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dissertation entitled

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

presented by
DAN-CRISTIAN DINCA

has been accepted towards fulfillment
of the requirements for the

Doctoral degree in Physics

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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 232,34 AND
APPROACHES TO SUB-SEGMENT INTERACTION-POINT
DETERMINATION IN 32-FOLD SEGMENTED HIGH-PURITY GERMANIUM
DETECTORS

By

Dan-Cristian Dinca

The even 52‘56Ti isotopes have been studied with intermediate-energy Coulomb
excitation and absolute B (E2; 0+ —» 2?) transition rates have been deduced. Our data
confirm the presence of a sub-shell closure at neutron number N = 32 in neutron-
rich titanium isotopes above the doubly-magic nucleus 48Ca 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 y-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“Z’6Ti” , was a
team effort, involving more than N SCL’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.

Contents

1 Introduction ............................... l

2 Reduced transition probabilities to the first 2+ state in 52*54’56Ti 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
A2 Gaussian smoothing using 9 points ................... 75
A3 Signal derivatives ............................. 77
A.4 Linear fit .................................. 79
A5 Threshold passing point ......................... 82
A.6 Trapezoidal shaper ............................ 84
A.7 A differentiator-integrator shaper .................... 88
A.8 Statistics on a waveform segment .................... 90
A9 RC-CR Filter ............................... 91
Bibliography ................................ 94

vi

List of Figures

1.1
1.2
1.3

2.1
2.2
2.3
2.4
2.5
2.6

2.7
2.8

2.9
2.10

2.11
2.12
2.13
2.14
2.15
2.16
2.17

2.18

A section of the nuclear chart showing the region of interest ..... 2
Simplistic view of the orbital structure for 56Ti above the 1f7/2 shell . 2
Example of 7-ray interaction with a segmented detector ........ 5
Particle ID, dispersive focal plane vs. time of flight .......... 10
Particle ID, time of flight vs. energy loss ................ 11
Particle ID, time of flight vs. energy loss ................. 11
Schematics of the projectile-target interaction ............. 13
The SeGA rings .............................. 15
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
Laboratory frame coincidence 'y—ray spectra for 197Au ......... 20
Laboratory frame 'y-ray spectra for 197Au in coincidence with 54Ti.
Background subtraction applied ...................... 22
Typical time spectrum for a SeGA detector ............... 22
Coincidence 'y-ray spectra for 76Ge Doppler—reconstructed event-by-
event in the projectile frame ....................... 24
Coincidence y-ray spectra for 52Ti Doppler—reconstructed event-by-
event in the projectile frame ....................... 24
Coincidence y-ray spectra for 54Ti Doppler-reconstructed event-by-
event in the projectile frame ....................... 25
Coincidence 'y-ray spectra for 56Ti Doppler-reconstructed event-by-
event in the projectile frame ....................... 25
Projectile frame 'y-ray energy in coincidence with 76Ge. Background
subtraction applied ............................ 26
Projectile frame 7-ray energy in coincidence with 52Ti. Background
subtraction applied ............................ 26
Projectile frame y-ray energy in coincidence with 54Ti. Background
subtraction applied ............................ 27
Projectile frame 'y-ray energy in coincidence with 56Ti. Background
subtraction applied ............................ 27

Comparison of the measured 21* excitation energies with the results
of large-scale shell model calculations using the GXPFl (dashed lines)
and GXPFIA (solid lines) effective interactions. ............ 30

vii

2.19

2.20

3.1

3.2

3.3
3.4
3.5
3.6
3.7
3.8
3.9

3.10
3.11
3.12
3.13
3.14
3.15
3.16
3.17
3.18
3.19
3.20

3.21
3.22

3.23

3.24

Comparison of the measured absolute B (E2; 0+ —> 2f) transition strengths
with the results of large-scale shell model calculations using the GXPF 1

and GXPFlA effective interactions ................... 31
Comparison of the measured absolute B (E2; 0+ —> 2?) transition strengths
with the results of large—scale shell model calculations using the GXPFl

and GXPFlA effective interactions using the effective charges calcu-

lated from Ref. [1] ............................. 34
The energy resolution dependence versus the detector angle for a typ-
ical beam ................................. 40
Laboratory frame (upper panel) and projectile frame (lower panel) 7-
ray energy spectra ............................ 41
Segment labeling scheme for SeGA crystals ............... 46
Weighting potential for segment E, transversal cut ........... 47
Weighting potential along the trajectories “1” and “2” ........ 48
Weighting potential for segment E, longitudinal cut .......... 49
Example of real and transient charge signals .............. 50
Experimental setup of the SeGA scanning stand ............ 52
Segment signals for an interaction closer to the E4 side of the D4
segment (Quadrant 3) .......................... 54
Segment signals for an interaction closer to the E4 side of the D4
segment (Quadrant 4) .......................... 55
Segment signals for an interaction closer to the C4 side of the D4
segment (Quadrant 3) .......................... 56
Segment signals for an interaction closer to the C4 side of the D4
segment (Quadrant 4) .......................... 57
The average miss of the algorithm in estimating the interaction position
for several positions of the collimator .................. 58
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

Amplitudes of the transient signals for segments F4 and D4. z = 0 mm. 60
Amplitudes of the transient signals for segments F4 and D4. z = 2 mm. 61
Amplitudes of the transient signals for segments F4 and D4. 2 = 4 mm. 62
Amplitudes of the transient signals for segments F4 and D4. 2 = 6 mm. 63
Amplitudes of the transient signals for segments F4 and D4. 2 == 8 mm. 64
Central contact waveforms corresponding to a photopeak event at

1332 keV .................................. 65
Segment waveforms corresponding to a photopeak event at 662 keV . 66
Time to reach 30% of the full amplitude (t30) plotted against the
position of the collimator in the radial direction ............ 67
Time to reach 60% of the full amplitude (t60) plotted against the
position of the collimator in the radial direction ............. 68

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). ...................................

