Missing-mass spectroscopy of short-lived nuclei
at low-momentum transfer region
opened by the MAIKo active target

Tatsuya FURUNO

A dissertation submitted in partial fulfillment of therequirements for the degree of Doctor of Philosophy in Science,
Department of Physics, Kyoto University
January, 2020

Abstract
A new active target system named MAIKo has been developed for the missing-mass
spectroscopies of unstable nuclei. The MAIKo active target is based on a time projection
chamber where the detection gas plays also as the target gas. This technique enables us
to detect low-energy recoil particles emitted at forward center-of-mass angles. As the firstphysics experiment with MAIKo, the elastic and inelastic alpha scatterings on 10 C were
measured using a 68-MeV/u radioactive 10 C beam at Research Center for Nuclear Physics,
Osaka University. As designed, we have succeeded to reduce the detection threshold for therecoil alpha particles down to 500 keV with MAIKo. The phenomenological α-N eﬀective
interaction and the point-nucleon density distribution in the ground state were determinedfrom the diﬀerential cross sections of the elastic alpha scattering. The distorted-wave Bornapproximation calculation was carried out by using the interaction and density distribution
to analyze the inelastic alpha scattering. The calculated cross sections were compared withthe measured cross sections, and the neutron transition matrix element Mn from the ground10
state to the 2+
1 state at the excitation energy of 3.35 MeV in C was determined to be Mn =6.9 ± 0.7 (fit) ± 1.2 (sys) fm2 . The obtained Mn value in 10 C is close to the proton transition
matrix element Mp in the mirror nucleus 10 Be, demonstrating that the isospin symmetry in
the A = 10 system is conserved. Combining the present result with the previously measured
proton transition matrix element in 10 C of Mp = 6.63 ± 0.11 fm2 , the ratio of the neutron
matrix element to the proton matrix element in 10 C was deduced as Mn /Mp = 1.05 ±0.11 (fit) ± 0.17 (sys). This ratio close to unity suggests that the quadrupole transition in
10 C is less neutron dominant compared to 16 C, and the Z = 6 shell closure which wasrecently proposed in the neutron-rich carbon isotopes like 16 C is less evident in proton-rich
10 C. The first physics experiment with MAIKo was successfully accomplished, demonstratingthat MAIKo has opened the missing-mass spectroscopy of unstable nuclei at low-momentum
transfer region.

Contents

1 INTRODUCTION

1

1.1

Inelastic scattering and its challenge in RI beam experiments . . . . . . . . .

1

1.2

Cooler ring method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.3

Active target method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.4

MAIKo active target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.4.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.4.2

Examples of application of MAIKo . . . . . . . . . . . . . . . . . . . .

6

1.4.3

The First experiment with MAIKo . . . . . . . . . . . . . . . . . . . .

8

1.5

Physics case in the present work: magicity at Z = 6 . . . . . . . . . . . . . .

8

1.6

Quadrupole transition matrix element . . . . . . . . . . . . . . . . . . . . . .

11

1.7

Previous experiments on 10 C . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

1.8

New experiment on 10 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

1.8.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

1.8.2

Kinematic condition . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

Contents of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

1.9

2 DESIGN OF MAIKO

17

2.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.2

TPC field cage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.3

Si detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

i

ii

CONTENTS
2.4µ-PIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.5

GEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.6

Readout electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

2.7

Selection of TPC gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

2.8

Gas handling system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

3 EXPERIMENT

27

3.1

Primary beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

3.2

Production of the 10 C secondary beam . . . . . . . . . . . . . . . . . . . . . .

27

3.3

Beamline detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

3.3.1

Low-pressure MWDC . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

3.3.2

F3 plastic scintillator . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

3.4

Data acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

3.5

Operation of MAIKo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

3.5.1

TPC gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

3.5.2

Measurement of the drift velocity . . . . . . . . . . . . . . . . . . . . .

36

Summary of the measurement . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

3.6

4 DATA REDUCTION
4.1

41

Examples of anode and cathode images . . . . . . . . . . . . . . . . . . . . .

41

4.1.1

Examples of α+10 C scattering events . . . . . . . . . . . . . . . . . . .

41

4.1.2

Examples of background events . . . . . . . . . . . . . . . . . . . . . .

44

4.2

Examples of the FADC data for the α+10 C scattering events . . . . . . . . .

46

4.3

Procedure of the TPC data analysis . . . . . . . . . . . . . . . . . . . . . . .

49

4.4

Optimization of analysis parameters . . . . . . . . . . . . . . . . . . . . . . .

55

4.5

Particle identification of the recoil particles . . . . . . . . . . . . . . . . . . .

57

4.6

Reconstruction of kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

4.7

Excitation-energy spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

CONTENTS

iii

4.8

Eﬃciency simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

4.9

Diﬀerential cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

4.10 Accuracy of the present analysis evaluated by the α+12 C elastic scattering . .

72

5 DWBA ANALYSIS

75

5.1

Single-folding model and eﬀective α-N interaction . . . . . . . . . . . . . . .

75

5.2

Determination of the range parameter . . . . . . . . . . . . . . . . . . . . . .

76

5.3

Analysis of α+10 C elastic scattering . . . . . . . . . . . . . . . . . . . . . . .

78

5.4

Analysis of α+10 C inelastic scattering to the 2+
1 state . . . . . . . . . . . . .

81

6 DISCUSSION

85

6.1

Comparison with the previous results . . . . . . . . . . . . . . . . . . . . . . .

85

6.2

Comparison with theoretical calculations . . . . . . . . . . . . . . . . . . . . .

85

6.3

Z = 6 shell closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

6.4

Future perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

6.4.1

Upgrade of the MAIKo detector . . . . . . . . . . . . . . . . . . . . .

89

6.4.2

Search for alpha cluster states . . . . . . . . . . . . . . . . . . . . . . .

90

7 SUMMARY95

Acknowledgement

98

A Hough transformation

99

B Relativistic kinematics

101

C Point proton density distribution of the ground state in 12 C

103

D Data tables of cross sections

107

References

108

1 INTRODUCTION
1.1 Inelastic scattering and its challenge in RI beam
experiments
Inelastic scatterings with light ions such as protons or alpha particles are excellent probes
to obtain the information on the structures of atomic nuclei. High-resolution spectroscopy
with the inelastic scattering unveils nuclear level structures and the cross section provides
transition matrix elements between the ground and excited states. The energy levels and thetransition matrix elements are the experimental observables which can be directly comparedwith the theoretical calculations. In inelastic scatterings, measurements at forward scattering angles in the center-of-mass (c.m.) frame with low-momentum transfer are especially
important because the diﬀerential cross section becomes larger and the reaction mechanism
is simple.
With the recent technical developments of the new facilities to provide rare isotope (RI)
beams [1–5], it is now possible to measure the inelastic scatterings oﬀ short-lived unstable
nuclei and to explore the structures of exotic nuclei far from the β-stability line. Studies on
the unstable nuclei allow us to test whether the present knowledge of the nuclear structures
established on stable nuclei are valid even in the exotic nuclei with large asymmetry in their
proton and neutron numbers.
Since we cannot prepare a target of a short-lived unstable nucleus, scattering spectroscopyof unstable nuclei must be performed in the inverse kinematic condition where an RI beam isincident on a target. There are two methods to determine the excitation energy of the beam
particle as presented in Fig. 1.1. One method is the invariant-mass spectroscopy in which
an excitation energy is deduced from an invariant mass reconstructed by detecting all decay
particles emitted from a beam particle such as light-ions, gamma rays, and neutrons. Thismethod enables us to use a thick liquid or solid target which ensures the highest yield. The
invariant-mass spectroscopy has been widely applied in many earlier RI beam experiments
using relatively high-energy (>50 MeV/u) beams. However, this method is very diﬃcult
both in elastic and inelastic scatterings exciting high-lying states. In case of the elastic
scattering, a scattered beam particle does not change its angle and energy a lot, and thus
it is diﬃcult to distinguish it from the unreacted beam particles. In the inelastic scattering
exciting high-lying states, the eﬃciency for detecting all of the fragments becomes small
because multiplicity of the decay fragments including neutral particles becomes high.
The other method is the missing-mass spectroscopy in which an excitation energy is determined by measuring the energy and angle of a recoil particle. Compared to the invariant-mass
1

1.2 Cooler ring method
spectroscopy, the missing-mass spectroscopy is eﬀective for the elastic scattering because therecoil particles are emitted around 90◦ in the laboratory frame, which makes it easier toseparate them from the beam particles. The missing-mass spectroscopy can be also applied
to the inelastic scattering exciting high-lying states because the detection eﬃciency for recoil
particles does not depend on the excitation energy.

decay particles

invariant mass
RI beamtargetγ

missing mass
recoil particle
Figure 1.1:
experiment.

Invariant-mass spectroscopy and missing-mass spectroscopy in an RI-beam

From the kinematics, the energy of the recoil particles become very small at forward
angles in c.m. frame, especially in alpha inelastic scattering. Typically, detection of lowenergy recoil particles below 1 MeV is required. However, it is almost impossible to detectsuch low-energy alpha particles with a thin target and an external detector because theseparticles stop in the target and cannot be detected by the detector. If one uses an extremely
thin target, it will reduce the luminosity. So far, there are two methods to detect low-energy
recoil particles without losing the luminosity: the cooler ring method and the active target
method. In the following sections, we review the two methods.

1.2 Cooler ring method
One method to detect low-energy recoil particles is the cooler ring method which is realized
at GSI as the project called “exotic nuclei studied in light ion induced reactions (EXL)”[6]. Figure 1.2 shows a schematic of the detector setup in the EXL experiment [7]. In
this method, the RI beams are stored in a cooler ring [8] and an internal gas-jet target–2–

CHAPTER 1. INTRODUCTION
of helium or hydrogen [9] is installed inside the ring. The recoil particles are detected bya dedicated telescope array which consists of double-sided silicon strip detectors (DSSD)
and lithium-doped silicon [Si(Li)] detectors [10]. This system can be operated under theultra-high vacuum condition for the cooler ring. The low-energy recoil particles are detected
because the mass thickness of the gas-jet target is extremely low (∼ 10 pg/cm2 ). At the sametime, high luminosity of 1026 cm−2 s−1 is achieved [7] because the unreacted beam particles
are impinged repeatedly onto the target. In addition, by using an electron cooler system[11], the emittance and momentum spread of the RI beams can be lowered as small as 0.1π
mm·mrad and δp/p ∼ 10−5 , which contributes to increase the luminosity due to the large
overlap between the beam particles and the gas-jet target.

Figure 1.2: Schematic detector setup of the EXL experiment. The figure is taken from Fig.
2 in Ref. [7].

Recently, the EXL project performed measurements of the proton elastic scattering on
unstable 56 Ni [12] and the alpha elastic and inelastic scatterings on 58 Ni [13, 14]. [14].
The experiments successfully lowered the detection threshold for the recoil alpha particles
down to 0.5 MeV. Although the EXL project is promising to carry out the missing-mass
spectroscopy in a low-momentum transfer region by detecting low-energy recoil particles, it
is only applicable to nuclei whose life times are longer than a minute because it takes about
one minute to store and cool the RI beams in the cooler ring [15].

1.3 Active target method
The use of a time projection chamber (TPC) as an active target is another solution to
detect the low-energy recoil particles. The TPC is a gaseous radiation detector with three–3–

1.3 Active target method

Table 1.1: Comparison of major active target systems.
Name
IKAR (1971) [16]MSTPC (2006) [19]MAYA (2007) [20]pAT-TPC (2012) [21]CAT (2015) [22]AT-TPC (2017) [23]ACTAR (2018) [25]TexAT (2020) [40]
MAIKo (2018) [41]

Lab.
GSI
CNS
GANIL
NSCL
CNS
NSCL
GANIL
Texas A&M
RCNP

e− ampl.

Volume (mm3 )

Pitch

Ch.

N.A.
Wire
Wire
µ-megasGEMµ-megasµ-megasµ-megasµ-PIC

600 × 2002 π

N.A.
11 mm
8.86 mm
2 mm
4.04 mm
3.33 mm
2 mm
1.67 mm
0.4 mm

18
128
1,024
256
400
10,24016,3841,024
512

700 × 150 × 200
300 × 280 × 280
500 × 1252 π
100 × 100 × 250
1000 × 2502 π
300 × 300 × 260
220 × 240 × 140
100 × 100 × 140

dimensional reconstruction capability for charged particle trajectories. The essential featureof the active target is that the detector gas of the TPC plays a role of a target gas. Typically,
helium, hydrogen, deuterium or hydrocarbon gas is used. Since the reaction occurs inside
the sensitive volume of the TPC, detection of low-energy recoil particles is possible overthe large solid angle. Moreover, the luminosity of the measurement can be increased byextending the length of the TPC along the beam axis, keeping the detection threshold forthe recoil particles low. While the cooler ring method can be applied only to relatively
long-lived nuclei with τ1/2 ≥ 1 minute, the active target method can be applied even to
short-lived nuclei because the RI beams are bombarded onto the active target immediately
after the production. The reconstructed trajectory of a recoil particle is used to determine
the recoil angle and recoil energy to obtain the excitation energy of the beam particle.
The first active target named IKAR was developed in 1970s in Leningrad Nuclear Physics
Institute, Gatchina, USSR [16]. The IKAR active target was a dedicated detector system tobe operated with the high-pressure hydrogen gas at 10 bar. The system was used for proton
elastic scattering measurements on neutron-rich isotopes to deduce nucleon distributions[17].
After IKAR, many active target systems were developed around the world [18–26], andmeasurements using these active targets have been reported [27–36]. Owing to the recent
developments of micro-pattern gaseous detectors (MPGD) and readout electronics such as
general electronics for time projection chamber (GET) [37], some of the active target systems
have large sensitive volumes and many readout channels over 10,000, which contributes tohigh statistics and high spacial resolutions. Major active targets are listed with the electronamplification method, sensitive volume of the detector, granularity of the readout (readout
pitch), and number of readout channels in Table 1.1. Comprehensive reviews on the recent
developments of the active targets are reported in Refs. [38, 39].
–4–

CHAPTER 1. INTRODUCTION

1.4 MAIKo active target
1.4.1 Overview
Recently, we developed a new active target system named MAIKo (mu-PIC based active
target for inverse kinematics◦ ) [41] at Research Center for Nuclear Physics (RCNP), Osaka
University to realize the detection of the low-energy recoil particles for the missing-mass
spectroscopy. In MAIKo, the micro-pixel chamber (µ-PIC) [42] was introduced for theamplification and readout of the electrons with the help of the cosmic ray group in Kyoto
University. A schematic view of MAIKo is shown in Fig. 1.3. Compared to the other active
targets, the sensitive volume of MAIKo is small, but the readout pitch is the finest (0.4 mm).
150 mm150

mm

Cathode plate
Field wire

140 mm

α

Si

10C

10C

y
Pillarz

Grid mesh
GEM, µ-PIC

x

Figure 1.3: Schematic view of MAIKo.

