspond to radii for which measurements were not performed. By plotting t30 on

one axis and t90 on the other axis and color coding the radius it can be seen that the
segment can be sub-segmented also along the radial dimension (see Figure 3.25).

Mostly for gating purposes, the deposited energy was reconstructed in software
from the central contact and segment waveforms. When using analog electronics,

the energy is usually determined by putting the pre-amplified signal into a shaping
amplifier, then using the output of the shaping amplifier as the input to a peak-
sensing analog-to-digital converter. With digital electronics, the digitized output of

the preamplifier is recorded. The rest of the spectroscopic chain is implemented in

software. Simple algorithms were used to accomplish this. The simplest one is to use

the height of the signal relative to the baseline, since in principle it is proportional
to the energy deposited. A baseline is calculated from the first 50 samples and it

is subtracted from the signal, making the result more accurate. To reduce the high

67
 50 120

45
100
40

35
80
Radius (mm)

30

25 60

20
40
15

10
20
5

0 0
0 50 100 150 200 250 300 350 400
t60 (ns)
Figure 3.23: Time to reach 60% of the full amplitude (t60) plotted against the position
of the collimator in the radial direction.

68
 50 70
45
60
40

35 50
Radius (mm)

30
40
25
30
20

15 20
10
10
5

0 0
0 50 100 150 200 250 300 350 400
t90 (ns)
Figure 3.24: Time to reach 90% of the full amplitude (t90) plotted against the position
of the collimator in the radial direction.

350 4500

4000

3500
300
3000
t90 (ns)

2500
250
2000

1500
200
1000

500
150 0
20 40 60 80 100 120 140
t30 (ns)

Figure 3.25: Time to reach 30% of the full amplitude plotted against the time to reach
90% of the full amplitude. The radius is color coded (arbitrary units).

69
frequency noise, smoothing techniques as running averages or Gaussian can be em-

ployed.
A better way to reconstruct the energy is to use a trapezoidal shaper [71, 72]. In

terms of digital signal processing, this shaper is an infinite impulse response filter,
with the equation:

yn = yn−1 + xn − xn−m − xn−m−k + xn−2∗m−k . (3.7)

applied to the smoothed signal shape. In the relationship above xn is the signal
before the filter is applied and yn is the resulting signal. The smoothing is done

with a centered running average in 7 points or with a Gaussian filter before the
trapezoidal filter is applied. A trapezoidal shaper corrects for the ballistic deficit
and it also has the advantage that is easily implementable in the digitizing board’s

hardware. Because the various time constants involved in the signal amplification
cannot be made infinitely large the signals associated with long charge collection

times experience losses in the amplification. This effect is known as ballistic deficit.
The filter has two independent parameters: integration time (m) and the flat top

duration (k ). From another perspective, it can be viewed as the difference between

two running sums separated by a certain amount of samples and acting on the same
trace. One drawback of this shaper is that in its simple form it does not apply pole-

zero corrections. Figure 3.26 shows the central contact signal corresponding to a
60
1332 keV event from a Co source, along with the same signal after a trapezoidal

shaper with integration time of 1 µs and flat-top duration of 0.5 µs is applied. Notice
the undershoot of the shaped signal on the right side of the trapezoid. The shape

can be straightened with a pole-zero correction algorithm, like the Moving Window
Deconvolution algorithm, for example Reference [73], leading to an improvement in
the energy resolution. For the trapezoidal filter without any additional corrections,

integration time 4 µs and flat-top duration 2 µs, the energy resolution attained was

70
 5000

4000
Amplitude (a.u.)

3000

2000

1000

0 1 2 3 4 5
Time (µs)
Figure 3.26: The central contact signal corresponding to a 1332 keV event and a
trapezoidal shaper with integration time of 1 µs and flat-top duration of 0.5 µs applied
on it.

about 4.6 keV Full width at half maximum (FWHM) for the central contact and
5.2 keV FWHM for segments for the 137 Cs 661.62 keV line. A previous measurement of

the central contact energy resolution using a shaping amplifier and an ADC produced
60
a value of 2.95 keV for the 1332 keV Co line. The other methods tried (a CR-RC

filter [74] and a trapezoidal shaper applied on a signal with the preamplifier decay
time corrected) marginally improved the resolution. The main factor in the loss of

energy resolution compared to the analog spectroscopic chain is the short integration
time. Details of the algorithms and filters used are presented in the Appendix.