3.26 The central contact signal corresponding to a 1332 keV event and a
trapezoidal shaper with integration time of 1 as and flat-top duration
of 0.5 [1,8 applied on it ..........................

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

ix

List of Tables

2.1 Summary of secondary beam properties. ................ 9
2.2 Summary of scattering angle cuts. The first entry for 52Ti corresponds

to the 256 mg/cm2 target and the second to the 518 mg/cm2 target. . 23
2.3 Summary of measured transition energies (2fr —> 0:8), number of par-

ticles detected and Coulomb excitation cross sections .......... 23
2.4 Comparison of measured B(E2;0+ -—+ 2?) values (labeled B(E2; T) in

the table) with shell model calculations using the GXPFl interaction

as well as the recently preposed GXPFlA interaction. The two 52Ti

entries correspond to separate measurements with Au targets of differ-

ent thicknesses; (a) 256 mg/cmz, (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

Chapter 1

Introduction

Beyond the valley of ,6 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 'y-ray spectroscopy following deep-inelastic scattering, Coulomb
excitation of fast projectiles or reaccelerated beams, fi-decay studies, or transfer re-
actions.

Beta-decay studies of 56Cr indicated a possible sub-shell gap at neutron number
N232 [8]. The nearest doubly-magic nucleus is 48Ca (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

21
20 C

 

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 fl-stable nuclei. The three titanium isotopes (52’54’56Ti) are circled.

 

 

 

 

 

 

 

199,2 10
1ng 5
291/2 H 2
2pm —H—O—O—— 4

 

H 1f7/2 W 8
TC V

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

 

 

identified as the proton-neutron monopole interaction [9]. As the number of protons
occupying the 7r f7/2 shell decreases, the 1r f7/2 — V f5/2 monopole pairing interaction
strength weakens. As a consequence, the neutron orbital uf5/2 is shifted up in energy.
Coupled with the rather significant spin-orbit splitting between the ”PI/2 and 1493/2
orbitals, this may lead to new sub-shell gaps (Figure 1.2). Shell-model calculations
using the effective interaction GXPFl [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, ,B-decay measurements
at the NSCL [11,12] in conjunction with prompt v-ray spectroscopy following deep—
inelastic reactions at Gammasphere [13,14] were performed for 54Ti and 56Ti. The

energies of the first excited states in these nuclei supported the notion of a shell gap at

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 52’54’56Ti 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 7-ray energy, giving the energy spacing between the two bound
states, the Coulomb excitation cross section is related to the electromagnetic matrix
elements B(EA) and B(MA) of multipolarity x\. 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 8Li radioactive beam
at 14.6 MeV on a "“‘N a target [21—23]. Gamma-rays emitted by exotic beams under-
going Coulomb excitation were measured first in an experiment involving scattering
of 76Kr on a 208Pb 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 52Ti, 54Ti and 56Ti 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-

surements in the regime of limited statistics would gain from an enhanced peak-to~
background ratio. Even for relatively large bin sizes of the y-ray energy spectra in
the projectile frame (Figures 2.10, 2.11, 212,213) 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
'y-ray transitions close in energy have to be resolved.

The concept of 7-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 7-ray tracking for nuclear physics. The method involves the determination of
the 'y-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 ’y-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 7-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:

Epmj = 7131011,“ — ,66086) (1.1)

where E10], and Ema]- are the y-ray energies in the laboratory frame and projectile

Detector

 

 

 

}.
Beam

 

 

 

Target

Figure 1.3: Example of y-ray interaction with a segmented detector

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

7 z .1 _ (1.2)

W
is the relativistic factor. The angle 0 is taken between the scattered beam and the
'y-ray direction. This angle can be calculated by knowing the position where the
nucleus emitted the photon and the point where the 7-ray first interacted with the
detector. Figure 1.3 depicts an example of a 7-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
y-ray spectroscopy. A sub-segment interaction position resolution offers the possibil-
ity of a better Doppler reconstruction for in-beam 7-ray spectroscopy, leading to an

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

can yield good performance, without involving a full ’y-ray tracking which requires

expensive hardware and software solutions.

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

 

lReprinted 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]

shape or the type of excitations characterizing the low-energy level schemes. One of
the pr0posed 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
of doubly-magic ggCagg. At present, experimental evidence for the existence of this
N = 32 gap rests solely on the low and medium spin (1 g 12) level sequences of the
52Ca [39], 52‘56Ti [11, 13,14] and 52‘58Cr [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
states of the 52"i34'56Ti 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" —> 21*) value in doubly-magic 56N i was found to be larger than
anticipated [16], while that measured for 136Te, with 2 protons and 2 neutrons outside
the doubly-magic 132Sn 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 p f-
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.

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/cmz)
76Ge 81 0.392 0.375 256
52Ti 89 0.408 0.399 256
52Ti 81 0.408 0.388 518
54Ti 88 0.406 0.398 256
56Ti 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 N a-
tional Superconducting Cyclotron Laboratory using secondary beams produced in
fragmentation of 76Ge primary beam at an energy of 130 MeV/ nucleon. Following
the 380 mg/cm2 9Be 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
76Ge primary beam was degraded to 81 MeV/nucleon and sent onto a 256 mg/cm2
197Au 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
89 MeV/ nucleon, were then selected in succession and directed onto 197Au targets of
256 mg/cm2 and 518 mg/cm2 thickness. The thinner Au target was used with the
52Ti and 54Ti fragments, the thicker with 52Ti and 56Ti. For a primary beam intensity
of 10 pnA, the three Ti settings resulted in average rates on target of 9000 Hz (52Ti),
600 Hz (54Ti) and 40 Hz(56Ti). 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.