The detailed design of MAIKo is described in Chap. 2. In the development, the following
several points were especially taken care.
• Fine readout pitch realized by the µ-PIC
By introducing the µ-PIC as the readout structure, MAIKo has the finest readout
pitch among the existing active target systems. The fine readout pitch enables theidentification of the short trajectory in the TPC, thus the low detection threshold for–5–

1.4 MAIKo active target
the energy of the recoil particles. By introducing the strip readout system in MAIKo,
the number of the readout channels is small, which made the cost of the development
relatively small.
• Purity of the target gas
As the first experiment using MAIKo, we planed the measurement of the inelastic
scattering with alpha particles. Therefore, MAIKo should be operated mainly with Hegas. To make the operation of the TPC stable, quenching gas is usually added to the
He gas. However, this quench gas might be a background source to the alpha inelastic
scattering, thus the fraction of the quenching gas should be as low as possible. Afterthe first beam test with MAIKo, we introduced a gas electron multiplier (GEM) to
make the operation more stable even with the smaller fraction of the quench gas.
• Installation of silicon detectors
Silicon (Si) detectors were used as ancillary calorimetric detectors for high-energy recoil
particles, which don’t stop in the sensitive volume of MAIKo. The Si detectors shouldbe placed as close to the TPC as possible to reduce the insensitive region. However,
eventual discharges between the Si detectors and the TPC took place which damaged
the Si detectors and the readout circuits. The discharge occurred especially when the
TPC was irradiated with high-rate beam particles. We developed a dedicated structure
to protect the Si detectors from the discharge.
• Track reconstruction algorithm
The MAIKo TPC provides the trajectories of the charged particles as two sets of
two-dimensional images from which the recoil alpha particles must be identified. We
developed a track reconstruction algorithm based on the Hough transformation.
• Reliability of the active target method
Compared to the cooler ring method, the detection and analysis of the scattering eventswith the TPC is complicated. These might cause considerable ambiguity in measuringcross sections. We verified the reliability of the MAIKo active target by comparing
the cross sections of the 12 C+α elastic scattering with those obtained in the previous
experiment.
During the development, several test experiments were performed at RCNP using alphasources, 4 He beam at 56 MeV, and 13 C beam at 60 MeV/u. The developments and resultsof the test experiments have been reported in Refs. [41, 43–49].

1.4.2 Examples of application of MAIKo
MAIKo was designed as a versatile detector system to perform various missing-mass spectroscopies using RI beams at several tens MeV/u at RCNP and RIKEN. In the following,some of the planed applications are presented.
–6–CHAPTER 1. INTRODUCTION
1.4.2.1 Shell structures in unstable nuclei
The magic numbers in nuclei were theoretically established by M. G. Mayer and J. H. Jensen
more than 50 years ago [50, 51], which resulted in the Novel Prize in Physics to them in
1963. The nuclear shell model is quite useful to describe various observables such as energy,
spin-parity, magnetic dipole and electric quadrupole moments, transition matrix elements,
etc. In principle, we must solve the A-body Schrödinger equation in a vast configuration
space to calculate these observables. However, the number of the configurations in thecalculation explosively increases with mass numbers. The nuclear shell model with a conceptof the magic numbers remarkably reduces the number of the configurations by truncating
the configuration space for the closed shells.
The nuclear shell structure is still of great interest because it has been reported that theshell structures change in unstable nuclei with unbalanced proton and neutron numbers andthe conventional magic numbers (2, 8, 20, 28, 50, 82, and 126) sometimes are not valid insuch nuclei. It was found that the magic numbers at N =8, 20, and 28 are less distinct
in the neutron-rich regions [52–59]. On the other hand, the emergence of the new magic
numbers at N = 16 in 24 O [60], N = 32 from Ar to Cr isotopes [61–66], and N = 34 in 54 Ca[67–69] were reported. Such appearance and disappearance of magic numbers are of great
importance to calculate the properties in unstable nuclei.
There exist many physical observables to pin-down the magic numbers in nuclei: binding
energy, nucleon separation energy, energy of the first excited state, and number of isotopes
or isotones. As we will discuss in Sec. 1.6, the quadrupole transition strength is also a usefulobservable. It can be deduced by measuring the inelastic scattering of unstable nuclei with
protons or alpha particles with the MAIKo active target.
1.4.2.2 Search for alpha molecular structures
One application is a search for alpha molecular structures in unstable nuclei. The groundstates in self-conjugate A = 4n nuclei can be reasonably described with the shell-model
picture and have compact structures as illustrated in the left side of Fig. 1.4. On the other
hand, well-developed alpha cluster structures are expected to emerge at excitation energies
close to their alpha-decay threshold energies. This speculation is based on the well-known“threshold rule” by Ikeda et al [70]. One of the most famous alpha cluster states is the 0+
2state at Ex = 7.65 MeV in 12 C. This state is known as the Hoyle state which located atonly 0.38 MeV above the 3α decay threshold in 12 C. The Hoyle state has a well-developed
3α cluster structure as illustrated in the middle of Fig. 1.4 and plays a crucial role for the
carbon synthesis in the universe [71].
Recently, several theoretical calculations predict that there exists a new type of cluster
structures called “molecular structures” when excess nucleons are added to the stable self
conjugate A = 4n system. For example, in 10 Be where two neutrons are added to 8 Be (=2α)
core, a unique α-2n-α structure is predicted by the antisymmetrized molecular dynamics(AMD) model [72, 73] as illustrated in the right side of Fig. 1.4. The excess neutrons play–7–

1.5 Physics case in the present work: magicity at Z = 6
a role of covalent particles in the molecular orbitals formed by two alpha cores just like twoelectrons shared by two hydrogen atoms to form a hydrogen molecule.
It is pointed out that the well-developed alpha cluster states are strongly excited byisoscalar monopole transitions [74, 75]. The strength of the isoscalar transition from the+ground (0+1 ) state to a excited (0ν ) state is defined as
B(IS) ≡ ⟨0+ν ||∑r2 ||0+
1 ⟩,

(1.1)

A

where the summation takes over all of the nucleons. To search for the alpha molecular structures in the Be isotopes, a new measurement of the monopole transition strengths is desired.
The alpha inelastic scattering is a useful prove to measure the monopole transition strengths
because of its selectivity to isoscalar natural-parity transitions in which the transferred spin
and isospin are zero. Since the diﬀerential cross section of the monopole transition becomesmaximum at θc.m. ∼ 0◦ , the measurement should be performed at the forward angles. It
requires the detection of low-energy recoil alpha particles from the kinematics. Therefore,
the MAIKo active target is indispensable for this measurement.

12C g.s.

12C Hoyle

10Be molecular

Figure 1.4: Structure of the ground state in 12 C (left), Hoyle state in 12 C (middle), andmolecular state in 10 Be (right).

1.4.3 The First experiment with MAIKo
In the present thesis, we report the results of the first physics experiment by using the
MAIKo active target [76]. We aimed to perform the missing-mass spectroscopy of the α+10 C
scattering to investigate the recently proposed new magicity at Z = 6. The quadrupole
10 C was measured. The physics
transition strength from the ground state to the 2+
1 state inmotivation of the present work is described in the next section.

1.5 Physics case in the present work: magicity at
Z=6Recently, a new magicity at Z = 6 was proposed in neutron-rich carbon isotopes [77]. The
proton radii in 14 Be, 12−17 B, and 12−19 C were systematically estimated by measuring the–8–CHAPTER 1. INTRODUCTION
charge-changing cross sections on a carbon target. The obtained proton radii of nuclei
around Z = 6 are shown in Fig. 1.5(a). Note that the proton radii in the figure werenormalized by the following empirical formula for the proton radii [78, 79]:
Rpcal =√
3/5(1.15 + 1.80A−2/3 − 1.20A−4/3 )A1/3 fm(1.2)

in order to eliminate the smooth mass-number dependence. The authors found that the
proton radii show a clear kink at Z = 6 which is similar to those observed at Z =20, 28, 50,
and 82 [80]. This indicates a possible major structural change, for example, emergence of ashell closure at Z = 6.
The authors also examined the reduced electric quadrupole transition rate B(E2; 0+
1 →
+
+
+
21 ) in neutron-rich carbon isotopes. The B(E2; 01 → 21 ) value represents the contributionof protons to the quadrupole excitation from the ground state to the 2+
1 state and should besmall in proton-closed shell nuclei. Figure 1.5(b) shows the systematics of the B(E2; 0+
1 →+21 ) values in the various isotonic chains as a function of proton number. It should be noted+that the B(E2; 0+
1 → 21 ) values become minimum at Z = 6 in the N =8, 10, 12, and 14
isotones. This fact suggests the magicity at Z = 6 in the neutron-rich nuclei.
To further examine the subshell closure at Z = 6, the authors investigated the second
derivative of binding energies defined as [81]∆p (N, Z) ≡ (−1)Z [Sp (N, Z) − Sp (N, Z + 1)]/2,(1.3)

where Sp (N, Z) represents the one-proton separation energy. 2∆p (N, Z) gives the proton
single-particle energy-level spacing or shell gap between the last occupied (ep ) and firstoccupied proton orbitals (ep+1 ). To eliminate the eﬀect of proton-proton (p-p) paring, theysubtracted out the p-p paring energies using the empirical formula: ∆p-p = 12A−1/2 MeV.
Figures 1.5(c) and (d) show the systematics of ep − ep+1 ≡ 2∆p (N, Z) − 2∆p-p for even-Z
nuclei. The cups at Z = N for all of the isotonic chains are due to the Wigner eﬀect [82].
Apart from the Z = N nuclei, sizable enhances were observed at Z = 6 in N =7–14 as well
as at Z = 8 in N =8–10 and N =12–16, which is also a signature of the magicity at Z = 6.
Combining the three results, the authors claimed the shell closure at Z = 6 in theneutron-rich 13−20 C isotopes. Actually, the idea of the possible magic number at six was
already pointed out by Mayer. In her Novel lecture in 1963 [83], she mentioned: “There
are two diﬀerent series of numbers, 2, 8, 20, 40..., of which 40 is no longer noticeable, andanother, 6, 14, 28, 50, 82, 126 of which the first two at 6 and 14 are hardly noticeable. The
second series is due to spin-orbit coupling.” Although Mayer pointed out that the magic
number six is hardly noticeable, the results in Ref. [77] shows that the eﬀect of the magicity
might be a reality, at least in the neutron-rich region.
It is a natural question whether the magicity at Z = 6 reported in the neutron-richisotopes persists even in the proton-rich side. Thus, experimental information on the protonrich carbon isotopes, for example 10 C, is of importance. To test the Z = 6 magicity in 10 C,
we investigate its quadrupole transition matrix elements from the ground state to the 2+
1–9–

1.5 Physics case in the present work: magicity at Z = 6
state. In the next section, we discuss the relation between the magicity and the quadrupoletransition matrix element.

Rp(Z )/Rpcal

a

1.05
1.00
0.95
0.90
0.85
25

B(E2) (W.u.)

b

20

c

10

15
10
5

ep–ep+1 (MeV)

8

3
4
5

6

N isotones
12
6
9
13
7
10
14
8
11

15
16

4
2
0

5

10

15

Z

20

5

25

nu
m

utr

15
10
Proton nu
mber

Ne

20

on

15
20

be

5
10

r

10
8
6
4
2
0

Z=
8
Z=
6

ep–ep+1 (MeV)d0

Figure 1.5: Proton-number dependencies of nuclear observables with diﬀerent isotones with
the neutron number N . (a): Proton radii deduced from the charge-changing cross section
which are normalized with the value in Eq. (1.2). (b): The reduced electric quadrupoletransition rate from the ground states to the 2+
1 states. (c) and (d): Second derivatives ofbinding energies subtracted by the p-p paring energies. The positions at Z = 6, 8, 20 (onlyin (c)) are indicated by the black dashed and dotted lines, respectively. The figures are takenfrom Fig. 3 in Ref. [77].

–10–CHAPTER 1. INTRODUCTION

1.6 Quadrupole transition matrix element
The quadrupole transitions between nuclear states provide valuable information on nuclear
structures. Their strengths play key benchmarks in testing theoretical models because theygive the overlaps of the wave functions between the ground state and the excited states. In
even-even nuclei, the neutron (proton) transition matrix element from the ground (0+1 ) state+
to the 21 state defined as∑
Mn(p) ≡ ⟨2+
r2 Y2 ||0+
(1.4)
1 ||
1 ⟩,
n(p)

is one of useful observables to investigate nuclear shell structures [84–86].
If one considers proton as a point particle, Mp can be related to the reduced electric+
quadrupole transition rate B(E2; 0+
1 → 21 ) by the following formula:+2
2
B(E2; 0+1 → 21 ) = e |Mp | .

(1.5)

+
+
+
The relationship between excitation (0+1 → 21 ) and deexcitation (21 → 01 ) reduced transition rates are given by an equation+
B(E2; 0+1 → 21 ) =2J ′ + 1+++
B(E2; 2+
1 → 01 ) = 5B(E2; 21 → 01 ),2J + 1(1.6)+
where J and J ′ are the spins of the 0+
1 and 21 states.

In a naive liquid-drop model, the ratio of the neutron and proton matrix elements isgiven as [85]
N δn
Mn
=
.
(1.7)
Mp
Zδp
Here, N and Z are neutron and proton numbers, respectively. δn(p) is the nuclear deformation
length of neutron (proton) and it is related to the deformation parameter βn(p) as3
δn(p) =
2√5β.
4π n(p)(1.8)

Assuming that the density distributions of proton and neutron in a nucleus are similar(δn = δp = δ), the ratio simply yields
Mn /Mp = N/Z.

(1.9)

+
In the nuclear shell model, the quadrupole transitions from the 0+
1 state to the 21 state
are described as rearrangements of particle-hole configurations in the valence shells. Whenthe neutron (proton) shell is closed, the transition of the neutrons (protons) is suppressed
remarkably because the intra-shell excitation in the closed shell is hard to occur energetically. As a consequence, the Mn /Mp ratio becomes significantly smaller (larger) than the
expectation of Eq. (1.7). After the construction of the RI beam facilities, numerous eﬀorts–11–

1.7 Previous experiments on 10 C
have been devoted to deduce transition matrix elements in unstable nuclei and clarify shellstructures [87–112].
The proton transition matrix element Mp can be determined directly by using the electromagnetic probes such as electron inelastic scattering or gamma decay measurement fromthe excited states to the ground states. However, there exists no direct way to determine Mn
because no probe is sensitive to the neutrons only. One method to obtain Mn in a nucleus
is to adopt Mp of its mirror nucleus relying on the charge symmetry. The other method
is to measure the cross section of the inelastic scattering to the 2+
1 state using a hadronic
probe such as proton or alpha particle. Since the cross section of the inelastic scattering issensitive to both Mn and Mp , the Mn value can be estimated from a combined analysis of
the measurements using the electromagnetic probe and hadronic probe.

1.7 Previous experiments on 10C
10 C locates at E = 3.35 MeV [113]. The B(E2; 0+ → 2+ ) value in 10 C had
The 2+x
1
1 state in1
been known to be 9.6 ± 1.6 Weisskopf units (W.u.) [114]. This value is not small comparedto those in the neutron-rich carbon isotopes, for example, 1.7 W.u. in 16 C [100] and 1.5 W.u.
in 18 C [101]. In order to investigate whether Z = 6 shell closure persists in the proton-richcarbon isotopes, the Mn /Mp ratio in 10 C should be determined first If the shell closure in
10 C is distinct, the ratio will be larger than the expectation of Eq. (1.7) or (1.9). In 16 C,
the Mn /Mp ratio of 3, which is much larger than N/Z = 1.7, was reported [92, 97, 101].
To deduce the Mn value in 10 C, the diﬀerential cross sections of the p+10 C elastic and
10 C beam
inelastic scattering exciting the 2+
1 state at Ex = 3.35 MeV were measured using a
at 45 MeV/u impinged on a polypropylene (CH2 CHCH3 )n target [94]. The cross section of
the inelastic scattering was measured at 20◦ < θc.m. < 50◦ as shown by the solid circles inFig. 1.6.
In this experiment, the missing-mass spectroscopy was performed by detecting the recoil
protons with a semiconductor detector system named MUST [115]. In case of the proton
inelastic scattering, the diﬀerential cross section becomes maximum around θc.m. ∼ 35◦ . The
recoil energy of the proton emitted to this angle is approximately 13 MeV, and they are easily
detected because they can escape from the target material and reach the MUST detector.
The authors performed a distorted-wave Born approximation (DWBA) calculation using
the Jeukenne-Lejeune-Mahaux (JLM) potential which is based on infinite nuclear matter
calculation [116]. By comparing the measured cross section with the DWBA calculation(solid line in Fig. 1.6), the authors in Ref. [94] determined the Mn /Mp ratio to be 0.70±0.08.
This small value suggests that the shell closure eﬀect at Z = 6 is not distinct because thelarge value is expected if the proton shell is closed.
+
The authors took the B(E2; 0+
1 → 21 ) value of 9.6 ± 1.6 W.u. from Ref. [114] in their
10 C, however, reported aanalysis. A more recent lifetime measurement of the 2+
1 state in+smaller value of B(E2; 0+
1 → 21 ) = 6.9 ± 0.2 W.u. [117] with much smaller uncertainty+
compared to the previous measurement. The smaller value of B(E2; 0+
1 → 21 ) might lead

–12–

CHAPTER 1. INTRODUCTION

Table 1.2: Ratios of the external-field interaction strengths bFn /bFp for various probes at
diﬀerent incident energies [84].
External field

EnergybFn /bFp

Electromagnetic
ProtonProtonProtonNeutron
Alpha
π−
π+

All
10-50 MeV
0.8 GeV
1 GeV
10-50 MeVAll
160-200 MeV
160-200 MeV

0
3
0.83
0.95
1/3
1
3
1/3

to the larger value of Mn , thus the Mn /Mp ratio can become larger.
To disentangle the neutron transition matrix element from the measured cross section of
the inelastic hadron scattering, the Bernstein prescription is widely used [84]:dσ∝ |bFn Mn + bFp Mp |2 .
dΩ

(1.10)

Here, bFn and bFp are external-field interaction strengths which reflect the eﬀective interaction
between an incident particle and a proton or neutron in nuclei. The values of bFn and bFp
depend on the incident energy of the probe and the target nucleus. The presently accepted
bFn /bFp ratios for various probes are presented in Table 1.2 [84]. In the case of the proton
inelastic scattering, the ratio bn /bp phenomenologically determined has a strong energy dependence varying from 0.83 to 3 in the incident-energy range of 10–1000 MeV, and it also
depends on nuclides. These dependencies cause a serious model ambiguity in the determination of the Mn /Mp ratio from the measured cross section [112]. On the other hand, in thecase of the alpha inelastic scattering, the bFn /bFp ratio is always 1 because of the isospin zeronature of an alpha particle. Thus, the alpha inelastic scattering is more suitable to deduce
Mn than the proton inelastic scattering.