3.5 Summary

The goal of this study was to provide a basis and a proof-of-principle argument to fu-
ture investigations. For single Compton interactions, the position resolution has been

71
increased from a theoretical 2.5 mm to 1.5 mm. Interaction positions for unrestricted

events within a segment can be qualitatively assigned to one side of the segment or the
other. Radial interaction position resolution can be achieved with rise time analysis.

A significant improvement in the scientific output of SeGA can be obtained with

modest improvements in the Doppler correction mechanisms. There is a balance be-
tween the γ-ray efficiency of the spectroscopic system and its energy resolution. By

doubling the effective segmentation along the z axis, the efficiency can be increased 4
times while the resolution is kept the same if the distance between target and detec-

tors is reduced by half. This can either increase the number of counts in photopeaks
four-fold, or reduce the beam time and cost of the beam time by a factor of four. This

adds more flexibility to SeGA. Depending of which factor is critical, efficiency for very
low rate beams or small cross sections or resolution for identification of doublets or
the need for high peak-to-background ratios, the layout of detectors in the array can

be modified to fit the experiment’s needs.

72
Chapter 4

Conclusions

Quantum mechanical observables B(E2; 0+ → 2+

1 ) have been measured for the neutron-

52 54 56
rich titanium isotopes Ti, Ti, and Ti via intermediate-energy Coulomb excita-
tion. Correlated with the energies of the first excited states of these nuclei, the reduced

transition elements suggest a shell gap at neutron number N=32. There is no indica-
tion of a shell gap at N=34. Theoretical studies are presently under way to investigate

further the energy spacings between the p3/2 , p1/2 and f5/2 neutron orbitals.
Doppler correction is essential in the determination of the de-excitation γ-ray
energies emitted by fast-moving nuclei. An important ingredient in the Doppler cor-

rection is the estimation of the point where the first interaction of a photon inside the
germanium detector occured. It has been demonstrated that for the SeGA detectors

a sub-segment position resolution can be attained by using quantities directly de-

rived (amplitudes, fractional rise times) from digitized central contact and segments
waveforms.

73
Appendix A

Algorithm description

The Appendix describes the algorithms used in Chapter 3 for processing digitized

waveforms taken from a SeGA detector’s central contact and segments.

A.1 Centered running average

The filter generates an average of the sample with three samples before and three
samples after.

xn+3 + xn+2 + xn+1 + xn + xn−1 + xn−2 + xn−3

yn = . (A.1)
7

// centered running average using 7 points

void CWaveform::centeredRunningAverage()

{
//first and last 2 elements remain in the initial state

for (unsigned int i = 3; i < m_iSize - 3; i++)

{

m_pWave[i] = m_pWave[i+3]+ m_pWave[i+2]

+ m_pWave[i+1] + m_pWave[i]
+ m_pWave[i-1] + m_pWave[i-2] + m_pWave[i-3];

74
 m_pWave[i] /= 7.;

// now set the first and last 3 values

m_pWave[0] = m_pWave[1]
= m_pWave[2]

= m_pWave[3];
m_pWave[m_iSize - 3] = m_pWave[m_iSize - 2]

= m_pWave[m_iSize - 1]
= m_pWave[m_iSize - 4];

A.2 Gaussian smoothing using 9 points

A weighted average rather than a simple average is used for the Gaussian filter. The
weighting factors form a Gaussian centered four samples before the current sample
(Figure A.1).

xn + 8xn−1 + 28xn−2 + 56xn−3 + 70xn−4 + 56xn−5 + 28xn−6 + 8xn−7 + xn−8

yn = .
256
(A.2)

