REACTION MECHANISM DEPENDENCE OF THE POPULATION

AND DECAY OF 10HE

By

Han Liu

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Physics - Doctor of Philosophy

2019

ABSTRACT

REACTION MECHANISM DEPENDENCE OF THE POPULATION AND DECAY OF

10HE

By

Han Liu

The two-neutron unbound nucleus 10He was studied with a 44 MeV/u 11Li beam and a 47

MeV/u 13B beam. Neutrons were measured in coincidence with 8He fragments, and the two-

body and three-body decay energies were reconstructed using invariant mass spectroscopy.

Due to low statistics and large decay contributions from the population of 9He, energies of

resonant states in 10He could not be extracted from the 13B beam. The 11Li beam data could

be described with a correlated background model, implying that the measured spectra are

strongly influenced by the initial halo configuration of 11Li. In addition, from comparisons

with previous data a target dependence of the data is suggested.

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Chapter 2 Background and Motivation . . . . . . . . . . . . . . . . . . . . .
2.1 Previous Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11Li(−p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10Be(14C,14O)10He . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3H(8He,1H)10He
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14Be(−2p2n)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11Li(2H,3He)10He . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Theoretical Efforts
2.3 Summary of Previous Works . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1.1
2.1.2
2.1.3
2.1.4
2.1.5

Chapter 3 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Decay Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Decay Energy Line Shapes
3.2.1 Breit-Wigner Line Shape . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Three-Body Dynamical Model . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Correlated Background . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 4 Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . .
4.1 Beam Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Charged Particle Detection . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
4.2.1.1 Timing Scintillators
4.2.1.2 Cathode Readout Drifting Chambers . . . . . . . . . . . . .
4.2.1.3
. . . . . . . . . . . . . . . . . . . . . .
4.2.2 Neutron Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 DAQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Invariant Mass Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ionization Chamber

4.3

Chapter 5 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Calibrations and Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Timing Scintillators . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.2 Cathode Readout Drifting Chambers . . . . . . . . . . . . . . . . . .
5.1.3
Ionization Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.4 MoNA-LISA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Light Calibration . . . . . . . . . . . . . . . . . . . . . . . .

5.1.4.1

iii

3
3
3
6
8
10
10
11
16

18
18
19
19
22
24

29
29
30
31
31
33
34
35
37
38

41
41
41
43
49
50
50

5.1.4.2 Timing Calibration and Correction . . . . . . . . . . . . . .
5.1.4.3 Position Calibration . . . . . . . . . . . . . . . . . . . . . .
5.2 Event Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Beam Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Element Identification . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.3
Isotope Identification . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.4 Event Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.4.1 CRDC Gates . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.4.2 Neutron Gates
. . . . . . . . . . . . . . . . . . . . . . . . .
5.2.4.3 Beam Gates . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Classification of Two-Neutron Events . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
5.4 Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Neutron 4-Momentum Reconstruction . . . . . . . . . . . . . . . . .
5.4.2 Fragment 4-Momentum Reconstruction . . . . . . . . . . . . . . . . .
5.4.3 Decay Energy Reconstruction . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3.1 Causality Cuts
5.3.2 Machine Learning Methods

5.5 Simulations

52
54
55
56
57
58
61
61
62
62
63
63
67
68
68
68
71
74

Chapter 6 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . .
13B beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11Li Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.1
6.2

78
78
81

Chapter 7

Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . .

90

APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

iv

LIST OF TABLES

Table 2.1: Summary of model predictions for 10He. . . . . . . . . . . . . . . . . . . .

17

Table 5.1: CRDC pads that were not used in fit . . . . . . . . . . . . . . . . . . . . .

46

Table 5.2: CRDC slopes and offsets

. . . . . . . . . . . . . . . . . . . . . . . . . . .

48

Table 5.3: Coefficients used for isotope identification . . . . . . . . . . . . . . . . . .

60

Table 5.4: Parameters used for causality gates . . . . . . . . . . . . . . . . . . . . . .

65

Table 6.1: Summary of 10He experiments with halo beams.

. . . . . . . . . . . . . .

86

v

LIST OF FIGURES

Figure 2.1: First observation of a 10He resonance from Ref. [14]. Line I indicates the
best fit. All other lines are other contributions considered which did not
explain the data.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2.2: Separation energy for p(11Li,2p)10He reactions from Ref. [15]. Solid lines
. . . . . . . . . . . . . .

indicate background from C in the CH2 target.

Figure 2.3: Two invariant mass spectra for 10He from Ref. [16]. The left panel shows
the 1.42 MeV g.s. plus a correlated background. The right panel shows a
1.54 MeV g.s. plus a 3.99 MeV excited state.
. . . . . . . . . . . . . . .

Figure 2.4: 10Be(14C,14O)10He spectra from Ref.

[19]. The lower panel shows the
spectrum before background subtraction. The scale of the decay energy
is shown in the upper-left corner of the upper panel.
. . . . . . . . . . .

Figure 2.5:

(a) 3H(8He,1H)10He spectrum from Ref. [20]. The shade histogram shows
data. The solid black line shows the prediction from Ref.
(b)
3H(8He,1H)10He spectrum from Ref.
[21]. The crosses shows data with
statistical error bars. The shaded histogram shows the missing mass spec-
tra with a correlation gate which is described in Ref. [21].
. . . . . . . .

[22].

Figure 2.6:

Invariant mass spectrum of 10He from Ref. [23]. The solid circles represent
experimental data. The solid red line represents a simulated 1.6 MeV
state. The blue dotted line represents simulated non-resonant background.
The solid black line shows the sum of the two simulations.
. . . . . . . .

Figure 2.7: Missing mass spectra of 10He from Ref. [24]. The solid blue histogram
represents the missing mass spectrum in coincidence with 8He fragments.
The dashed red histogram represents the missing mass spectrum in coin-
cidence with 6He fragments
. . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2.8:

Invariant mass spectrum of 10He from Ref. [38]. The solid circles represent
data from 14Be(−2p2n). The solid red line represents the calculation from
[39]. The blue dotted line represents the calculations from from [39] folded
with the experimental response function. The causality cuts applied to
. . . . . . . . . . . . . . . .
this spectrum are discussed in Section 5.3.1.

Figure 3.1: Coordinate schemes for the sudden proton-removal calculations from Ref.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

[39].

4

5

6

7

9

11

12

15

22

vi

Figure 3.2: Coordinate scheme for one of the translation invariant Jacobi coordinates.
y, X, lx in Fig. 3.1, respec-
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Labels (cid:126)y3, ly, (cid:126)x3, lx are corresponding to Y(cid:48), l(cid:48)
tively.

Figure 3.3: Measured and predicted spectra of 10He from Ref. [39]. The line shape of
the correlated background is denoted as “11Li Fourier”. The prediction
of the three-body dynamical model is denoted as “Total 10He”.
. . . . .

25

27

Figure 4.1: Schematic of the experimental area of the NSCL.

. . . . . . . . . . . . .

30

Figure 4.2: Schematic of the experimental area. LISA was placed on two separate
tables with LISA 2 in the front table and LISA 1 in the back table. MoNA
was placed on one table behind LISA.
. . . . . . . . . . . . . . . . . . .

31

Figure 4.3: Schematic of the last thin scintillator viewed from upstream to downstream. 32

Figure 4.4: Schematic of a CRDC from Ref. [56]. The z-direction has been expanded.

34

Figure 4.5: Schematic of the ionization chamber from Ref. [57]. . . . . . . . . . . . .

35

Figure 4.6: Abbreviated schematic of the electronics and DAQ. The electronics and
DAQ of both MoNA and LISA are represented by the lable “MoNA”. . .

Figure 5.1: CRDC2 TAC versus sample width. (a) before the Nsw overflow fix; (b)
. . . . . . . . . . . . . . . . . . . . . . . . . .

after the Nsw overflow fix.

Figure 5.2: Raw (left) and calibrated (right) CRDC1 pad charge summary from a
. . . . . . . . . . . . . . . . . . . . . . . . . . . .

continuous sweep run.

Figure 5.3: Raw (left) and calibrated (right) CRDC2 pad charge summary from a
. . . . . . . . . . . . . . . . . . . . . . . . . . . .

continuous sweep run.

38

44

45

46

Figure 5.4: TAC versus CRDC2 X fit from a mask run.

. . . . . . . . . . . . . . . .

47

Figure 5.5: Calibrated CRDC2 Y versus CRDC2 X spectrum from a mask run.

. . .

48

Figure 5.6: Raw (left) and calibrated (right) ionization chamber pad energy loss sum-
. . . . . . . . . . . . . . . . . . . . .

mary from a continuous sweep run.

Figure 5.7: Original (left) and corrected (right) ionization chamber energy loss versus
. . . . . . . . . . . . . . . . . .

CRDC2 X from a continuous sweep run.

49

50

vii

Figure 5.8: Example light spectra. The left panel shows the raw spectrum, where the
peak of cosmic muon is at around channel 900. The right channel shows
the calibrated light spectrum where the cosmic-ray peak appears at about
20 MeVee. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

Figure 5.9: Corrected TOF and velocity spectra for MoNA-LISA. A coincidence with
the front layer is required. Events in the first sharp peak in time are
gamma-rays originating from the target; the peak was correctly placed at
30 cm/ns. The second sharp peak in time is due to gamma-rays produced
by the beam hitting the Sweeper chamber. The velocities of the sweeper
gamma-rays peak at less than the speed of light since in the calculation
the TOFs from the target to the sweeper chamber for the beam particles
were erroneously treated as part of the gamma-ray TOFs. The broad peak
corresponds to the neutrons. . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 5.10: Example time difference spectrum and X position spectrum for one MoNA-
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

LISA bar.

Figure 5.11: TOF for incoming beams from the A1900 scintillator to the target scintil-
lator. Events between solid red lines were selected as in coincidence with
interested beams. The left panel shows the 13B beam gate, and the right
panel shows the 11Li beam gate. . . . . . . . . . . . . . . . . . . . . . . .

53

54

56

Figure 5.12: Energy loss in the ionization chamber vs TOF from the target scintillator

to the thin scintillator for the 13B beam. Neutron coincidence was required. 57

Figure 5.13: Example S800 ionization chamber energy loss versus TOF used for par-
ticle identification. The fragmentation of an 85 MeV/u 36Ar beam was
simulated using LISE++ [62].
. . . . . . . . . . . . . . . . . . . . . . .

Figure 5.14: 3D correlation and 2D projection of He isotopes from the 13B beam. The
color in the right panel represents TOF (the z-axis in the left panel). In
the 3D plot, the cluster that is in between 60 ns and 75 ns, and away from
the other three bands, corresponds to He particles that hit the Sweeper
chamber. They were excluded from further analysis. . . . . . . . . . . . .

Figure 5.15: xtx versus TOF for He isotopes from the 13B beam. The most intense
. . . . . . . . . . . . . . . . . . . . . . . . . .

band is attributed to 6He.

Figure 5.16: xtx tof used for isotope identification in the 13B beam data set. . . . . .

Figure 5.17: CRDC Padsum versus X sigma for 8He fragments from the 13B beam.
CRDC quality gates are shown as red circles. Each gate selects approxi-
mately 80% of the events registered by a CRDC.
. . . . . . . . . . . . .

58

59

60

61

62

viii

Figure 5.18: MoNA-LISA light yield versus TOF for 8He fragments from the 13B beam
(left) and the 11Li beam (right). Events between the vertical lines and
above the horizontal line are selected as good neutron events. 8He frag-
ments originating from the 11Li beam (right panel) do not have a light
. . . . . . . . . . . . .
yield gate (thus the absence of a horizontal line).

Figure 5.19: Target scintillator light output versus TOF from the A1900 to the target
scintillator for 8He fragments using the 13B data from a beam down center
run (left) and a production run (right). Events between outside the red
lines were excluded from data. The vertical scales are different since the
voltage of the target scintillator PMT changed between the beam down
. . . . . . . . . . . . . . . . . . . . . . .
center and the production run.

63

64

Figure 5.20: Schematic of a 2-neutron event.

. . . . . . . . . . . . . . . . . . . . . . .

64

Figure 5.21: 2-neutron efficiency (dashed red) and gated 2-neutron efficiency (solid
blue). A simulation with phase space decay model and realistic beam
parameters for 11Li beam was used for estimating these efficiencies.
. . .

66

Figure 5.22: Cut efficiencies estimated from simulated 11Li beam data.

. . . . . . . .

67

Figure 5.23: Reconstructed kinetic energy for 13B (left) and 11Li (right) beam using
beam down center runs. Expected beam energies (47 MeV/u and 44
MeV/u, respectively) are well-reproduced.
. . . . . . . . . . . . . . . . .

Figure 5.24: Decay energy spectra from the 13B beam. The upper left shows the 8He
+ n two-body decay energy. The upper right shows the 8He + n two-body
decay energy with a neutron multiplicity=1 gate. The lower left shows
8He + 2n three-body decay energy. The lower right shows causality-gated
8He + 2n three-body decay energy.
. . . . . . . . . . . . . . . . . . . . .

Figure 5.25: Decay energy spectra from the 11Li beam. The left-upper shows the 8He
+ n two-body decay energy. The upper left shows the 8He + n two-
body decay energy. The upper right shows the 8He + n two-body decay
energy with a neutron multiplicity=1 gate. The lower left shows 8He +
2n three-body decay energy. The lower right shows causality-gated 8He
+ 2n three-body decay energy. . . . . . . . . . . . . . . . . . . . . . . . .

Figure 5.26: Comparison of simulated fragment to data for 8He from the 11Li beam. .

70

72

73

77

ix

Figure 6.1:

(a) Two-body decay energy spectrum. (b) Ungated three-body decay en-
ergy spectrum. (c) Decay energy spectrum gated on causality cuts. Data
are presented as crosses and solid circles. The black solid line shows the
sum of simulations. The purple dot-dash line is the thermal background.
The p-state and d-state in 9He are shown as the green solid and dark blue
solid lines, respectively. The light blue dot-dash line shows the state in
10He. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 6.2:

(b) Ungated three-body decay
(a) Two-body decay energy spectrum.
energy spectrum.
(c) Decay energy spectrum gated on causality cuts.
Data are presented as crosses and solid circles. The black line shows the
. . . . . . . . . . . . . .
simulation of the correlated background model.

Figure 6.3:

Invariant mass spectrum for 10He from Ref. [16]. The data are presented
as blue solid circles. The correlated background is shown as black solid
lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 6.4:

(b) Ungated three-body decay
(a) Two-body decay energy spectrum.
energy spectrum.
(c) Decay energy spectrum gated on causality cuts.
Data are presented as crosses and solid circles. The shaded area shows
the simulation of the correlated background model. The purple dot-dash
line represents the resonant state at 1.6 MeV. The resonant contribution
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
is fixed at 10%.

Figure 6.5: Decay energy spectra of 7He from the H target (left) and the C target
. . . . . . . . . . . . . . . . . . . . . . . . . . . .

(right) from Ref. [78].

Figure 6.6: Decay energy spectra on the C target (red) and the D target (black) for
the first observation in 10He. The D target spectrum is digitized from Ref.
[14]. The C target spectrum is deduced from the CD2 and the D target
spectra in Ref. [14], with estimated statistical error bars. . . . . . . . . .

79

82

83

85

86

87

Figure .1:

Screenshot of the TMVA GUI. . . . . . . . . . . . . . . . . . . . . . . . .

97

Figure .2:

Screenshot of GUI (1a), input variables.

. . . . . . . . . . . . . . . . . .

98

Figure .3:

Screenshot of GUI (4b), classifier output. . . . . . . . . . . . . . . . . . .

98

Figure .4:

Screenshot of GUI (5a), classifier cut efficiencies.

. . . . . . . . . . . . .

99

Figure .5:

Lines of the macro for setting variables.

. . . . . . . . . . . . . . . . . . 101

Figure .6:

Lines of the macro for cuts.

. . . . . . . . . . . . . . . . . . . . . . . . . 102

Figure .7:

Screenshot of GUI (1a), input variables with st mona. . . . . . . . . . . . 103

x

Figure .8:

Screenshot of GUI (4b), classifier output with st mona. . . . . . . . . . . 104

Figure .9:

Screenshot of GUI (5a) classifier, cut efficiencies with st mona. . . . . . . 105

Figure .10: Lines of the macro for setting the TMVA reader.

. . . . . . . . . . . . . 107

Figure .11: Typical BDT responses towards data. Gated on valid events. . . . . . . . 108

xi

Chapter 1

Introduction

Nuclear structure and nuclear reactions are two important subfields of modern nuclear

physics. While nuclear reactions study the process that changes nuclei from one kind to

another, the investigation of nuclear structure is focused on understanding properties of a

given nuclear species. One tremendous success in nuclear structure is the Shell Model. Dif-

ferent from its atomic counterpart in which electrons are attracted by the central charge, the

nucleonic shell model assumes nucleons feel a mean field caused by all constituent nucleons of

a nucleus. With a phenomenological Wood-Saxon potential including spin-orbit terms, the

Shell Model predicts states of a large range of isotopes with satisfactory accuracy. Notably,

the magic numbers, where atomic nuclei with those numbers of neutrons and protons are

more tightly bound than others, are reproduced by Shell Model predictions. Those numbers

are 2, 8, 20, 28 and 50 for the light nuclei.

However, as modern rare isotope facilities push nuclear structure studies to the nucleon

driplines, challenges have appeared. For example, the breakdown of shell closures [1, 2], the

emergence of the islands of inversion [3, 4] and the existence of halo nuclei as weakly-bound

systems [5] were observed near the neutron dripline. An island of inversion is a region in the

chart of nuclei where the ordering of energy levels of nuclei are different from the Shell Model

prediction, and a halo nucleus is an atomic nucleus with extended wave-functions of valence

neutrons or protons surrounding the core. These phenomena are enhanced as we move closer

1

to the dripline. Most extreme cases are expected beyond the neutron dripline, where nuclei

are unbound and exist as resonances. These resonances decay by emitting neutrons in an
extremely short time, on the order of 10−22 s. These nuclides are not only shorter-lived than

unstable bound nuclei which have lifetimes usually more than 1 ms, but also significantly

shorter-lived than excited bound states, since the lifetime of electromagnetic decays is usually
on the order of 10−15 s. 10−22 s is comparable to the timescale of nucleon motion within a

nucleus. With such short lifetimes, questions arise. Can we investigate those unbound nuclei

independently from reaction mechanisms? Do these nuclei have sufficient time to “forget”

how they were produced? And can we even call those unbound systems nuclei?