1.8 New experiment on 10C
1.8.1 Overview
In the present work, the cross sections of the elastic and inelastic alpha scattering on 10 C
were measured with the MAIKo active target. The eﬀective α-N interaction and the pointnucleon density distribution in the ground state in 10 C were determined to reproduce thecross sections of the alpha elastic scatterings. Using the eﬀective interaction and the density–13–

dσ/dΩ(mb/sr)

1.8 New experiment on 10 C

10,2+ 3.35 MeV

p( C,p )
10
9
8
7
6

JLM
fixed Mp = exp Mp

54

Mn/Mp = 0.70

3

Mn/Mp = N/Z = 0.66
10

15

20

25

30

35

40

45

50

55

60

Θc.m.(deg)Figure 1.6: Diﬀerential cross section of the proton inelastic scattering on 10 C exciting the2+
1 state at Ex = 3.35 MeV at a beam energy of 45 MeV/u. The solid and dashed lines
represent the DWBA calculation with the JLM potential [116], assuming the Mn /Mp valuesof 0.70 and 0.66 (=N/Z), respectively. The figure was taken from Fig. 17 in Ref. [94].

distribution, the DWBA calculation was performed to deduce the Mn value from the measured cross section of the alpha inelastic scattering without ambiguity of the parameters in
the Bernstein prescription.
Although the cross section of the inelastic scattering could be measured by detecting the
decay gamma rays from the 2+
1 state to the ground state, this technique cannot be applied tomeasure the cross section of the elastic scattering which is needed to determine the eﬀective
α-N interaction and the point-nucleon density distribution. Therefore, the missing-mass
spectroscopy should be performed to measure both the elastic and inelastic scattering crosssections.

1.8.2 Kinematic condition
Figure 1.7 shows the diﬀerential cross sections of the alpha inelastic scattering on 10 C at
68 MeV/u with diﬀerent angular momentum transfers ∆L calculated with the DWBA. The
adopted energy of the 10 C beam was set at the highest energy available at RCNP to make theintensity as high as possible. The DWBA calculation was carried out with the single-folding
model potentials (see Chap. 5) assuming that an excited state at Ex = 3.35 MeV exhausts–14–CHAPTER 1. INTRODUCTION
100% of the energy-weighted sum-rule strengths. This figure suggests that the measurement◦of the ∆L = 2 transition exciting the 2+
1 state should cover θc.m. ∼ 7 where the diﬀerential
cross section takes the maximum. The energy of the recoil alpha particle in the inverse
kinematic scattering is plotted as a function of the recoil angle in the laboratory frame in
Fig. 1.8. The blue, orange, green, and red lines show the correlation between the energy andangle at the excitation energies of 0, 5, 10, and 15 MeV, respectively. The c.m. angles areplotted by the black solid circles. At θc.m. ∼ 7◦ , the energy of the recoil alpha particles is as
low as 2 MeV and it is hard to detect such a low energy alpha particles in the conventional
experimental setup. Since the half life of 10 C is only 19 seconds, the cooler ring method
cannot be applied. Therefore, MAIKo is a unique device to realize the present measurement.

α+10C 68 MeV/u, E x =3.35 MeV
∆L =0
∆L =1
∆L =2
∆L =3
∆L =4

d σ /d Ω (mb/sr)

103

102

10

1
0

2

4

6

8

10 12 14 16 18 20

θ c.m. (deg)Figure 1.7: Diﬀerential cross sections of the α+10 C inelastic scattering at 68 MeV/u calculated by the DWBA. The blue, orange, green, red, and purple lines represent the crosssections with diﬀerent transferred angular momenta ∆L.

1.9 Contents of the thesis
The present thesis is organized as follows. Firstly, the detailed design of the MAIKo system
is described in Chap. 2. The experimental details such as experimental apparatus, dataacquisition, and summary of the measurement are described in Chap. 3. The analysis of
the MAIKo TPC data is presented in detail in Chap. 4. The DWBA analysis to deduce Mnfrom the measured cross section is described in Chap. 5. The results are discussed in Chap.
–15–

1.9 Contents of the thesis

θ c.m. = 21°

θ c.m. = 15°

10

θ c.m. = 9°

1

10

Me

−1

0

=5

Ex

V

0
=1

Ex = 0 MeV

5 MeV

Ex = 1

MeVθ c.m. = 3°

Ex

Recoil α energy (MeV)

α+10C 68 MeV/u

10 20 30 40 50 60 70 80 90 100

Recoil α angle (deg)Figure 1.8: Energy of recoil alpha particle as a function of the recoil angle in the α+10 C
scattering at 68 MeV/u with diﬀerent excitation energies in 10 C.

6. Finally, the conclusions and future outlooks are given in Chap. 7.

–16–

2 DESIGN OF MAIKO
2.1 Overview
The schematic view of the MAIKo active target is shown in Fig. 1.3. The system is based ona cubic-shape TPC detector . The volume of the TPC field cage is 150 × 150 × 140 mm3 andthe sensitive volume is 102.4 × 102.4 × 140 mm3 . The detector is installed in a stainless-steel
vacuum chamber with a volume of approximately 30 L. The chamber is filled with heliumgas which plays a role of a helium target. A small fraction of quenching gas is added to thehelium gas for the stable operation of the TPC. The pressure of the TPC gas can be changed
from 100 to 2000 hPa depending on the experimental requirements.
The charged particles ionize the mixture gas along their trajectories in the TPC and theionization electrons drift vertically downwards along the electric field formed by the TPC field
cage. At bottom of the TPC, the electrons are multiplied first by a gas electron multiplier(GEM) and then by the µ-PIC. The µ-PIC is also used to determine the two-dimensional
position of the electrons. The electron drift time multiplied by the drift velocity determine
the vertical position of the primary electrons. These information was used to determine thethree-dimensional position of the charged particles.
The angle of the recoil particles is determined from the reconstructed trajectory. If therecoil particles stop inside the TPC sensitive volume, the recoil energy is determined fromthe length of the trajectory in the gas. High-energy recoil particles which pass through the
TPC are detected with four silicon detectors installed at the left and right sides of the TPC.

2.2 TPC field cage
The electric field inside the sensitive volume of the TPC is formed by applying negative highvoltages on the stainless-steel cathode plate and the nickel grid mesh. Figure 2.2 shows theelectric circuit of the field cage. The cathode plate has a dimension of 150 × 150 × 3 mm2 .
The surface of the plate is buﬀ-polished to avoid an electric discharge. The wires of the gridmesh have a diameter of 150 µm and its pitch size is 0.85 mm (30 wires per inch). The
grid mesh is glued on a frame made of Macor. Macor is a machinable glass-ceramic and it
is a low-outgassing material. The high voltage applied to the grid mesh is tuned so that itis transparent to the electrons but opaque for the positive ions generated by the avalancheat the GEM and the µ-PIC. The distance between the cathode plate and the grid mesh is
kept at 140 mm with four pillars made from Macor. Field wires with diameters of 125 µm
are wound doubly around the pillars to make the drift field uniform. The pitch between the17

2.2 TPC field cage
wires is 5 mm. They are connected via 10-MΩ metal-film resistor1 chain to make a voltage
divider. The precision of the resistance value is 0.1%. The tension of the field wires is kept
at 2 N.
A finite element simulation with the computer codes Garfield [118] and neBEM [119] wasperformed to calculate the uniformity of the electric field. The calculation was carried out
including the Si detectors which may worsen the uniformity of the electric field. Figure 2.1
shows the result of the calculation. The color map in the figure shows the distortion of thevertical electric field at the middle height of the TPC field cage (y = 70 mm). The distortion∆Ey at a point (x, y, z) is defined as:∆Ey (x, y, z) =Ey (x, y, z) − Ey (0, y, 0),
Ey (0, y, 0)(2.1)

where the origin of the coordinate is the center of the µ-PIC sensitive surface. The boundaryof the µ-PIC sensitive region is shown by the black solid line. The Si detectors are indicatedby the blue solid lines. The direction of the incident beam is shown by the black dashedarrow. According to the simulation, the present field cage ensures that the distortion of theelectric field is kept smaller than 0.4% within the sensitive volume.

150

0

100

−0.2

z (mm)

50

−0.4

0
−50

−0.6
Si

Si
−0.8

−100
−150
−150 −100

−50

0

50

100

150

Field distortion (%)

Beam

−1

x (mm)
Figure 2.1: Distortion of the vertical electric field with respect to the field at the centerof the TPC. The distortion is defined by Eq. (2.1). The calculation was performed using
computer codes Garfield [118] and neBEM [119].
1

SM5 by JAPAN FINECHEM COMPANY, INC., Tokyo, Japan–18–

CHAPTER 2. DESIGN OF MAIKO
The negative high voltage of the cathode plate is applied via a low-pass filter which
consists of a 1000 kΩ resistor and a 1000 pF capacitor. This low-pass filter suppresses
the ripple of the high voltage module2 for the cathode plate to reduce the noise on the Si
detectors. Since the measured ripple frequency of the high voltage module was about 60
kHz, the cutoﬀ frequency of the low-pass filter was determined to be fc = 1/(2πCf Rf ) = 1.6
kHz.

Low-pass filter

Cathode plate

100 kΩ

Vp

10 MΩ

140 mm1000 pF

Field wire

10 MΩ

~
~

10 MΩ

10 MΩ

5 mm

Vg

Grid mesh

VGEMt

10 MΩ
10 MΩ

5 mm

GEM

10 MΩ

1 MΩ
1 MΩ

VGEMb

Cathode

μ-PIC
Anode
Vμ

Figure 2.2: Electric circuit of the field cage, GEM, and µ-PIC.

2.3 Si detectors
The four Si detectors are placed at 30 mm away from the TPC field cage to detect highenergy recoil particles. Two Si detectors are installed at the left side of the TPC and theother two Si detectors are installed at the right side. Figure 2.3 shows a photograph of the
two Si detectors. Each Si detector has a sensitive area of 90 × 60 mm2 and a thickness of 500µm. These Si detectors can stop alpha particles with energies up to 33 MeV if the particles
hit the detectors perpendicularly.
2

HARb-15N by Matsusada Precision Inc., Shiga, Japan

–19–

2.4 µ-PIC
The surfaces of the Si detectors are grounded while the cathode plate is applied highvoltage (typically −5 kV). Since the shortest distance from the Si detectors to the cathodeplate is as short as 30 mm, we should take care with the electric discharge between them. If
an electric discharge occurs, a large current flows in the Si detectors which may damage the
Si detectors and the readout circuits. Thus the Si detectors are protected from the electric
discharge by putting an electrostatic shield between the cathode plate and the Si detectors.
Since the electrostatic shield is grounded, the discharge occurs not to the Si detectors but
to the shield. The electrostatic shield is made of the Cu-Be wires with a diameters of 125µm soldered on a glass epoxy (grade FR4) frame with a pitch of 5 mm as shown in Fig.
2.4. These wires are connected to the vacuum chamber through the red cable shown in the
figure. During the measurement with the 10 C beam, several discharges occurs, but none of
the Si detectors were damaged.

Wire

Figure 2.3: Photograph of two Si detectors. Figure 2.4: Electrostatic shield made of Cu-Be
wires to protect the Si detectors.

2.4 µ-PIC
The drift electrons are multiplied by the µ-PIC placed 10 mm below of the grid mesh. Theµ-PIC also provides the two-dimensional position information of the electrons.
The µ-PIC was developed in 2001 [120] and continuously under improvement by thecosmic-ray group at Kyoto University. It is one of micro pattern gaseous detectors (MPGD).
It achieves high position resolution owing to a fine readout pitch while the number of thereadout channels is relatively small. The µ-PIC has been successfully employed in Compton
cameras for gamma-ray imaging [121–124], in the dark matter search experiment [125, 126],
and in neutron imagings [127, 128].
A schematic structure of the µ-PIC is shown in Fig. 2.5. The µ-PIC has a sensitive
area of 102.4 × 102.4 mm2 . It is constructed using a print circuit board (PCB) technology.
The µ-PIC consists of 256 cathode electrode strips, each with a width of 314 µm. They are
fabricated on a 100-µm-thick polyimide base at 400-µm intervals. Each cathode strip has–20–

CHAPTER 2. DESIGN OF MAIKO
256 holes along the strip, each with a 256-µm diameter, arranged at equal intervals of 400µm. At the center of each cathode hole, an anode pixel of 50-µm diameter is fabricated.
The anode pixels are extended to the back side of the base and connected to form the anodestrips which are perpendicular to the cathode strips. A positive high voltage is applied tothe anode strips while the cathode strips are grounded. A strong electric field, enough tooccur an electron avalanche, is formed around the anode pixels.
Each strip of the anode or cathode is connected to the readout board to provide a twodimensional image of the charged particles. The number of the readout channels is 256 forthe anode strips and 256 for the cathode strips.
314 µm400µ

m400 µm

100 µm256 µAnode Pixeld = 50 µmm

Cathode Strip
Anode Strip

Figure 2.5: Schematic structure of the µ-PIC.

2.5 GEMA GEM is installed between the grid mesh and the µ-PIC for the pre-amplification of theelectrons. Figure 2.6 shows an photograph of a GEM [129]. The GEM is a thin polymer
sheet whose from and back sides are coated with thin copper layers. The sheet is chemically
perforated by holes with a high density, typically 100 holes per square mm [130]. A highvoltage of typically 300 V is applied between the front and the back sides of the GEM whichforms a strong electric field in the holes. The electrons enter the holes and are multiplied by
the strong electric field. The GEM with a sensitive area of 100 × 100 mm2 , which is almost
same as that of the µ-PIC, was commercially purchased3 and mounted on a frame made ofMacor to install in MAIKo. The hole geometry of the GEM is the so-called standard GEM:
diameter of 70 µm and 140 µm apart in a triangular pattern. The thickness of the GEM
is 100 µm, which is twice as thick as the standard one and achieves higher gas gain. The
3

Manufactured by SciEnergy Inc., Kanagawa, Japan.