// Gaussian smoothing using 9 points

// y(n) = x(n) + 8*x(n-1) + 28*x(n-2) + 56*x(n-3) + 70*x(n-4) +
// 56*x(n-5) + 28*x(n-6) + 8*x(n-7) + x(n-8)

// The sum of all coefficients is 256.

void CWaveform::gaussianSmooth9()

{
//first 8 elements remain in the initial state

75
 80

70

60
Filter coefficient

50

40

30

20

10

-8 -7 -6 -5 -4 -3 -2 -1 0
Sample order

Figure A.1: Gaussian filter coefficients.

76
 for (unsigned int i = 8; i < m_iSize; i++)

{
m_pTest[i] = m_pWave[i] + 8.*m_pWave[i-1] + 28.*m_pWave[i-2] +

56.*m_pWave[i-3] + 70.*m_pWave[i-4] + 56.*m_pWave[i-5] +

28.*m_pWave[i-6] + 8.*m_pWave[i-7] + m_pWave[i-8];
m_pTest[i] /= 256.;

}
//set the calculated elements

for (unsigned int i = 8; i < m_iSize; i++)

{

m_pWave[i] = m_pTest[i];
}
// now set the first 8 elements

for (unsigned int i = 0; i < 8; i++)

{

m_pWave[i] = m_pTest[8];
}

A.3 Signal derivatives

Derivatives are important in estimating the signal before it is integrated by the charge-
sensitive preamplifier. However, due to the noise present in the system, the following

relationships cannot be used for real waveforms without smoothing.

The first derivative is calculated with:

−xi+2 + 8.xi+1 − 8.xi−1 + xi−2

yi = (A.3)
12

77
and the second derivative with:

−xi+2 + 16.xi+1 − 30.xi + 16.xi−1 − xi−2

yi = (A.4)
12

void CWaveform::calculateDerivatives()
{

//first and last 2 elements remain in an indefinite state

for (unsigned int i = 2; i < m_iSize - 2; i++)

{
m_pDeriv1[i] = -m_pWave[i+2] + 8.*m_pWave[i+1]

-8.*m_pWave[i-1] + m_pWave[i-2];
m_pDeriv1[i] /= 12.* m_dStep;

m_pDeriv2[i] = -m_pWave[i+2] + 16.*m_pWave[i+1] -30.*m_pWave[i]

+16.*m_pWave[i-1] - m_pWave[i-2];

m_pDeriv2[i] /= 12.* m_dStep * m_dStep;

// zero the undefined points

m_pDeriv1[0] = m_pDeriv1[1]
= m_pDeriv1[m_iSize - 2]

= m_pDeriv1[m_iSize -1] = 0.0;

m_pDeriv2[0] = m_pDeriv2[1]

= m_pDeriv2[m_iSize - 2]
= m_pDeriv2[m_iSize -1] = 0.0;
}

78
A.4 Linear fit

Function to perform a linear regression on a portion of the waveform. It was used to

estimate slopes of waveform rising edges and to measure in a first approximation the
preamplifier decay time constant. The relations in the code work only for sampling
of type (xi , yi ) where xi is defined as:

xi = x0 + iδx (A.5)

where δx is the spacing between samples. The fit function is:

Y = a + bX (A.6)

The routine returns the (a, σa ), (b, σb ). With a extra pass it calculates χ2 .

struct LinRegressCoef

CWaveform::fitDataSegmentWithLine(unsigned int iBegin,

unsigned int iNrSamples)
{

double dSumX, dSumX2, dSumY= 0.0,

dSumY2 =0.0 , dSumXY = 0.0, dDiscr;

double dDeltaX = SAMPLING_DELTA;

double a, b, Sig_a, Sig_b, dSigmaY, dChi2 = 0.0;

struct LinRegressCoef Coeff;

//check for possible out of boundary errors

if (iBegin+iNrSamples >= WAVEFORM_SIZE)

{

cerr << "---------------------------------------------------"

<< endl;

79
cerr << "WARNING -- The operation as requested would require"

<< " out of boundary access"

<< endl;

cerr << "WARNING -- A truncation will be performed"

<< endl;
cerr << "--------------------------------------------------"

<< endl;
iNrSamples = WAVEFORM_SIZE - iBegin;

}
if (iBegin >= WAVEFORM_SIZE)

{
cerr << "------------------------------------------------"
<< endl;

cerr << "ERROR -- The operation as requested would require"

<< " out of boundary access"

<< endl;
cerr << "ERROR -- Returning zeroes..."