As studies of nuclear structure move further away from the driplines, more nuclei with

large widths and short lifetimes are expected to be encountered. One such example is 10He,

which has an expected lifetime that is so short that it is possible to probe the assumption that

the extracted nuclear structure is independent of the nuclear reaction. 10He is two-neutron

unbound and decays to the last bound helium isotope, 8He, plus two neutrons. It is nominally

a doubly magic nucleus since it consisted of 8 neutrons and 2 protons. The resonance energy

of 10He is controversial. However, the measured widths are on the magnitude of MeV,
corresponding to a half-life on the order of 10−22 s, making 10He an ideal test case for

investigating the interplay between nuclear structure and nuclear reactions.

2

Chapter 2

Background and Motivation

2.1 Previous Measurements

Since 10He is nominally doubly-magic, earlier experimental searches for 10He attempted to

find bound 10He fragments [6, 7, 8, 9, 10, 11, 12]. It was not until 1988 that the particle-

instability of 10He was widely accepted through the result of a projectile fragmentation

experiment with a 30 MeV/u 18O beam [13]. Later on, 10He resonances were reported from

eight experiments. In the following subsections, those experiments are discussed in detail,

grouped by production method.

2.1.1 11Li(−p)

Three out of the eight 10He measurements were conducted using one-proton knockout (sud-

den removal of one proton) reactions, with different beam energies and targets.

The first 11Li(−p) experiment, which was the first observation of a 10He resonance, was

performed at RIKEN (Rikagaku Kenkyujo, Wako, Japan) and published in 1994 [14], and

the measured spectrum can be seen in Fig. 2.1. In this experiment, a 61 MeV/u (MeV per

nucleon kinetic energy) 11Li secondary beam produced from a 18O primary beam impinged

on a 390 mg/cm2 CD2 target and a 280 mg/cm2 C target. The four-momenta of neutrons

and 8He fragments were measured, and the 8He + n + n invariant mass spectrum was

3

Figure 2.1: First observation of a 10He resonance from Ref. [14]. Line I indicates the best
fit. All other lines are other contributions considered which did not explain the data.

reconstructed. Background subtraction was achieved by subtracting scaled data obtained

with the C target from the data obtained from the CD2 target. The corresponding spectrum

is shown in Fig. 2.1 which exhibits a strong 1.2 MeV peak. The authors considered line

shapes calculated from multi-particle phase space and the 11Li fragmentation process, with

or without final state interactions (FSIs). None of these calculations could explain their

spectra. Therefore, they concluded that the observed 1.2 MeV peak could be a resonance in

10He. They performed Monte Carlo simulations, and extracted a decay energy for the 10He
ground state resonance, a 1.2 ± 0.3 MeV state above the 2n threshold with a width less than

1.2 MeV.

Another 11Li(−p) experiment was also performed at RIKEN in 1997 [15]. In this exper-

iment, a 83 MeV/u 11Li secondary beam was produced from a 15N primary beam, and a

200 mg/cm2 CH2 target was used to populate 10He via the 11Li(p,2p) reaction. The recoil

proton and the knocked out proton were detected in coincidence and the separation energy

4

Figure 2.2: Separation energy for p(11Li,2p)10He reactions from Ref. [15]. Solid lines indicate
background from C in the CH2 target.

spectrum of the removed proton was reconstructed from the four-momenta of the incident,

the scattered and the knocked-out proton in inverse kinematics (the projectile is heavier

than the target nucleus), as shown in Fig. 2.2. The authors deduced a decay energy of 10He
of 1.7 ± 0.3 (stat.) ± 0.3 (syst.) MeV from the measured separation energy spectrum and

the 11Li one-neutron separation energy. The authors indicated that the absolute value of

the decay energy is preliminary, and the width was not reported. The authors observed no

peak structures in the separation energy spectra in coincidence with 6He or 4He, suggesting

that the 10He resonance they observed does not have decay modes to 6He + n + n or 4He

+ n + n.

The third 11Li(−p) experiment was carried out at GSI (Gesellschaft fur Schwerionen-

forschung, Darmstadt, Germany). A 280 MeV/u secondary 11Li beam produced from a 18O

primary beam was used. The target in this experiment was a liquid hydrogen target with

a thickness of 350 mg/cm2. Neutrons and 8He fragments were measured, and the decay

energies of 10He were reconstructed using invariant mass spectroscopy. The results of this

experiment were reported in two papers [16, 17]. Instead of fitting the data with simula-

tions, the authors converted the observed decay energy spectra to the absolute decay energy

5

Figure 2.3: Two invariant mass spectra for 10He from Ref.
[16]. The left panel shows the
1.42 MeV g.s. plus a correlated background. The right panel shows a 1.54 MeV g.s. plus a
3.99 MeV excited state.

differential cross section and then fit the differential cross section with theoretical distribu-

tions. In the first paper [16], they explained the measurement with two possibilities without

preferences. Their spectrum could be fit by a 1.42(10) MeV ground state resonance on top

of a correlated background [18], which originated from the neutron halo wave-function of the

11Li beam, or with a 1.54(11) MeV ground state plus a 3.99(26) MeV excited state, as shown

in Fig. 2.3. In the later paper [17], however, Jacobi coordinates were analyzed in addition

to decay energy spectra. The authors concluded that since the correlated background did

not fit Jacobi coordinates, only the interpretation involving the a 1.54(11) MeV ground state

plus a 3.99(26) MeV excited state was plausible.

2.1.2 10Be(14C,14O)10He

Shortly after the first 10He observation, the analysis of a double charge-exchange reaction

10Be(14C,14O)10He was reported [19]. This experiment was performed at HMI (Hahn-

6

Figure 2.4: 10Be(14C,14O)10He spectra from Ref. [19]. The lower panel shows the spectrum
before background subtraction. The scale of the decay energy is shown in the upper-left
corner of the upper panel.

7

Meitner Institut, Berlin, Germany) with a 14C beam with energy of Elab = 334.4 MeV.

The target consisted of a 4.9 mg/cm2 Pt-backing, a 600 µg/cm2 BeO layer (230 µg/cm2 Be,

with 94% enrichment 10Be) and a 400 µg/cm2 Au cover. A 600 µg/cm2 V2O5 target and a

500 µg/cm carbon target were also used for background estimation. The spectra before and

after background subtraction are shown in Fig. 2.4. A peak at 1.07(7) MeV was identified

as a ground state in the missing mass spectrum with 28 counts, with a 80% confidence level.

The width of the ground state was reported to be 0.3 MeV. In addition to the ground state,

3.23(20) MeV and 6.80(7) MeV excited states were also reported. A mass excess M.E. =

48.81(7) MeV was deduced from the measured Q-value.

2.1.3 3H(8He,1H)10He

Two subsequent 3H(8He,1H)10He experiments were performed at JINR (Joint Institute for

Nuclear Research, Dubna, Russia) [20, 21] and the missing mass spectra of the two exper-

iments are shown in Fig. 2.5a and Fig. 2.5b, respectively. Gaseous tritium targets were

used for transferring two neutrons to the 8He beam, and empty target chambers were used

for background estimation. In both experiments, the momenta of the recoil protons were

measured in order to derive the missing mass spectra.

In the first experiment [20], a 34 MeV/u 11B primary beam produced a 27.4 MeV/u 8He

secondary beam. The gaseous tritium target was operated at a temperature of 28 K. The

statistics of this experiment were limited. No events with decay energy lower than 2.5 MeV

were observed. 10 events were distributed between 2.5 MeV to 5.5 MeV, and the authors
of this experiment identified those events as a resonant state at ∼ 3 MeV. They made cross

section estimations and claimed that if a ground state below 2.5 MeV exists, they expect 8
counts, and the probability of a non-observation is less than e−8. The authors suggested the

8

(a)

(b)

Figure 2.5: (a) 3H(8He,1H)10He spectrum from Ref. [20]. The shade histogram shows data.
The solid black line shows the prediction from Ref. [22]. (b) 3H(8He,1H)10He spectrum from
Ref. [21]. The crosses shows data with statistical error bars. The shaded histogram shows
the missing mass spectra with a correlation gate which is described in Ref. [21].

∼ 3 MeV state to be the ground state of 10He and it is the same state measured by previous

11Li experiments [14, 15] which concluded the ground state is below 1.7 MeV. They argued

the extended wave-functions of the halo neutrons in the 11Li beam shifts the peak of the

observed decay energy down, according to reference [22]. This effect is discussed in Section

2.2 in detail.

The following refined 3H(8He,1H)10He experiment [21] produced a 21.5 MeV/u 8He sec-

ondary beam from a 36 MeV/u 11B primary beam. A tritium target running at a temperature

of 26 K was used. Statistics were improved and angular correlations were reconstructed in
this experiment. The authors claimed to observe a broad ground state at 2.1 ± 0.2 MeV
with spin assignment 0+, a 1− excited state at a maximum energy between 4 - 6 MeV, and

a 2+ state above 6 MeV. They argued that their result agreed with previous experiments

and the same 11Li initial state effect was used to explain why their new measured state is

higher than of the 11Li experiments.

9

2.1.4 14Be(−2p2n)

The first 10He experiment performed at NSCL (National Superconducting Cyclotron Lab-

oratory, East Lansing, USA) used a 14Be 2n2p-removal reaction [23]. The experimental

setup, data analysis, and simulations of this experiment were similar to the current exper-

iment, which is described in the following chapters in detail. A 59 MeV/u 11Li secondary

beam was produced from a 120 MeV/u 18O primary beam, and a 435 mg/cm2 deuterated

carbon target was used as the reaction target. The 8He fragment and neutrons from 10He

decays were detected and the invariant mass spectrum of 10He was reconstructed from these

decay products. The decay energy spectrum is shown in Fig. 2.6.The authors evaluated

an energy for the ground state of 1.60(25) MeV above the 8He + n + n decay threshold
with a width of 1.8(4) MeV. This ground state energy agreed with previous 11Li(−p) ex-

periments, although it disagreed with 3H(8He,1H)10He. The authors of this analysis argued

the initial state effect cannot explain the discrepancy, because despite the halo structure of

14Be, the dispersive 2p2n-removal reaction should disturb the halo neutrons and eliminate

the initial state effect. The authors argued that because the decay energy measured from
14Be(−2p2n) experiment disagreed with the 3H(8He,1H)10He experiments, it disproved the

theory of initial state effects in reference [22].

2.1.5 11Li(2H,3He)10He

The most recent 10He measurement was carried out at RIKEN with the 11Li(2H,3He)10He

reaction [24] and the measured missing mass spectrum is shown in Fig. 2.7. A 50 MeV/u

11Li beam was produced from a 100 MeV/u 18O beam. A 1.9 mg/cm2 CD2 target was used

for the proposed reaction and a 1 mg/cm2 natural carbon target was used for background

10

Figure 2.6: Invariant mass spectrum of 10He from Ref.
[23]. The solid circles represent
experimental data. The solid red line represents a simulated 1.6 MeV state. The blue dotted
line represents simulated non-resonant background. The solid black line shows the sum of
the two simulations.

subtraction. 3He recoils were measured to reconstruct the missing mass spectra of 10He. 8He,

6He, and 4He from 10He decays were detected to determine the final states of 10He decays.

The measured spectra were fit by Breit-Wigner distributions, convoluted with Gaussian

resolution functions. The authors of the article concluded the ground state of 10He to be

1.4(3) MeV with a 1.4(2) MeV width. One excited state was reported to be at 6.3(7) MeV

with a 3.2 MeV width. The authors also determined the 6He + 4n decay channel to be

stronger than the 8He + n + n channel. Decays to 4He + 6n were not observed.

2.2 Theoretical Efforts

There is also a debate about 10He among theorists. After it was generally accepted that 10He

is unbound, theoretical works regarding the structure of 10He were focused on calculating

11

Figure 2.7: Missing mass spectra of 10He from Ref. [24]. The solid blue histogram represents
the missing mass spectrum in coincidence with 8He fragments. The dashed red histogram
represents the missing mass spectrum in coincidence with 6He fragments

12

resonances in 10He. Korsheninnikov et al. predicted with hyperspherical calculations that

10He exists as a narrow three-body resonance with a resonance energy less than 1 MeV, and

a width of 150 - 300 keV [25], before Korsheninnikov et al. reported the first observation of

10He [14]. Shortly after the discovery of 10He, Kato et al. calculated 10He with three-body

models using basis functions of the cluster orbital shell model (COSM) and the complex

scaling method (CSM) [26]. They concluded that the 10He ground state is at 2.14 MeV with

a width of 1.63 MeV and an excited state at 5 MeV. Aoyama and Kato, et al. reported

with the complex scaling method that the 10He ground state resonance is at 1.8 MeV with

a 1.4 MeV width and the two valence neutrons were considered to occupy the p1/2 orbital

[27]. Later, however, Aoyama, Kato, et al. reported with both the complex scaling method

and the analytical continuation in the coupling constant (ACCC) method that the earlier

calculated [p1/2p1/2]0+ is not the ground state of 10He, but that the ground state has a

[s1/2s1/2]0+ configuration near 0 MeV, which had not been observed so far [28]. Aoyama

repeated similar ACCC calculations and reiterated that the ground state of 10He is a three-

body s-wave resonance ([s1/2s1/2]0+) with a decay energy smaller than 0.05 MeV, and that

this state had not been observed [29, 30, 31]. Kamada et al, calculated the energy of ground

0+ state at 0.803 MeV with a core-excitation three-body model [32]. The most recent 10He

calculation reported by Fossez et al. was the only structure calculation using a many-body

model [33]. The authors predicted a narrow double-halo 10He ground state very close to

threshold, and the predicted configuration was almost pure s-wave.

Rather than predicting states in 10He, theoretical efforts were also specifically made to

reconcile the conflicting results of 10He experimental results, i.e.

the discrepancy in the

g.s. energy between the 3H(8He,1H)10He experiments with other measurements. Fortune

suggested that the “ground state” measured so far might be two overlapping 0+ states [34].

13

He argued that the relative population of the two 0+ states changes with the reaction types

so different experiments observed different energies of the “ground state”. He calculated

the relative ratio between the two states using simple reaction models [35]. Fortune then

extracted that the p-shell 0+ ground state is lower than 1.4 MeV and the sd-shell 0+ excited

state is higher than 2.1 MeV [36]. In a later refined work, Fortune indicated that the 1.07(7)

MeV resonance measured by the experiment described in Section 2.1.2 is the ground state,

and the first excited state is in the region between 2.1 to 3.1 MeV [37].

On the other hand, Grigorenko and Zhukov argued the states observed in the 3H(8He,1H)10He

transfer experiment might be the same states measured in 11Li knockout experiment, if

the reaction dynamics were considered [22]. Their calculation suggests the extended wave-

functions of the two halo neutrons in 11Li beam “shifts” the observed decay energies down.

The authors also indicated that for their explanation be correct, 9He cannot have a virtual

state, and they were unable to make 9He and 10He data consistent with their model. Grig-

orenko and Zhukov also predicted an s-state below 0.3 MeV according to the same theory

[22]. Fortune commented that such an s-state would locate the mixed ground state below the

two-neutron separation threshold [35, 34]. As discussed earlier in section 2.1.4, the result of
the 14Be(−2p2n) experiment did not favor the theory proposed by Grigorenko and Zhukov.
A later theoretical prediction [39] indicated the result from the 14Be(−2p2n) experiment

might still agree with 3H(8He,1H)10He experiments if the 2p2n-removal reaction were con-

sidered as an α removal. If that assumption were true, a similar initial state effect existed,
therefore the “shifted” down 14Be(−α) decay energy still agrees with the 3H(8He,1H)10He

transfer experiments. In Ref.

[39], Sharov et al. also extended the calculations performed

by Grigorenko and Zhukov [22], and claimed that the ground state of 10He observed from
a 11Li beam might be a superposition of 1−, 0+ and 2+ excitations, and the 1− excita-

14

Figure 2.8: Invariant mass spectrum of 10He from Ref.
[38]. The solid circles represent
data from 14Be(−2p2n). The solid red line represents the calculation from [39]. The blue
dotted line represents the calculations from from [39] folded with the experimental response
function. The causality cuts applied to this spectrum are discussed in Section 5.3.1.

15

tion is actually the lowest excitation. However, Sharov et al. used the three-cluster model

[22], and they required the scattering length of 9He to be
and potential developed in Ref.
positive, which is not supported by experiments. The authors of the 14Be(−2p2n) paper

responded in Ref.

[38]. The response pointed out that the authors of Ref.

[39] compared

pure theoretical line shapes with experimental spectra which were folded with the experi-

mental response function, as shown in Fig. 2.8. The authors of the follow-up paper showed
that after the experimental conditions were considered, the 14Be(−α) calculation did not fit

data, and concluded no evidence for initial state effects were found.

2.3 Summary of Previous Works

Previous studies of 10He are conflicted in both experiment and theory. The effect of extended

wave-functions of the halo neutrons in incoming beams were proposed to reconcile conflicting

experimental measurements. Although the quantitative conclusions have been disproven by

multiple theoretical and experimental analyses, the initial state effect itself remains intrigu-

ing. Measurements with other initial states are necessary to expand the understanding of

10He and the initial state effect. Therefore, the current experiment was proposed to study

10He using a compact non-halo 13B beam, and a halo 11Li beam.

16

Table 2.1: Summary of model predictions for 10He.