–21–

2.6 Readout electronics
electric circuit to apply high voltages on the GEM electrodes is shown in Fig. 2.2.
The two-stage electron amplification with the GEM and µ-PIC contributes to the stable
operation of the detector system because it reduces the discharge probabilities. The electricfields formed around the GEM or µ-PIC can be smaller compared to that in the amplification
system with the µ-PIC only, while keeping the total gas gain high. As reported in Ref. [41],the operation of the TPC without the GEM was unstable during a test measurement with
the He(93%)+CO2 (7%) gas at 430 hPa due to the electric discharges of the µ-PIC. After thetest measurement, we introduced the GEM to the MAIKo TPC and the operation without
any discharge of the GEM or µ-PIC became possible even with the smaller fraction of theCO2 gas.

Figure 2.6: Photograph of a GEM. Taken from Ref. [129].

2.6 Readout electronics
The anode and cathode strips of the µ-PIC are connected to the capacitor and resistor (CR)circuit boards. The CR circuits consist of 1-GΩ resistors and 100-pF capacitors. Figure 2.7
shows a photograph of the CR circuit board. The CR circuit boards are mounted on the
TPC chamber and directly connected to the µ-PIC board in the TPC chamber via a hole on
the top flange of the chamber. The CR circuit boards are sandwiched between the top flange
and another blind flange with O-rings to seal the TPC gas, i.e., these CR circuit boards
work also as the vacuum feed-through for the µ-PIC analog signals.
The analog signals from the µ-PIC are processed by the dedicated readout electronics
boards called Iwaki board [131]. The Iwaki boards are connected on the CR circuit boards
as shown in Fig. 2.7. Each Iwaki board processes signals from 128 channels, thus the four
Iwaki boards are used in MAIKo: two boards for the 256 anode strips and the other twoboards for the 256 cathode strips.
A block diagram of the signal processing in the Iwaki board is shown in Fig. 2.8. The–22–

CHAPTER 2. DESIGN OF MAIKO

Iwaki
board
Anode

Iwaki
board
CathodeBeam
CR circuit
CathodeFigure 2.7: Photograph of the CR circuit board and the Iwaki boards. The CR circuitboard for the cathode strips is circled by the green dashed lines and the Iwaki boards arecircled by the blue solid lines.

Table 2.1: Performances of the FE2009bal ASIC.
Number of channels
Peaking time
Gain of the amplifier
Dynamic range
Cross talk
Noise
Power consumption

16
30 ns
800 mV/pC−1 to +1 pC< 0.5%∼ 6000 electrons
18 mW/ch

input signals from 128 strips on the µ-PIC are preamplified, shaped, and discriminatedby the eight application specific integrated circuit (ASIC) chips named FE2009bal [131].
Each FE2009bal chip processes 16 channels of the analog signal. The performances of the
FE2009bal chip are summarized in Table 2.1. The threshold levels for the individual channelsof the discriminators on the ASIC chips are remotely controlled via the SiTCP connection.
The high or low status of the discriminators is synchronized with a 100-MHz clock toprovide timing information of drift electrons. The ASIC chips also output the shaped analog
signals of the sum of the adjacent 32 strips. These sum of the analog signals can be read out
from the 4 LEMO connectors for the signal check or making the internal triggers for the dataacquisition. These analog signals are also digitized by 25-MHz 8 bit flash analog-to-digitalconverters (FADCs). The clock-synchronized status of the discriminators and the digitizedwaveforms of the analog signals are continuously recorded in a ring buﬀer implemented in–23–

2.7 Selection of TPC gas
a field-programmable gate-array (FPGA) chip for a duration of 10.24 µs. When a trigger
signal is provided to the Iwaki boards, the digitized data stored in the ring buﬀer are writtenout to two VME memory modules after the data formatting in the FPGA. The TPC data
are acquired event-by-event together with the data from the Si detectors and the otherbeamline detectors via the VME bus using the standard data acquisition system for RI
beam experiments at RCNP, which was originally developed at RIKEN [132].
Trigger

Ethernet
16 ch analog
signals

Thresholdcontrol...

FE2009 bal#0
128 ch digital
hit data

16 ch analog
signals

FE2009 bal#7

4 ch analog
sum signals

FPGA

SiTCP
Ring
buffer

FIFO

Dataformat

FIFO

VME
memoryFlash ADCs

LEMO
analog output

Figure 2.8: Block diagram of the signal processing in the Iwaki board.

By plotting the clock numbers when the discriminators are in the high status as a functionof the strip number, two black-and-white images with 256 × 1024 pixels are obtained. Each
image represents the particle trajectories projected onto the plane perpendicular to the anodeor cathode strips as shown in Fig. 2.9. Since the anode strips are perpendicular to the beamaxis and the cathode strips are parallel to the beam axis, the anode image gives the sideview of the particle trajectories while the cathode image gives the front view.

2.7 Selection of TPC gas
The TPC gas should be composed mainly of the He gas to perform the alpha inelastic
scattering inside the TPC. The operation without any quenching gas is ideal from a viewof the backgrounds, however, the MAIKo TPC could not be operated stably with the pure
helium gas due to the discharges at the cathode plate or the µ-PIC. Therefore, a quenchinggas should be added to the gas for the stable operation. Hydrocarbon gasses such as isobutane or methane gas have been widely used as a quenching gas in various gaseous radiation
detectors, but these gasses cause the background events due to the (p, p′ ) scattering. Because
the (α, α′ ) and (p, p′ ) scatterings are kinematically similar, distinguishing the (α, α′ ) eventsfrom the (p, p′ ) events is not easy although the background events due to the carbon are easily
distinguished from the (α, α′ ) scattering owing to the kinematic eﬀect. Another candidate
for the quenching gas is CO2 or CF4 . Considering that CO2 contains less number of atoms–24–

CHAPTER 2. DESIGN OF MAIKO

e

lan
ep

10

d
tho
a
C

10

Anode plane

Drift Time (µs)

8

6

4

t
Ca

Anode Strips

de
ho

St

s
ri p

Drift Time (µs)

8

6

4

15b0er
m
0
u
0
1 pN

2

200

250

ri
50 St
0

0

2

0

0

50

100

150

Strip Number

200

250

Figure 2.9: Three-dimensional particle trajectories in the MAIKo TPC and their projections
on the anode and cathode planes.

than CF4 , we chose the CO2 gas for the quenching gas. The fraction of the CO2 gas should
be as small as possible. We carried out test measurements and found that 4% is the minimum
mixing ratio to operate the TPC stably.

2.8 Gas handling system
We developed a gas handling system to flow the mixture gas and suppress the gas impurity
inside the TPC chamber. A schematic diagram of the MAIKo TPC gas handling system
is shown in Fig. 2.10. The route of the gas flow is indicated by the red thick solid lines.
The He and CO2 gasses were individually supplied into the TPC chamber with two mass
flow controllers4 (MFC) to keep the mixture ratio of 96:4 and the flow rate constant. The
total flow rate during the present α+10 C experiment was 100 cm3 /min. The pressure andtemperature inside the chamber were monitored with a diaphragm gauge5 and a Pt-100
thermometer6 , respectively. The humidity inside the TPC chamber was also measured using
a capacitive dew-point transmitter7 . The measured gas pressure, temperature, humidity,
4

HM1000 by TOKYO KEISO CO., LTD., Tokyo, Japan
VHR(G3) by VALCOM CO., LTD., Osaka, Japan
6
TN2531 by ifm electronics gmbh, Essen, Germany
7
Easidew Online by Michell Instruments Ltd., Cambridgeshire, England
5–25–

2.8 Gas handling system
and gas flow rate at the MFCs were digitized by analog-to-digital converter (ADC) modules
and periodically read out by a programmable logic controller8 (PLC). The mixture gas inside
the TPC chamber was continuously exhausted using a oil-free scroll pump9 . A piezo valve10
was installed between the pump and the TPC chamber. The PLC calculated the density of
the TPC gas from the pressure and temperature and a digital-to-analog converter (DAC)
module output a voltage level according to the gas density to the piezo valve controller. The
aperture of the piezo valve was automatically tuned by a proportional-integral-diﬀerential(PID) controller to keep the gas density constant referring to the DAC output.

He

MFC

CO2

MFC

Scrollpump

TPC chamber

Piezo valve

Pressure gauge
Dew-point gauge
Thermometer

Figure 2.10: Schematic diagram of the MAIKo TPC gas handling system. The gas flowroute is indicated by the red thick solid lines.

8

F3RP61 by Yokogawa Electric Corporation, Tokyo, JapanFOSSA FO by Kerr Pump & Supply Inc., Michigan, USA
10
PV-1501MC and PCU-2100 by HORIBA, Ltd., Kyoto, Japan
9–26–

3 EXPERIMENT
3.1 Primary beam
The first physics experiment (E463) to measure the α+10 C scattering was carried out atthe cyclotron facility of Research Center for Nuclear Physics (RCNP), Osaka University.
Figure 3.1 shows an overview of the RCNP cyclotron facility. The 10 C secondary beam
was produced through the projectile fragmentation reaction from a 12 C primary beam. The
primary beam from the super-conducting electron cyclotron resonance (SCECR) ion source
was first accelerated to 22 MeV/u with the K140 azimuthally varying field (AVF) cyclotron
and then further accelerated to 96 MeV/u with the K400 six-sector ring cyclotron. This
energy is the maximum energy with the ring cyclotron. The intensity of the 12 C beam was
50–100 pnA. The 12 C beam was transported to the exotic nuclei (EN) beamline to obtainthe 10 C secondary beam.

3.2 Production of the 10C secondary beam
The 12 C primary beam bombarded on a 9 Be production target with a thickness of 450
mg/cm2 at the F0 focal plane. The size of the beam spot at F0 was typically 1 mm indiameter. The 10 C nucleus was separated from other nuclei with the fragment separator at
the EN course.
Figure 3.2 shows a schematic layout of the EN course fragment separator. The beamline
was originally developed in 1990s [133, 134]. It was upgraded in 2013 [135] in which thebeamline was extended from the F2 focal plane to the F3 focal plane to improve the purityof the secondary beam. The fragment separator consists of two dipole magnets (D1 and
D2) and nine quadrupole magnets (Q1–Q9). Some of the basic properties of the EN course
fragment separator are listed in Table 3.1.
The separation of the secondary particles was performed by the following principles. The
secondary particles produced at the F0 focal plane were first analyzed by the D1 magnet.
After the D1 magnet, the particles satisfying Eq. (3.1) were selected at the first focal plane(F1).
Z
pnucleon
,
(3.1)
=
A
e(Bρ)D1
where Z and A are the proton and mass numbers, pnucleon is the momentum per nucleon,
and (Bρ)D1 is the magnetic rigidity of the D1 magnet. Here we note that the ions from the
projectile fragmentation are fully ionized in the present energy region, thus the electric chargeof the ions is equal to its atomic number Z. After the momentum selection, the fragments27

3.2 Production of the 10 C secondary beam

Table 3.1: Properties of the EN course fragment separator.

Configuration
Maximum magnetic rigidity
Bending radius
Bending angle
Energy degrader at F1
Energy dispersion at F1
Momentum dispersion at F1
Momentum acceptance to F2
Horizontal angular acceptance to F2
Vertical angular acceptance to F2
Horizontal magnification at F2
Vertical magnification at F2

Q-Q-Q-D-SX (F1, dispersive focus)
SX-Q-D-Q-Q-Q (F2, double achromatic focus)
Q-Q (F3, double achromatic focus)
3.4 Tm2.2 m
60◦
uniform thickness
8.66 mm/%
17 mm/%±4%±20 mrad±14 mrad2.0
1.4

Table 3.2: Parameters of the beamline during the 10 C measurement.
Parameter

Value

Magnetic rigidity of D1F1 degrader
F1 collimator
Magnetic rigidity of D2
F2 collimator

2.3 TmAluminum 2 mm
±6 mm
2.1 Tm±15 mmpenetrated a degrader at the F1 focus point. The energy loss through the degrader depends
on A and Z, thus the the magnetic rigidities of the particles are diﬀerent according to A and
Z. The particles are spatially separated after the D2 magnet. The particles at the second
focal plane (F2) were purified by a pair of horizontal collimators.
After the separation, the purified beam was focused onto the MAIKo active target by theQ8 and Q9 magnets. The Q8 and Q9 magnets were used to focus vertically and horizontally,
respectively. The energy spread of the secondary beam was determined by the width of theslits at F1. The parameters of the fragment separator during the present 10 C measurement
are summarized in Table 3.2. The average energy of the 10 C beam before MAIKo was 68
MeV/u and the energy spread was less than ±1%.
During the tuning of the secondary beam, the beam was detected, identified, and monitored with a 320-µm-thick silicon detector (F3 Si) installed at upstream of MAIKo at F3–28–

CHAPTER 3. EXPERIMENT
and a 1-mm-thick plastic scintillator (F3PL) placed downstream of MAIKo. The details of
the F3PL are described in Sec. 3.3.2. Figure 3.3 shows the correlation between the energy
loss of the beam particles in the F3 Si detector and the time of flight from the F0 target to
the F3PL. The time of flight was obtained from the timing information of the F3PL and theradio-frequency signal from the AVF cyclotron. This correlation was used for the particle
identification (PID) of the secondary beam. It is seen that 10 C is clearly separated from 12 C
and 11 C. Because the purity of the 10 C particle was as high as 96%, event-by-event PID was
not necessary during the measurement, and we did not use the F3 Si detector after the beamtuning.

–29–

3.2 Production of the 10 C secondary beam

EN
LAS

F0
WS

Grand Raiden

BLP-2

BLP-1

Ring
Cyclotron
N-BLP

0

50 m
superconducting
solenoid magnets

N
AVF
Cyclotron

Figure 3.1: Overview of the RCNP cyclotron facility.

–30–

CHAPTER 3. EXPERIMENTF2 PPACs

F2

Q7
Q6
Q5

Q4
2
SX

D2

F3

Q9
Q8

F2PL

10

C secondary beam
68 MeV/uF2 collimator

F1
1
SX

Al degrader
F1 collimator

F3 Si

MAIKo

D1

Q3
Q2

α

Q1

F0

MWDC

Aramid50 µm

9Be 450 mg/cm2

Si

collimator

F3PL

12C primary beam

96 MeV/u

Figure 3.2: Schematic view of the EN course fragment separator. The setup near the F3
focal plane is shown in the inset.

40
35

12

120

Energy loss (MeV)

C

30
25

11
10

C

C

100
80

20
60
15
40

10

20

5
0

110 120 130 140 150 160 170 180 190 200

0Time of flight (ns)Figure 3.3: Correlation between the energy loss in the F3 Si and the time of flight from theF0 target to the F3 plastic detector.

–31–

3.3 Beamline detectors

Table 3.3: Specifications of each MWDC.
Configuration
Sensitive area
Drift length
Gap
Anode wire
Potential wire
Cathode foil

X-X’-Y-Y’-X-X’-Y-Y’
77.5 mm (X)×77.5 mm (Y)
2.5 mm (Anode-Potential)
4.8 mm (Cathode-Cathode)Au plated W-Re (d =16 µm)
Au plated Al (d =80 µm)
Alminized Kapton (t =7.5 µm)

3.3 Beamline detectors
The setup at F3 is shown in the inset of Fig. 3.2. The incident trajectory of the 10 C
was monitored with the two low-pressure multi-wire drift chamber (MWDC) installed at
upstream of MAIKo. After the MWDCs, the 10 C beam was horizontally collimated by apair of tungsten collimators with thicknesses of 10 mm. The aperture of the collimators was
set at 20 mm. The intensity of the 10 C beam was measured with the F3PL installed atdownstream of MAIKo. Each detector is described in detail in the following sections.