 

l'r‘tr'rl—IIIII-IIIIII'IIUIII

I

'IIIlllllllIllllllllllllllllllllllllllll'

55V

  

-. 54Ti

<—TIme of flight (au)

.IIIIIIIIIIIIIIllllllllllllllllllllllllI.

N‘s-q.

.l;...l....l....l....l....ll

 

 

 

Dispersive focal plane position (au)‘

Figure 2.1: Particle identification spectrum for the 54Ti 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 54Ti 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 8800

Spectrograph see References [42] and [43].

10

 

Energy loss (au)

 

 

 

 

{-— Tlme of flight (au)

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

Counts

Energy loss (au)

IlllllllllllllllllJlllllII

 

 

(— Time of flight (au)

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

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 6cm in the following way (Figure 2.4, Equation 2.1).

a 0m

where '7 is the relativistic factor (1.2) and a is dependent on the projectile and target
atomic numbers (mej and Ztar respectively) and the reduced mass of the target-

projectile system (m0) via:

 

 

Zpr'ontare2
= —— 2.2
m0 C2fi2 ( )

1 I 1
— = (2.3)

m0 mproj + mm.

Based on the kinematics of the process a relationship can be established between
the scattering angles in the center—of-mass frame (66",) and laboratory frame (6,01,).

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).

7T0.

b—+b .
+27

(2.4)

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

jectile to the cross section via:

12

 

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

 

 

2 —) A _ 1 -1 OTA Z 2
a“ z (22:; ) Bpm(1r)\,20 A) vrbiiffi’ ( ) f (2.5)
e 2 111(1),...” /b,,,,-,,) fem = 1

where 7r is the parity of transition and A 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 "y rays emitted in flight (on an event-by-event basis).

2.3 Analysis method

The reduced transition matrix elements B(E2;0‘L ——> 2]") 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 152Eu 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 8800 spectrograph.
The Coulomb excitation cross section is given by the relationship
N7

0 = 2.6
Clot Ntarget Nbeam ( )

 

14

Backward ring Forward ring

 

 

 

 

 

 

 

Incoming ...................................................
—>

 

 

 

 

 

 

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

15

N, 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. 6:0: 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

NAXp
A )

 

(2.7)

N target :

where N A is Avagadro’s number (N A = 6.022 x 1023 particles/mol), p is the target
density (thickness) in g/cmz, 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
scaled 2 (D/ S) particle identification spectrum NpartD/g times the D/ S factor set in

the trigger electronics fD/s-

Nbeam = partD/S X fD/S- (28)

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

Ctot 2 £370 + €900 (2.9)

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

GEANT

effing : €(E7 lab X (SCEANT X Af. (2.10)

2To 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 7-ray detector (SeGA) trigger in coincidence

with 8800. 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
'y-rays.

16

where 6(E.,)ff,'§‘,"NT is the efficiency simulated with GEANT in the projectile frame
assuming an isotropic emission, 60““ is the GEANT scaling coefficient, and A f is
an angular factor that accounts for the anisotropic angular distribution of the emitted
7—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 efliciency in the projectile frame is
calculated, a scaling coefficient 66mm is defined for the energy range of the 7-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 'y-ray

angular distribution (W(0p,.oj)) can be found in References [20,25,46,47]. N9 is the

proj

number of y-rays detected by the detector array at a given angle 0. The index “pro j”

indicates that the quantity is considered in the reference frame of the projectile.

Z Noproj W(6proj) Sin Oproj

A aproj
f = 1 .
E 2 New”. Sin 0pm,-

aproj

 

(2.11)

The uncertainty of the experimental cross section is

AN 2 ANb.3 2 AN: t 2 A5: t 2
A = 7 +( am) +( urge) +( 0) 2.12
U 0 ¢( N '7 ) N beam N target 6tot ( )

 

 

 

 

 

where
AN, = ,/N, (2.13)
Ap 2 ANA 2
Ajlvtarget = Ntarget\/(7> + ( NA ) (214)

17

A f and e(E,)fi,EANT 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 (N7) and the simulation

scaling coefficient 60mm (5 - 6%).

2.4 Results

Inelastic scattering of the primary 76Ge beam on a 197Au target was used to validate
the experimental technique. In 76Ge, 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]) = 2923(346) ezfm4
that compares well with the adopted one of 2780(30) e"’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”r —> 7/2+) = 4472(951) versus 4494(409) e2fm4.

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.
The first value labeled 52Ti is for the 256 mg/cm2 197Au target thickness and the
second is for the 518 mg/cmz. The value labeled “adopted” is the one from [49]. The
velocity of the excited 197'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 197Au nuclei from the

target can be considered at rest.

18

 

I ' I ' I ' I ' I 1 I

8°°°“ ] 197Au (547.5 keV, 7033/23...) I " ‘

 

 

 

 

 

60004 i -
$4000---£--- ----[—---¥---«[----..—-4
6?
9i . .
m

2000- _

0 I r I ' I ‘ I ' I ' I
adopted 760a 52Ti 52Ti 54Ti 56Ti

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

 

01
O

 

 

197Au (547.5 keV, mus/2:3)

 

 

b
O

511

N
O

(A)
O
IIIITTIIIIIIIIIIIIIIII

Counts / keV

 

 

    

 

_\
o

 

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

9150 500 550 600 650 700 750
Energy (keV)

Figure 2.7: Laboratory frame coincidence 7-ray spectra for 197Au. The nucleus is
Coulomb excited by the electromagnetic field of the incoming 54Ti 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
from background lines (e.g. 40K from the environment, 60Co 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 'y-ray spectrum is subtracted from the spectrum
obtained for the prompt time. The result is a background subtracted spectrum. As an
illustration of the results, a background subtracted spectrum for 197Au is displayed
in Figure 2.8. Compared to the Figure 2.7 the background around the 197Au peak is
more flat (e.g. the 511 keV line disappeared).