Model Type Assumption(s)

Main Prediction(s)

Three-body Hyperspherical
harmonic bases

Eg.s. <1 MeV,
150 keV<Γg.s<300 keV

Three-body COSM bases
CSM method

Three-body COSM bases
CSM method

Three-body COSM bases

ACCC method

Three-body COSM bases

ACCC method

Three-body COSM bases

ACCC method

Three-body Core-excitation,

Three-body
Dynamical

Three-body
Dynamical

AGS bases

10He source
wave-functions from
11Li; sudden removal

10He source
wave-functions from
11Li; sudden removal

Many-body α-core, single particle

Berggren bases

Eg.s. = 2.14 MeV, Γg.s = 1.63 MeV;
Eexc. ∼ 5 MeV
Eg.s. = 1.8 MeV, Γg.s = 1.4 MeV,
p-wave
Eg.s. ∼ 0 MeV, s-wave
Eexc. = 1.68 MeV, p-wave
Eg.s. ∼ 0 MeV, s-wave

Eg.s. ∼ 0.05 MeV, Γg.s = 0.21 MeV,
s-wave

Eg.s. = 0.803 MeV, Γg.s = 0.67 MeV, 0+;
Eexc. = 1.25 MeV, Γexc. = 0.21 MeV, 1−
Eg.s. ≥ 2 MeV, p-wave; the observed peak
shifted to lower energies with halo sources;
three-body virtual s-state less than 0 MeV

2.0 MeV < Eg.s. < 2.3 MeV, p-wave;
the observed peak shifted to lower energies
with halo sources;
Eg.s. ∼ 0 MeV, s-wave, “double halo”
structure

Ref.

[25]

[26]

[27]

[29]

[28]
[30]

[31]

[32]

[22]

[39]

[33]

17

Chapter 3

Theoretical Background

This chapter will discuss the decay models and line shapes used in the simulations. The

full derivation is beyond the scope of this chapter. Therefore, only the framework will be

provided. For more details, the readers are referred to the original works.

3.1 Decay Models

Given a decay energy, the decay model for two-body decays is straightforward since kine-

matics are determined by energy and momentum conservation. The two decay products are

distributed uniformly in their center-of-mass frame.

Three-body decays or n-body decays (n>3), however, cannot be easily determined by the

conservation laws because there are additional degrees of freedom in the final state. Decay

models are needed to decide the angles of the decay products or the energy partition among

different two-body pairs. For example, in the phase space decay model decay products uni-

formly fill the phase space in the center-of-momentum frame. In this model, the kinematics

are governed by the n-body phase integral, defined as [40]

(cid:90)

Rn =

δ4(P0 − n(cid:88)

Pi)

n(cid:89)

δ(P 2

j − m2

j )d4Pj

(3.1)

i=1

j=1

where P0 is the 4-momentum of the unbound nucleus, Pi and Pj are the 4-momenta of the

18

decay products, mi and mj are the masses of the decay products, and the statistical factor,

δ(P 2

j − m2

j )d4Pj, in the spherical coordinates is written as

δ(P 2

j − m2

j )d4Pj =

d|(cid:126)pi|d cos (θi)dφi,

(cid:126)pi
Ei

(3.2)

where (cid:126)pi is the three momentum. Then, distributions of any kinematic parameters, such as

momenta or angles, are given by

σ(α) =

dRn
dα

.

(3.3)

The full description of the phase space model is given in Ref. [40].

3.2 Decay Energy Line Shapes

3.2.1 Breit-Wigner Line Shape

When the final state interaction is not influenced by the reaction mechanism, the process

of a 1-neutron decay can be treated like a neutron scattering off the residual fragment. For

this problem, R-matrix phenomenology can be used to derive Breit-Wigner distributions

[41] which are widely used for the description of resonant states. This section provides a

brief summary of the derivation given in Thompson and Nunes [42]. More details about

R-matirx theory can also be found in the work of Lane and Thomas [43]. For 2-neutron

decays, such as 10He, the two neutrons can be thought of as coupling to the same orbital.

Then a Breit-Wigner line shape can be used to describe the decay of 10He.

The decay of a resonance involves entrance and exit channels. Therefore, multi-channel

19

R matrices are considered and they can be written as [42, p. 296]

P(cid:88)

p=1

Rα(cid:48)α(E) =

γpαγpα(cid:48)
ep − E

(3.4)

where α(cid:48) and α represent the entrance and the exit channels, respectively, ep is a pole in the

R matrix, γ is a reduced width, and E is essentially the decay energy. Then, the scattering

S matrix can be written in terms of the R matrix:

(cid:16)

t1/2H+(cid:17) 1 − aR(cid:0)H−(cid:48)/H− − β(cid:1)

1 − aR (H+(cid:48)/H+ − β)

S =

(3.5)

where H± has only diagonal elements H±
α = Gα ± iFα, where Gα and Fα are Coulomb
functions [42, p. 61]. The t matrix also has non-zero elements on the diagonal as tα ≡ ¯h2/2µα,

where µα is the reduced mass. Here β is the logarithmic derivative at an arbitrary radius a

where the nuclear interaction is negligible. A ‘logarithmic’ L matrix can be defined as [42,

p. 306]

L = H+(cid:48)

/H+ − β =

(S + iP − aβ)

1
a

(3.6)

where the penetrability P and shift function S are diagonal matrices and their matrix ele-

ments are

,

Pα =

kαa
α + G2
F 2
α

(cid:16) ˙FαFα + ˙GαGα

(cid:17)

Pα.

(3.7)

(3.8)

Sα =

Here dots represent derivatives with respect to ρ = kR, where k is the quantum mechanical

20

wave number and R is the radial coordinate. Then the S matrix can be written as

S = Ω

1√
tH−H+

1 − aRL∗
1 − aRL

tH−H+Ω,

(cid:112)

(3.9)

where Ω is defined as a diagonal matrix with elements Ωα = eiφα, with Ωα a hard-sphere

phase shift.

The transformation v1/2Sv−1/2 is used to construct a symmetric matrix(cid:101)S ≡ v1/2Sv−1/2,

(cid:104)

1 + 2iP 1/2(1 − aRL)−1RP 1/2(cid:105)

(cid:101)S = Ω

Ω.

(3.10)

Suppose there are two channels and one pole. Then, S simplifies to:



˜S12 = eiφ1

ep − E − γ2

1 (S1 − aβ) − iγ2

2iP

1/2

α γαγα(cid:48)P

1/2

α(cid:48)
1P1 − γ2

2 (S2 − aβ) − iγ2

2P2

 eiφ2.

(3.11)

To organize Equation 3.11, the formal width Γα, the energy shift ∆a, the total energy

shift ∆T , and the total formal width ΓT are defined as

Γα = 2γ2
∆α = −γ2

αPα
α(Sα − aβ)
(cid:88)
∆α = −γ2
(cid:88)

α

∆T =

ΓT =

1 − γ2

1S0

2S0
2

= 2γ2

1P1 + 2γ2

2P2,

α

where aβ can be set to any constant.
It is suggested in Ref. [42, p. 299] to set aβ so
Sα − aβ = 0 at the pole, known as the natural boundary condition. Then, the cross section

21

Figure 3.1: Coordinate schemes for the sudden proton-removal calculations from Ref. [39].

is

σ12 ∝(cid:12)(cid:12)(cid:12)(cid:101)Sαα(cid:48)

(cid:12)(cid:12)(cid:12)2
(cid:0)E − ep + ∆T

Γ2

=

(cid:1)2 + Γ2

T /4

.

(3.12)

Since decay through the entrance channel is less likely, Γ2 >> Γ1, approximations can be
made such as ΓT ∼ Γ2 and ∆T ∼ ∆2. Then, with the natural boundary condition, the line

shape for the neutron decay is:

(cid:0)E; ep, Γ0

(cid:1) ∝

σl

(cid:2)ep − E + ∆l

where

Γ0 = 2γ2Pl

Γl

(cid:0)E; ep; Γ0
(cid:1)
(cid:0)E; ep, Γ0
(cid:1)(cid:3)2 + 1
(cid:1) .
(cid:0)ep

4

(cid:2)Γl

(cid:0)E; ep, Γ0

(cid:1)(cid:3)2

(3.13)

(3.14)

3.2.2 Three-Body Dynamical Model

Predictions including reaction dynamics from Refs.

[22, 39] were used as the line shapes

for the 10He three-body decay energy where the state of the daughter nuclei are affected by

their halo (11Li) parents. These predictions are based on the sudden removal of a proton

from 11Li, and an outline of the original works is provided below.

22

The model starts by constructing the 11Li cluster wave-function in the form

(cid:0)X, Y(cid:48), r(cid:48)

(cid:1) =

p

(cid:104)

Ψ

(3b)
11Li

(cid:0)X, Y(cid:48)(cid:1)(cid:79)

(cid:1)(cid:105)

(cid:0)r(cid:48)

p

Ψ9Li

Ψ

JiMi
11Li

where the coordinates are defined in Fig. 3.1, Ψ

,

(3.15)

JiMi

(cid:0)X, Y(cid:48)(cid:1) is the three-body cluster wave-
(cid:1) is the 8Li-p single particle wave-function. The readers are

(3b)
11Li

(cid:0)r(cid:48)

function of 11Li, and Ψ9Li
referred to Refs. [22, 39] for more details about how this 11Li cluster wave-function was

p

numerically constructed.

The reaction populating 10He is modeled as the sudden removal of a proton from the

9Li core and a momentum transfer to the remaining 8He cluster. To describe that process,
a Raynal-Revai transformation [44] is performed taking {Y(cid:48), r(cid:48)
p} to {Y, rp}, which connects
the proton coordinate. Then, the 10He source wave-function is obtained by applying the

annihilation operator

(cid:90)

(cid:0)X, Y, rp

(cid:1) ,

d3rpeiqrpΨ11Li

Φq(X, Y) =

(3.16)

where for different J π the source wave-functions are given by the angular momentum de-

composition

ΦJM

q,γ,lp

(X, Y ) =

(cid:90)
×(cid:104)(cid:104)(cid:104)

(cid:105)
dΩxdΩydΩqΦq(X, Y)
Ylx( ˆX) ⊗ Yly ( ˆY )

(cid:105)

J

(cid:105)
⊗ Ylp(ˆq)

,

JM

⊗ χS

(3.17)

L

where γ is a multi-index (γ = {LSlxly}) that defines the complete set of angular momentum

quantum numbers.

The 10He source wave-function is used for solving the inhomogeneous Schrodinger equa-

23

tion

(cid:16) ˆH3 − Edecay

(cid:17)

Ψ

JM (+)
Edecay

(X, Y ) = ΦJM

q

(X, Y ),

(3.18)

where Edecay is the decay energy and Ψ

JM (+)
Edecay

(X, Y ) is the outgoing 10He wave-function.

Then, the decay energy line shape predicted by this model is proportional to the flux with

the outgoing asymptotic

∼ j(cid:0)Edecay

(cid:1) =

dσ

dEdecay

(cid:90)

1
M

Im

dΩ5Ψ

(+)†
ET

ρ5/2 d
dρ

ρ5/2 Ψ

(+)
ET

(cid:12)(cid:12)(cid:12)ρmax

(3.19)

3.2.3 Correlated Background

In another model, in which 10He is even more influenced by the incoming halo beams, the

decay is described as a correlated background [18, 45]. In this model, the 10He system is the

remnant of the 11Li halo neutrons without any final state interactions, and mathematically

the decay energy distributions are the Fourier transformations of the wave-function of the

11Li halo neutrons. The correlated background was originally developed to describe the

break up of halo beams. Then the use was extended to unbound nuclei with the assumption

that the center-of-mass of the fragment in the unbound three-body system coincides with

the center-of-mass of the core in the halo beam. This section summarizes the main results of

the correlated background based on the original work of Forssen et al. [18]. The readers are

also referred to the appendix of Ref. [45] for an advanced treatment of the function χ

lxly
KL (ρ)

which is discussed below.

24

Figure 3.2: Coordinate scheme for one of the translation invariant Jacobi coordinates. Labels
(cid:126)y3, ly, (cid:126)x3, lx are corresponding to Y(cid:48), l(cid:48)

y, X, lx in Fig. 3.1, respectively.

Translation invariant Jacobi coordinates are defined first as

(cid:113)
(cid:113)
Aij ((cid:126)ri − (cid:126)rj)

(cid:18) Ai(cid:126)ri + Aj(cid:126)rj

A(ij)l

Ai + Aj

(cid:126)xl =

(cid:126)yl =

(cid:19)

,

− (cid:126)rl

(3.20)

where l ∈ (1, 2, 3), (cid:126)ri is the Cartesian coordinates of the i-th particle, Ai is the mass number

of the i-th particle ,Aij is the reduced mass number of the particle pair ij, and A(ij)l =

(cid:1). One set of the Jacobi coordinates is shown in Fig. 3.2. Note

(Ai+ Aj)Al/(cid:0)Ai + Aj + Al

this particular set is similar to coordinates used in the previous subsection.

A three-body wave-function can be expanded in hyperspherical harmonics [46] without

spin and isospin

Ψ ((cid:126)xl, (cid:126)yl) = ρ−5/2 (cid:88)

KLlxly

χ

lxly
KL (ρ)Γ

lxly
KL

(3.21)

where K is the extra quantum number hypermomentum defined as K = lx + ly + 2µ (µ =

0, 1, 2, . . .), ρ is the hyperradius defined as ρ = (x2

harmonics basis,(cid:0)Ωρ

(cid:1) ≡(cid:8)θρ, ˆxl, ˆyl

5

(cid:1) is the hyperspherical
(cid:9) represents the five angles parameterizing a hypersphere

l )(1/2), Γ

l +y2

lxly
KL

5

5

(cid:1) ,
(cid:0)Ωρ
(cid:0)Ωρ

25

y3ly→x3→lxcore1n32where Jacobi momenta are defined as

(cid:32) (cid:126)ki
(cid:33)
(cid:32) (cid:126)ki + (cid:126)kj

− (cid:126)kj
Aj

Ai

Ai + Aj

Aij

A(ij)l

(cid:33)

,

− (cid:126)kl
Al

(3.24)

(cid:90)

(cid:113)
(cid:113)

(cid:126)ql =

(cid:126)pl =

(cid:32) (cid:126)k2

1
A1

with a hyperangle θρ ≡ arctan (xl/yl) and angular coordinates {ˆxl, ˆyl} corresponding to (cid:126)xl

and (cid:126)y, and the asymptotics of χ

lxly
KL (ρ) is

lxly

KL (ρ) ∼ exp (−κ0ρ) .

χ

(3.22)

Ψ ((cid:126)xl, (cid:126)yl) can be expressed in momentum representation as

Ψ ((cid:126)ql, (cid:126)pl) =

1

(2π)3

Ψ ((cid:126)xl, (cid:126)yl) exp [i ((cid:126)ql · (cid:126)xl + (cid:126)pl · (cid:126)yl)] d(cid:126)xld(cid:126)yl

(3.23)

where ki is the momentum of the i-th particle. A variable κ2 = (cid:126)q2

l + (cid:126)p2

l can be defined so

that

E =

¯h2
2m

(cid:33)

(cid:16)

(cid:17)

+

(cid:126)k2
2
A2

+

(cid:126)k2
3
A3

=

¯h2
2m

(cid:126)q2
l + (cid:126)p2
l

=

¯h2
2m

κ2,

(3.25)

where m is the nucleon mass, E is the internal energy of the beam particles as well as the
decay energy of unbound systems, and the relation dE ∝ κdκ can be obtained.

Substituting Equation 3.21 into Equation 3.23, the wave-function in momentum space is

26

Figure 3.3: Measured and predicted spectra of 10He from Ref. [39]. The line shape of
the correlated background is denoted as “11Li Fourier”. The prediction of the three-body
dynamical model is denoted as “Total 10He”.

obtained

Ψ ((cid:126)ql, (cid:126)pl) =

=

χ

lxly
KL (κ) =

(cid:90)

(2π)3

(cid:88)
(cid:90) ∞

KLlxly
iK
κ2

0

1

exp [i ((cid:126)ql · (cid:126)xl + (cid:126)pl · (cid:126)yl)] Ψ ((cid:126)xl, (cid:126)yl) d(cid:126)xld(cid:126)yl

χlxly
KL (κ)Γ

lxly
KL (Ωκ
5 )

(3.26)

χlxly
KL (ρ)JK+2(κρ)ρ1/2dρ

For K = 0, the hyperspherical harmonic is a constant, and the wave-function of 11Li has a

simple form,

Ψ ((cid:126)xl, (cid:126)yl) ∝ exp (−κ0ρ)

ρ5/2

.

(3.27)

Here κ0 can be related to the binding energy Eb by κ2

0 = 2mEb/¯h2. Substituting Equation

3.27 into Equation 3.26 and performing the radial integral gives

(cid:32)

Ψ ((cid:126)ql, (cid:126)pl) ∝

(cid:0)κ2
0 + κ2(cid:1)7/4

1

F

7
4

,

3
4

, 3,

κ2
κ2
0 + κ2

(cid:33)

,

(3.28)

where the function F is the standard hypergeometrical function [47]. Then the momentum

27

distributions are given by:

d6N
d(cid:126)qld(cid:126)pl

∝ |Ψ ((cid:126)ql, (cid:126)pl)|2 ∝

(cid:32)

0 + κ2(cid:1)7/2
(cid:0)κ2

1

F 2

7
4

,

3
4

, 3,

κ2
κ2
0 + κ2

(cid:33)

(3.29)

Combining the phase space factor d(cid:126)qld(cid:126)pl = κ5dκdθκdΩqdΩp and the relation dE ∝ κdκ
from Equation 3.25, the decay energy distribution of a 11Li correlated background is

dN

dEdecay

∝

E2

(cid:0)Eb + Edecay

decay

(cid:1)7/2

F 2

(cid:32)

(cid:33)

7
4

,

3
4

, 3,

Edecay

Eb + Edecay

.

(3.30)

The line shapes predicted from the correlated background model and the three-body

dynamical model are shown in Fig. 3.3.

28

Chapter 4

Experimental Techniques

4.1 Beam Production

The experiment was carried out at the National Superconducting Cyclotron Laboratory

(NSCL) at Michigan State University in August 2015. A non-halo 13B beam and a halo 11Li

beam were used for populating 10He or 9He systems. Both 13B and 11Li are unstable against

β decay with half-lives of 17.33(17) ms and 8.75(14) ms [48], respectively. They cannot be

directly accelerated due to these short half-lives. Therefore, the fast fragmentation method

[49] was used to produce these short lived species. The half-lives are long enough for them

to be delivered from the fragment separator to the experimental vault.