<< endl;
cerr << "------------------------------------------------"
<< endl;

Coeff.a = 0.0;
Coeff.b = 0.0;

Coeff.siga = 0.0;
Coeff.sigb = 0.0;

Coeff.chi2 = 0.0;
return(Coeff);
}

//sum of x

80
dSumX = 0.5*dDeltaX*(iNrSamples - 1)*iNrSamples +iNrSamples*iBegin;

//sum of x^2
dSumX2 = (1./6.)*iNrSamples*(dDeltaX*dDeltaX*\

(2*iNrSamples*iNrSamples-3*iNrSamples+1)+\
6.*dDeltaX*(iNrSamples-1)*iBegin+6.*iBegin*iBegin);
// discriminant

dDiscr = dDeltaX*dDeltaX*iNrSamples*iNrSamples
*(iNrSamples*iNrSamples -1)/12.;

// for calculating Sum_y, Sum_y^2 ,and Sum_xy we need a loop

for (unsigned int i = iBegin; i < iBegin + iNrSamples; i++)

{
dSumY += m_pWave[i];
dSumY2 += m_pWave[i]*m_pWave[i];

dSumXY += i*m_pWave[i];
}

// calculate the coefficients and their uncertainties

a = (dSumX2*dSumY - dSumX*dSumXY)/dDiscr;

b = (iNrSamples*dSumXY - dSumX*dSumY)/dDiscr;
// this pass is for calculating Chi-squared
for (unsigned int i = iBegin; i < iBegin + iNrSamples; i++)

{
dChi2 += (m_pWave[i] - (b*i+a))*(m_pWave[i] - (b*i+a));

}
dSigmaY = sqrt(dChi2/(iNrSamples - 2));

Sig_a = dSigmaY*sqrt(dSumX2/dDiscr);
Sig_b = dSigmaY*sqrt(iNrSamples/dDiscr);
Coeff.a = a;

Coeff.b = b;

81
 Coeff.siga = Sig_a;

Coeff.sigb = Sig_b;
Coeff.chi2 = dChi2;

return (Coeff);
}

A.5 Threshold passing point

The function returns the point on the waveform where the amplitude is higher than
a given threshold. If the threshold is given as a fraction (f ) of the amplitude of the

waveform, then the point is:

ythreshold = ylow (1 − f ) + yhi ∗ f (A.7)

where ylow is the waveform baseline and yhi is the maximum amplitude. It is between
a simple leading-edge discriminator but the performance is lower than of a constant-
fraction discriminator (CFD). In the case of a CFD, the zero-crossing point (the actual

trigger point) is more accurate.

struct PairXY

CWaveform::getThresholdPassPoint(double dLowest,
double dHighest,

double dFraction,
enum DetectorType DetType)
{

struct PairXY ThresholdPoint;

if (DetType == GE_CENTRAL_CONTACT)
{

82
for(unsigned int i = 0; i < m_iSize; i++)

{
if( m_pWave[i] >= dLowest*(1. - dFraction) + dHighest*dFraction)

{
ThresholdPoint.x = (double)i;
ThresholdPoint.y = m_pWave[i];

return (ThresholdPoint);
}

}
}

if (DetType == GE_SEGMENT)
{
for(unsigned int i = 0; i < m_iSize; i++)

{
if( m_pWave[i] <= dHighest*(1. - dFraction) + dLowest*dFraction)

{
ThresholdPoint.x = (double)i;