Figures 2.10, 2.11, 2.12, and 2.13 show representative coincidence 7-ray spectra
for 76Ge and the even 52‘56Ti isotopes Doppler reconstructed event-by—event in the
Projectile frame. The energy at mid-target for 76Ge 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, 06", < 3.1°. For 52Ti the
corresponding values for the 256 mg/cm2 and 518 mg/cm2 Au targets were 82.4 and
79-1 MeV/ nucleon, respectively, with 66m < 3.1° and < 3.3° and similar minimum im-

vgct parameter of 13.9 fm. The spectrum measured with the thinner target is shown

in the Figure 2.11. For 54Ti and 56Ti, the respective projectile energies were 83.3

21

 

40

 

 

197Au (547.5 keV, fires/2;.)

 

 

30

20

 

IIIIITIIIIIIIIIIIIII

10

Counts / keV

 

 

0

-10
IIILIIIII IJII

I I I
450 500 550 600
Energy (keV)

I I L I I I I I

I I
650 700 750

I I I

Figure 2.8: Laboratory frame 'y-ray spectra for 197Au in coincidence with 54Ti. Back-
ground subtraction applied.

 

 

 

 

 

I I I I
800 — I I
| | . :
I I s a
I I i
..-.. 600 — I z :
3 ' ' : :
a Prompt time gatel | I I Off-prompt time gate

‘3' I | : :
E ' I : :
3 I I
8 I | . .
200 — ' . '
' 5 :
I I a :

-.I._I_ I I I III I I I III I L I I I I I I J L L I‘ll-LIA‘ 4- I.—

QOO 200 300 400 500 600 700

Time (ns)

Q‘xgure 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 52Ti 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) (°) (°)
76Ge 17.6 3.06 2.18
52Ti 13.9 3.10 2.42
52Ti 13.9 3.29 2.57
“Ti 14.0 3.20 2.48
56Ti 14.1 3.58 2.75

 

 

 

 

 

 

Table 2.3: Summary of measured transition energies (2] —-> 05“,), number of particles

 

 

 

detected and Coulomb excitation cross sections.
Beam Energy Particles detected Cross section Target thickness
21“ -+ 0;...
(keV) (mb) (mg/cm?)
76Ge 562.6(6) 26 x 106 394(47) 256
52Ti 1050(2) 130 x 106 119(16) 256
52Ti 1049(2) 67 x 106 125(16) 518
“Ti 1497(4) 92 x 106 83(15) 256
56Ti 1123(7) 6 x 106 155(51) 518

 

 

 

 

 

 

 

and 78.6 MeV/nucleon, with minimum impact parameters of 14.0 fm and 14.1 fm
computed from 9cm < 3.2° and < 36°. The arrows indicate the expected location
of transitions deexciting the 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 (2fr —-> 0%.) the
number of particles detected (with down sealer 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-7 coincidence spectra for Doppler corrected
’y-ray energy with background subtraction applied.

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

23

500

400

00
O
O

Counts l 8 keV
N
O
O

100

0

200

Figure 2.10: Coincidence y-ray spectra for 76Ge Doppler-reconstructed event-by-event
in the projectile frame.

.3
O
O

Counts / 8 keV
8

 

 

21—0“

.1.
600

g.s.

.800 '
Energy (keV)

v/c = 0.356

I I l l l 7
1000 1200

 

 

 

Figure 2.11: Coincidence 7-ray spectra for 52Ti Doppler-reconstructed event-by-event
in the projectile frame. The possible location of the 2;L —> 2? transition is shown by

an arrow

24

52Ti
v/c=0.385
2143;...

2;»2;
.In.1l.1|...l.
800 1000 1200 1400

Energy (keV)

 

 

 

 

 

 

2+_’2+ 54TI
40.] / 2 1 v/c=0.397
> |
330* + +
2 21 9.5.
.59.
g20—
O
0
10]-
i...|.L.l...1...I..
1000 1200 1400 1 600 1800 2000

Energy (keV)

Figure 2.12: Coincidence y-ray spectra for “Ti Doppler-reconstructed event-by-event
in the projectile frame. The possible location of the 22; —-> 2? transition is shown by
an arrow.

 

I 56Ti

v/c = 0.362
‘10-”- 1 21-‘0-1- .

 

 

 

 

 

 

 

Counts/ 12 keV

 

 

a, J,
. ' ' .I

; I I I l l I I III I . o I
Energy (keV)

 

 

 

 

 

 

Figure 2.13: Coincidence 7—ray spectra for 56Ti Doppler-reconstructed event-by-event
in the projectile frame.

25

 

 

 

 

     

250
: 76Ge
; v/C = 0.356
> 200 T
3 : + +
<13 - —>
B 150 :- 21 0g.s.
S E
8 100 :-
50 {-
0 b . l . . L 1 Militant]

 

 

. I . .
200 400 600 800 1 000 1200
Energy (keV)

Figure 2.14: Projectile frame 'y-ray energy in coincidence with 76Ge. Background
subtraction applied.

 

 

 

 

 

 

70 Iil 52TI
60 2:»038 V/C = 0.385
> . .
E2 50
co
m 40
§ :
o 30'
0 ::
20 5:
‘° Nil
0 600 800 1000 1200 1400
Energy (keV)

Figure 2.15: Projectile frame '7-ray energy in coincidence with 52Ti. Background sub-
traction applied.

26

 

30

 

 

 

 

 

 

 

 

54Ti

5 25 v/c=0.397
:5 20 21-7038.
E
g 15
8

10

5 l IIIIIII ill

0 . . . J i

 

. 4 l .
1000 1200 1400 1600 1800 2000
Energy (keV)

Figure 2.16: Projectile frame y—ray energy in coincidence with “Ti. Background sub-
traction applied.