A stable 18O primary beam was accelerated to 120 MeV/u with the coupled K500 and

K1200 cyclotrons at NSCL [50]. Targets of 3196 mg/cm2 9Be and 2609 mg/cm2 9Be were

then bombarded by the primary beam for the production of 11Li and 13B, respectively. The

thicknesses were different for optimizing the yields of the secondary beams. The fragmenta-

tion reactions produce a large variety of nuclei from which 11Li or 13B were selected by the

A1900 Fragment Separator [51] based on the magnetic rigidities (Bρ = p/q, where p is the

momentum and q is the charge). A 1050 mg/cm2 aluminum wedge was additionally inserted

after the second dipole for better separation. The momentum slit for 13B was set to 0.5 % to

improve purity but for the more exotic 11Li it was set to 2 % to increase intensity. The 11Li

29

Figure 4.1: Schematic of the experimental area of the NSCL.

secondary beam was delivered to the experimental vault with a magnetic rigidity 3.55763

Tm, corresponding to 44.03 MeV/u. The 13B beam was delivered to the experimental vault

with a magnetic rigidity 2.60827 Tm, corresponding to 47.24 MeV/u. The layout of the

NSCL experimental area is shown in Fig. 4.1.

4.2 Experimental Setup

After the A1900 Fragment Separator, the secondary beams entered the experimental area,

which is shown as Fig. 4.2. The beams were focused by a quadrupole triplet magnet and

then impinged on a 405 mg/cm2 thick 9Be reaction target. After leaving the reaction target,

charged reaction fragments and unreacted beam particles were bent by a superconducting

dipole magnet called Sweeper [52] into a vacuum box containing various charged particle

detectors, including a pair of Cathode Readout Drift Chambers (CRDCs), an ionization
chamber and a thin timing scintillator. The dipole magnet had a 43.3◦ bending angle and
a 1 meter bending radius, and it could be operated at up to a magnetic rigidity of 4 T·m.
For the 13B beam, the current was set to 340 A, corresponding to a 3.6923 T·m magnetic

30

05 mECR IonSourcesK500A1900K120018OExperimental Area 11Li or 13BFigure 4.2: Schematic of the experimental area. LISA was placed on two separate tables
with LISA 2 in the front table and LISA 1 in the back table. MoNA was placed on one table
behind LISA.

rigidity. For the 11Li beam, the current was set to 347 A, corresponding to 3.7498 T·m. The

Sweeper magnet has a 14 cm vertical gap through which neutrons traveled straight without

being diverted by the magnetic field. They were then detected by two plastic scintillator

arrays called the Modular Neutron Array (MoNA)[53] and the Large-area multi-Institutional

Scintillator Array (LISA) [54].

4.2.1 Charged Particle Detection

4.2.1.1 Timing Scintillators

Two out of the three timing scintillators were located in the beamline: the A1900 scintilla-

tor and the target scintillator. The A1900 scintillator was 1000 µm thick, located 10.88 m

upstream from the reaction target. The target scintillator was 420 µm thick, located 1.04

m upstream from the reaction target. Each of the two scintillators was made of the organic

plastic scintillator material BC-404 [55] and coupled to one photomultiplier tube (PMT).

Combining the timing information provided by the A1900 scintillator and the target scin-

31

Sweeper MagnetFocusing Quadrupole TripletLISA 2Reaction TargetTarget Scint.5 mBeamsMoNA{{A1900 Scint.CRDCsIonization ChamberThin Scint.LISA 1{Sweeper Vacuum BoxFigure 4.3: Schematic of the last thin scintillator viewed from upstream to downstream.

tillator, the time-of-flight (TOF) of beam particles could be measured. Energy deposited

in the target scintillator was also measured as an auxiliary method for beam identification.

Additionally, the radio-frequency (RF) of the cyclotrons was recorded, providing correlations

with the primary beam that helped with beam separation.

The last timing scintillator was located immediately behind the ionization chamber in

the Sweeper vacuum box. The size of this scintillator is 55 cm by 55 cm by 5 mm. Four

PMTs were coupled to this scintillator through four trapezoidal light guides, as illustrated

by Fig. 4.3. The materials used for the thin scintillator was organic plastic EJ-204, which is

similar to BC-404 used for the A1900 and target scintillators.

Typical organic scintillators utilize the π-electron structure in the molecules polymerized

into plastics. The energy interval between the first (S0) and second (S1) singlet states of

32

yxzLight GuidesPMT 0PMT 1PMT 3PMT 2Scintillatorthe π-electron are 3 to 4 eV for materials of interest. Each singlet configuration can be

further divided into a series of levels with spacing on the order of 0.15 eV. Since the room

temperature is equivalent to 0.025 eV which is much smaller than the spacing between those

levels, most of the π-electrons are in the ground state. When a charged particle passes

through, part of its energy lost in the plastic scintillators excites electrons to excited states.

While higher-lying states quickly decay to S1 through radiationless internal conversion, S1

electrons de-excite to S0 with prompt fluorescent light emitted. This light can be collected

and multiplied by PMTs. The decay time of BC-404 and EJ-204 is 1.8 ns, and therefore fast

counting is possible with these materials.

4.2.1.2 Cathode Readout Drifting Chambers

The first CRDC was located 1.73 m behind the reaction target, calculated along the central

path of the Sweeper magnet. The distance between the two CRDCs was 1.54 m. The working

gas used in the CRDCs was a mixture of 25% isobutane and 75% CF4 at a pressure of 40 Torr,

and the two CRDCs were physically connected by a single gas handling system. A 1000 V

drift voltage was applied to each CRDC. When a charged particle passed through, it created

electron-hole pairs along its path. Due to the applied drift electric field, those electrons

moved towards the anode wire and 128 cathode pads distributed along the X-direction, as

indicated in Fig. 4.4. The electrons caused an avalanche near the Frisch grid used for

eliminating the position dependence of the drift electric field. The avalanche electrons then

were collected by pads. The Y-position of the charged particle was determined by the drift

time of the electrons. The X-position was determined by the charge distribution on the pads,

which are further described in the next chapter. The centroid of the CRDC was used as the

Z-position of the charged particle measurement. Combing the spatial measurements made

33

Figure 4.4: Schematic of a CRDC from Ref. [56]. The z-direction has been expanded.

by the pair of CRDCs, the angular distribution of the charged particles could be derived.

4.2.1.3 Ionization Chamber

The ionization chamber was located immediate behind the second CRDC. The mechanism

of the ionization chamber is similar to the CRDCs. However, the fill gas in the ionization

chamber was a mixture of 90 % argon and 10 % percent methane (P-10). The operating

pressure during the experiment was 520 Torr and a drift voltage of 800 V was applied.

The ionization chamber was used for measuring the energy loss of a charged particle in

the gas, which can be used for charge number identification. Therefore 16 pads of the

34

Figure 4.5: Schematic of the ionization chamber from Ref. [57].

ionization chamber were distributed along the Z-direction to increase the accuracy of the
energy measurements, as shown in Fig. 4.5. The active volume was 40 × 40 × 65 cm3.

4.2.2 Neutron Detection

Neutrons were detected by the Modular Neutron Array (MoNA) [53] and the Large-area

multi-Institutional Scintillator Array (LISA) [54]. They were designed to be identical mod-

ular arrays in terms of number of detector bars, electronics and DAQ. LISA was set on two

tables during the experiment. The front table (LISA2) consisted of 5 layers of bars with the

front face of the first layer at 6.05 m behind the reaction target. Each layer consisted of 16

bars with the exception of the first layer, where the top bar was missing since it was under

repair at the time. There was a gap between the third layer and the fourth layer, because

pre-experiment simulations showed that gaps help improve the multi-neutron detection effi-

35

ciency. The LISA 2 table hosted 4 layers of bars and there was a gap in the middle for the

same reason. In total LISA had 143 scintillator bars. MoNA was positioned on one single

table. There were eight layers with each layer consisting of 16 bars, and two gaps were after

the third and fifth layer, as shown in Fig. 4.2. Originally MoNA had 9 layers of 16 bars each,

but at the time of the experiment, the ninth layer was used at another experimental facility.

In total there were 271 active bars in the experimental setup. Each bar was coupled to two

photomultiplier tubes (PMTs) through light guides at the end of the bars. The Y and Z

coordinates of a neutron hit were determined by the physical location of the bar, while the

X position is derived from the difference of the TOFs measured by the PMTs attached to

the two end of this bar. The TOF of the hit was the arithmetic average of the TOFs from

the two PMTs while the light deposited was calculated as the geometric average of the light

from each PMT.

MoNA consists of BC-408 and LISA consists of EJ-204 and the two materials are equiva-

lent except they were made by different companies. Scintillation light was produced through

similar mechanisms in those materials and is described in the Section 4.2.1.1. However, since

neutrons do not carry charges, they cannot directly transfer energies to electrons. They must

first undergo a (in)elastic scattering with a charged particle and transfer recoil energy. For

elastic scattering, the recoil energy is:

ER =

4A

(1 + A)2 (cos 2θ)En

(4.1)

where A is the mass number of the charged particle, En is the energy of the neutron and θ is

the scattering angle in the lab frame. The formula indicates that the amount of detectable

energy deposited decreases with increasing mass number. Therefore, a high hydrogen ra-

36

tio is desired for plastic scintillators aiming to detect neutrons. BC-408 and EJ-204 are

polymerized from C9H10 in order to achieve a high detection efficiency.

4.2.3 DAQ

Ref. [58, 59, 60, 56, 57] discussed the electronics and data acquisition (DAQ) for the setup

in detail. This section provides a brief overview of the timing part and highlights the major

temporary changes made to the electronics and DAQ during the experiment.

The trigger logic was processed by Xilinx Logic Modules (XLMs) with different levels.

“Level 1” determined if there was a valid event in MoNA or LISA and passed the result

to “Level 2”. A valid event was defined as at least one detector bar having a valid signal

from both constant fraction discriminators (CFD) receiving input from the PMTs at its two

ends. Level 2 then waited for a system trigger. If a system trigger was received, Level 2

communicated with the DAQ and the event was recorded. Otherwise Level 2 sent a fast

clear signal to Sweeper and MoNA-LISA electronics. The system trigger was provided by

the PMT 0 in Fig. 4.3.

MoNA-LISA ran in common stop mode. Typically, the common stop for MoNA-LISA was

provided by the CFD connected to the target scintillator. However, during the experiment,

data with this setting could not simultaneously record timing and light deposited signals due

to the high beam rates. Temporarily using the thin 0 signal as the common stop for MoNA-

LISA fixed the issue. The timing of the new common stop depended on the time-of-flights of

the charged reaction products. Thus, event by event corrections were necessary and details

are discussed in the next chapter. The change is reflected in the abbreviated schematic Fig.

4.6.

37

Figure 4.6: Abbreviated schematic of the electronics and DAQ. The electronics and DAQ of
both MoNA and LISA are represented by the lable “MoNA”.

4.3

Invariant Mass Spectroscopy

Direct measurements of neutron unbound states are not possible due to their very short
lifetimes of about 10−22 s. However, the energies released from the decays of those states

which are called decay energies can be measured through an indirect method named invariant

mass spectroscopy. Considering a four-momentum of an unbound nucleus

Pν

i = (EA, (cid:126)pA)

(4.2)

where EA is the total energy and (cid:126)pA is the three-momentum. Relativistic momentum con-

servation gives

38

CFDA1900CFDTargetCFDThin1, 2, 3CFDThin 0SweeperTDCcommon startDAQMoNATDCCFDMoNAoriginal common stopLevel2Level1fast clearfast clearcommon stopn(cid:88)

i=1

Pν
i

Pν

A =

(4.3)

where the right hand side is the sum of four-momenta over all n decay products of the

nucleus.

The decay energy of the above mentioned unbound nucleus is

Edecay = MA − n(cid:88)

Mi

(4.4)

i=1

where MA is the invariant mass of the unbound nucleus, defined as M 2

A = (Pν

A)2, and Mi is

the invariant mass of one of the decay products. Since MA, Mi are Lorentz invariants, they

can be calculated in any frame. Then in the rest frame of each decay product we have

Mi = mi

(4.5)

where mi is the rest mass of the decay product. MA can be derived by

M 2

A = (Pν

i=1

j=1

A)2

n(cid:88)
n(cid:88)
n(cid:88)
(cid:118)(cid:117)(cid:117)(cid:117)(cid:116) n(cid:88)

M 2

i=1

i=1

=

=

MA =

Pν

i Pνi

n−1(cid:88)
n(cid:88)
(EiEj − (cid:126)pi · (cid:126)pj)
n(cid:88)
n−1(cid:88)

j=i+1

i=1

(EiEj − (cid:126)pi · (cid:126)pj)

i + 2

M 2

i + 2

i=1

j=i+1

39

(4.6)

Explicitly, for a one-neutron decay, the decay energy is

(cid:113)

Edecay =

m2

A−1 + m2

n + 2(EA−1En − (cid:126)pA−1 · (cid:126)pn) − mA−1 − mn

(4.7)

where the subscript A − 1 represents the fragment and n represents the neutron from the

decay. With two or more neutrons emitted, the expression of Edecay can be demonstrated

with cross terms added.

40

Chapter 5

Data Analysis

This chapter first discusses how the raw data recorded from each electronic module were

calibrated and how the events of interest were selected from the whole data set; then, the

process of reconstructing physical observables will be described. Observables, like decay

energy, cannot be directly compared with theoretical calculations since the experimental

measurements include the resolutions, efficiencies, and acceptance of the experimental setup.

Monte Carlo simulations that were used for extracting the physics of interest will be described

at the end of the chapter.

5.1 Calibrations and Corrections

5.1.1 Timing Scintillators

As discussed in the previous chapter, timing scintillators were coupled to PMTs. The raw

timing data from those PMTs were processed by time-to-digital converters (TDCs) and

must be converted to nanoseconds. Because the FPGAs were operated at 50 MHz, the raw

timing data from timing scintillators also contain a 20 ns jitter originating from the FPGA

processing. Since a TDC measures the time interval between a scintillator and the master

trigger (provided by “thin PMT 0”, as illustrated in Fig. 4.3), the raw timing value from

“thin PMT 0” was used for jitter subtraction event-by-event. The calibrated time for a

41

timing scintillator is:

tcal = (traw − tthin0

raw ) × 0.0625

ns

channel

+ toffset

(5.1)

where 0.0625 ns/channel is the conversion from channel to nanosecond, determined by the

manufacturer, and verified using a time calibrator module. The time offset toffset was cali-

brated using data taken from a run for which the reaction target was out and the unreacted

beam was centered (named a beam down center run hereafter). The time offset toffset was

chosen so that tcal corresponded to the time that a particle traveling at the beam velocity

passed through the scintillator, taking t=0 to be the time the beam passed through the

target. Since the thin scintillator was coupled to four PMTs, the calibrated time was the

average of the four PMTs’ calibrated times.

For the target scintillator and the thin scintillator, the scintillator light output was also

processed and recorded by analog-to-digital converters (ADCs). In the current analysis, light

signals were only used for event quality checks, so they were left in arbitrary units. Because

the thin scintillator had four PMTs, the PMT light signals were scaled so their centroids

were aligned. The thin scintillator light output is given by

(cid:113)

q2
top + q2

bottom

qthin =

2

(5.2)

where qtop and qbottom are the average of the two top (PMT 0 and PMT 3) and the two

bottom (PMT 1 and PMT 2) PMTs, respectively. For the A1900 scintillator, only timing

information was recorded.

42

5.1.2 Cathode Readout Drifting Chambers

Multiple steps were taken to transform raw data from the CRDCs to spatial information of

charged particle tracks.

As discussed in Section 4.2.1.2, charge distribution on CRDC pads were used to determine

the X position of a charged particle passing through a CRDC. Therefore, the charge collected

on each pad has to be correctly counted.

The charge collected on each pad was divided into one to four samples. A Riemann-sum

used those samples to determine the charge integral on a pad. Therefore, the raw total

charge on a pad is given by

Nsw(cid:88)

i=0

qi

qpad =

1

Nsw

(5.3)

where Nsw is the number of samples (often referred to as the sample width) of the pad and

qi is the charge collected in a sample.

Two issues related to Equation 5.3 were encountered during data analysis. The first issue

was the original code omitted the normalization by the sample width. The second issue was

an overflow of the sample width. The sample width is calculated according to

Nsw = Nend − Nbegin + 1

(5.4)

where Nend and Nbegin are the number of the first sample and the number of the last

sample, respectively. Those numbers were assigned by the electronics, and they continuously

increased from 0 to 31. If Nbegin was large enough, Nend might exceed 31 and start counting

from 0 again. If that happened, several samples were discarded since their sample number

was smaller than Nbegin, and 31 was used as Nend, making the calculated Nsw smaller than

43

(a)

(b)

Figure 5.1: CRDC2 TAC versus sample width. (a) before the Nsw overflow fix; (b) after the
Nsw overflow fix.

what it should be. The overflow happened in coincidence with specific values of CRDC

TAC, as can be seen in Fig. 5.1a. This issue was fixed by adding 32 to Nend when Nend

was smaller than Nbegin. A plot of CRDC2 TAC versus sample width after fixing the Nsw

overflow issue can be seen in Fig. 5.1b.

Leakage currents in the CRDCs were also included in the pad charge readout; these are

referred as pedestals. The pedestals had to be subtracted to correctly determine the total

charge. Pedestal values for each pad were extracted from data taken when the beam was off.

After a Riemann-sum was completed correctly and the pedestal was subtracted, each

pad needed to be gain-matched to account for differences in the electronics response of each

channel to the same amount of charge. A continuous sweep run was used for the gain match.

During this run, the reaction target was moved out, and the current of the sweeper magnet

was gradually changed so the beam spot smoothly moved across the entire CRDC2.

A pad-max was identified on an event-by-event basis as the index of a pad with the

largest pedestal-subtracted charge among all 128 pads of a CRDC. Pad 70 of CRDC1 and

44

Sample Width01234CRDC2 TAC [channel]020040060080010001200020004000600080001000023Sample Width14CRDC2 TAC [channel]00200400600800100012000200040006000800010000Sample Width1234CRDC2 TAC [channel]00200400600800100012000200040006000800010000Figure 5.2: Raw (left) and calibrated (right) CRDC1 pad charge summary from a continuous
sweep run.