ThresholdPoint.y = m_pWave[i];
return (ThresholdPoint);
}

}
}

cerr << "-----------------------------------------------------"

<< endl;

cerr << "ERROR -- The wave form never passed the set threshold."
<< "Recheck your numbers." << endl;
cerr << "ERROR -- Returning zeroes..." << endl;

cerr << "-----------------------------------------------------"

83
 << endl;

ThresholdPoint.x = 0.0;
ThresholdPoint.y = 0.0;

return (ThresholdPoint);
}

A.6 Trapezoidal shaper

The trapezoidal shaper equation is [71, 72]:

yn = yn−1 + xn − xn−m − xn−m−k + xn−2∗m−k . (A.8)

With a shorter time constant it can pe used as a trigger or for pile-up inspection
purposes.

double CWaveform::trapezoidalShaper(enum DetectorType DetType,

unsigned int m,
unsigned int k,

struct PairXY TriggerPoint)

{

unsigned int n;
double elem_n_m, elem_n_m_k, elem_n_2m_k;

double dEnergy, dMin = +1.E+10, dMax = -1.E+10;

if (DetType == GE_CENTRAL_CONTACT)

{
for (n = 0; n < m_iSize; n++)

{
// element x(n-m)

84
if((int)n -(int)m >= 0)

{
elem_n_m = m_pWave[n-m] - TriggerPoint.y ;

}
else
{

elem_n_m = 0.0;
}

// element x(n-m-k)
if((int)n - (int)m - (int)k >= 0)

{
elem_n_m_k = m_pWave[n-m-k] - TriggerPoint.y;
}

else
{

elem_n_m_k = 0.0;
}

// element x(n-2m-k)
if((int)n - 2*(int)m - (int)k >= 0)
{

elem_n_2m_k = m_pWave[n-2*m-k] - TriggerPoint.y;

}

else
{

elem_n_2m_k = 0.0;
}
m_pTrapezoid[n] = m_pTrapezoid[n-1]

+ (m_pWave[n] - TriggerPoint.y)

85
 - elem_n_m - elem_n_m_k + elem_n_2m_k;

// calculate the extremes

if ( m_pTrapezoid[n] > dMax)

{
dMax = m_pTrapezoid[n];
}

if ( m_pTrapezoid[n] < dMin)

{
dMin = m_pTrapezoid[n];

}
}
}

else if (DetType == GE_SEGMENT)

{

for (n = (unsigned int) TriggerPoint.x; n < m_iSize; n++)

{

// element x(n-m)
if((int)n -(int)m >= 0)
{

elem_n_m = TriggerPoint.y - m_pWave[n-m] ;

}

else
{

elem_n_m = 0.0;
}

// element x(n-m-k)

86
if((int)n - (int)m - (int)k >= 0)

{
elem_n_m_k = TriggerPoint.y - m_pWave[n-m-k];

}
else
{

elem_n_m_k = 0.0;
}

// element x(n-2m-k)
if((int)n - 2*(int)m - (int)k >= 0)

{
elem_n_2m_k = TriggerPoint.y - m_pWave[n-2*m-k];
}

else
{

elem_n_2m_k = 0.0;
}

m_pTrapezoid[n] = m_pTrapezoid[n-1]
+ (TriggerPoint.y - m_pWave[n])
- elem_n_m - elem_n_m_k + elem_n_2m_k;

// calculate extremes

if ( m_pTrapezoid[n] > dMax)

{

dMax = m_pTrapezoid[n];
}
if ( m_pTrapezoid[n] < dMin)

87
 dMin = m_pTrapezoid[n];

}
}

}
dEnergy = dMax;//-dMin;
return(dEnergy);

A.7 A differentiator-integrator shaper

A shaper using three parameters R (integration constant), D (differentiation con-

stant), and τ (preamplifier decay time constant). It is a four-pass filter with an inter-

mediary waveform (an ).

an = xn + (1 − τ )xn (A.9)

yn = an − an−R − an−(R+D) + an−(2R+D) (A.10)

an = an−1 + yn (A.11)

yn = yn−1 + an (A.12)