 

 

 

 

 

 

 

    

 

 

 

 

 

 

 

 

103. _ 56TI
- VIC-0.362
- + +
E 8 _— 21.095.
x .
3 6 '_
3 u-
c
8 4
O
2 I—Ih III
0 I...1..Il.1...1..-II
800 1 000 1 200 1400 1 600

Energy (keV)

Figure 2.17: Projectile frame y-ray energy in coincidence with 56Ti. Background sub-
traction applied.

27

Table 2.4: Comparison of measured B(E2;0+ -—) 21‘) values (labeled B(E2;T) in
the table) with shell model calculations using the GXPFI interaction as well as the
recently proposed GXPFlA interaction. The two “Ti entries correspond to sepa-
rate measurements with Au targets of different thicknesses; (a) 256 mg/cmz, (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;T) B(E2;T) B(E2;T) B(E2;T)
(e2fm4) (e2fm4) (e2fm4) (e2fm4)
expt. GXPFI GXPFIA 547 keV Au

“Ti (0) 593(80) 427 435 4114(627)

52Ti “0 548(70) 427 435 4063(455)

52Ti (C) 567(51) 427 435

“Ti 357(63) 453 446 4279(672)

56Ti 599(197) 483 448 6356(2227)

 

 

 

 

 

 

 

three even-mass Ti isotopes of interest. The analysis was carried out following the
prescription given above for 76Ge. In each case, the cross section for the excitation of
the first 2+ level was extracted from the 'y-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:) values. In the case of “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 “Ti. The two “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+ —+ 21“) = 66513]? e2fm4. 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
of 197Au 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 21* levels of interest

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

28

lying 2+ states to the measured 2] —> 0+ y-ray yields. In “Ti, a number such higher
2+ states are known [51], and a 22" level has also been proposed tentatively in “Ti [13].
In the data for both nuclei, none of the 'y rays associated with decays from these levels
towards the ground state and the yrast 2] level was observed, as illustrated in Figures
2.11 and 2.12 where the location of the 2'; —-> 2'] transitions in “’“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?) values under
discussion here. The largest such strengths is predicted to occur for the 2;; level in
“Ti. In this case, the upper limit for the 2; —> 2;“ intensity obtained from the data
translates into a maximum correction to the B(E2; 0+ —» 2?) value of 34 ezfm“, z'.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(2f) energies increase significantly
at N = 28 and N = 32 (Figure 2.18), the B(E2; T) strengths are lowest for these

two neutron numbers (Figure 2.19). Furthermore, both these physical quantities also

29

 

26 28 3'0 32 34M

1600— -.-‘ -

'0' ""

800

1
l

 

0 Experiment
' --- GXPF1 ‘
— GXPF1A

48h 50h 52h 54Ti 56Ti

Figure 2.18: Comparison of the measured 23‘ excitation energies with the results of
large-scale shell model calculations using the GXPF1 (dashed lines) and GXPF1A
(solid lines) effective interactions.

E(2;‘) Energy (keV)
I

 

 

 

 

 

 

0

differ markedly from the corresponding values at neutron numbers N = 26,30 and
34. For “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 21" 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
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 21+ excitation energies lower
by several hundreds of keV and B (E2; 0+ —> 2]) values higher by a factor of ~ 2 can

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

30

2'6 2'8 3 0 3'2 314 N

 

 

 

 

 

 

 

 

 

 

4" _ .
E 800 I
N0)
+: i
N k vvvvvvvvvvvvvvvvvvvvv l
1 400- [ -
05 o
+ d) -
o“ - ep = 1.5e .:.. Eggggment .
N en = 0.5e
H", — GXPF1A
m 0

48Ti “Ti “Ti “Ti 56Ti

Figure 2.19: Comparison of the measured absolute B(E2;0‘L —> 2]”) transition
strengths with the results of largescale shell model calculations using the GXPF1
(dashed lines) and GXPF1A (solid lines) effective interactions. The B(E2; 0+ —) 21+)
value for “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 p f-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 2121/2 — 2123/2
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 z 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, 121/2 and f5/2 neutron orbitals, as well
as the degree of admixture between these states in the wavefunctions of the “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
121/2 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(2f) 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
54Ti 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
(epAp + enAn)2

BUB“): 2.7-+1 ’

 

(2.15)

where Ap and A, 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. J.- is the spin of the initial state, 0 in the case of
the ground states of 52““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
(f 7/2)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 A, 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 (88,154),
(10.7,9.5), (90,144), (107,106), and (11.8,8.7) for the even 4“‘“Ti. The theoretical
shell gaps for neutrons at N = 28,32 and 34 result in reduced An amplitudes and
in excitation spectra for “I“'“Ti that most closely reflect the (f7/2)2 proton struc-
ture. The deviation of the experimental “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 (Am/1,.) = (103,114) reflect a larger neu-
tron admixture. With this interaction, the calculated 191/2 — f5/2 shell gap at N = 34
is still significant, i.e., 2.5 MeV. Furthermore, this gap is calculated to increase to
3.5 MeV for “Ca [53], so that a neutron sub-shell closure is still predicted in this
Case. The B(E2, 1) rates computed from the (Ap,A,,) values (Equation 2.15) with
conventional effective charges of ep 2 1.5e and e. = 0.5e overestimate the measured

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

33

 

2'6 2'8 3 0 3'2 34 N

1

CI)
0
O

.h
C
O

 

_-» 2T)(e2fm4)

 

 

 

 

 

 

 