3.3.1 Low-pressure MWDC
In the previous experiments at the EN course [136–140], the incident position of the beam wasmeasured using the parallel-plate avalanche counters (PPAC) [141]. However, the detection
eﬃciency of the PPACs for 10 C at 68 MeV/u was not good enough to monitor the beam
position. Therefore, we developed a low-pressure MWDC system at RCNP.
The system is an imitation of the similar system developed for the ESPRI experiment tomeasure the elastic scattering of protons with RI beams at RIKEN [142]. The system consistsof two sets of MWDCs, which are installed in a vacuum chamber. The relative distance ofthe two detectors is 600 mm and the distance from the downstream MWDC to the entranceof MAIKo TPC is 733 mm. The vacuum chamber of the MWDCs and the MAIKo chamber
was connected without any separation window films, thus the MWDCs were operated withthe same gas as the MAIKo TPC.
The specifications of the MWDC are summarized in Table 3.3. Each MWDC consistsof four planes (X-X’-Y-Y’-X-X’-Y-Y’). Structure of each plane is shown in Fig 3.4. Eachplane consists of 16 anode wires with a pitch of 5 mm. Between the anode wires, potential
wires are placed. These wires are sandwiched by the cathode planes with a gap of 4.8 mm.
Negative high voltages are applied to the potential wires and the cathode planes. Signals
from the 256 anode wires are amplified, shaped, and discriminated with 16 ASD cards. The
discriminator output from the ASD cards are recorded with two multi-hit time to digitalconverters (TDC).
–32–

CHAPTER 3. EXPERIMENT
CathodePotential wire
5.0 mm
4.8 mmAnode wire

CathodeFigure 3.4: Structure of each MWDC plane.

3.3.2 F3 plastic scintillator
We developed a plastic scintillator system (F3PL) to measure the intensity of the 10 C beam at
downstream of MAIKo. Figure 3.5 shows a photograph of the F3PL. The plastic scintillator
has a dimension of 100 mm in width, 110 mm in height, and 1 mm in thickness. The plastic
scintillator is installed inside a vacuum chamber which is connected to the MAIKo vacuum
chamber. It can be moved out from the beamline by hand. The lights emitted from thescintillator pass through the light guides installed inside the chamber and converted to anelectric signal by two photo-multiplier tubes (PMT) placed at the left and right sides. The
light guides are also used to separate the vacuum chamber volume from the outside air.
While the plastic scintillator is installed in the TPC gas atmosphere, the PMTs are placedoutside of the vacuum chamber to avoid exposure to the TPC gas containing He.

Movingsystem

PMT

PMT

Beam

Figure 3.5: Photograph of F3PL.

–33–

3.4 Data acquisition

3.4 Data acquisition
Since the recoil alpha particles from the α+10 C elastic and inelastic scattering at q ∼ 0.5
fm−1 are emitted at θlab ∼ 90◦ , the trigger signal for the data acquisition was made with
the Si detectors or the cathode strips of MAIKo. Figure 3.6 shows a schematic drawing of
the trigger circuit. As described in Sec. 2.6, the Iwaki readout boards output the shaped
analog signals of the sum of the 32 strips (8 channels for the anode strips and 8 channels forthe cathode strips). The trigger signal induced by the cathode strips was generated by to aleading edge discriminator (L.E.D.) module for the shaped analog signals. To suppress thetriggers due to the beam particles, Ch. 2, 3, and 4 of the shaped analog signals from thecathode strips were excluded from the trigger as shown in Fig. 3.7.
Iwaki
Cathode#0,1,5,6,7

L. E. D.

Si detectors

Pre amp.

5.2 µsDelay
TriggerIwaki boards

Shaper amp. trig. 10 µs
Delay
+ CFD
gate

TDC

ADC

VME memory

trig.
Transfer end

VME IRQ

Latch G.G.

Figure 3.6: Schematic diagram of the trigger circuit.

The analog signals from the Si detectors were amplified and shaped by a preamplifier
module and a shaping amplifier module, respectively The shaping amplifier module was also
equipped with a constant fraction discriminator module to output a trigger signal. The
trigger signal from the Si detectors and the cathode strips were summed and transmitted
to the Iwaki boards for the data acquisition (DAQ). After receiving the trigger signal, theIwaki boards transferred the TPC data to the VME memory modules. This procedure took
typically 100 µs depending on the data size. Each Iwaki board outputted a signal to latch agate generator (GG) module when it finished transferring the data. When all of the Iwaki
boards finished transferring the data, all of the four channels of the GG module were latched
and an interrupt signal was sent to the VME system.
The DAQ system read the data from the TPC and the other detectors event-by-event.
The VME memory module for the TPC had double-buﬀer memories. After the interruption,
the DAQ system first switched the buﬀer for writing data to the another one, and then readthe data of the Si detectors and the beamline detectors without reading the data of the
TPC. The used readout modules are listed in Table 3.4. After the readout, the DAQ system–34–

CHAPTER 3. EXPERIMENT

Si
0

255

0

Anode strips

Cathode#0

Beam

Cathode#2
Cathode#3
Cathode#4
Cathode#5
Cathode#6

Anode#7

Anode#6

Anode#5

Anode#4

Anode#3

Anode#2

Anode#1

Cathode#7
Anode#0

255

Cathode strips

Cathode#1

Si
Figure 3.7: Layout of the µ-PIC anode and cathode strips. The broad orange arrow shows
the direction and approximate width of the 10 C beam. The cathode analog signals in theshaded region were used to generate the trigger.

enabled the next interrupt signal. During waiting for the next interruption, the DAQ systemread out the TPC data. This treatment helped to reduce the dead time of the readout. Whenthe 10 C beam intensity was 77 kHz and the gas pressure was 500 hPa, the total trigger rate
was 270 Hz. The live time ratio of the data acquisition was 88%.

3.5 Operation of MAIKo
3.5.1 TPC gas
The MAIKo TPC was operated with the He(96%)+CO2 (4%) mixture gas. The recoil alphaenergy was determined from the range in the gas or the Si detectors. However, when therecoil alpha particles stop between the sensitive volume of the TPC and the Si detectors,
the recoil energy cannot be determined. Because the insensitive energy region depends onthe gas pressure, the TPC was operated at diﬀerent gas pressure, 500 and 1000 hPa. Whenthe gas pressure is 500 hPa, alpha particles with an energy of Eα < 1.0 MeV stop in thesensitive volume of the TPC and alpha particles with Eα > 4.0 MeV reach the Si detectors
with their energies high enough above the detection threshold. When the gas pressure is–35–

3.5 Operation of MAIKo

Table 3.4: List of the readout modules.
Module type

Module nameSi pre-amplifier
Si shaping amplifier
VME CPU
TPC memory
Si ADC
Si TDC
Plastic scintillator ADC
Plastic scintillator TDC
MWDC TDC
VME scaler
VME IRQ

Mesytec MSI-8
Mesytec MSCF-16
Abaco V7768
ARKUS TMB-2
Mesytec MADC-32
CAEN V1190A
QTC+V1190A
V1190A
V1190ACAEN V830
CAEN V977

Table 3.5: Operating conditions of the TPC at 500 and 1000 hPa.
Pressure (hPa)

Vp (V)Vg (V)VGRM t (V)VGEM b (V)Vµ (V)v (cm/µs)

500
1000

−4300
−7500

−820
−820

−616
−576

−276
−276

420
350

1.69(4)
1.69(7)1000 hPa, alpha particles with Eα < 3.6 MeV stop inside the TPC.
The vacuum chambers of the MAIKo TPC, MWDCs, and the F3PL detectors wereconnected without any separation materials and they were filled with the TPC gas. The gas
volume was separated from the upstream beamline with a 50-µm-thick aramid foil which was
installed at the entrance of the MWDC chamber. As described in Sec. 2.8, the gas densityof the chamber was kept constant. Figure 3.8 shows the monitored temperature (a), density(b), and concentration of moisture (c) during the operation at 500 hPa. The temperature of
the gas fluctuated about ±0.3% (±1 K) during the measurement. The gas density variation
due to this temperature fluctuation was compensated by adjusting the gas pressure inside
the chamber by the piezo valve. As the result, the variation of the gas density was only
0.08%. Although the moisture of the gas was gradually increased during the measurement,
it was kept as low as 220 ppm.

3.5.2 Measurement of the drift velocity
The applied high voltages to the cathode plate (Vp ), grid mesh (Vg ), top and bottom sidesof the GEM (VGEM t and VGEM b ), and µ-PIC anode (Vµ ) (see Fig. 2.2) in the measurements
at 500 and 1000 hPa are listed in Table 3.5.
–36–

CHAPTER 3. EXPERIMENT

Table 3.6: Measurement time for each beam particle and TPC gas pressure
Beam

Pressure (hPa)

Measurement time (hours)

10 C at 68 MeV/u

500
1000
500
1000

81
28
6
2

10 C at 68 MeV/u
12 C at 94 MeV/u
12 C at 94 MeV/u

Because the vertical position of the charged particles are determined by the electron drift
time multiplied by the electron drift velocity, the drift velocities at the conditions listed in
the table must be measured. The drift velocities were measured by using alpha particles
from an 241 Am standard alpha source. The alpha particles were collimated at 30 degrees
from the horizontal line, and injected to the TPC as illustrated in Fig. 3.9.
An example of the obtained anode images is shown in Fig. 3.10. The horizontal length(Lh ) of the trajectory was determined from the number of the hit strips multiplied by 400µm, since the horizontal length (Lh ) of the trajectory in the sensitive volume was determinedfrom the number of the hit strips multiplied by 400 µm. The vertical length (Lv ) should be
Lv = Lh tan 30◦ . Thus, the drift velocity was determined to be v = Lv /Dt = Lh /Dt tan 30◦
from the measured horizontal length and drift time (Dt ). For each event, the ratio Lh /Dt
was obtained by fitting a straight line to the hit pixels in the anode image. Figure 3.11 shows
the measured drift velocity when the gas pressure was at 500 hPa. The resolution of the drift
velocity was limited by the hole size of the collimator. We fitted a Gaussian function to the
distribution and took the central position of the peak for the drift velocity. The obtained
drift velocities are listed in Table 3.5.

3.6 Summary of the measurement
Table 3.6 summarize the present measurements. We measured the α+10 C scattering at two
diﬀerent TPC gas pressures. In addition to the α+10 C measurements at 68 MeV/u, wealso carried out similar measurements using the 12 C primary beam in order to check theconsistency between the present measurement with MAIKo at 94 MeV/u and the previous
measurement under the normal kinematic condition [143].

–37–

3.6 Summary of the measurement(a)Temperature (K)

299.5

299

298.5

298

297.5
09:00
10/08

2.0055

21:00
10/08

09:00
10/09

21:00
10/09

09:00
10/10

21:00
10/10

09:00
10/11

21:00
10/11

09:00
10/12

21:00
10/08

09:00
10/09

21:00
10/09

09:00
10/10

21:00
10/10

09:00
10/11

21:00
10/11

09:00
10/12

21:00
10/08

09:00
10/09

21:00
10/09

09:00
10/10

21:00
10/10

09:00
10/11

21:00
10/11

09:00
10/12

(b)

Density (×10-5 mol/cm3)

2.005
2.00452.004
2.00352.003
2.00252.002
2.0015
09:00
10/08

240

(c)

Moisture (ppm)

230
220
210
200
190
180
170
09:00
10/08

Figure 3.8: Monitored temperature (a), density (b), and moisture concentration (c) of the
TPC gas during the measurement at 500 hPa from October 7 to 12, 2017.

–38–

CHAPTER 3. EXPERIMENTα

100 MHz clock number

1000

30°

241Am

Anode image

800
600

Lv

400

Lh
200
0
0

50

100

150

200

250

Anode strip number

Figure 3.9: Schematic view of a trajectoryof collimated alpha particle in the TPC.

Figure 3.10: Example of the anode image acquired for a collimated alpha particle
from 241 Am.

160
140

Counts

120
100
80
60
40
20
0
0

0.5

1

1.5

2

Drift velocity (cm/ µs)

2.5

3Figure 3.11: Measured drift velocity when the gas pressure in the TPC chamber was at 500
hPa.

–39–

4 DATA REDUCTION
In order to determine energies and angles of the recoil alpha particles for the missing-mass
spectroscopy, we need to reconstruct the trajectories of the recoil alpha particles from theacquired data. As explained in Sec. 2.6, the acquired data contains two black-white images
with 256 × 1024 pixels and 2 × 8 waveforms recorded by the 25-MHz FADCs. Using theseinformation, we first distinguish the α+10 C scattering events and background events. After
selecting the α+10 C scattering events, the trajectories of the recoil alpha particles wereextracted to determine the recoil energies and angles. In this chapter, details of the data
analysis procedures are described.

4.1 Examples of anode and cathode images
4.1.1 Examples of α+10 C scattering events
Two examples of the α+10 C scattering events are presented in Figs. 4.1 and 4.2. Figs.
4.1(a) and 4.2(a) represent the anode images, and Figs. 4.1(b) and 4.2(b) represent thecathode images. The pressure of the TPC gas was 500 hPa. Both of the examples record
the trajectories of the incident 10 C particles and the recoil alpha particles. The trajectoriesof the unreacted 10 C particles were also recorded accidentally.
Since the beam axis was perpendicular to the anode strips and parallel to the cathodestrips, the trajectories of the 10 C particles, which flew from the right side of the anode images,
look horizontal lines in the anode images whereas they look elliptical shapes in the cathodeimages. On the other hand, the trajectories of the recoil alpha particles have large angles
from the beam axis in the anode images and have longer lengths than the 10 C trajectories in
the cathode images. The trajectories of the recoil alpha particles are thicker than those of
the 10 C particles because dE/dx of the recoil alpha particles is larger than that of the 10 C
particles.
After the scattering points, the trajectories of the scattered particles look diﬀerent according to the excitation energy of 10 C. The known excited states and particle decay thresholds
in 10 C are shown in Fig. 4.3. When 10 C was in the ground or the 2+
1 (3.35 MeV) state
below the particle decay threshold at Ex = 3.73 MeV, the trajectory of the scattered 10 C isrecorded in the anode image with a similar thicknesses to that of the incident 10 C as shownin Fig. 4.1(a), since dE/dx of the scattered 10 C is almost same as that of the incident 10 C.
On the other hand, when the scattered 10 C was excited above the particle decay thresholdat 3.73 MeV, 10 C was not observed after the scattering point as shown in Fig 4.2(a) becausesuch highly excited 10 C immediately decay into 2α + 2p. Even if 10 C decays to 8 Be, 9 B, and41

4.1 Examples of anode and cathode images

1000

(a)

100 MHz clock number

100 MHz clock number

1000

800

600

α
10

C

400

200

10

(b)

800

α

600
10

400

C

10

C

200

C

0
0

50

100

150

200

0
0

250

50

Anode strip number

100

150

200

250

Cathode strip number

Figure 4.1: Example of acquired track images of the α+10 C event. (a) Anode image. (b)Cathode image.

1000

(a)

800

100 MHz clock number

100 MHz clock number

1000

α
10

C

600

400

10

C

200

0
0

50

100

150

200

Anode strip number

α

800

600
10

C

400
10

C

200

0
0

250

(b)

50

100

150

200

250

Cathode strip number

Figure 4.2: Same as Fig. 4.1, but track images of an inelastic scattering event exciting ahighly excited state above the 2α + 2p decay threshold at Ex = 3.73 MeV.

–42–

CHAPTER 4. DATA REDUCTION
6 Be, they are unbound nuclei and eventually decay into protons and alpha particles
8

Be → 2α

(4.1)

B →

(4.2)

9
6

8

Be + p → 2α + p

Be → α + 2p.

(4.3)

Considering that dE/dx is proportional to Z 2 /β 2 , where Z is the charge of the particle andβ is the velocity, the energy loss per unit length of the 2α + 2p particles is about 1/3 of theincident 10 C. As a consequence, the observed horizontal trajectories in the anode image after
the scattering point become thinner than before the scattering point as seen in Fig 4.2(a).

6.58
5.10
6Be+α
3.73
2α+2p

3.82
8Be+2p5.22 5.38
4.01
2+ (3.35)

0+ (g.s.)
10C
Figure 4.3: Known energy levels in 10 C taken from Ref. [113].

–43–

9B+p

4.1 Examples of anode and cathode images

1000

(a)

100 MHz clock number

100 MHz clock number

1000

800

600
10

C

400

200

0
0

50

100

150

200

(b)

800

600
10

200

0
0

250

C

400

Anode strip number

50

100

150

200

250

Cathode strip number

Figure 4.4: Example of background events in which the beam particle passed through thecathode trigger region. The gray areas show the trigger region.