Pad 81 of CRDC2 were selected as the reference pads. The charge spectrum of the reference

pad was drawn using only events where the pad registering the maximum charge was close to

the reference pad. By requiring this, it can be guaranteed that the charged particle passed

nearby the reference pad. The mean value of this gated spectrum, mref , was extracted from

a Gaussian fit. A similar procedure was used to obtain the mean value for all other pads.

The gated spectra of other pads should be aligned with the gated spectrum of the reference

pad since they were the response to the same amount of charge due to the same beam.

Therefore, the gain-matched pad charge is given by:

qcal =

mref
mpad

(qpad − qped)

(5.5)

where qped is a charge pedestal of the pad. The raw and calibrated spectra for CRDC1 and

CRDC2 can be seen in Fig. 5.2 and Fig. 5.3, respectively.

For each event, the gain-matched charge registered by each pad plotted versus pad number

was fit with a Gaussian to determine the interaction point in a CRDC in units of pad index.

45

02040 60 80 CRDC1 Pad Number100120 [arb.]padq0100200300400500600700800900100011021002040 60 80 CRDC1 Pad Number100120 [arb.]calq01002003004005006007008009001000110210Figure 5.3: Raw (left) and calibrated (right) CRDC2 pad charge summary from a continuous
sweep run.

Table 5.1: CRDC pads that were not used in fit

Device
CRDC 1
CRDC 2

Pad Number
0-48, 95-127

0-13, 24, 89, 124

Several pads exhibited irregular responses, so they were excluded from the fit. Since CRDC1

was closer to the magnet, areas far away from its center cannot be illuminated. Consequently,

pads in those areas cannot be gain-matched and the charge collected on them cannot be

correctly counted. Therefore, those ungain-matched pads were also excluded from the fit.

The summary of all excluded pads are listed in Table 5.1.

As explained in Section 4.2.1.2, the Y position of a charged particle interacting with a

CRDC was determined from the drift time of electrons. This drift time was measured by

a time to amplitude converter (TAC). Slopes and offsets were needed to linearly transform

the TAC data and the fitted pad number to Y position and X position, respectively. The X

slopes were given by the 2.54 mm/pad spacing between pads. To determine X offsets and Y

slopes, mask runs were used. For a mask run, a Tungsten mask with known holes and slits

was inserted in front of a CRDC to block the beam, and only the areas right behind the holes

46

02040 60 80CRDC2 Pad Number100120 [arb.]padq0100200300400500600700800900100011021031041002040 60 80CRDC2 Pad Number100120 [arb.]calq01002003004005006007008009001000110210310Figure 5.4: TAC versus CRDC2 X fit from a mask run.

and slits can be illuminated. An example spectrum of TAC vs. interacting point from a mask

run is shown in Fig. 5.4, and it can be seen that holes and slits can be clearly identified. The

known locations and distances between holes on the mask were used to calculate X offsets

and Y slopes. However, since a mask might not be fully inserted, Y offsets obtained from

a mask run might not be correct. Therefore, the Y centroid of a beam from a beam down

center run was used to define Y=0. As a result, Y offsets for different beams were slightly

different. The final offsets and slopes for the CRDCs are shown in Table 5.2. Note the sign

change between devices for the X offset and slopes. This is because the pads are numbered

from the opposite direction for the two CRDCs. Fig. 5.5 shows an example calibrated

positions spectrum for the CRDC2.

47

CRDC2 X Fit [Pad Number]020406080100120CRDC2 TAC [channel]020040060080010001200Figure 5.5: Calibrated CRDC2 Y versus CRDC2 X spectrum from a mask run.

Table 5.2: CRDC slopes and offsets

Device

X slope

[mm/pad]

X offset Y slope Y offset (13B beam) Y offset (11Li beam)
[mm]

[mm/ch]

[mm]

[mm]

CRDC 1
CRDC 2

2.54
-2.54

-185.2
186.7

-0.20
-0.20

114.4
112.2

115.9
118.4

48

CRDC2 X [mm]150-100-50-050100150CRDC2 Y [mm]150-100-50-050100150Figure 5.6: Raw (left) and calibrated (right) ionization chamber pad energy loss summary
from a continuous sweep run.

5.1.3 Ionization Chamber

As discussed in Section 4.2.1.3, the energy loss of a charged particle that passed through

the ionization chamber was measured by 16 pads along the Z direction. Signals from those

pads were processed by analog-to-digital converters (ADCs). An ADC measures the height

of a signal rather than integrating over the signal. Therefore, pedestals were not present in

ionization chamber data so pedestal subtraction was not necessary. The raw energy loss of

a pad was simply scaled so that the centroid of this pad was aligned with the centroid of

the reference pad which was chosen to be Pad 7. A beam down center run was used for this

calibration. Fig. 5.6 shows the energy loss of each pad before and after calibration. It also

can be seen in Fig. 5.6 that Pad 1, Pad 5, and Pad 8 exhibited abnormal responses, so they

were excluded from the rest of the analysis.

The average energy loss (∆E) as measured by good pads, showed some dependence

on the position of beam. Therefore, each calibrated pad was corrected using a fifth order

polynomial, so that the energy loss of this pad as a function of CRDC2 X position was flat.

49

Ionization Chamber Pad Number0246810121416Energy Loss [arb.]0100200300400500600110210310410Ionization Chamber Pad Number0246810121416Calibrated Energy Loss [arb.]0100200300400500600110210310Figure 5.7: Original (left) and corrected (right) ionization chamber energy loss versus CRDC2
X from a continuous sweep run.

A beam down center run was used for this correction. The corrected energy loss, ∆Ecorr, is

given by the equation

∆Ecorr =

A
N

i=15,i(cid:54)=1,5,8(cid:88)

i=0

5(cid:80)

n=0

qi

ki,nxn

(5.6)

where A is an arbitrary normalization factor, N is the number of good pads, qi is the

calibrated pad charge, ki,n is a correction coefficient, and x is the CRDC2 X position. The

energy loss versus CRDC2 X position before and after applying the position correction can

be seen in Fig. 5.7.

5.1.4 MoNA-LISA

5.1.4.1 Light Calibration

As described in Section 4.2.2, for one hit, scintillation light was produced in a bar of MoNA-

LISA. This light signal was processed and recorded by charge-to-digital converters (QDCs)

which were connected to the anodes of the two PMTs at the end of the bar. It was necessary

50

CRDC2 X [mm]150-100-50-050100150E [arb.]D1201401601802002202400510152025303540CRDC2 X [mm]150-100-50-050100150 [arb.]corrED891011121314151605101520253035Figure 5.8: Example light spectra. The left panel shows the raw spectrum, where the peak
of cosmic muon is at around channel 900. The right channel shows the calibrated light
spectrum where the cosmic-ray peak appears at about 20 MeVee.

to convert raw QDC data to light yield in units of MeVee (MeV electron equivalent) so that

the experimental light yield could be compared to simulations.

The linear QDC calibration was carried out using several cosmic-ray runs taken before the

experiment. For a cosmic-ray run, each PMT of MoNA-LISA was operated in self-trigger

mode, so data for cosmic muons could be recorded. The high voltage of each PMT was

adjusted first to put the cosmic muon peak for all PMTs at about 900 channels in each raw

QDC spectrum, as shown in the left panel of Fig. 5.8. Once this gain match was finished,

a new cosmic-ray run was taken. The light yield of cosmic muons traveling a bar is about

20.5 MeVee. Therefore, offsets and slopes were chosen so that pedestals were placed at 0,

and the cosmic muon peak was placed at 20.5 MeVee, as shown in the right panel of Fig.

5.8. The calibrated light yield of a bar was the geometrical average of the signals from its

two PMTs, as described in Section 4.2.2.

Once the QDC calibration was finished, hardware QDC thresholds were set to suppress

51

Raw Light Yield [channel]0500100015002000Counts10210310410510610Light Yield [MeVee]05101520253035404550Counts110210310410510pedestals. The values of these thresholds were determined by

QDCthresh =

qped
16

+ 2

(5.7)

where qped is the pedestal value. These thresholds imposed cutoffs in 16-channel increments

in the QDC spectra and above pedestals, since QDC raw data were stored in 12-bit while

QDC thresholds were stored in 8-bit. Since hardware thresholds as well as QDC slopes, were

different, those thresholds translated to different values in the unit MeVee. To achieve uni-

form thresholds in all calibrated spectra, a 0.91 MeVee post-experimental software threshold

was applied to each PMT. The same light threshold was used for simulations.

5.1.4.2 Timing Calibration and Correction

The timing information of MoNA-LISA was processed and recorded by TDCs. The slopes

converting TDC channel to nanosecond was calibrated with a time calibrator that provided

pulses at 40 nanosecond intervals.

The time of an interaction in a bar was determined by the averaged time of both PMTs

and is referred to as tmean. The reference time when the beam particles hit the target

(tmean=0) reference was chosen as described in Section 5.1.1. A few steps were necessary

to determine the offsets for all detectors.

First, tmean offsets for each bar were calibrated with a cosmic-ray run. This run was

taken after the light calibration and the TDC slope calibration were completed. For each

layer, the top bar was chosen as the reference bar. Then, muons that passed all 16 bars of

the layer were used to calculate the expected time of each bar, since the velocity of 1 GeV

cosmic muons is known to be 29.8 cm/ns [60, 56]. The tmean offsets for single bars were the

52

Figure 5.9: Corrected TOF and velocity spectra for MoNA-LISA. A coincidence with the
front layer is required. Events in the first sharp peak in time are gamma-rays originating
from the target; the peak was correctly placed at 30 cm/ns. The second sharp peak in time
is due to gamma-rays produced by the beam hitting the Sweeper chamber. The velocities
of the sweeper gamma-rays peak at less than the speed of light since in the calculation
the TOFs from the target to the sweeper chamber for the beam particles were erroneously
treated as part of the gamma-ray TOFs. The broad peak corresponds to the neutrons.

values that shifted the tmean of each these bar to the expected time.

Once the offsets for the bars were applied, muons that passed the top bar of one layer

and the bottom bar of another layer were used to set offsets for one layer relative to another.

The tmean offsets for layers were determined in a similar way as described above.

Due to the orientation of the detector tables, muons did not pass through bars on separate

tables. Therefore, the global tmean offsets for each table could not be determined using

cosmic-ray runs. Instead, a collimator run was taken before the production runs. A thick

target was placed in the target chamber, and it was bombarded by the beams from the A1900

to produce a large number of gamma-rays. The expected γ-ray flight time was compared to

the measured value to extract global offsets for each table.

Usually these steps provided accurate tmean values for MoNA-LISA. However, as de-

scribed in section 4.2.3, during the experiment, the common stop for MoNA-LISA was

53

TOF [ns]020406080100120140160180200Counts210310410Velocity [/ns]0510152025303540Counts110210310410gamma-rays from target gamma-rays from sweeperneutronsgamma-rays from target gamma-rays from sweeperneutronsFigure 5.10: Example time difference spectrum and X position spectrum for one MoNA-LISA
bar.

changed to the thin 0 signal from the target scintillator signal. This change happened

after the collimator run. This change meant the common stop was delayed by an amount of

time that varied event-by-event,

T OFcorr = T OF − (tthin0

cal − ttarget

cal

) + offset,

(5.8)

where TOF is the original MoNA-LISA time of flight. An offset was needed since delay

cables were added to the new common stop. To determine this offset, TOF and velocity

spectra were plotted with all 13B beam production runs chained together. The offset was

chosen so the gamma-rays from the target peaked at 30 cm/ns, as shown in Fig. 5.9.

5.1.4.3 Position Calibration

The X position of an interaction along a bar was determined by the time difference between

the left and right PMT signals. A linear calibration was used, and the time difference spectra

of bars were plotted with a long cosmic-ray run. This cosmic-ray run was taken after time

54

Time Difference [ns]30-20-10-0102030Counts02004006008001000120014001600180020002200X Position [cm]200-150-100-50-050100150200Counts020040060080010001200140016001800calibration was finished, and it required that both PMTs of a bar registered a signal. Fermi

functions were used to find the edges. The slope of a bar is given by:

slope =

200 cm

Redge − Ledge

where 200 cm is the physical length of a bar. The offset is given by

offset = 100 cm − slope × Redge

(5.9)

(5.10)

where 100 cm is used to put the center of the time difference at zero. The raw time difference

spectrum and the X position spectrum of a bar are shown in Fig. 5.10.

5.2 Event Selection

Although the experiment was designed to measure 10He from specific secondary beams, these

desired 10He events accounted for a small portion of the experimental data. Other isotopes

produced by the beams, or reaction products from other beam components, were recorded

as well. On the other hand, events might contain inaccurate or invalid information. For

example, if the charge collected by a CRDC in an event was incomplete, the X position from

the fit of the pad charge distribution, which is described in Section 5.1.2, might not be valid.

This section covers how events of interest and of good quality were selected from the whole

data set.

55

Figure 5.11: TOF for incoming beams from the A1900 scintillator to the target scintillator.
Events between solid red lines were selected as in coincidence with interested beams. The
left panel shows the 13B beam gate, and the right panel shows the 11Li beam gate.

5.2.1 Beam Identification

Even though in the A1900 Separator devices such as wedges and slits were used to limit

transmission of other beams, secondary beams were not pure. However, since the A1900

Separator selected charged particles with the same rigidity, other beam species arrived at

the experimental area at a different time since their A/Z ratios were different from the beam

of interest. Therefore, events originating from different beams can be separated by their

TOFs between the A1900 scintillator and the target scintillator.

Fig. 5.11 shows beam gates for the 13B and 11Li beams. While the 13B beam was almost

pure, the 11Li beam was not. In order to increase the 11Li rate, a momentum slit had to

be opened large, allowing additional beam contaminants to leak through. Nevertheless, the

right panel of Fig. 5.11 shows that 11Li were well separated from the contaminants.

56

TOF [ns]100102104106108110112114Counts020040060080010001200310·TOF [ns]859095100105110115Counts0100200300400500600310·Figure 5.12: Energy loss in the ionization chamber vs TOF from the target scintillator to
the thin scintillator for the 13B beam. Neutron coincidence was required.

5.2.2 Element Identification

The energy loss of a charged particle in matter is determined by the Bethe formula [61],

− dE
dx

= A

Z2
v2 B(v),

(5.11)

where A is a constant, Z is the charge number of the particle, v is the velocity and B(v)

changes slowly with particle energy for non-relativistic particles. That means ∆E measured

by the ionization chamber was proportional to Z2/v2. Element identification (Z separation)

was achieved by plotting the ionization chamber ∆E versus the TOF from the target scin-

tillator to the thin scintillator. An example plot used for element identification is shown in

Fig. 5.12. Element bands can be clearly seen and identified. 2D gates can be drawn around

these bands to select individual elements.

57

TOF [ns]45505560657075808590E [arb.]D024681012110210HHeLiBeFigure 5.13: Example S800 ionization chamber energy loss versus TOF used for particle
identification. The fragmentation of an 85 MeV/u 36Ar beam was simulated using LISE++
[62].

5.2.3 Isotope Identification

For a spectrometer used for bound nuclei experiments, such as the S800 at NSCL, element

identification and isotope identification are simultaneously achieved, as shown in Fig. 5.13.

However, the same separation was not observed in Fig. 5.12 because the Sweeper magnet is

just a bending magnet and not a full spectrometer. Also the short flight distance reduced

the TOF resolution. In addition, the energy and angular spreads of the charged particles

for the current neutron-unbound experiment were larger due to neutron decays. As a result

the individual isotope overlapped with the element bands. Nevertheless, for a given element,

different isotopes follow different trajectories, and if appropriate variable transformations are

made, different isotopes can be separated.