U) I
+03 _ Ex . t
q - ep~1.15e ......Gx°.:::‘°" I
L”,
m 0

I I I I L
48Ti “Ti “Ti 54Ti 55Ti
Figure 2.20: Comparison of the measured absolute B(E2;0+ ——> 21+) transition

strengths with the results of large-scale shell model calculations using the GXPF1 and
GXPF 1A 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
6,, effective charge needs to be increased, while keeping the isoscalar effective charge
6,, + e,, constant. Such a modification in the e, and en charges could result in a better
agreement with experiment. Recent data [1] on analogue states in A = 51, T, = i1/2
mirror nuclei suggest that this may well be the case and values of e, N 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 p f -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
number N = 32 in neutron-rich Ti nuclei above 48Ca, 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])
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 fromy-ray spectroscopy following fi-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 fi-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 y-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 I (a detector is hit
by a single 'y-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 'y 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 'y-ray emission point. Due to the relativistic velocities at which the nuclei travel,
the emitted 'y-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 7—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 7-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 7-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 'y-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 y-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:

AE, 2_ fisinl? 2 2 —-fl+cos€ 2 2 AEint 2
(E) ‘ (1-30089) (M) +((1-i32)(1—flc080)) (M) "Ll E. l '

(3.1)

 

 

 

where 0 is the velocity of the nucleus that emitted the y-ray at the time when the
nuclear transition occurred, 6 is the emission angle with respect to the velocity of
the beam and E, is the ’y-ray energy. In a first approximation, the uncertainty in the
determination of the target position is neglected. The intrinsic contribution AEint 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 A6 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 7-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 ’y-ray emission angle by a significant amount. Figure 3.2
shows an example of a ry-ray energy spectrum from the Coulomb excitation of “Ti
on a 197Au 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-
rection the “Ti appears and the background lines are smeared out. Only the first
interaction of the 'y-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 y—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 'y-ray tracking methods [29, 30]

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

39

 

 

I, , . L.
0 T— ——.- I E . L . A
0 60 120 180
9 (deg)

Figure 3.1: The energy resolution dependence with the detector angle for a 5“Ti beam
with ,8 = 0.385 and uncertainty A6 = 0.024. The energy of the y-ray is considered
ECM = 1.049 MeV (projectile frame). The opening angle of the detector is considered
A0 = 3 deg, corresponding to a SeGA detector places at 20 cm from target. The
intrinsic constant is AEW = 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

 

 

 

 

 

 

 

 

 

> 6:0.
2 40 1173 1332 1460
N
E
c I
3 20 :I. .
0 “WHEN
0 . . . I . . . I . . A 1 . - . I
_ 0:03:35
3100 1049
x
Q
E
g 50 _
O
0
600‘ ‘ iaoogi A1000; ‘ ‘1200‘ ‘1400‘ i ‘1600
Energy(keV)

Figure 3.2: Laboratory frame (upper panel) and projectile frame (lower panel) 7—ray
energy spectra. In the laboratory frame, the 1173 keV and 1332 keV lines correspond
to the 60Co decay and the 1460 keV line to the 40K decay. In the projectile frame, the

1049 keV line corresponds to the “Ti 2] ——> 0;, 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
'y-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”, the Compton effect to Z, and the pair produc-
tion with Z“. 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 (Hglg, 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 um) and the central contact region is lithium
diffused (500 pm). 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 7—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 (p(r)) can be calculated from the distribution

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

W“) = eNed?) (3-2)

where e is the electron charge. The electric potential (<I>(7‘)) throughout the crystal

can be obtained by solving the Poisson equation:

V2<I>(7‘) : _p_(fl (3.3)

6

where e 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) = —V<I>(r) (3.4)

The charge carrier drift velocity is then:

vdrift,c(rc) : #cE(Tc) (35)

The subscript 0 stands for charge carrier, since the electron and hole mobilities (lie
and #h) 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:

2.c = QC-fidriftfiiw (36)

where E", 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

I

{—>

 

J

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l'___—_T_———_"I
I
I
I
l
I
I
I
______.'_______
I
I
I
I
I
I
I
I
I
I
L.______I__.____I

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

 

 

IIIFIIIIIIII

 

“y (mm)

JIIIIIIIIIII

 

 

 

 

IIIIIIIIII IllIIlIIIIIl 0
-30 -20 -10 U 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

0.2

0.15

0.1

Weighting potential (V)

lIITIIIIIIIIIIIIIIIIIIIIII

 

 

 

 

IIIllllIllIlIllIlIIlllIlIl

25

 

CO

Radius (mm)

Figure 3.5: Weighting potential along the trajectories “1” and “2” in Figure 3.4.

48

 

 

 

 

 

 

 

80_ V
70} 0'8

E 0.7
60—

l.

E 0.6

A50?

E : 0.5

£40:—

I : 0.4
30} 03
10:- 0.1

:lllllllllIIIlIllllllllJlllllllllLl
00 0
R(mm)

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

49

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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 pr0portional to the charge created in the segment, which in turn is

proportional to the energy deposited in the interaction.

3.3 Experimental setup

A 7-7 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 'y-rays from a collimated 137Cs 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
N aI(Tl) scintillation detector was placed underneath the SeGA detector to allow
detection of 7-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 8183300 12-bit and two 3133301 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 as) to 1024 samples per trace(~10ps).

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

and cosmic rays, only the 622 keV '7-rays from the 137Cs 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 'y-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

2025

 

 

 

2020

 

2015
2010

2005
2015

 

Signal amplitude (channel)

 

2010

 

2005-J

CB

0. 0.5 1.0 1.5 2.0 2.5
Time (us)

2000

 

1 995

Figure 3.9: Segment signals for an interaction closer to the E4 side of the D4 segment
(Quadrant 3).

54

 

2025
2020

'IIIIITTIII

2015

 

2010

 

ilIIIIIT
CITI _
.$>.

2005

 

2010

1 990

1 970

 

_x
CD
(TI
0

 

 

Signal amplitude (channel)

 

' 2025
2020
2015
2010

 

C4
2005

L l l I . AL A I 1 L; A A l IIIIIIIII l

0. 0.5 1.0 1.5 2.0 2.5
Time (us)

 

Figure 3.10: Segment signals for an interaction closer to the E4 side of the D4 segment
(Quadrant 4).