4.1.2 Examples of background events
Not only the α+10 C scattering events, background events were also acquired in the present
measurement. The background events were mainly caused by the 10 C beam particles thatjust passed through the cathode trigger region without a scattering in the sensitive volumeof the TPC as shown in Fig. 4.4. The trigger region is indicated with the gray areas in the
figure. Hereafter, we call these events “beam-particle events”.
The background events were also caused by the quenching CO2 gas. Figure 4.5 shows anexample of the 10 C + CO2 scattering events. Because the mass of carbon and oxygen nuclei
in the CO2 gas is close to that of the incident 10 C, 10 C was scattered at a large angle. The
trajectories of the scattered 10 C and the recoil carbon or oxygen after the scattering wererecorded. Another example of the 10 C + CO2 events is presented in Fig. 4.6. In this event,carbon or oxygen nucleus was highly excited above particle decay thresholds and decayed
into lighter particles. Multiple trajectories due to the scattered 10 C and the decay products
from carbon or oxygen are observed.
In case of the beam-particle events, no trajectory with a large angle is observed in theanode images. In case of the 10 C + CO2 events, two or more trajectories with large angles
from the beam axis are observed in the anode images.

–44–

CHAPTER 4. DATA REDUCTION

1000

(a)

100 MHz clock number

100 MHz clock number

1000

800

600
10

C

400

200

(b)

800

600

400
10

C

200

10

C

0
0

50

100

150

200

0
0

250

50

Anode strip number

100

150

200

250

Cathode strip number

Figure 4.5: Example of the 10 C + CO2 event. The incident 10 C particle was scattered fromcarbon or oxygen contained in the CO2 gas. Both the scatted 10 C and recoiled carbon or
oxygen trajectories are observed.

1000

(a)

100 MHz clock number

100 MHz clock number

1000

800
10

C

600

400

200

0
0

50

100

150

200

800

600

400

200

0
0

250

Anode strip number

(b)

50

100

150

200

250

Cathode strip number

Figure 4.6: Another example of the 10 C + CO2 event. The carbon or oxygen nucleus was
excited above the particle decay thresholds and immediately decayed into lighter particles.

–45–

4.2 Examples of the FADC data for the α+10 C scattering events

4.2 Examples of the FADC data for the α+10C scattering events
The waveform information obtained by the FADCs is also useful for the analysis. Fig. 4.7
presents typical recorded waveforms in the α+10 C event shown in Fig. 4.1. The verticalaxes represent the recorded pulse heights. The horizontal axes represent the 25-MHz clocknumbers which are synchronized with the 100-MHz clock numbers used as the vertical axes in
the anode and cathode images. While the polarity of the anode signals is positive, that of thecathode signals is negative. The baselines of the recorded waveforms should be subtracted
in the analysis of the charge collected. The baseline values were assumed to be constant foreach FADC channel.
The waveforms after the baseline subtraction are presented in Fig. 4.8. As seen in Ch.
7 of Fig. 4.8(a), the analog signals due to the incident 10 C particles are observed as sharp
peaks. Because the trajectories of the recoil alpha particle has vertical component, thedistribution of the electron drift time of the electrons spreads and the analog signals become
wider than those due to the beam particles as seen in Ch. 2 and Ch. 3.
The integration of the pulses, which corresponds to dE/dx of the particles, would be
helpful to distinguish the recoil alpha particles and the 10 C particles.

–46–

CHAPTER 4. DATA REDUCTION(a) Anode FADC before baseline subtraction
Ch. 0 (strip 0-32)

50

10

C

10

Ch. 1 (strip 32-64)
10
C C

10

C

Ch. 2 (strip 64-96)Cα10

Ch. 3 (strip 96-128)
10

C

α

Pulse height (ch)

40
30
20
10
Ch. 4 (strip 128-160)

50

10

C

Ch. 5 (strip 160-192)

10

Ch. 6 (strip 192-224)
10

C

C

10

40

C

Ch. 7 (strip 224-256)

10

C

10

10

10

C

C

C

30
20
10

0

100

200 0

100

200 0

100

200 0

100

200

25 MHz clock number(b) Cathode FADC before baseline subtraction
Ch. 0 (strip 0-32)

200

Ch. 1 (strip 32-64)

Ch. 2 (strip 64-96)

Ch. 3 (strip 96-128)

180

Pulse height (ch)

160
10

C, α

140
120
100

10

C

200 Ch. 4 (strip 128-160)

Ch. 5 (strip 160-192)

Ch. 6 (strip 192-224)

Ch. 7 (strip 224-256)

180
α

160

α

α

140
120
100

0

100

200 0

100

200 0

100

200 0

100

200

25 MHz clock number
Figure 4.7: Waveforms acquired by the FADCs before the baseline subtraction in the event
shown in Fig. 4.1. (a) Anode waveforms. (b) Cathode waveforms.

–47–

4.2 Examples of the FADC data for the α+10 C scattering events(a) Anode FADC after baseline subtraction25

Ch. 0 (strip 0-32)

Ch. 1 (strip 32-64)

20

Pulse height (ch)

15

C

10

C

Ch. 3 (strip 96-128)

10

C

10
10

Ch. 2 (strip 64-96)

C

α

10

C

α

10

C

10
5
0
25

Ch. 4 (strip 128-160)

Ch. 5 (strip 160-192)

Ch. 6 (strip 192-224)

Ch. 7 (strip 224-256)

20
10

10

15

10

C

C

C
10

C

10

10

C

C

10

10

C

C

10
5
0

0

100

200 0

100

200 0

100

200 0

100

200

25 MHz clock number

Pulse height (ch)(b) Cathode FADC after baseline subtraction
90
80
70
60
50
40
30
20
10

Ch. 0 (strip 0-32)

Ch. 2 (strip 64-96)

Ch. 3 (strip 96-128)
10

C
10

C, α

90 Ch. 4 (strip 128-160)
80
70
60
50
40
α
30
20
10

0

Ch. 1 (strip 32-64)

100

200 0

Ch. 5 (strip 160-192)

Ch. 6 (strip 192-224)

Ch. 7 (strip 224-256)

α
α

100

200 0

100

200 0

100

200

25 MHz clock number
Figure 4.8: Same with Fig. 4.7, but after the baseline subtraction.

–48–

CHAPTER 4. DATA REDUCTION

4.3 Procedure of the TPC data analysis
The α+10 C events exhibit the following two features. First, the energy loss per unit lengthof the recoil alpha particle at Eα ∼ 0.5 MeV is about 7 times larger than that of theincident 10 C beam at 68 MeV/u. Second, because of the large mass ratio between 10 C
and alpha particle, 10 C hardly changes its angle in the scattering with alpha particles but
alpha particles are recoiled at large angles from the beam axis. Therefore, just one tilted
trajectory from the horizontal line due to the recoil alpha particle is observed in the anodeimage as seen in Figs 4.1(a) and 4.2(a). On the other hand, in a beam-particle event, notilted trajectory is observed in the anode image as seen in Fig. 4.4(a). In a CO2 + 10 C
background event, multiple tilted trajectories are observed as seen in Figs 4.5(a) and 4.6(a).
Considering the above two features, the analysis of the TPC data was carried out with thefollowing procedures. As an example, we present the data analysis of the event shown in
Figs. 4.1 and 4.8.
i) Count the number of pulses in the baseline subtracted FADC data, and integrate eachpulse over time to obtain the energy loss of the particle. The threshold for the pulse
height was determined to be 2. Figure 4.9 shows a typical pulse shape of the anodeanalog signal from Ch. 2 [see Fig. 4.8(a)] after the baseline subtraction for the FADC
data. The threshold level is indicated by the black dashed line. In this example, twopulses (Pulse 1 and Pulse 2) were extracted which are indicated by the blue and green
regions in Fig. 4.9.
25

Pulse 2
Pulse height (ch)20

Pulse 1FADC anode Ch. 2
15
105

Threshold
0
−5
0

50

100

150

200

250

25 MHz clock number

Figure 4.9: Typical pulse shape of the anode analog signal from Ch. 2 [see Fig. 4.8(a)]after the baseline subtraction for the FADC data. The extracted two pulses are indicatedby the blue and green regions.

ii) If the integrated values of the FADC pulses were lower than a certain threshold (ppulse ),the corresponding pixels in the anode and cathode images were eliminated from the–49–

4.3 Procedure of the TPC data analysis

FADC Ch. 7 region

FADC Ch. 6 region

FADC Ch. 5 region

FADC Ch. 4 region

FADC Ch. 3 region

FADC Ch. 2 region

800

FADC Ch. 1 region

100 MHz clock number

1000

FADC Ch. 0 region

analysis. This procedure was necessary to exclude hit pixels due to the 10 C beam
particles and to facilitate the extraction of the recoil alpha particle trajectories. Since
the energy loss of the 10 C beam is much smaller than those of the recoil alpha particles,
the pixels due to the 10 C beam were selectively eliminated using the FADC integral
information. The threshold value was determined to 80 as described in Sec. 4.4. Figure
4.10 shows the same anode image of Fig. 4.1(a) but the pixels which correspond to
the analog signals of Pulse 1 and Pulse 2 in Fig. 4.9 are indicated by the blue andgreen pixels, respectively. In this example, the integration of Pulse 1 was lower than thethreshold value but the integration of Pulse 2 was higher than the threshold. Therefore,
only the green pixels in Fig. 4.10 remained after the pixel elimination for FADC Ch.
2. The images after the pixel elimination are shown in Figs. 4.11(a) and (b).

600

400

200

0
0

50

100

150

200

250

Anode strip number
Figure 4.10: Same with Fig. 4.1(a) but the pixels which correspond to the analog signalsof Pulse 1 and Pulse 2 in Fig. 4.9 are indicated by the blue and green pixels, respectively.

–50–

CHAPTER 4. DATA REDUCTION

1000(a) Anode after 1st elimination
100 MHz clock number

100 MHz clock number

1000
800
600
400
200
0
0

50

100

150

200

800
600
400
200
0
0

250

(b) Cathode after 1st elimination

50

Anode strip number
1000(c) Anode after 2nd elimination
100 MHz clock number

100 MHz clock number

1000
800
600
400
200
0
0

50

100

150

100

150

200

250

Cathode strip number

200

800
600
400
200
0
0

250

Anode strip number(d) Cathode after 2nd elimination

50

100

150

200

250

Cathode strip number

Figure 4.11: (a) and (b) Anode and cathode images after eliminating the pixels using the
FADC integral information [see the procedure ii)]. (c) and (d) Anode and cathode images
after eliminating the pixels near the straight lines extracted by the Hough transformation[see the procedure v)]. The straight lines extracted by the Hough transformation are plottedby the red solid lines.

–51–

4.3 Procedure of the TPC data analysisiii) Transform the anode and cathode images after the pixel elimination to the anode andcathode Hough histograms, according to the Hough transformation which is explained
in detail in Appendix A. The Hough histograms transformed from the anode andcathode images in Figs. 4.11(a) and (b) are shown in Figs. 4.12(a) and (b), respectively.
1000

1000

(a) Anode Hough(b) Cathode Hough

600
800

800

600

400
300

400

r (pixels)r (pixels)

500
600

500
600
400
400

300

200
200

100

0
0
0 20 40 60 80 100 120 140 160 180

700

200
200
100
0
0
0 20 40 60 80 100 120 140 160 180

θ (deg)θ (deg)Figure 4.12: Two-dimensional histograms in the Hough spaces obtained from Figs 4.11(a)and (b). The red circles show the positions of the bins with the maximum entries in the
Hough histograms.

iv) After the Hough transformation, find the bins of the maximum entries in the Hough
histograms. The bins of the maximum entries are indicated by the red circles in Fig.
4.12. From the coordinates of the maximum bins, one straight line is extracted fromthe anode and cathode images. The extracted straight lines are drawn by the red solidlines in Figs. 4.11(a) and (b).
v) Eliminate the pixels near the straight lines determined in the previous procedure fromthe anode and cathode images in Figs. 4.11(a) and (b). The anode and cathode images
after the elimination are shown in Figs. 4.11(c) and (d), respectively. In this event, allpixels were eliminated except for the small spots around (80, 400) in the anode image.
vi) Transform the remaining pixels again into the Hough spaces.
vii) Repeat iv), v), and vi) until the maximum bin counts in the anode and cathode Hough
histograms become smaller than thresholds. The threshold values are denoted as Canodeand Ccathode for the anode and cathode Hough histograms, respectively, and they are
optimized in Sec. 4.4. In this event, the Hough transformation was carried out onlyonce.
viii) Count the number of straight lines in the anode image whose angle from the horizontal
axis is larger than a certain threshold value θtilt (tilted line). In Fig. 4.11(a), this–52–

CHAPTER 4. DATA REDUCTION
number was one. The optimization procedures of θtilt values are described in Sec. 4.4.
ix) Count the number of extracted straight lines in the cathode image. In Fig. 4.11(b),this number was one.
x) If the number of the tilted lines in the anode image and the number of the straightlines in the cathode image were both one, the event was regarded as a candidate of the
α+10 C scattering event and the extracted lines were regarded as the trajectory of therecoil alpha particle. If the number of the tilted lines in the anode image was zero, theevent was classified as a beam-particle event. If the number of the tilted line was morethan one, the event was classified as a 10 C + CO2 event.
xi) For a candidate of the α+10 C scattering event, find edge hit-pixels along the extracted
straight lines by the Hough transformation. In the anode image, the edge points weresearched by the following procedure. At every clock number, calculate the anode strip
number on the extracted line according to the equation of the line. If the calculated
point is a black pixel, search the left and right edges of the trajectory at the same clocknumber, and calculate the middle point between the left and right edges. In a cathodeimage, the edge points were extracted by the similar procedure to the anode ones, but
the up and down edges at every strip were searched, and the middle point between
the two edges was calculated. The extracted edge points are shown in Fig 4.13. In
the anode image (a), the left and right edges and the middle points are plotted by thegreen, blue, and red solid circles, respectively. In the cathode image (b), the up anddown edges and the middle points are plotted by the green, blue, and red solid circles,
respectively.
xii) Fit straight lines to the extracted middle points in the anode and cathode images,
respectively, to determine the angle of the recoil alpha particle with better precisionbecause the angular resolution was limited by the bin size of the Hough histogram.
The extracted middle points tend to deviate from the straight line near the track stop
point because charged particles significantly deflects just before they stop. Therefore,
the second fitting was performed for the anode image after excluding the points whichwere more than 3 pixels away from the first fitted line. In the cathode image, thesecond fitting was not necessary because the number of the extracted middle points
was much larger than that in the anode image, and the fitting was not aﬀected by the
points near the stop point. The straight lines determined by the first fit are plottedby the red solid lines in Figs. 4.13(a) and (b), and the other line determined by thesecond fit is plotted by the blue dashed line in Fig. 4.13(a).
xiii) Find the edge pixels in the anode and cathode images along the fitted lines. These
points are the scattering point and track stop point of the recoil alpha particle. They
are plotted by the cyan solid circles in Fig. 4.13.
–53–

4.3 Procedure of the TPC data analysis

550
500
450

600

(a)
Left edge
Right edge
Middle point
1st fit
2nd fitScattered & Stop

100 MHz clock number

100 MHz clock number

600

400
350
300
50

60

70

80

90

100

110

120

550
500
450

(b)
Up edge
Down edge
Middle point
1st fitScattered & Stop

400
350
300
60

80 100 120 140 160 180 200 220

Anode strip number

Cathode strip number

Figure 4.13: (a) Zoomed anode image for the event shown in Fig. 4.1. The extracted left
and right edges and the middle points are plotted by the green, blue, and red solid circles.
The lines determined by the first and second fit are plotted by the red solid and blue dashedlines, respectively. The scattered and track stop points are plotted by the cyan solid circles.
(b) Zoomed cathode image for the event shown in Fig. 4.1. The line determined by the first
fit is plotted by the red solid line. The extracted up and down edges and the middle points
are plotted by the green, blue, and red solid circles.

xiv) From the straight lines in the anode and cathode images, the polar angle in the laboratory frame (θlab. ) and the azimuthal angle (ϕ) of the recoil alpha particle were
calculated assuming the beam axis was parallel to the cathode strips of the µ-PIC as()|∆z|θlab. = arccos
(4.4)
,
∆r
)
(
∆y
ϕ = arctan
,
(4.5)
∆x
where
∆x =
∆y =
∆z =
tan θa′ =tan θc′ =∆y,tan θc′1,∆y−,tan θa′v tan θa,0.40
v tan θc
.
0.40

(4.6)
(4.7)
(4.8)
(4.9)
(4.10)

Here, tan θa and tan θc are the slope parameters of the fitted lines in the anode andcathode images, and v is the measured electron drift velocity.
–54–CHAPTER 4. DATA REDUCTION
xv) Calculate the range of the recoil alpha particle in the TPC gas from the distance
between the scattered and stop points.
xvi) Integrate the analog signals from the cathode FADCs to calculate the induced charges.
The integration was performed between the scattered and stop points to avoid thecontributions from the accidental-coincidence beam particles.
We selected the α+10 C events in which the scattered point locates between the 33–224th
strip in the anode image. Thus, the eﬀective thickness of the He target gas was 76.8 mm.