The energy is given by the maximum of the resulting yn signal.

double CWaveform::shaper(double dDecayTimeConst,

double dR, double dD)

{
double a, b, c;

double dEnergy, dMin = +1.E+10, dMax = -1.E+10;

int n;

88
a = 1.0-dDecayTimeConst;

for (n = 1; n < m_iSize; n++)

{

m_pTrapezoid[n] = m_pWave[n] -a*m_pWave[n-1];

}
for (n = 0; n < m_iSize; n++)

{
m_pDeriv1[n] += m_pTrapezoid[n];

if (n > (int)dR)
{

m_pDeriv1[n] -= m_pTrapezoid[n-(int)dR];
}
if (n > ((int)dR+(int)dD))

{
m_pDeriv1[n] -= m_pTrapezoid[n-((int)dR+(int)dD)];

}
if (n > (2*(int)dR+(int)dD))

{
m_pDeriv1[n] += m_pTrapezoid[n-(2*(int)dR+(int)dD)];
}

}
m_pTrapezoid[0] = 0.0;

for (n = 1; n < m_iSize; n++)

{

m_pTrapezoid[n] = m_pTrapezoid[n-1]+m_pDeriv1[n];
}
m_pDeriv1[0] = 0.0;

for (n = 1; n < m_iSize; n++)

89
 {

m_pDeriv1[n] = m_pTrapezoid[n]+m_pDeriv1[n-1];
}

//calculate extremes
for (n = 1; n < m_iSize; n++)
{

if (m_pDeriv1[n] > dMax)

{

dMax = m_pDeriv1[n];
}

if ( m_pDeriv1[n] < dMin)

{
dMin = m_pDeriv1[n];

}
}

dEnergy = dMax;// - dMin;

return (dEnergy);

A.8 Statistics on a waveform segment

Extracts the mean and standard deviation for a section of the waveform. It is suitable
for baseline calculations and simple noise amplitude estimation.

struct StatCoefs

CWaveform::calculateStatistics(unsigned int iBeginSample,

unsigned int iEndSample)

{
double dMean = 0.0, dStdDev = 0.0, dVariance = 0.0;

90
 double dSum =0.0, dSumSquares = 0.0;

struct StatCoefs Statistics;

unsigned int i;

unsigned int iIntervalSize;

iIntervalSize = iEndSample - iBeginSample;

for (i = iBeginSample; i < iEndSample; i++)

{

dSum += m_pWave[i];
dSumSquares += m_pWave[i]*m_pWave[i];

}
dMean = dSum/iIntervalSize;
dVariance = (dSumSquares

- dSum*dSum/iIntervalSize)/(iIntervalSize - 1.);
dStdDev = sqrt(dVariance);

Statistics.mean = dMean;
Statistics.stdev = dStdDev;

return (Statistics);
}

A.9 RC-CR Filter

Described in detail in References [75] and [74], it is a succession of two single pole
filter, a low-pass (integrator) followed by a high-pass (differentiator). It also applies

a pole-zero correction to the shaped signal.

double CWaveform::shaperRC_CR(double dDecayTimeConstDiff,

double dDecayTimeConstInt,
double dPoleZeroCorrection)

91
{

double dDecayFactorDiff, dDecayFactorInt;

double dDF;

double dEnergy = 0.0;

double dBase = 0.0;

dDecayFactorDiff = exp(-1./dDecayTimeConstDiff);
dDecayFactorInt = exp(-1./dDecayTimeConstInt);

dDF = 0.5*(1+dDecayFactorDiff);
// apply the differential filter (RC)

for (unsigned int i = 2; i < m_iSize; i++)

{
m_pTrapezoid[i] = dDF * m_pWave[i] +

(dPoleZeroCorrection - dDF)*m_pWave[i-1] +
dDecayFactorDiff * m_pTrapezoid[i-1];

}
// apply the integral filter (CR)

for (unsigned int i = 2; i < m_iSize; i++)