55

 

2025
2020

llll'T7171

 

2015
2010

E3

2005 . . . . I . . . . l . A . A l . . . . l . . .
2025

IIIIIIIIU

 

 

2020

 

2015
2010

D3

2005 . -11 ... -...... ... ... ... ..1
2015

Signal amplitude (channel)

IIIIIII- I—Firl'lTITI

 

 

2010

lllfll—ll

2005
2000

 

 

IIIIIIIITII

C3 ..e..

(15 11) ‘L5 2L0 215
Time (us)

1995 . .

 

9’

Figure 3.11: Segment signals for an interaction closer to the C4 side of the D4 segment

(Quadrant 3).

56

 

2025
2020

 

2015
2010

IIIIIII _r]lllerTTIl

E4

2005‘-n-l-4--l....l....l.LLAl

 

 

 

N
O
..s
O

1 990

 

Signal amplitude (channel)

.3 .3
(O (9
U1 V
O O

I U
4:.

h
-

L J A L L l l A I L L

 

2020
2015 '

2010
2005

 

 

C4

0.5 1.0 1.5 2.0 2.5
Time (us)

2000

 

.o IIIIIIIIIIIII

Figure 3.12: Segment signals for an interaction closer to the C4 side of the D4 segment
(Quadrant 4).

57

 

’E‘

a 2- -
a)

.g 0 . .
m I .0

O) O

‘3 1_ O .. _
2

<1:

0 . 2 I 4 I 6 l 8
Linear position (mm)

 

 

O

 

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

 

0.2

0.15

0.

—L

llllllllllllllllllllllll

Fraction of events

0.05

 

 

 

-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 'y-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
E 0.9
7o_—
E : 0.8
a 60:—
5 I 0.7
g 50:— 0.6
‘5 :
E 40:— 0.5
g) : 0.4
,7, 30_—
5 E 0.3
2 20.—
(fi _ I
1: : _ 0.2
10;" 0.1
o Illllllllllllllllll l l l l l I l lllllllllll o
0 10 20 30 40 50 60 70 m

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

 

 

 

 

3°: . 1
E ' 0.9
70_—
i I 0.8
v 60:—
0 _
E : 0.7
a, _
5 5°:— 0.6
'73 :
g 40_— 0.5
E Z 04
g 30 _— '
.6 E 0.3
g 20—
1: E - 0.2
1°;— 0.1
9 II I I I l I l I I I I l I l I I I LII I I I l l I l I l I I I I I 14 I o
0 10 20 30 40 50 so 70 so

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
E - 0.9
A 70 _— '
fa, * ' 0.3
g 60;—
c C 0.7
"7’ 50E
2 T 0.6
a E
g 4o__ 0.5
E t
% 30 j— 0'4
.5 E 0.3
g 20_—
|: '_' I 0.2
1° :— 0.1
0 LI l l l I l l l l I l J J l I I l l l I l l l l I l l l l l l l l l I I l l L o
0 10 20 30 40 90 70

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

 

1

N
O

 

 

 

E 0.9
A 70 .—
é : 0.8
V 60 L
D
5 g 0.7
a) _
Ig’ 5° :— 0.6
3. _
g 40 L 0.5
E E
g 30 :_ 0.4
5 : 0.3
820;
.9 g 0.2

1° :— 0.1
0 L I l l I l I I I I I l I I I l I l l I I I l I I I I I I I I I I I I I I I l o
o 10 20 30 40 50 60 70 90

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
E 0.9
A70.—
52 : ._ 0.8
V60; .
2 E .. 0.7
0) ' II I
"550? I a I:' 0.6
a : . -'
$40} 0.5
E :
§ 3 0.3
220‘
E E 0.2
10;
E II. I. 0.1
OLIIIIIIIIIIIlIIIIIIIIIIIIIIIIIIIIIIIIIII
0 80 0

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

 

2500-

2000

1500

Voltage (a.u.)

1000

500

 

 

 

 

900 1000 1100 1200 1300 1400
time (ns)

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 'y-ray and from a segment inwhich the energy from a 662 keV ’7-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 'y-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

 

400-

Voltage (a.u.)

-1200

 

 

-1600

 

 

-2000 ‘ ‘
700 800 900 1000 1100 1200 1300 1400

time (ns)

 

Figure 3.21: Segment waveforms corresponding to a photopeak event at 662 keV.
Only the interactions in which the ’y-ray energy was fully deposited in the segment
are selected.

66

U!
C

 

45;— 140
402—- I 120
m.
g : —
10253— 80
:3 E — I

.3
01
H

_\
O
lllllllllllill
A
O

 

 

5 I I I 20
0 I I_IIL I41 I I I l I l I I l l I I I I I l Li I I I l I l l l I I l I l l 0
0 50 1 00 150 200 250 300 350 400
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:—
35:—
: g—80
E30:— ‘
g : i. i
.3252— $160
g 20;— "
15:— "40
10? — 20
5:—
0: LILIIIIIIIJ I I I I I I I I I I l I I I I ILI I I I I I I I II I 0
o 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:—
E 60
40': —
C I —
I I I I I
”‘30:— BEE-l—
E I — I :- 40
- L" _—
s 20:. __'. ' 30
DC :
E. _—
15: 2o
,0; In .
E I."— ' ' 10
5}. I I I
0IIllIlIlIIIIIIIIJIIIlIIIIIIIIIIIlIIIIII 0
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
I _‘ _ 4000
_ __- - .-
300 — _ 3500
. .-_- I _
1? _ - 3000
5 ._
8 250 l 250°
.. : 2000
1 1500
200 _ 1000
: 500
150 . . - I .