4.4 Optimization of analysis parameters
In the analysis procedures described in the previous section, the values of the following
parameters must be determined:• The threshold value ppulse for the integrals of the FADC pulse in the procedure ii).
• The threshold values Canode and Ccathode for the number of the maximum entry in the
Hough histograms described in the procedure vii).
• The angular threshold θtilt for the “tilted line” in the anode image described in theprocedure viii).
In order to optimize the above four parameters, we sampled 2,000 events from the acquired data and labeled these events as α+10 C scattering or background by eyes. Before
the sampling, the main background events “beam-particle events” were roughly excluded
from the analysis by selecting the events in which at least one FADC pulse integral higher
than 150 was recorded in the anode data. The sampled 2,000 events were composed of the889 α+10 C scattering events and the 1,111 background events. Assuming certain values ofthe four parameters, the 2,000 events were classified into the α+10 C scattering or the background. The goodness of the parameters was evaluated comparing the classification of the
2,000 events with the answers labeled by eyes.
The result of the comparison was classified into four types which are known as confusion
matrix shown in Fig. 4.14. True positive (TP) means that the model correctly predicted
the α+10 C event (positive) as an α+10 C event. False positive (FP) means that the model
mis-predicted the α+10 C event as a background event (negative). False negative (FN) meansthat the model mis-predicted the background event as an α+10 C event. True negative (TN)means that the analyzer correctly predicts the background event as a background event.
In order to estimate the goodness of the parameters, we used the Jaccard coeﬃcientwhich is defined as
NTP
J=,(4.11)
NTP + NFP + NFN
where NTP , NFP , and NFN are the numbers of the TP, FP, and FN events in the analysis of
the 2,000 events. The Jaccard coeﬃcient falls into the range 0 ≤ J ≤ 1. This value decreases–55–

4.4 Optimization of analysis parameters

ActualNegative Positive

Prediction

Positive Negative
True
False
Positive Positive
(TP)
(FP)
False
True
Negative Negative
(FN)
(TN)

Figure 4.14: Confusion matrix.

Table 4.1: Optimized parameters in the present analysis tabulated with the Jaccard coeﬃcient, the signal eﬃciency, and the background rejection ratio when these optimized parameters were used.
Parameter

Value

ppulseCanodeCcathode
θtilt

80
160
200
12 degJaccardSignal eﬃciency
Background rejection

0.79
0.93
0.86

when the model mis-predict the events and increases when the model correctly extract the
α+10 C events.
The four parameters were optimized by a mesh-search method to maximize the Jaccard
coeﬃcient. Figure 4.15 shows correlations of the Jaccard coeﬃcient versus the four parameters. The optimized parameters and their Jaccard coeﬃcients are listed in Table 4.1. The
signal eﬃciency and background rejection ratio, defined as
Signal eﬃciency =Background rejection =NTP,
NTP + NFPNTN
NTN + NFN

(4.12)(4.13)

are also listed in the table. Here, we note that the achieved background rejection ratio is
not high compared to the signal eﬃciency. However, this ratio was improved to 0.99 after
applying the PID to select only the Z = 2 recoil particles as described in Sec. 4.5.
–56–

CHAPTER 4. DATA REDUCTION

0.85

0.85

140

(a)

180

(b)

0.8

160

120
0.8

0.75

120

0.7

0.75

0.65

100

J

J

80
60

80

0.7

0.6

60

40

0.55

40

0.65

0.5

20

0.45

0

40

60

80

p

100

120

140

20
0.6

120

140

160

p

pulse

0.85
0.8

180

200

220

90

(d)

80

0.8

100

0

anode

0.85

120

(c)

70

0.75

80

0.75

0.7

60

0.7

J

J

140

100

60
50
40

0.65

40

0.65

0.6

20

0.6

0

0.55
2

30
20

0.55

160

180

200

p

220

240

260

cathode

10
4

6

8

10 12

14

θ tilt

16 18

20 22

24

0

Figure 4.15: Correlations of the Jaccard coeﬃcient versus (a) ppulse , (b) panode , (c) pcathode ,and (d) θtilt .

4.5 Particle identification of the recoil particles
As seen in Table 4.1, the present analysis with the optimized parameters cannot reject all of
the background events. Further background rejection was done by the recoil PID.
The range of charged particles in the TPC gas is proportional to E 2 /(AZ 2 ), and the total
charge collected by the µ-PIC is proportional to the kinetic energy of the charged particles
if the charged particles stop inside the sensitive volume of the TPC. Figure 4.16 shows thecorrelations between the total charge collected by the µ-PIC and the range of the recoil
particle in the measurements with the 10 C beam. In these figures, only the the events inwhich the stop point locates inside the TPC sensitive volume are presented. Figure 4.16(a)is the correlation in the measurement at a gas pressure of 500 hPa, while Fig. 4.16(b) is at1000 hPa. In both of the plots, the loci due to the Z = 2 particles are clearly separated
from those due to Z = 1 and Z > 2 particles. The Z = 1 and Z > 2 particles are the decay
products of the carbon or oxygen nuclei in the CO2 quenching gas. We selected the Z = 2
events enclosed by the red solid lines in the figures. For the reliable track reconstruction, we
set the minimum range for the recoil alpha particles at 25 mm.
If the recoil particles escaped from the sensitive volume of the TPC and hit one of the–57–

4.6 Reconstruction of kinematics

3030(a) 10C 500 hPa

25
102

20

Z >2

15

Z =2
10

10

Total charge (nC)

Total charge (nC)25(b) 10C 1000 hPa

5

102

20

Z >2

15
10

10

Z =2
5

Z =1
0
1
0 10 20 30 40 50 60 70 80 90 100

01
0 10 20 30 40 50 60 70 80 90 100

Range (mm)Range (mm)Figure 4.16: Scatter plot of the total charge measured by µ-PIC and the range of the recoil
particle. The labels of “Z = 1”, “Z = 2”, and “Z = 3” are drawn near the loci due to theZ = 1, Z = 2, and Z = 3 particles, respectively. (a) PID plot for the measurement with the
10 C beam and at a gas pressure of 500 hPa. (b) PID plot for the measurement with the 10 C
beam and at 1000 hPa.

Si detectors, PID can be performed by the correlation between the energy measured by the
Si detector and the charge collected by the µ-PIC as shown in Fig. 4.17. The horizontal
axis represents the charge obtained by integrating the analog pulse from Ch.7 of the cathode
FADC and the vertical axis represents the energy measured by the left down side Si detector.
The loci due to the Z = 2 particles are clearly separated from the Z = 1 loci. The Z = 2
particles were selected by gating events which locate in the regions enclosed by the red solidlines.

4.6 Reconstruction of kinematics
If the recoil alpha particle stopped inside the TPC sensitive volume, the recoil energy wasdetermined from the range in the gas using the computer program SRIM [144]. The calculated recoil energies as a function of range in the TPC gas are plotted in Fig. 4.18. The redsolid line and blue dashed lines represent the recoil energies when the gas pressure is at 500
and 1000 hPa, respectively.
When the recoil alpha particle hit one of the Si detectors, the recoil alpha energy Eα was
calculated by integrating the energy loss in the TPC gas from the the position where therecoil alpha particle hit the Si detector (r Si ) to the scattered point (r s ) as
∫ rs
Eα = ESi +dE
dr,r Si dx E=E(r )–58–(4.14)

CHAPTER 4. DATA REDUCTION

10C 500 hPa

Si energy (MeV)

30
25

10

20

Z =2

15
10
5
0

Z =1
0.2

0.4

0.6

0.8

1

1.2

1

Charge in TPC (nC)

Figure 4.17: Correlation between the energy measured with the left down Si detector andthe charge obtained by integrating the analog pulse from Ch. 7 from the cathode. The gaspressure was at 500 hPa.

where ESi is the energy measured by the Si detector. The energy loss in the TPC gas as afunction of the energy of the alpha particle was calculated by the SRIM code. The calculated
dE/dx values as a function of energy of the alpha particle are shown in Fig. 4.19.
The two dimensional scatter plot of the measured kinetic energy versus recoil angle in the
10
α+ C events is shown in Fig. 4.20(a). The red and blue dots represent the events in which
the recoil alpha particles stopped inside the TPC sensitive volume in the measurements at
500 and 1000 hPa (denoted as “500 hPa events” and “1000 hPa events”), respectively. The
green dots represent the events in which the recoil alpha particles hit one of the Si detectors
in the measurement at 500 hPa (denoted as “Si events”). The kinematic correlation between
energies and angles of the recoil alpha particles at diﬀerent excitation energies (Ex ) in 10 C
are shown by the black solid lines and the c.m. angles (θc.m. ) are plotted by the black solid
circles.
Owing to the active target technique, the detection threshold for the recoil alpha particles
was successfully lowered down to 0.5 MeV. This detection threshold of the recoil energy isdetermined by the minimum range of 25 mm in the PID procedure (See Sec. 4.5).

–59–

4.6 Reconstruction of kinematics

6

α range in He(96%)+CO2(4%)
500 hPa
1000 hPa

Energy (MeV)

5
4
3
2
1
0
0

20

40

60

80 100 120 140 160 180 200

Range (mm)Figure 4.18: Energies of alpha particles as a function of range in the He(96%)+CO2 (4%)
mixture gas calculated by SRIM [144]. The red solid line and blue dashed lines represent
the recoil energies when the gas pressure is at 500 and 1000 hPa, respectively.

0.06

α energy loss in He(96%)+CO2(4%)
500 hPa
1000 hPadE / dx (keV/ µm)

0.05
0.04
0.03
0.02
0.01
0
0

5

10

15

20

α energy (MeV)

25

30

Figure 4.19: dE/dx values of the alpha particle through the TPC gas as a function of energyof the alpha particle calculated by the SRIM code [144]. The red solid line and blue dashedlines were calculated at a gas pressure of 500 and 1000 hPa, respectively.

–60–

CHAPTER 4. DATA REDUCTION

Figure 4.20: Two dimensional scatter plot of kinetic energy versus angle of the recoil alphaparticles. The red, blue, and green dots represent the “500 hPa events”, “1000 hPa events”,
and “Si events”, respectively. The kinematic correlation between energies and angles of therecoil alpha particles at diﬀerent excitation energies in 10 C are shown by the black solid lines
and the c.m. angles are plotted by the black solid circles. (a) All of the α+10 C scattering
events. (b) The elastic and inelastic scattering events to low excited states selected by theenergy-loss cut condition (see Sec. 4.7).

–61–

4.7 Excitation-energy spectra

4.7 Excitation-energy spectra
The excitation energy Ex of 10 C and θc.m. were calculated from the measured angle andenergy of the recoil alpha particle and the central energy of the 10 C beam (68 MeV/u)
using the missing-mass formulae described in Appendix B. Fig. 4.21 shows the excitationenergy spectrum obtained from the measurement when the gas pressure was at 1000 hPa. A
prominent peak due to the elastic scattering was observed at Ex = 0 MeV as well as small
contribution from inelastic scatterings.

α+10C 68 MeV/u
1000 hPa

Counts / 0.25 MeV

1400
1200
1000
800
600
400
200
0
−10

−5

0

5

10

15

20

Excitation energy (MeV)

Figure 4.21:
Excitation-energy spectrum of the α+10 C scattering at 68 MeV/u. The
spectrum was obtained from the measurement when the gas pressure was at 1000 hPa.

In the present work, we focus on the analysis of the elastic scattering and the inelastic
scattering to the 2+
1 state at Ex = 3.35 MeV to deduce the neutron transition matrix
element from the ground state to the 2+
1 state. Therefore, contributions from the highly
excited states above the particle decay threshold at Ex = 3.37 MeV, which may overlap
onto the 2+
1 region, should be excluded. As discussed in Sec. 4.1.1, the elastic scatteringand the inelastic scattering exciting the 2+
1 state can be distinguished from the inelastic
scattering exciting the highly excited states at Ex > 3.73 MeV by using the informationon the energy loss obtained from the anode FADCs. Since both the incident and scatteredparticles are 10 C in the elastic scattering and the inelastic scattering exciting the 2+
1 state,the energy losses of these particles per unit length in the TPC gas are almost same. On
the other hand, highly excited states above the threshold immediately decay to 2α + 2p
particles. Since the energy loss of the 2α + 2p particles is about 1/3 of the incident 10 C,
the energy loss after the scattering point become smaller than before the scattering point
when Ex > 3.73 MeV. Figure 4.22 shows the correlation between the pulse heights of theanalog signals from the most downstream channel (Ch. 0) and the most upstream channel(Ch. 7) in the anode. Figures 4.22(a) and (b) are the correlations at 500 hPa and 1000 hPa,
–62–

CHAPTER 4. DATA REDUCTION
respectively. We selected the low-excitation events at Ex < 3.73 MeV by gating the region
enclosed by the red solid lines. On the other hand, the high-excitation events at Ex > 3.73
MeV were selected by gating the region enclosed by the cyan dashed lines. Figure 4.20(a) is
the scatter plot before applying the gate and Figure 4.20(b) is that after applying the gate to
select the low-excitation events. With the above gate condition, only low-lying states below
the particle decay threshold were successfully selected. The excitation-energy spectra of thelow-excitation events at diﬀerent θc.m. and the high-excitation events are shown in Figs. 4.23
and 4.24, respectively.

(a) 500 hPa

35

30

103

25
20

102

15
10

10

5
0
0

5

10

15

20

25

30

35

Pulse height anode Ch. 0

Pulse height anode Ch. 0

35

30

Pulse height anode Ch. 7

103

25
20

102

15
10

10

5
0
0

1

(b) 1000 hPa

5

10

15

20

25

30

35

1

Pulse height anode Ch. 7

Figure 4.22: Correlation between the pulse heights of the analog signals from the mostdownstream (vertical axis) and the most upstream (horizontal axis) channels in anode. (a)Measurement at 500 hPa with 10 C beam. (b) Measurement at 1000 hPa with 10 C beam.
The excitation-energy spectra at 4.0◦ < θc.m. < 5.0◦ and 5.8◦ < θc.m. < 7.2◦ in Fig.
4.23 were obtained by selecting the low-energy recoil alpha particles, which stopped inside
the sensitive volume of the TPC, when the gas pressure was 500 and 1000 hPa (“500 hPa
events” and “1000 hPa events”). The excitation-energy spectra at 9.0◦ < θc.m. < 16.0◦ wereobtained by selecting the high-energy recoil alpha particles, which were detected by one of
the Si detectors, when the gas pressure was 500 hPa (“Si events”).
The prominent peaks due to the ground state are observed with a small contributions
from the 2+
1 state at Ex = 3.35 MeV. The excitation-energy resolution for the ground stateis 0.9 MeV in sigma at 4.0◦ < θc.m. < 4.5◦ . The resolution becomes worse to 2.0 MeV at
14.0◦ < θc.m. < 16.0◦ . This is because the kinematic condition depends on the c.m. angles.
As seen from Fig. 4.20, the distances between the black solid lines at diﬀerent excitationenergies are smaller at backward c.m. angles, and thus the excitation-energy resolution
becomes worse. The kinematic curves are almost vertical at 3◦ < θc.m. < 20◦ . This meansthat the excitation-energy resolution is determined mainly from the angular resolution for
the recoil alpha particle.
–63–4.7 Excitation-energy spectra
By fitting two Gaussians to each spectrum, the yields of the ground and 2+
1 states at
each c.m. angle were obtained. The distances between the centers of the Gaussians were
fixed at 3.35 MeV. In Fig. 4.23, the Gaussians for the ground and 2+
1 states are plotted by
the blue dashed and the green dotted lines, respectively. The sum of the two Gaussians areplotted by the red solid lines. The values of the reduced chi square (χ2 /ndf) of the fitting◦◦
are given in each histogram. The yields for the 2+
1 state at 4.0 < θc.m. < 5.0 were notreliably determined because cross sections for the 2+
1 state are very small compared to those+for the ground state. Therefore, the yields of the 21 state were deduced at θc.m. > 5.8◦ only,
and the yields of the ground state at 4.0◦ < θc.m. < 5.0◦ were obtained neglecting the tiny
contributions from the 2+
1 state.