{
m_pTrapezoid[i] = (1.- dDecayFactorInt)*m_pTrapezoid[i] +

dDecayFactorInt*m_pTrapezoid[i-1];
}

// energy is the max of the trapezoid shaped pulse

for (unsigned int i = 0; i < m_iSize; i++)

{
if (m_pTrapezoid[i] > dEnergy)
{

dEnergy = m_pTrapezoid[i];

92
 }

if (m_pTrapezoid[i] < dBase)

{

dBase = m_pTrapezoid[i];
}
}

return dEnergy - dBase;

}

93
Bibliography

[1] R. du Rietz et al. Phys. Rev. Lett., 93:222501, 2004.

[2] A. Navin et al. Phys. Rev. Lett., 85:226, 2000.

[3] O. Sorlin et al. Phys. Rev. Lett., 88:092501, 2002.

[4] B.V. Pritychenko et al. Phys. Rev. C, 62:051601, 2000.

[5] B.V. Pritychenko et al. Phys. Rev. C, 63:047308, 2001.

[6] B.V. Pritychenko et al. Phys. Rev. C, 65:061304, 2002.

[7] H. Scheit et al. Phys. Rev. Lett., 77(19):3967–3970, 1996.

[8] J.I. Prisciandaro et al. Phys. Lett., B510:17, 2001.

[9] T. Otsuka et al. Phys. Rev. Lett., 87:082502, 2001.

[10] M. Honma et al. Phys. Rev. C, 65:061301, 2002.

[11] S.N. Liddick et al. Phys. Rev. Lett., 92:072502, 2004.

[12] S.N. Liddick et al. Phys. Rev. C, 70:064303, 2004.

[13] R.V.F. Janssens et al. Phys. Lett., B 546:55, 2002.

[14] B. Fornal et al. Phys. Rev. C, 70:064304, 2004.

[15] K.L. Yurkewicz et al. Phys. Rev. C, 70:054319, 2004.

[16] G. Kraus et al. Phys. Rev. Lett., 73:1773, 1994.

[17] T. Glasmacher. Annu. Rev. Nucl. Part. Sci., 48:1, 1998.

[18] E.Z. Fermi. Z. Phys., 29:315, 1924.

[19] C.F. Weizsäcker. Z. Phys., 88:612, 1934.

[20] K. Alder et al. Rev. Mod. Phys., 28:432, 1956.

[21] F.D. Becchetti et al. Phys. Rev. C, 40:1104, 1989.

[22] J.J. Kolata et al. Nucl. Instr. and Meth. B, 503:503, 1989.

94
[23] J.A. Brown et al. Phys. Rev. Lett., 66:2452, 1991.

[24] M. Oshima et al. Nucl. Instr. and Meth. A, 312:445, 1992.

[25] A. Winther and K. Alder. Nucl. Phys., A319:518, 1979.

[26] T. Glasmacher. Nucl. Phys. A, 693:90–104, 2001.

[27] A. Gade et al. Phys. Rev. C, 68:014302, 2003.

[28] K. L. Miller. Study of Neutron-Deficient Nickel Isotopes Via One-Neutron Knock-

out and Intermediate-Energy Coulomb Excitation. PhD thesis, Michigan State
University, 2003.

[29] M.A. Deleplanque et al. Nucl. Instr. and Meth. A, 430:292–310, 1999.

[30] I.-Y. Lee et al. Nucl. Phys. A, 746:255C–259C, 2004.

[31] D. Bazzacco. Nucl. Phys. A, 746:248C–254C, 2004.

[32] D.-C. Dinca et al. Phys. Rev. C, 71:041302, 2005.

[33] B. A. Brown. Prog. in Part. and Nucl. Phys., 47:517, 2001.

[34] F. Tondeur. Z. Phys., A 288:97, 1978.

[35] P. Haensel and J. L. Zdunik. Astron. Astrophys., 222:353, 1989.

[36] J. Dobaczewski et al. Phys. Rev. Lett., 72:981, 1994.