 

 

l l I I I I I l I I I I l I I I l l I
20 40 60 80 100 120 140 0
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 2 300—1 + xn _ xn—m _ xn—m—k + $n—2tm—k- (37)

applied to the smoothed signal shape. In the relationship above 92,, 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 (16). 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
1332 keV event from a 60C0 source, along with the same signal after a trapezoidal
shaper with integration time of 1 as and fiat-top duration of 0.5 as 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 as and flat-top duration 2 ,us, the energy resolution attained was

70

 

 

 

 

4000- -
3°
£3000- -]
a)
'0
3
52000 ..
Q.
E
<

1000 -

 

 

I n I l I I

0 1 2 3 4 5

 

 

 

Time (us)

Figure 3.26: The central contact signal corresponding to a 1332 keV event and a
trapezoidal shaper with integration time of 1 as and flat-top duration of 0.5 as 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 137Cs 661.62 keV line. A previous measurement of
the central contact energy resolution using a shaping amplifier and an ADC produced
a value of 2.95 keV for the 1332 keV 60Co 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 y-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]) have been measured for the neutron-
rich titanium isotopes 52Ti, 54Ti, and 56Ti 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 193/2, 101/2 and f5/2 neutron orbitals.
Doppler correction is essential in the determination of the de—excitation 'y-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.
_ $n+3 + $n+2 + $n+l + xn + 1171—1 + (En—2 + (En—3

yn — 7 . (A-l)

 

// 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_pWavefi-2] + m_pWaveEi-B];

74

m_pWaveEi] /= 7.;
}

// now set the first and last 3 values

m_pWave[O] 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).

 

-_ xn4—8xn_14-28$n_24—56xn_34-70$n_4+—56xn_5+-28zn_64—8$n_7+-xn_3

y" 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

Filter coefficient

80

70

60

50

40

30

20

10

 

 

     

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_pWavefi] = 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:

flmn+8mn1—8%4+102

 

11,-: 12

77

and the second derivative with:

—-, 1 -i —- 0.; 16.;_ -— ,_
yr: $+2+ 611M izx-+ x 1 x 2 (A4)

 

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_pWaveEi-2];

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

// zero the undefined points

m_pDeriv1[O] 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 (mi, 31,-) where :73,- is defined as:

1:,- = $0 + 2°61: (A-5)

where 6,, is the spacing between samples. The fit function is:

Y = a + bX (A.6)

The routine returns the (a, a“), (b, ab). With a extra pass it calculates X2-

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*(iNrSamp1es - 1)*iNrSamp1es +iNrSamples*iBegin;
//sum of x‘2
dSumX2 = (1./6.)*iNrSamp1es*(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_pWaveIi];
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/(iNrSamp1es - 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.chiZ = 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 (A7)

where glow is the waveform baseline and yh, 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_pWaveIi];
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_pWaveIi];
return (ThresholdPoint);
}
}
}
cerr << " ----------------------------------------------------- "
<< endl;
cerr << "ERROR -- The wave form never passed the set threshold."
<< "Recheck your numbers." << endl;

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

CGI‘I' << " _____________________________________________________ u

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—l + In — xn—m _ xn—m—k + xn—2tm—k- (A8)

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 e1em_n_m, elem n m k, e1em_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)

{
e1em_n_m = m_pWave[n-m] - TriggerPoint.y ;
}
else
{
e1em_n_m = 0.0;
}

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

{

elem_n_m_k - m_pWaveIn-m-k] - TriggerPoint.y;
}

else

{
0.0;

elem_n_m_k
}
// element x(n-2m-k)

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

{
e1em_n_2m_k = m_pWaveIn-2*m-k] - TriggerPoint.y;
}
else
{
e1em_n_2m_k = 0.0;
}

m_pTrapezoid[n] = m_pTrapezoid[n-I]

+ (m_pWave[n] — TriggerPoint y)

85

- elem_n_m - e1em_n_m_k + e1em_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_pWaveIn-m] ;
}
else
{
elem_n_m = 0.0;
}

// element x(n-m-k)

86

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

{

e1em_n_m_k

}

TriggerPoint.y — m_pWavefn-m-k];

else

{

e1em_n_m_k 0.0;
}

// element x(n-2m-k)

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

{

e1em_n_2m_k

}

TriggerPoint.y - m_pWavefn-2*m-k];

else

{

e1em_n_2m_k

}

0.0;

m_pTrapezoid[n] = m_pTrapezoid[n-i]
+ (TriggerPoint.y - m_pWave[n])

- elem_n_m - e1em_n_m_k + e1em_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 7' (preamplifier decay time constant). It is a four-pass filter with an inter-

mediary waveform (an).

a,, = 03,, +(1 — T):c,,
yn==an-an—R-an—uu¢n-tan—QR+D)
an : an—l + yn

yn : yn—l + an

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.9)
(A.10)
(A.11)

(A.12)

a = 1.0-dDecayTimeConst;

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

{

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

for (n = O; 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[O] 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, dSthev = 0.0, dVariance = 0.0;

90

double dSum =0.0, dSuquuares = 0.0;
struct StatCoefs Statistics;
unsigned int i;

unsigned int iIntervalSize;

iIntervalSize = iEndSample - iBeginSample;
for (i = iBeginSample; i < iEndSample; i++)
{
dSum += m_pWaveEi];
dSuquuares += m_pWave[i]*m_pWave[i];
}
dMean = dSum/iIntervalSize;
dVariance = (dSuquuares
- dSum*dSum/iIntervalSize)/(iIntervalSize - 1.);
dSthev = sqrt(dVariance);
Statistics.mean = dMean;
Statistics.stdev = dSthev;

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(doub1e 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_pWaveIi] +

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

dDecayFactorDiff * m_pTrapezoid[i-l];

}

// 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

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