The TOFs from the target scintillator to the thin scintillator, X position in the focal

58

020408010060TOF [ns]1030507090110130150170Energy loss [MeV] 1234610152335558513320632150037Cl37Cl37Cl37Cl37Cl37Cl37Cl37Cl37Cl36Cl36Cl36Cl36Cl36Cl36Cl36Cl36Cl36Cl36S36S36S36S36S36S36S36S36S35S35S35S35S35S35S35S35S35S34S34S34S34S34S34S34S34S34S33S33S33S33S33S33S33S33S33S35P35P35P35P35P35P35P35P35P34P34P34P34P34P34P34P34P34P33P33P33P33P33P33P33P33P33P32P32P32P32P32P32P32P32P32P31P31P31P31P31P31P31P31P31P34Si34Si34Si34Si34Si34Si34Si34Si34Si33Si33Si33Si33Si33Si33Si33Si33Si33Si32Si32Si32Si32Si32Si32Si32Si32Si32Si31Si31Si31Si31Si31Si31Si31Si31Si31Si30Si30Si30Si30Si30Si30Si30Si30Si30Si29Si29Si29Si29Si29Si29Si29Si29Si29Si28Si28Si28Si28Si28Si28Si28Si28Si28Si33Al33Al33Al33Al33Al33Al33Al33Al33Alllllllll31Al31Al31Al31Al31Al31Al31Al31Al31Al30Al30Al30Al30Al30Al30Al30Al30Al30Al29Al29Al29Al29Al29Al29Al29Al29Al29Al28Al28Al28Al28Al28Al28Al28Al28Al28Al27Al27Al27Al27Al27Al27Al27Al27Al27Al26Al26Al26Al26Al26Al26Al26Al26Al26Al32Mg32Mg32Mg32Mg32Mg32Mg32Mg32Mg32Mg31Mg31Mg31Mg31Mg31Mg31Mg31Mg31Mg31MgMggMgMgMggMgMgMg29Mg29Mg29Mg29Mg29Mg29Mg29Mg29Mg29Mg28Mg28Mg28Mg28Mg28Mg28Mg28Mg28Mg28Mg27Mg27Mg27Mg27Mg27Mg27Mg27Mg27Mg27Mg26Mg26Mg26Mg26Mg26Mg26Mg26Mg26Mg26Mg25Mg25Mg25Mg25Mg25Mg25Mg25Mg25Mg25Mg24Mg24Mg24Mg24Mg24Mg24Mg24Mg24Mg24Mg31Na31Na31Na31Na31Na31Na31Na31Na31Na30Na30Na30Na30Na30Na30Na30Na30Na30Na29Na29Na29Na29Na29Na29Na29Na29Na29Na28Na28Na28Na28Na28Na28Na28Na28Na28Na26Na26Na26Na26Na26Na26Na26Na26Na26Na25Na25Na25Na25Na25Na25Na25Na25Na25Na24Na24Na24Na24Na24Na24Na24Na24Na24Na23Na23Na23Na23Na23Na23Na23Na23Na23Na22Na22Na22Na22Na22Na22Na22Na22Na22Na29Ne29Ne29Ne29Ne29Ne29Ne29Ne29Ne29Ne28Ne28Ne28Ne28Ne28Ne28Ne28Ne28Ne28Ne27Ne27Ne27Ne27Ne27Ne27Ne27Ne27Ne27Ne26Ne26Ne26Ne26Ne26Ne26Ne26Ne26Ne26NeNeNeNeNeNeNeNeNeNe24Ne24Ne24Ne24Ne24Ne24Ne24Ne24Ne24Ne23Ne23Ne23Ne23Ne23Ne23Ne23Ne23Ne23Ne22Ne22Ne22Ne22Ne22Ne22Ne22Ne22Ne22Ne21Ne21Ne21Ne21Ne21Ne21Ne21Ne21Ne21Ne20Ne20Ne20Ne20Ne20Ne20Ne20Ne20Ne20Ne19Ne19Ne19Ne19Ne19Ne19Ne19Ne19Ne19Ne27F27F27F27F27F27F27F27F27F26F26F26F26F26F26F26F26F26F25F25F25F25F25F25F25F25F25F24F24F24F24F24F24F24F24F24F23F23F23F23F23F23F23F23F23F22F22F22F22F22F22F22F22F22F21F21F21F21F21F21F21F21F21F20F20F20F20F20F20F20F20F20F19F19F19F19F19F19F19F19F19F18F18F18F18F18F18F18F18F18F17F17F17F17F17F17F17F17F17F24O24O24O24O24O24O24O24O24O23O23O23O23O23O23O23O23O23O22O22O22O22O22O22O22O22O22O21O21O21O21O21O21O21O21O21O0O0O0O0O0O0O0O0O0O19O19O19O19O19O19O19O19O19O18O18O18O18O18O18O18O18O18O17O17O17O17O17O17O17O17O17O16O16O16O16O16O16O16O16O16O15O15O15O15O15O15O15O15O15O23N23N23N23N23N23N23N23N23N22N22N22N22N22N22N22N22N22N21N21N21N21N21N21N21N21N21N20N20N20N20N20N20N20N20N20N19N19N19N19N19N19N19N19N19N18N18N18N18N18N18N18N18N18N17N17N17N17N17N17N17N17N17N16N16N16N16N16N16N16N16N16N15N15N15N15N15N15N15N15N15N14N14N14N14N14N14N14N14N14N13N13N13N13N13N13N13N13N13N12N12N12N12N12N12N12N12N12N20C20C20C20C20C20C20C20C20C19C19C19C19C19C19C19C19C19C18C18C18C18C18C18C18C18C18C17C17C17C17C17C17C17C17C17C16C16C16C16C16C16C16C16C16C15C5C5C5C15C5C5C5C5C14C14C14C14C14C14C14C14C14C13C13C13C13C13C13C13C13C13C12C12C12C12C12C12C12C12C12C11C11C11C11C11C11C11C11C11C10C10C10C10C10C10C10C10C10C19B19B19B19B19B19B19B19B19B17B17B17B17B17B17B17B17B17B15B15B15B15B15B15B15B15B15B14B14B14B14B14B14B14B14B14B13B13B13B13B13B13B13B13B13B12B12B12B12B12B12B12B12B12B11B11B11B11B11B11B11B11B11B10B10B10B10B10B10B10B10B10B 8B 8B 8B 8B 8B 8B 8B 8B 8B14Be14Be14Be14Be14Be14Be14Be14Be14Be12Be12Be12Be12Be12Be12Be12Be12Be12Be11Be11Be11Be11Be11Be11Be11Be11Be11BeBeBeBeBe0BeBeBeBeBe 9Be 9Be 9Be 9Be 9Be 9Be 9Be 9Be 9Be 7Be7Be7Be7Be 7Be7Be7Be7Be7Be11Li11Li11Li11Li11Li11Li11Li11Li11Li 9Li 9Li 9Li 9Li 9Li 9Li 9Li 9Li 9Li 8Li 8Li 8Li 8Li 8Li8Li 8Li 8Li 8Li 7Li7Li7Li7Li 7Li7Li7Li7Li7Li 6Li 6Li 6Li 6Li 6Li 6Li 6Li 6Li 6Li 8He8He8He8He 8He8He8He8He8He 6He 6He 6He 6He 6He 6He 6He 6He 6He 4 4 4 4 4 4 4 4 4 337Cl37Cl37Cl37Cl37Cl37Cl37Cl37Cl37Cl36Cl36Cl36Cl36Cl36Cl36Cl36Cl36Cl36Cl36S36S36S36S36S36S36S36S36S35S35S35S35S35S35S35S35S35S34S34S34S34S34S34S34S34S34S33S33S33S33S33S33S33S33S33S35P35P35P35P35P35P35P35P35P34P34P34P34P34P34P34P34P34P33P33P33P33P33P33P33P33P33P32P32P32P32P32P32P32P32P32P31P31P31P31P31P31P31P31P31P34Si34Si34Si34Si34Si34Si34Si34Si34Si33Si33Si33Si33Si33Si33Si33Si33Si33Si32Si32Si32Si32Si32Si32Si32Si32Si32Si31Si31Si31Si31Si31Si31Si31Si31Si31Si30Si30Si30Si30Si30Si30Si30Si30Si30Si29Si29Si29Si29Si29Si29Si29Si29Si29Si28Si28Si28Si28Si28Si28Si28Si28Si28Si33Al33Al33Al33Al33Al33Al33Al33Al33Al32Al32Al32Al32Al32Al32Al32Al32Al32Al31Al31Al31Al31Al31Al31Al31Al31Al31Al30Al30Al30Al30Al30Al30Al30Al30Al30Al29Al29Al29Al29Al29Al29Al29Al29Al29Al28Al28Al28Al28Al28Al28Al28Al28Al28Al27Al27Al27Al27Al27Al27Al27Al27Al27Al26Al26Al26Al26Al26Al26Al26Al26Al26Al32Mg32Mg32Mg32Mg32Mg32Mg32Mg32Mg32Mg31Mg31Mg31Mg31Mg31Mg31Mg31Mg31Mg31Mg30Mg30Mg30Mg30Mg30Mg30Mg30Mg30Mg30Mg29Mg29Mg29Mg29Mg29Mg29Mg29Mg29Mg29Mg28Mg28Mg28Mg28Mg28Mg28Mg28Mg28Mg28Mg27Mg27Mg27Mg27Mg27Mg27Mg27Mg27Mg27Mg26Mg26Mg26Mg26Mg26Mg26Mg26Mg26Mg26Mg25Mg25Mg25Mg25Mg25Mg25Mg25Mg25Mg25Mg24Mg24Mg24Mg24Mg24Mg24Mg24Mg24Mg24Mg31Na31Na31Na31Na31Na31Na31Na31Na31Na30Na30Na30Na30Na30Na30Na30Na30Na30Na29Na29Na29Na29Na29Na29Na29Na29Na29Na28Na28Na28Na28Na28Na28Na28Na28Na28Na27Na27Na27Na27Na27Na27Na27Na27Na27Na26Na26Na26Na26Na26Na26Na26Na26Na26Na25Na25Na25Na25Na25Na25Na25Na25Na25Na24Na24Na24Na24Na24Na24Na24Na24Na24Na23Na23Na23Na23Na23Na23Na23Na23Na23Na22Na22Na22Na22Na22Na22Na22Na22Na22Na29Ne29Ne29Ne29Ne29Ne29Ne29Ne29Ne29Ne28Ne28Ne28Ne28Ne28Ne28Ne28Ne28Ne28Ne27Ne27Ne27Ne27Ne27Ne27Ne27Ne27Ne27Ne26Ne26Ne26Ne26Ne26Ne26Ne26Ne26Ne26Ne25Ne25Ne25Ne25Ne25Ne25Ne25Ne25Ne25Ne24Ne24Ne24Ne24Ne24Ne24Ne24Ne24Ne24Ne23Ne23Ne23Ne23Ne23Ne23Ne23Ne23Ne23Ne22Ne22Ne22Ne22Ne22Ne22Ne22Ne22Ne22Ne21Ne21Ne21Ne21Ne21Ne21Ne21Ne21Ne21Ne20Ne20Ne20Ne20Ne20Ne20Ne20Ne20Ne20Ne19Ne19Ne19Ne19Ne19Ne19Ne19Ne19Ne19Ne27F27F27F27F27F27F27F27F27F26F26F26F26F26F26F26F26F26F25F25F25F25F25F25F25F25F25F24F24F24F24F24F24F24F24F24F23F23F23F23F23F23F23F23F23F22F22F22F22F22F22F22F22F22F21F21F21F21F21F21F21F21F21F20F20F20F20F20F20F20F20F20F19F19F19F19F19F19F19F19F19F18F18F18F18F18F18F18F18F18F17F17F17F17F17F17F17F17F17F24O24O24O24O24O24O24O24O24O23O23O23O23O23O23O23O23O23O22O22O22O22O22O22O22O22O22O21O21O21O21O21O21O21O21O21O20O20O20O20O20O20O20O20O20O19O19O19O19O19O19O19O19O19O18O18O18O18O18O18O18O18O18O17O17O17O17O17O17O17O17O17O16O16O16O16O16O16O16O16O16O15O15O15O15O15O15O15O15O15O23N23N23N23N23N23N23N23N23N22N22N22N22N22N22N22N22N22N21N21N21N21N21N21N21N21N21N20N20N20N20N20N20N20N20N20N19N19N19N19N19N19N19N19N19N18N18N18N18N18N18N18N18N18N17N17N17N17N17N17N17N17N17N16N16N16N16N16N16N16N16N16N15N15N15N15N15N15N15N15N15N14N14N14N14N14N14N14N14N14N13N13N13N13N13N13N13N13N13N12N12N12N12N12N12N12N12N12N20C20C20C20C20C20C20C20C20C19C19C19C19C19C19C19C19C19C18C18C18C18C18C18C18C18C18C17C17C17C17C17C17C17C17C17C16C16C16C16C16C16C16C16C16C15C15C15C15C15C15C15C15C15C14C14C14C14C14C14C14C14C14C13C13C13C13C13C13C13C13C13C12C12C12C12C12C12C12C12C12C11C11C11C11C11C11C11C11C11C10C10C10C10C10C10C10C10C10C19B19B19B19B19B19B19B19B19B17B17B17B17B17B17B17B17B17B15B15B15B15B15B15B15B15B15B14B14B14B14B14B14B14B14B14B13B13B13B13B13B13B13B13B13B12B12B12B12B12B12B12B12B12B11B11B11B11B11B11B11B11B11B10B10B10B10B10B10B10B10B10B 8B 8B 8B 8B 8B 8B 8B 8B 8B14Be14Be14Be14Be14Be14Be14Be14Be14Be12Be12Be12Be12Be12Be12Be12Be12Be12Be11Be11Be11Be11Be11Be11Be11Be11Be11Be10Be10Be10Be10Be10Be10Be10Be10Be10Be 9Be 9Be 9Be 9Be 9Be 9Be 9Be 9Be 9Be 7Be7Be7Be7Be 7Be7Be7Be7Be7Be11Li11Li11Li11Li11Li11Li11Li11Li11Li 9Li 9Li 9Li 9Li 9Li 9Li 9Li 9Li 9Li 8Li 8Li 8Li 8Li 8Li8Li 8Li 8Li 8Li 7Li7Li 7Li 7Li 7Li7Li 7Li 7Li 7Li 6Li 6Li 6Li 6Li 6Li 6Li 6Li 6Li 6Li 8He8He8He8He 8He8He8He8He8He 6He 6He 6He 6He 6He 6He 6He 6He 6He 4 4 4 4 4 4 4 4 4 3Figure 5.14: 3D correlation and 2D projection of He isotopes from the 13B beam. The color
in the right panel represents TOF (the z-axis in the left panel). In the 3D plot, the cluster
that is in between 60 ns and 75 ns, and away from the other three bands, corresponds to He
particles that hit the Sweeper chamber. They were excluded from further analysis.

plane (FP), and X angle in the FP were selected for isotope identification.

As shown in the left panel of Fig. 5.14, isotopes were well-separated in the 3D plot of

the TOFs from the target scintillator to the thin scintillator, X position in the focal plane

(FP), and X angle in the FP. Then, auxiliary variables were constructed to decorrelate each

isotope. The first is used to decorrelate the FP position and angles. It is defined as

xtx = θx,F P − (a · xF P + b · x2

F P )

(5.12)

where xF P is the focal plane X position, θx,F P is the focal plane X angle, and a and b are

constant coefficients. The two coefficients were extracted from fitting the boundary between

the light blue and dark blue in the 2D projection with a second order polynomial. Then,

xtx could be drawn against TOF, as shown in Fig. 5.15. Now the isotope bands could be

identified in a 2D variable space. A final variable was constructed that projected the three

bands into a 1D plot. It is defined as

59

FP X position [mm]100-50-050100FP X angle [mRad]100-50-050100TOF [ns]55606570758085FP X position [mm]100-80-60-40-20-020406080100FP X angle [mRad]100-80-60-40-20-02040608010055606570758085Figure 5.15: xtx versus TOF for He isotopes from the 13B beam. The most intense band is
attributed to 6He.

Table 5.3: Coefficients used for isotope identification

Beam
13B
11Li

a

b

c

d

0.717268
0.659213

0.00281395
0.00411903

-35.538
-9.08

0.231951

0

xtx tof = xtx − (c · T OF + d · T OF 2)

(5.13)

where TOF is the time-of-flight from the target scintillator to the thin scintillator, and c and

d are constant coefficients. The two coefficients were extracted from fitting an isotope band

in the xtx vs. TOF plot with a second (first) order polynomial.

An xtx tof spectrum for He isotopes is shown in Fig. 5.16. Similar spectra for heavier

elements might display large overlaps between isotope peaks. However, since 5He and 7He

are unbound, He isotopes can be clearly separated and identified in Fig. 5.16. All coefficients

used for He isotope identification are listed in Table 5.3.

60

50556065TOF [ns]707580xtx [arb.]200-150-100-50-050100150200110210Figure 5.16: xtx tof used for isotope identification in the 13B beam data set.

5.2.4 Event Gates

Event gates were used to improve the accuracy and validity of selected events. Not all event

gate were necessary for all isotopes form the different beams.

5.2.4.1 CRDC Gates

The accuracy of position information measured with the CRDCs was crucial to extracting

information about the fragments leaving the target.

In Fig. 5.17, the CRDC padsum is

plotted against the sigma of a Gaussian fit to the charge distribution across CRDC pads.

The plots show a band where the width increases with padsum. Events deviating from this

band were excluded by applying 2D gates as shown in the figure.

61

xtx_tof [arb.]1100115012001250130013501400Counts020004000600080001000012000140001600018000He4He6He8Figure 5.17: CRDC Padsum versus X sigma for 8He fragments from the 13B beam. CRDC
quality gates are shown as red circles. Each gate selects approximately 80% of the events
registered by a CRDC.

5.2.4.2 Neutron Gates

MoNA-LISA recorded signals not only from neutrons, but from gamma-rays, charged parti-

cles, and random events. Light yield was plotted against TOF throughout the whole 400-ns

TDC window, as shown in Fig. 5.18. The time range between 50 ns and 150 ns corresponds

to beam velocity neutrons and events outside this range were excluded. A uniform band

of events below 3 MeVee distributed throughout the time window for both beams was due

to uncorrelated random evens (mostly gamma-rays). The random rate for 13B was 7 times

bigger than for 11Li. Therefore, a 3 MeVee light gate was applied to data from the 13B

beam, and no light gate was applied to 11Li data, as a trade-off between reducing noise and

keeping good events.

5.2.4.3 Beam Gates

As shown in Fig. 5.19, the 13B beam production run contained background events in the

target scintillator light output spectrum. These might be caused by random events due to

62

CRDC1 X Sigma [Pad Number]012345678910CRDC1 Padsum [arb.]0200040006000800010000120000100200300400500CRDC2 X Sigma [Pad Number]012345678910CRDC2 Padsum [arb.]02000400060008000100001200002004006008001000Figure 5.18: MoNA-LISA light yield versus TOF for 8He fragments from the 13B beam
(left) and the 11Li beam (right). Events between the vertical lines and above the horizontal
line are selected as good neutron events. 8He fragments originating from the 11Li beam
(right panel) do not have a light yield gate (thus the absence of a horizontal line).

the high beam rate. Therefore, a horizontal gate in the target scintillator light spectrum was

applied to data from the 13B beam to remove those events. The light output from the 11Li

beam did not exhibit such events, thus no gates were not applied to the 11Li beam.

5.3 Classification of Two-Neutron Events

5.3.1 Causality Cuts

The reconstruction of three-body decay energies requires the correct identification of two-

neutron events. Even if MoNA-LISA records two interactions in coincidence with a 8He

fragment, it is possible that a neutron scatters in one bar and then subsequently interacts in

another, creating two interactions that can be incorrectly identified as two neutrons. These

cases are called “false 2-neutron events”. In contrast, if two different neutrons are detected,

the event is a “true 2-neutron event”. Contributions from false 2-neutron events can be

63

TOF [ns]50-050100150200250300350Light Yield [MeVee]01020304050607080051015202530354050-050100150200250300350Light Yield [MeVee]01020304050607080010203040506070TOF [ns]Figure 5.19: Target scintillator light output versus TOF from the A1900 to the target
scintillator for 8He fragments using the 13B data from a beam down center run (left) and a
production run (right). Events between outside the red lines were excluded from data. The
vertical scales are different since the voltage of the target scintillator PMT changed between
the beam down center and the production run.

Figure 5.20: Schematic of a 2-neutron event.