[37] J.M. Pearson et al. Phys. Lett. B, 387:455, 1996.

[38] G. A. Lalazissis et al. Phys. Lett., B 418:7, 1998.

[39] A. Huck et al. Phys. Rev. C, 31:2226, 1985.

[40] D.C. Radford et al. Phys. Rev. Lett., 88:222502, 2002.

[41] D.J. Morrissey et al. Nucl. Instr. and Meth. B, 204:90, 2003.

[42] D. Bazin et al. Nucl. Instr. and Meth. B, 204:629, 2003.

[43] J. Yurkon et al. Nucl. Instr. and Meth. A, 422:291, 2001.

[44] W.F. Mueller et al. Nucl. Instr. and Meth. A, 466:492, 2001.

[45] GEANT. CERN library long writeup. Technical Report W5013, CERN, 1994.

[46] H. Scheit. Low-Lying Collective Excitations in Neutron-Rich Even-Even Sulfur

and Argon Isotopes Studied via Intermediate-Energy Coulomb Excitation and
Proton Scattering. PhD thesis, Michigan State University, 1998.

[47] H. Olliver et al. Phys. Rev. C, 68:044312, 2003.

95
[48] S. Raman et al. Atomic Data and Nuclear Data Tables, 78:1, 2001.

[49] Zhou Chunmei. Nucl. Data Sheets, 76:399, 1995.

[50] B.A. Brown et al. Phys. Rev. C, 14:1016, 1976.

[51] H. Junde. Nucl. Data Sheets, 90:1, 2000.

[52] M. Hjorth-Jensen et al. Phys. Rep., 261:125, 1995.

[53] M. Honma et al. Eur. Phys. J. A, Proc. Fourth Int. Conf. on Exotic Nuclei and
Atomic Masses (ENAM04), 2005.

[54] B. Fornal et al. :to be published., 0000.

[55] B. A. Brown and B. H. Wildenthal. Phys. Rev. C, 21:2107, 1980.

[56] B.A. Brown et al. Nucl. Phys. A, 277:77–108, 1977.

[57] T. Mizusaki. RIKEN Accel. Prog. Rep., 33:14, 2000.

[58] P.F. Mantica et al. Phys. Rev. C, 67:014311, 2003.

[59] P.F. Mantica et al. Phys. Rev. C, 68:044311, 2003.

[60] L.A. Riley et al. Phys. Rev. C, :to be published., 2005.

40
[61] C.M. Campbell et al. First measurement of an excited state in Si, :to be
published., 2005.

[62] D. Bazin et al. Phys. Rev. Lett., 91:012501, 2003.

[63] A. Gade et al. Phys. Rev. C, 69:034311, 2004.

[64] S. Ramo. Proc. IRE, 27:584, 1939.

[65] W. Shockley. J. Appl. Phys., 9:635, 1938.

[66] G. F. Knoll. Radiation Detection and Measurement Nuclear Reactions, 3rd ed.
John Wiley & Sons, Inc., 2000.

[67] L. Mihailescu et al. Nucl. Instr. and Meth. A, 447:350–360, 2000.

[68] R. Fox et al. IEEE Trans. Nucl. Sci., 2004.

[69] R. Fox et al. 13th IEEE-NPSS Real Time Conference, 2003.

[70] Th. Kroell et al. Nucl. Instr. and Meth. A, 371:489–496, 1996.

[71] V. Radeka. Nucl. Instr. and Meth., 99:525–539, 1972.

[72] Valentin T. Jordanov and Glenn F. Knoll. Nucl. Instr. and Meth. A, 345:337–
345, 1994.

96
[73] A. Georgiev and W. Gast. IEEE Trans. Nucl. Sci., 40:770–779, 1993.

[74] T. Kihm et al. Nucl. Instr. and Meth. A, 489:334–339, 2003.

[75] Steven W. Smith. The Scientist and Engineer’s Guide to Digital Signal Process-
ing. California Technical Pub., 1997.

97