64

TOF [ns]106106.5107107.5108 [arb.]targetq01002003004005006007008000100200300400500600700TOF [ns]106106.5107107.5108 [arb.]targetq02004006008001000120014001600180020000100200300400500600700800First HitSecond HitTargetDn1n2Vn1n2TOFn1TOFn2Table 5.4: Parameters used for causality gates

Beam A [cm/ns] B [cm]
13B
11Li

30
30

14
11

significantly reduced by applying causality cuts. This technique assumes that two hits in

the detector arrays are caused by the same neutron. Then, a hypothetical velocity for the

neutron from the first interaction to the second interaction can be calculated as

Vn1n2 =

Dn1n2

T OFn2 − T OFn1

(5.14)

where T OFn1 and T OFn1 are corrected TOFs measured by MoNA-LISA, and Dn1n2 is the

distance between the two hits, as illustrated in Fig. 5.20. If the event is a false 2-neutron

event, Vn1n2 tends to be smaller than the beam velocity, and it’s more likely that Dn1n2 is

small. Therefore, causality cuts require

Vn1n2 > A

Dn1n2 > B

(5.15)

where A and B are causality parameters. A is usually chosen as the high limit of the beam

velocity, and B is usually the distance between a few adjecent MoNA-LISA bars. The actual

values used in the current analysis are listed in Table 5.4.

Causality cuts have previously been applied to extract three-body decay energy spectra

in other two-neutron unbound systems [23, 63, 64, 65, 66]. The two-neutron detection

efficiencies with and without causality cuts are shown in Fig. 5.21.

The gated efficiency drops near 0 MeV because the kinematics required by causality cuts

65

Figure 5.21: 2-neutron efficiency (dashed red) and gated 2-neutron efficiency (solid blue).
A simulation with phase space decay model and realistic beam parameters for 11Li beam
was used for estimating these efficiencies.

66

 [MeV]decayE012345678efficiency00.10.20.30.40.50.60.7n_effEntries  16Mean    2.426RMS     2.165Figure 5.22: Cut efficiencies estimated from simulated 11Li beam data.

reject more events at low energy. However, the efficiency still has a finite value at zero decay

energy.

5.3.2 Machine Learning Methods

The classification of 2-neutron events was also attempted using machine learning. This

approach was accomplished with TMVA (Toolkit for Multivariate Data Analysis) [67] built-in

to the data analysis framework ROOT [68]. For this approach, several variables in simulated

data were used first to train and test a classifier. Then, the classifier was used to evaluate

experimental data, based on the corresponding variables. A probability for each event to

be a false 2-neutron event can be estimated from the classifier. A detailed description is

provided in Appendix A.

67

Machine learning methods show the potential to achieve higher efficiency of the true 2-

neutron event selection, compared to causality cuts. For the 11Li beam data, causality cuts

return a signal purity = 0.78 and a signal efficiency = 0.49. For the same purity, the boosted

decision trees achieve a signal efficiency = 0.87, as shown in Fig. 5.22, increasing the signal

efficiency by 78%. However, machine learning methods also add complexity to analysis,

since simulations must be performed first for training a classifier, and the dependence of the

classifier distribution on simulation parameters. To avoid complexity and to be consistent

with the previous analysis, the rest of the analysis was performed with causality cuts.

5.4 Reconstruction

5.4.1 Neutron 4-Momentum Reconstruction

Neutron 4-momenta, as well as fragment 4-momenta, were reconstructed in the lab frame in

the current work. The magnitude of the neutron velocity is

vn =

| (cid:126)D|

T OFcorr

(5.16)

where T OFcorr is the corrected neutron TOF and (cid:126)D is the vector from the reaction target

to a neutron interaction in MoNA-LISA as (cid:126)D. Then, the Lorentz factor, the momentum,

and the energy are calculated according to relativistic mechanics.

5.4.2 Fragment 4-Momentum Reconstruction

The fragment 4-momenta to be reconstructed are the momenta at the reaction target, where

the unbound nuclei decayed. However, the CRDCs measure the fragment positions and

68

angles after the Sweeper magnet. Therefore, the fragments need to be tracked back to the

target. This was achieved by the ion-optics program COSY INFINITY (referred as COSY

hereafter) [69].

To use COSY, a magnetic field map was generated using IGOR PRO [70] first, based

on previous Hall probe measurements and the current of the Sweeper magnet. Then, the

magnetic field map is used as an input to COSY, which generates an ion-optical matrix M

relating the beam parameters at the target to the parameters at CRDC1



xCRDC1

θCRDC1
x

yCRDC1

θCRDC1
y

L





xtarget

θtarget
x

ytarget

θtarget
y

δE



= M

(5.17)

where x and y stand for positions, θ stands for corresponding angles, CRDC1 or target

represents the location of the coordinates, L is the tracking length, and δE is defined by

K.E.frag = Eref (1 + δE)

(5.18)

where K.E.f rag is the kinetic energy of the fragment and Eref is the reference energy

determined by the rigidity of the field map.

This matrix uses target information as the input, but since the coordinates are measured

69

Figure 5.23: Reconstructed kinetic energy for 13B (left) and 11Li (right) beam using beam
down center runs. Expected beam energies (47 MeV/u and 44 MeV/u, respectively) are
well-reproduced.

at CRDC1 a transformed matrix M’ [69] has to be used such that



θtarget
x

ytarget

θtarget
y

δE

L





xCRDC1

θCRDC1
x

yCRDC1

θCRDC1
y

xtarget



= M(cid:48)

;

(5.19)

xtarget was not measured, however, a constant representing the x centroid of the beam can

be substituted for a measured value. Reconstructed beam kinetic energies are shown in Fig.

5.23 for verification.

The relation between the kinetic energy and the Lorentz factor is:

K.E.frag = Mf (γf − 1)

(5.20)

70

K.E. [MeV/u]36384042444648505254Counts010002000300040005000K.E. [MeV/u]36384042444648505254Counts050100150200250300350400where Mf is the mass of the fragment, γf is the Lorentz factor of the fragment. Using

angles at the target, the fragment 4-momenta can be reconstructed similarly according to

relativistic mechanics in the previous section.

5.4.3 Decay Energy Reconstruction

Using the 4-momenta of the fragments and neutrons, the decay energy of the unbound system

can be reconstructed according to Equation 4.7 in Section 4.3. Explicitly, the two-body decay

energy of 8He + n is

Edecay =

(cid:113)

m2

f + m2

n + 2(Ef En − (cid:126)pf · (cid:126)pn) − mf − mn

(5.21)

where f stands for the 8He fragment. The three-body decay energy of 8He + 2n decay is

(cid:113)

Edecay =

Hhere T is written as:

m2

f + 2m2

n + 2T − mf − 2mn.

(5.22)

T = Ef En1 + Ef En2 + En1En2 − (cid:126)pf · (cid:126)pn1 − (cid:126)pf · (cid:126)pn2 − (cid:126)pn1 · (cid:126)pn2

(5.23)

where n1 and n2 stands for the two neutrons. The three-body and two-body decay energy

spectra from the two beams are shown in Fig. 5.24 and Fig. 5.25.

71

Figure 5.24: Decay energy spectra from the 13B beam. The upper left shows the 8He + n
two-body decay energy. The upper right shows the 8He + n two-body decay energy with a
neutron multiplicity=1 gate. The lower left shows 8He + 2n three-body decay energy. The
lower right shows causality-gated 8He + 2n three-body decay energy.

72

decayE         [MeV]0123456Counts501001502002503000123456Counts050100150200250012345678Counts0510152025303540012345678Counts024681012decayE         [MeV]decayE         [MeV]decayE         [MeV]Figure 5.25: Decay energy spectra from the 11Li beam. The left-upper shows the 8He + n
two-body decay energy. The upper left shows the 8He + n two-body decay energy. The upper
right shows the 8He + n two-body decay energy with a neutron multiplicity=1 gate. The
lower left shows 8He + 2n three-body decay energy. The lower right shows causality-gated
8He + 2n three-body decay energy.

73

0123456Counts02004006008001000120014000123456Counts01002003004005006007000123456Counts0501001502002503000123456Counts020406080100120140160180decayE         [MeV]decayE         [MeV]decayE         [MeV]decayE         [MeV]5.5 Simulations

Monte Carlo simulations including theoretical decay energy distributions, decay modes, re-

action mechanisms, beam parameters, magnetic fields, as well as detector acceptances, ef-

ficiencies, and resolutions were performed for comparison with data. The same simulation

program has been used for many previous experiments, and the details and verification of

this program were thoroughly discussed in Refs. [60, 57, 56]. Thus, this section only briefly

describes the results of the simulations of 10He.

The simulation starts by creating the beam particles at the surface of the target. The

positions and angles of the beam are modeled by Gaussian distributions. The parameters

were extracted from the CRDC position and angle measurements of a beam down center

θtarget
x

run, using the transformation matrix

ytarget

θtarget
y

xtarget

L



= M(cid:48)(cid:48)



xCRDC1

θCRDC1
x

yCRDC1

θCRDC1
y

δE



,

(5.24)

where δE was calculated from the beam energy.

Once created, the beam particles are drifted to reaction positions, which were randomly

chosen throughout the thickness of the target. Energy loss and straggling are applied before

the reaction. The reaction is treated in two steps. First, nucleons are removed from the beam

particle to form 10He, while the kinetic energy per nucleon is kept the same. Then, a parallel

momentum kick based on the work of Goldhaber [71] and a transverse momentum kick based

74

on the work of Bibber [72] are added to the 10He. The 10He then decays according to a line

shape described in Section 3.2. The energies and momenta of the three decay products (8He

+ n + n) are determined by the phase space decay model (TGenPhaseSpace [40] in ROOT).

The charged reaction fragments are drifted to the back of the target, taking into account

the energy loss and straggling. Then, a forward COSY matrix transports the fragments from

the target to CRDC1. A free drift is applied between CRDC1 and CRDC2. Positions of

the fragments at CRDC1 and CRDC2 were folded with Gaussian resolutions to simulate the

detector response. Combining with angles calculated from these folded positions, a fragment

trajectory through the Sweeper is determined as discussed in Section 5.4.2. A comparison of

simulated fragments to data is shown in Fig. 5.26 where the measured positions, measured

angles, and tracked angles are plotted.

It should be mentioned that, for the data, the

measured Y angles, and tracked X and Y angles are narrower. However, they have minor

effects on decay energy spectra.

Neutrons from the decays are handled by GEANT4 [73, 74] using the neutron physics

package MENATE R [75]. The Sweeper chamber is modeled as steel to simulate the geo-

metric cut on the neutrons. For the neutrons that interact with MoNA-LISA, the deposited

energy is converted to light output. The light propagates to the two ends of the detector

bar according to Birks’ law [76]. An event is recorded only if both of the ends receive a

light signal greater than 0.91 MeVee. This is used to simulate the software light threshold

described in Section 5.1.4.1. Gaussian distributions were added to the neutron X position

and the neutron TOF to account for the detector resolutions.

The neutron 4-momentum reconstruction, fragment 4-momentum reconstruction, and

decay energy reconstruction were performed for data and simulated events in the same way.

The same gates were used as causality cuts. By treating simulations and data the same way,

75

spectra from simulations could be used to fit data, and the parameters of the theoretical line

shape for 10He could be extracted.

76

Figure 5.26: Comparison of simulated fragment to data for 8He from the 11Li beam.

77

CRDC1 X [mm]150-100-50-050100150Counts02000400060008000100001200014000CRDC1 Y[mm]150-100-50-050100150Counts020004000600080001000012000 [mRad]XqCRDC1 150-100-50-050100150 Counts02000400060008000100001200014000160001800020000 [mRad]YqCRDC1 150-100-50-050100150 Counts050001000015000200002500030000 [mRad]XqTarget 150-100-50-050100150 Counts020004000600080001000012000140001600018000200002200024000 [mRad]YqTarget 150-100-50-050100150 Counts05000100001500020000250003000035000Chapter 6

Results and Discussion

6.1

13B beam

Fig. 6.1 shows the measured and simulated decay energy spectra from the 13B beam. The

8He + n two-body decay energy is broad, and there are no prominent structures at higher

decay energies. In the ungated three-body decay energy, a peak seems appear at around

1 MeV. However, with the causality cuts are applied, the greatly reduced statistics do not

show any peaks. The highest bin at about 0.8 MeV in the gated spectrum cannot be a

narrow resonance since it is narrower than the resolution. This 9-count spike may be caused

by statistical fluctuations.

The simulations consist of four components. A state at 1.6 MeV from Ref. [23] is used as

the ground state of 10He, and a 1.0 MeV p-state and a 3.0 MeV d-state are used for simulating

resonant states in 9He. The resonance energies and widths come from an experiment per-

formed in the same experimental area, the analysis of which is still in progress. A Maxwellian

distribution with temperature 4.5 MeV was used as a phenomenological description for the

de-excitation from high-lying 9He continuum.

Although 1.6 MeV was used as the resonance energy of the 10He ground state in Fig.

6.1, changing it to 2.1 MeV (which was extracted from a 3H(8He,p)10He transfer experiment

[21]) results equally good fit. This is understandable since the causality-gated spectrum has

78

Figure 6.1:
(a) Two-body decay energy spectrum. (b) Ungated three-body decay energy
spectrum. (c) Decay energy spectrum gated on causality cuts. Data are presented as crosses
and solid circles. The black solid line shows the sum of simulations. The purple dot-dash
line is the thermal background. The p-state and d-state in 9He are shown as the green solid
and dark blue solid lines, respectively. The light blue dot-dash line shows the state in 10He.

79

Counts050100150200250300Counts0510152025 [MeV]decayE0123456Counts0246810(a)(b)(c)low statistics, so the fit is not sensitive to the states in 10He which decay by emitting two

neutrons. Therefore, results from the compact 13B beam cannot resolve the two 10He ground

state energies extracted from reactions of different types.

In knockout reactions, the cross sections for populating the ground state of final nucleus

decreases with the number of removed nucleons. However, the fit in Fig. 6.1 indicates

the population of 10He is significantly weaker than that of 9He. This suggests that the

population of 9He contains contributions other than direct knock-out. Simple combinatorics

can be used to estimate the relative populations of 9He and 10He. Assume multi-nucleon

removal reactions only remove nucleons above the s-shell (i.e., on top of the α core), and the

possibility of removing each nucleon is the same. Then, 3 protons and 6 neutrons can be

removed from 13B. Given a 3-nucleon removal reaction, the probability of populating 10He

from 13B beam (13B(−3p)10He) is(cid:0)3
removal reaction, the probability of populating 9He (13B(−3p1n)9He) is (cid:0)3

(cid:1)/(cid:0)9
(cid:1) = 1.2%. On the other hand, given a 4-nucleon
(cid:1)/(cid:0)9
(cid:1) ×(cid:0)6
(cid:1) =

3

3

3

1

4

4.8%. Also, one might calculate the probability of 8He direct population (13B(−3p2n)8He)

is 11.9%. Those simple estimations indicate that the spectrum of 8He in coincidence with

neutrons is dominated by the decay of 9He.

More insights can be gathered from studying the components of the 9He contribution.

Even though 9He resonant states extracted from another experiment fit data, the non-

resonant thermal background accounts for 90% of the observed spectrum. This suggests

that the 3p1n-removal reaction is dispersive and excites 9He to the high-lying continuum.

Therefore, the 3-proton knockout reaction is not ideal for investigating 2-neutron decays

since the spectrum is dominated by the 1-neutron decay competition, and the 2-neutron

information will be usually embedded in the 1-neutron thermal background. In addition,

the 8He direct population channel discussed above also contributes to the background. For

80

these reasons, the current experiment cannot resolve the ground state of 10He using a com-

pact 13B beam. Even increasing the measurement time to improve the statistics might not

be sufficient to extract the 10He resonance parameters. These factors should be taken into

consideration in the design of future experiments.

6.2

11Li Beam

Fig 6.2 shows the measured and simulated decay energy spectra from the 11Li beam. All

measured spectra from the 11Li beam are significantly narrower than the spectra from the 13B

beam. The measured three-body decay energy spectrum with the causality gates shows that

a large number of counts survived the cuts, suggesting substantial 2n contributions. In fact,

the best fit to data contains only one component, which is the 10He correlated background

described in Section 3.2.3. The 11Li binding energy Eb = 369 keV [77] is used for the

correlated background calculation. The observation of a dominant correlated background

means that the observed 10He spectra are heavily influenced by the initial state.

In contrast to the result from the 13B beam, 9He contributions are not observed in the

11Li data. Usually, the 1p1n-removal is expected in addition to the 1p-removal. However,

the absence of 9He can be understood from the structure of 11Li beam. For population of
9He, a 11Li(−1p1n) reaction is necessary. Knocking out a neutron and a proton from the 9Li

core is more plausible than knocking out a proton from the core and a neutron from the halo.

The former leads to a neutron hole state in 9He since neutrons in the core are deeply-bound

compared to the valence neutrons in the halo. Since the two-neutron separation energy of

8He is only 2.1 MeV [77], it possible the high-lying neutron hole state in 9He directly decays

to 6He + 3n. Therefore, 9He contributions are not significant in the coincidence spectra of

81

(a) Two-body decay energy spectrum. (b) Ungated three-body decay energy
Figure 6.2:
spectrum. (c) Decay energy spectrum gated on causality cuts. Data are presented as crosses
and solid circles. The black line shows the simulation of the correlated background model.

82

Counts02004006008001000120014001600Counts050100150200250300 [MeV]decayE0123456Counts020406080100120140160180(b)(a)(c)Figure 6.3:
blue solid circles. The correlated background is shown as black solid lines.

Invariant mass spectrum for 10He from Ref. [16]. The data are presented as

8He fragments.

Simulations of resonances and/or a thermal background in 10He were fit to data, but

they resulted in unphysically large width and poor χ2 values. Fitting with a 10He resonant

state with a correlated background was also attempted. However, the scale factor of any

resonant state added was optimized to 0 by the χ2 minimization procedure. This was in

contrast with the similar GSI experiment for which a 11Li beam was used [16], where a

resonant state could be fit on top of the correlated background, even though in a follow-up

paper the authors concluded that their data should be interpreted as a 1.54 MeV ground

state plus a 3.99 MeV excited state [17], instead of a 1.42 MeV ground state plus a correlated

background. Sharov, Egorova and Grigorenko later argued that these conclusions were not

83

valid from the point of view of full reaction dynamics [39]. Still, there are inherent differences

between the GSI and the current 11Li beam experiments, since the correlated background

that fits the current data did not fit GSI experiment, as shown in Fig. 6.3. Besides why the

3H(8He,p)10He transfer experiments observed different ground state energies of 10He than

all other experiments, there is one more interesting question to be answered: why the current

experiment observed different line shapes although the same 11Li beam was used.

Although the χ2 minimization optimized the contribution of the 10He resonant state to 0,

considering the resolutions of the experiment setup, resonant contributions can account for

up to 10% of the 10He spectra without significantly reducing the quality of the fit, as shown

in Fig. 6.4. This indicates that some resonant contributions cannot be ruled out, however,

not as dominant as in the GSI experiment.

This discrepancy might be explained with the the difference in time scales, since Zhukov

and Grigorenko’s theory predicts that the “shift” of the resonance energy is a consequence

of the large size of the 11Li halo and the short lifetimes of neutron-unbound states. The GSI

experiment ran at 280 MeV/u whereas the current experiment at 44 MeV/u, therefore, the

timescale for the target and the beam particles being in contact with each other was shorter

for GSI than for the current experiment. Simply estimations can be made by dividing the
sum of the diameters by the beam velocities: the contact time is 1.0×10−22 s for the GSI
experiment and 2.5×10−22 s for the current experiment. The extracted width from the GSI
experiment was 1.91 MeV [17] which would corresponds to a halflife of 2.4×10−22. While the

contact time in the GSI experiment is significantly shorter, the contact time in the present

experiment is about the same. Thus, there was not sufficient time to form 10He and the

decay might be still influenced by the structure of the initial 11Li halo state.

However, as summarized by Table 6.1, there were 11Li beam experiments ran at lower

84

Figure 6.4:
(a) Two-body decay energy spectrum. (b) Ungated three-body decay energy
spectrum. (c) Decay energy spectrum gated on causality cuts. Data are presented as crosses
and solid circles. The shaded area shows the simulation of the correlated background model.
The purple dot-dash line represents the resonant state at 1.6 MeV. The resonant contribution
is fixed at 10%.

85

Counts02004006008001000120014001600Counts050100150200250300 [MeV]decayE0123456Counts020406080100120140160180(b)(a)(c)Table 6.1: Summary of 10He experiments with halo beams.

Reaction
11Li(-p)
11Li(p,2p)
11Li(-p)

14Be(-2p2n)
2H(11Li,3He)

11Li(-p)

Target

Ebeam

Eg.s.

Reference

2H
1H
1H
2H
2H
9Be

61 MeV/u

1.2(3) MeV

84 MeV/u

1.7(6) MeV

280 MeV/u 1.54(10) MeV

59 MeV/u

1.60(25) MeV

50 MeV/u

1.4(3) MeV

[14]

[15]

[17]

[23]

[24]

44 MeV/u

N.A.

current

Figure 6.5: Decay energy spectra of 7He from the H target (left) and the C target (right)
from Ref. [78].

energies, which were closer to the current one. Those experiments also reported resonant

states even without a correlated background, so this contradicts the timescale argument.

Table 6.1 nevertheless shows one crucial difference and feature of the current experiment: it

was the only experiment that used a heavy ion target.

Indeed target dependence has been observed between hydrogen and heavy ion targets used

for the population of unbound nuclei. Aksyutina et al. [78] showed the target dependence of

7He populated from the 8He beams. The measured decay energy spectra are different for the

H target and for C target, as shown in Fig. 6.5. The line shape of the well-known 7He deviated

86

Figure 6.6: Decay energy spectra on the C target (red) and the D target (black) for the
first observation in 10He. The D target spectrum is digitized from Ref. [14]. The C target
spectrum is deduced from the CD2 and the D target spectra in Ref.
[14], with estimated
statistical error bars.

87

decayE012345Counts020406080100from the Breit-Wigner distribution when a carbon target was used at relativistic energy. The

authors attributed this effect to target-dependent re-scattering, and the destruction of the

carbon nucleus. The first observation of the resonance in 10He also used two different targets:

CD2 and C [14]. The comparison of the 10He spectra on the Deutron nucleus and on the C

nucleus is shown in Fig. 6.6. It can be seen that the C target spectrum is shifted to lower

energies compared to the D target spectrum, which agrees with the current experiment. Note

8He is also a halo nucleus and the width of the 7He ground state is 125 keV [77] which is

smaller than 10He roughly by a factor of 10, meaning the lifetime of 7He is 10 times longer.

Both of the two cases of target dependence might be related to timescales. For heavier

targets, short lifetimes and smaller beam energies (which means longer contact time) result

in a larger distortion in the measured decay energy spectra. However, it’s not yet clear why

there is target-dependence.

In principle, the correlated background corresponds to a 0+ configuration of the 8He

+ n + n three-body system. There are theoretical calculations predicting an s-wave 0+

resonant state at low energies. Aoyama predicted a three-body s-wave 0+ state at 50 keV

using analytic continuation in the coupling constant method and complex scaling method

[29, 30, 31]. He claimed that the state was not observed so far without indicating why.

Fossez et al. made the first many-body prediction that the 10He ground state is a double-

halo 0+ very close to the 2n threshold with almost pure s-wave valence neutrons [33]. It is

understandable that such a state is hard to populate from a 11Li beam since the inner halo

does not exist in the 9Li core. However, the very low-lying narrow 0+ was still not observed

from 3H(8He,p)10He transfer experiments, even if the inner halo was provided by 8He beam.

Therefore, the existence of a very low-lying s-wave resonant state is questionable.

Alternatively, Grigorenko and Zhukov predicted a three-body virtual state might exist

88

in 10He with s-wave valence neutrons if 11Li beams were used [22]. They predicted that a

three-body virtual state corresponds to a sharp increase in the decay line shape near zero

decay energy. Their quantitative results also disagrees with the current observation.

89

Chapter 7

Summary and Outlook

In this work, 10He was studied using a 44 MeV/u halo 11Li beam and a 47 MeV/u non-

halo 13B beam at the NSCL. Neutrons were measured by MoNA-LISA; charged particles

were measured by the Sweeper detectors; and the decay energies were reconstructed using

invariant mass spectroscopy. Monte Carlo simulations were performed to compare theoretical

models to experimental data. Energies of resonant states in 10He could not be extracted from

the 13B beam due to low statistics and competition from the population of 9He. The 11Li

beam data could be described with the correlated background model, implying the initial

state has a significant influence on the measured spectra. Interpretations of decay energy

spectra should be made with extra caution if the expected widths are greater than 1 MeV

and halo beams are involved. In addition, an apparent target dependence was observed from

the 11Li beam.

Since resonant states are not resolved from the 13B(-3p) reaction, the question of the

10He ground state is not solved yet. Experiments using different reactions are encouraged,
notably the 12Be(−2p) reaction and fragmentation with more balanced neutron and pro-

ton numbers to be removed. Most importantly, the 3H(8He,p)10He transfer experiment [21]

should be repeated with significantly more statistics. Different transfer experiments with

better statistics and resolutions are also desired. On the theory side, complete reaction dy-

namical calculations including the structure of the beam and target nuclei will be necessary

90

to explain the different observations. To investigate the target dependence, 11Li(−p) exper-

iments are suggested to be performed at 50 MeV/u to 250 MeV/u with 50 MeV/u intervals,

on both Hydrogen and heavy ion targets.

91

APPENDIX

92

Appendix

Guide for TMVA with MoNA

.1

Introduction

This document provide a basic, step-by-step guide for using the TMVA (Toolkit for Mul-

tivariate Data Analysis) with MoNA. TMVA is a built-in component since ROOT version

5.11/06, providing machine learning environments. Sections for the TMVA setup and run-

ning simple examples are basically rephrasings of corresponding chapters of the more detailed

official guide listed at the end. This document assumes a basic understanding of ROOT and

st mona simulations, but you don’t have to know details about multivariate methods.

TMVA can be used for classification and regression, but this document only considers the

classification of 2-neutron events. An event where the first two hits are due to two different

neutrons is referred as a true two neutron event, or “Signal” for consistency with the official

guide, while an event where the first two hits are from the same neutron is referred to as a

scattered one neutron event or “Background”. Simulated Signal trees and Background trees

are provided separately for training and testing a specific classifier. The trained classifier

can be used for evaluating against simulations and data at the evaluation phase, resulting

classifier responses that can be used for determining the classes of events.

93

.2 TMVA Setup

.2.1 Build Your Local TMVA Against Your ROOT (optional)

.2.1.1 Downloading TMVA Source Code

Building a local TMVA against ROOT might be beneficial, especially for ROOT users with

an older version of TMVA. If needed, the source code of the latest TMVA can be downloaded

at

https://sourceforge.net/projects/tmva/files/ as a gzipped tar file(.tgz).

.2.1.2 Unzipping and Building the TMVA Source Code

Move the downloaded .tgz file to the directory where you want to install, and unzip it in

that directory by typing

˜> t a r −x v z f YourFileName . t g z

on a Linux terminal.

You can then build the source code by typing the following commands.

˜> cd tmva

˜/tmva> make

˜/tmva> cd t e s t

˜/tmva/ t e s t /> s e t u p . sh

Note “tmva” used above is the name of the directory generated by unzipping. By default

it could be something like “TMVA-v4.2.0”. The command “make” typically takes about 10

94

minutes.

Once built, by default you run your TMVA macros at /tmva/test/ directory. More

frequently, if you want to work with TMVA somewhere else, additionally you should type

˜> cd YourWorkingDirectroy

˜/ YourWorkingDirectroy> cp ˜/tmva/ t e s t / s e t u p . sh

˜/ YourWorkingDirectroy> s o u r c e s e t u p . sh <Your

i n s t a l l a t i o n path>

where < Y our installation path > is the absolute path to where you make TMVA. Note

setup.sh should be sourced for EVERY new terminal session. Also, note the path to your

working directory should NOT contain white spaces.

According to the official guide, sourcing the setup.sh file is sufficient on a Unix/Linux

machine. However, if you run ROOT on a Mac, additional steps might be necessary, and

you should read the official guide.

.3 Running the Simple Example

.3.1 Run the Macros

Before dealing with simulations and data, you can run examples for pedagogical purposes.

A simple example of classification is /tmva/test/TMVAClassification.C. Copy it to your

working directory, and it can be run at the working directory by typing

r o o t −l YourWorkingDirectory / TMVAClassification .C

That macro automatically downloads a toy ROOT file containing 4 variables with signal

95

and background saved in different trees. The macro uses half of the events for training and

another half for testing for about 10 classifiers. You can also run through a specific type of

classifier, for example, boosted decision trees (BDTs), by typing:

r o o t −l YourWorkingDirectory / TMVAClassification .C\(\”BDT\”\)

If the macro ran successfully, a GUI like Fig. .1 pops up in the end. If errors occur, try

to re-source setup.sh file as instructed by the previous section.

.3.2 Outputs of Interests

Each item of the GUI can be clicked. (1a) displays all input variables used for training and

it displays background and the signal separately, as shown in Fig. .2. (4b) displays classifier

responses which are outputs used for identifying the signal, as shown in Fig. .3. (5a) displays

signal efficiency, background efficiency, signal purity, and signal significance, as a function of

classifier output, based on the test set, as shown in Fig. .4.

.4 Prepare Trees for Training & Testing Simulations

If only “original” variables (x, y, z, t and q of the two hits) are used for training & testing,

a GEANT tree of st mona can be used directly, and this section can be skipped.

You can use aliases as input variables without causing any errors, but the actual values

fed in TMVA might not be the values calculated from the aliases. Therefore, if you would

like to work with “tuned” variables (variables derived from the original variables that make

more physical sense) such as scattering angles, distances between two hits or hypothetical

velocities between two hits, a ROOT file containing those variables should be prepared in

96

Figure .1: Screenshot of the TMVA GUI.

97

Figure .2: Screenshot of GUI (1a), input variables.

Figure .3: Screenshot of GUI (4b), classifier output.

98

Figure .4: Screenshot of GUI (5a), classifier cut efficiencies.

99

advance.

An example macro for preparing such a file, called prepare root for TMVAtraining.C , is

provided with this document. You could simply use it by changing the input ROOT file path

and output file path. In addition to x, y, z, t, and q of two hits of GEANT simulations, the

sample macro also calculates the Lorentz factor, the scattering angle, the distance between

two hits and the hypothetical velocity between two hits, and saves them to the output ROOT

file.

The provided macro assumes the branches of neutron are beginning with “b13p”. Differ-

ent versions of st moma might have different naming conventions, and the names of variables

of this macro should be changed accordingly. More “tuned variables“ can also be added but

the details are beyond the scope of this document.

A file called “prepared sim for training.root” generated by the above-mentioned macro

is provided, and it can be directly used for testing the macro mentioned below.

.5 Training and Testing Against MoNA Simulations

This section focuses on how you can modify a macro called TMVAClassification2Neutron.C,

provided with this document. A ROOT file directly from st mona or prepared by the previous

section will be used. In both cases, only two neutron simulations should be considered since

TMVA must know what is Signal, and what is Background, while 1n simulations only provide

Background.

.5.1 Change paths

Two paths need to be changed:

100

Figure .5: Lines of the macro for setting variables.

ˆ change “YourOutput.root” to a name you want for saving results of training & testing

ˆ change “PathToYourRootFile.root” to the actual path of the root file you want use for

training & testing

.5.2 Change Variables

In the provided macro, line 192 to 206 are commands for setting variables, as shown in Fig.

.5. Four items are listed within the parentheses. The first one is a variable to be added for

training. Branches of the input tree should be used, and simple expressions are supported.

The second is the name of the variable, which will appear in a GUI later. The remaining

are the unit and the type of the variable, where ‘D’ means double-precision floating-point

format.

More variables can be added similarly, and the existing one’s can be commented if you

don’t want them to be part of training.

101

Figure .6: Lines of the macro for cuts.

.5.3 Change Cuts

In the simple macro, signal and background are placed in separated trees. However, due to

the nature of st mona simulations, true 2 neutron events (Signal) and 1 neutron scattered

twice events (Background) are mixed in the same tree.

Although it is possible to split simulated Signal and Background into different trees

during the pre-training phase, for convenience one can set the same tree as both signal tree

and background tree but with different cuts.

Fig.

.6 shows the applied cuts from line 294 to line 295. An event is guaranteed to be

signal if the first hit is the first neutron and the second hit is the second neutron or the

first hit is the second neutron and the second hit is the first neutron. Otherwise, an event is

background. b13pg1light and b13pg2light are required to be greater than the light threshold

used for simulating the input file so that only events with more than 2 valid hits are used.

.5.4 Training & Testing Results

Running the macro you just modified is similar to running a simple example. Just enter:

r o o t −l YourWorkingDirectory / TMVAClassification2Neutron .C

or

r o o t −l YourWorkingDirectory / TMVAClassification2Neutron .C\(\”BDT\”\)

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Figure .7: Screenshot of GUI (1a), input variables with st mona.

if a specific classifier is desired.

The same GUI as Fig. .1 pops up after a successful execution. Typical training&testing

plots are given below as Fig. .7, Fig. .8, and Fig. .9.

.6 Evaluating MoNA Data or Simulations

In the evaluation phase we get classifier responses for events we want to know their identity,

according to information contained in those events. Weight files inside “weights” directory

generated by the training procedure are used, so the working directory of the evaluating

phase should be the same as the training and testing phase.

Again, since aliases do not work properly, files to be evaluated might have to be prepared

in advance if any “tuned” variables are involved. Data, 1n simulations or 2n simulations need

103

Figure .8: Screenshot of GUI (4b), classifier output with st mona.

104

Figure .9: Screenshot of GUI (5a) classifier, cut efficiencies with st mona.

105

to be prepared in slightly different ways, and changes might also be necessary due to the

software version or the experimental setup. Therefore, one simple macro that can deal with

all cases cannot be provided. Instead, this section mentions key concepts of the evaluating

phase and shows how one can evaluate a provided ROOT file “data to be evaluated.root”

by using the provided macro “TMVAClassificationApplication2Neutron.C”. Only the BDT

method is used in that macro. If ROOT files with the same branches can be generated, the

provided application macro can be used for evaluating them.

.6.1 Set the TMVA Reader

Fig. .10 shows line 143 to line 157 of the macro “TMVAClassificationApplication2Neutron.C”.

There are two arguments for each command. The first is the NAME of the variable, and

should be exactly the same as the name used in the training phase (Fig.

.5). The second

argument is the corresponding variable. It should originate from the root file to be evaluated.

Note, the type given in the weight files is float so the variables used as the second

argument should also be declared as float type variables.

.6.2 Getting Responses from a Classifier

To get a response of one entry, you should first assign the correct values of the entry into

the variables used as the second argument in Fig. .10. For example, you should assign the

scattering angle in degrees to n0 n1 scat angle, and you should assign the kinetic energy of

the first hit in MeV to n0 gamma. Those steps are accomplished by lines 267 to 280 and

lines 306 to 320.

Once the assignment is finished, the classifier response can be reached by:

106

Figure .10: Lines of the macro for setting the TMVA reader.

bdt=r e a d e r−>EvaluateMVA ( ‘ ‘BDT method ’ ’ )

as shown by line 333 in the application macro. Then the value of the response can be

saved in a tree for future use.

.6.3 Use the Example for Data Classification

Copy “data to be evaluated.root” and “TMVAClassificationApplication2Neutron.C” to your

working directory.

Before using the example macro for the evaluating phase, you should run the example

macro for the training & testing phase. Then at the same working directory, the evaluating

macro can be run by typing

r o o t −l T M V A C l a s s i f i c a t i o n A p p l i c a t i o n 2 N e u t r o n .C\(\”BDT\”\)

where the ending is NOT optional.

107

Figure .11: Typical BDT responses towards data. Gated on valid events.

Once finished, a ROOT file “TMVApp2nda.root” is generated in the working directory.

The only leaf should be “bdt” and it is the responses of the BDT classifier trained from

previous phases. Typical BDT responses are shown in Fig. .11. The tree in the file can be

added as a friend to the original tree in your data file. Then a cut like “bdt > 0” can be used

as a true 2n gate.

108

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