–64–

CHAPTER 4. DATA REDUCTION

2500 500 hPa
22000 χ /ndf=4.9

2500
0+1 (g.s.)
2+1 (3.35)
Total

1500

2000 500 hPa
χ2 /ndf=3.4
1500
1000

1000
500
0

5

0
− 10

10

240
220 6.1° < θ c.m. < 6.5°
200 1000 hPa
1802160 χ /ndf=0.9
140
120
100
80
60
40
20
0
− 10
−5
0

5

10

Counts / 0.50 MeV

Counts / 0.50 MeV

9.0 ° < θ c.m. < 10.0 °200 Siχ2 /ndf=1.0
150
100
50
0

5

10

Excitation energy (MeV)
180
14.0 ° < θ c.m. < 16.0 °
160
Si
140
2
120 χ /ndf=1.3
100
80
60
40
20
0
− 10
−5
0
5

5

250

150
100
50
0
− 10

10

−5

120 6.9° < θ

5

0

5

10

Excitation energy (MeV)
c.m. < 7.2 °100 1000 hPa
χ2 /ndf=0.9

80
60
40
20
0
− 10

10

−5

0

5

10

Excitation energy (MeV)
400

500 10.0 ° < θ c.m. < 12.0 °Siχ2 /ndf=1.3

400

5.8° < θ c.m. < 6.1°
1000 hPa

200 χ2 /ndf=1.3

Excitation energy (MeV)

250

−5

0

200 6.5° < θc.m. < 6.9°
1801000 hPa160
2140 χ /ndf=1.0
120
100
80
60
40
20
0
− 10
−5
0

Excitation energy (MeV)

0
− 10

−5

Excitation energy (MeV)Counts / 0.25 MeVCounts / 0.25 MeV

Excitation energy (MeV)Counts / 0.25 MeV

−5

500

Counts / 0.50 MeV

0− 10Counts / 0.50 MeV

4.5° < θ c.m. < 5.0 °Counts / 0.25 MeV

c.m. < 4.5°

Counts / 0.25 MeVCounts / 0.25 MeV

3000 4.0 ° < θ

300
200
100

350
300

12.0 ° < θ c.m. < 14.0 °Si
χ2 /ndf=2.5

250
200
150
100
50

0
− 10

−5

0

5

Excitation energy (MeV)

10

0
− 10

−5

0

5

10

Excitation energy (MeV)

α+10C 68 MeV/u
10

Excitation energy (MeV)

Figure 4.23: Excitation-energy spectra of the low-excitation events at diﬀerent c.m. angles
fitted by the two Gaussian functions. The blue dashed and the green dotted lines are theGaussians due to the ground and 2+
1 states at Ex = 3.35 MeV, respectively. The sums ofthe two Gaussians are plotted by the red solid lines.

–65–

4.7 Excitation-energy spectra

2α+2p

140

Counts / 0.50 MeV

120

α + 10C 68 MeV/u
500 hPa

100
80
60
40
20
0

0

5

10

15

20

25

30

35

40

45

Excitation energy (MeV)
2α+2p

180

Counts / 0.50 MeV

160

α + 10C 68 MeV/u
1000 hPa

140
120
100
80
60
40
20
0

0

5

10

15

20

25

30

35

40

45

Excitation energy (MeV)
2α+2p

220
200

Counts / 0.50 MeV

180

α + 10C 68 MeV/u
Si

160
140
120
100
80
60
40
20
0

0

5

10

15

20

25

30

35

40

45

Excitation energy (MeV)

Figure 4.24: Excitation-energy spectra of the high-excitation events. The spectra of “500hPa events”, “1000 hPa events”, and “Si events” are plotted in the top, middle, and bottom
panels, respectively. The blue vertical lines show the 2α + 2p threshold at Ex = 3.73 MeV.

–66–

CHAPTER 4. DATA REDUCTION

4.8 Eﬃciency simulation
The geometrical detection eﬃciency and the track reconstruction eﬃciency of MAIKo were
estimated from a Monte Carlo simulation. In the simulation, α+10 C events were generated.
Positions and angles of incident 10 C particles, positions of scattering points, polar (θlab. )and azimuthal (ϕ) angles, and energies (Eα ) of recoil alpha particles in the laboratory frametwere randomly generated. “True values” of scattering angles in the c.m. system θc.m.
andt10
excitation energies Ex were calculated from θlab. , Eα , and the C energy. The spot size ofthe beam at the entrance of the TPC was set at 5 mm in sigma for both the horizontal andvertical positions. The energy of incident 10 C particles was fixed at 68 MeV/u. The incident
angular spread of the beam were set at 6 mrad (horizontal) and 15 mrad (vertical) in sigma.
These parameters were determined according to those measured by the beamline MWDCs.
Primary electrons were generated along the trajectories of the incident and scattered 10 C
particles and the recoil alpha particles according to the SRIM calculation. The stragglingof the recoil alpha particles through the TPC gas was considered. These primary electrons
drifted towards the µ-PIC taking into account the transverse and longitudinal diﬀusions of
the electrons. The diﬀusion coeﬃcients were calculated with the computer code Magboltz[145]. An example of the generated trajectories of the incident and scattered 10 C and recoilalpha particle are plotted by the red lines in Fig. 4.25. In this example, θlab. and ϕ of therecoil alpha particle are 85.4◦ and 60◦ , respectively. The recoil energy is 0.7 MeV.
1000(a) Anode simulation

100 MHz clock number

100 MHz clock number

1000

800

600

400

200

0
0

50

100

150

200

800

600

400

200

0
0

250

Anode strip number(b) Cathode simulation

50

100

150

200

250

Cathode strip number

Figure 4.25: The simulated (a) anode and (b) cathode images of an α+10 C scattering. In
this event, θlab. = 85.4◦ , ϕ = 60◦ , and Eα = 0.7 MeV. The generated trajectories are plottedby the red lines.
The analog signal from the µ-PIC for a single electron including the gas amplificationon the electrodes and the response of the readout circuit was simulated using the computercode Garfield++ [146]. Figure 4.26 shows the average of the simulated waveforms over 2,000
events divided by the gas gain as a function of time [147]. The arrival time of the electrons at–67–

4.8 Eﬃciency simulation
the µ-PIC was folded by the analog signal for a single electron in Fig. 4.26, multiplied by thegas gain to simulate the induced signals in the scattering event. In the present simulation, thegas gain was assumed to have the Polya distribution according to Ref. [147]. The simulated
signals were virtually processed and the anode and cathode images were generated as shownin Fig. 4.25.
×10−6
40
35

-

Pulse height (mV/e /gain)

45

30
25
20
15
10
5
0

0

500

1000

1500

2000

2500

Time (ns)Figure 4.26: Averaged analog signal of the µ-PIC in the simulation for a single electron
including the gas amplification processes on the electrodes and the response of the readoutboard [147]. The pulse height was divided by the gas gain.

The generated images were analyzed in the same manner as done for the real data, andthe excitation energies Ex and the scattering angles θc.m. were determined. The reconstructedtexcitation-energy spectrum for −0.3 < Ext < 0.3 MeV at 6.5◦ < θc.m.
< 6.9◦ is shown in Fig.
4.27. The excitation-energy resolution at Ext = 0 MeV is 1.2 MeV in sigma, which is almost
same with the experimental result of 1.1 MeV in Fig. 4.23. According to the simulation,
this resolution is limited by the angular straggling of the recoil alpha particles.
t
At each (Ext , θc.m.
), the number of events reconstructed as a scattering event was dividedtby the number of the generated events at (Ext , θc.m.
) to estimate the detection and trackttt
reconstruction eﬃciencies ϵ(Ex , θc.m. ). The obtained eﬃciencies as a function of Ext and θc.m.
are shown in Fig. 4.28. The top, middle, and bottom panels shows the eﬃciencies for the“500 hPa events”, “1000 hPa events”, and “Si events”, respectively.

–68–CHAPTER 4. DATA REDUCTION

Simulationt
6.5° < θ c.m.
< 6.9°Counts / 0.25 MeV

1201000 hPaσ = 1.2 MeV

100
80
60
40
20
0
−10 −8

−6

−4

−2

0

2

4

6

8

10

Excitation energy (MeV)
Figure 4.27:
Reconstructed excitation-energy spectrum for −0.3 < Ext < 0.3 MeV att
6.5◦ < θc.m.
< 6.9◦ in the simulation.

–69–

4.8 Eﬃciency simulation

18(a) 500 hPa events

0.25

16

θ c.m. (deg)

12
0.15

10
8

0.1

Efficiency

0.2

14

6
4

0.05

2
0
−5

0

5

10

15

20

25

30

35

40

45

0

Excitation energy (MeV)18(b) 1000 hPa events

0.1

16

θ c.m. (deg)

12
0.06

10
8

0.04

Efficiency

0.08

14

6
4

0.02

2
0
−5

0

5

10

15

20

25

30

35

40

45

0

Excitation energy (MeV)

θ c.m. (deg)

16

0.14

14

0.12

12

0.1

10

0.08

8
0.06

Efficiency

0.16

18

6
0.04

4
2
0
−5

0.02

(c) Si events
0

5

10

15

20

25

30

35

40

45

0

Excitation energy (MeV)

Figure 4.28: Geometrical detection and track reconstruction eﬃciency as a function of Exttand θc.m.
. (a) “500 hPa events”, (b) “1000 hPa events”, and (c) “Si events”.

–70–

CHAPTER 4. DATA REDUCTION

4.9 Diﬀerential cross sections

Finally, the diﬀerential cross sections for the elastic scattering and the inelastic scattering to
the 2+
1 state at Ex = 3.35 MeV were obtained bydN (Ex , θc.m. )1dσ(Ex , θc.m. ) =
×
.
t
t
dΩ
2π sin θc.m. dθc.m. ϵ(Ex , θc.m. )Nbeam Ntarget tlive(4.15)

Here, dN (Ex , θc.m. ) is the yield of the ground or 2+
1 state. ϵ(Ex , θc.m. ) is the detection andtrack reconstruction eﬃciency estimated by the simulation. Nbeam is the number of theincident 10 C beam measured by the F3PL. Ntarget is the number of the 4 He atom per unit
area. tlive is the live time ratio of the DAQ. Figure 4.29 shows the obtained diﬀerential crosssections of the elastic scattering and the inelastic scattering to the 2+
1 state. The numericalvalues of the diﬀerential cross sections are listed in Appendix D.

α+10C 68 MeV/u
0+1 (g.s.)
2+1 (3.35)

d σ /d Ω (mb/sr)

104

103

102

10
2

4

6

8

10

θ c.m. (deg)

12

14

16

Figure 4.29: Diﬀerential cross sections of the α+10 C elastic (solid circles) and inelastic
scattering to the 2+
1 state at Ex = 3.35 MeV (open squares). The cross section of the elastic
scattering calculated with the optical-model potential is plotted by the solid line. The crosssection of the inelastic scattering obtained by the DWBA calculation is plotted by the dashed
line.

–71–

4.10 Accuracy of the present analysis evaluated by the α+12 C elastic scattering

4.10 Accuracy of the present analysis evaluated by
the α+12C elastic scattering
The diﬀerential cross sections of the α+12 C elastic scattering were measured with the present
experimental setup and the 12 C primary beam in order to check the accuracy of the present
analysis by comparing the present result with the previous result reported in Ref. [143].
In the present measurement, the 12 C primary beam was slowed down from 96 MeV/u
to 94 MeV/u before entering of MAIKo due to the beamline detectors. The present crosssection was obtained with the same analysis procedures with the α+10 C scattering describedin the present chapter.
The previous data was measured at RCNP using a 4 He beam at 386 MeV (96.5 MeV/u)
and the magnetic spectrometer Grand Raiden (GR) [148]. The two results are compared inFig. 4.30. The solid circles and open squares represent the cross sections from the present andprevious measurements, respectively. The uncertainties of the cross section were statistical
only.

C(α, α)12Present (94 MeV/u)Previous (96 MeV/u)d σ /d Ω (mb/sr)

104

103

102

10
2

4

6

8

10

θ c.m. (deg)

12

14

16

Figure 4.30: Diﬀerential cross sections for the α+12 C elastic scattering. The solid circles are
the cross sections measured with MAIKo in the present work and the open squares represent
the cross sections measured with the 4 He beam in the previous work [143]. The solid line
connecting the open squares is drawn for guiding eyes.
The fractional diﬀerences between the present and previous results [(dσ/dΩ)MAIKo −(dσ/dΩ)GR ]/(dσ/dΩ)MAIKo at various c.m. angles are plotted in Fig. 4.31. The present
result agrees with the previous result qualitatively but it is systematically smaller than theprevious result by 10% in average. The fractional diﬀerences fluctuate ±16% around the av–72–

CHAPTER 4. DATA REDUCTION[(d σ /d Ω )

MAIKo

GR

− (d σ /d Ω ) ] / (d σ /d Ω )MAIKoerage value, which is much larger than the typical statistical uncertainties of a few percent.
The uncertainty of the target density is smaller than 1% and the uncertainty in counting
the incident beam particles using a plastic scintillator is typically 1%. Therefore, this fluctuation is considered to be mainly because of the uncertainty in estimating the detection
and track reconstruction eﬃciency with the simulation. To incorporate uncertainty from thesimulation, we simply added the 16% fractional uncertainty to the statistical uncertainty inquadrature in the following analysis. The error bars in Fig. 4.29 are plotted according toboth the statistical and fractional uncertainties.

1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1

2

4

6

8

10

θ c.m. (deg)

12

14

16

18

Figure 4.31: Fractional diﬀerences of the α+12 C elastic scattering cross sections between
the present and previous results [143].

–73–

5 DWBA ANALYSIS
The measured α+10 C cross sections were compared with distorted wave Born-approximation(DWBA) calculations to extract the neutron transition matrix element Mn . First, the crosssection of the elastic scattering was analyzed to obtain the eﬀective α-N interaction. The
transition potential to the 2+
1 state was constructed by folding the obtained eﬀective interaction with the macroscopic transition density. The Mn value was then determined toreproduce the cross section of the inelastic scattering. The DWBA calculations were performed using a computer code ECIS-95 [149].

5.1 Single-folding model and eﬀective α-N interaction
The optical-model potential U (r) for the α+10 C elastic scattering at 68 MeV/u was obtainedby integrating a phenomenological α-N eﬀective interaction u over the 10 C nucleus (singlefolding model):∫U (r) =

Φ∗c (r1 , r2 , ..., rA )A∑u(|r − ri |)Φc (r1 , r2 , ..., rA )dr1 dr2 · · · drA

(5.1)

i=1

Here, Φc (r1 , r2 , ..., rA ) is the wave function of the ground state in 10 C with A = 10 and r
represents the position of the alpha particle from the center of the 10 C nucleus (see Fig. 5.1).
ri is the position of the i-th nucleon in 10 C with respect to the center of the 10 C nucleus.
Using delta functions, Eq. (5.1) yields∫
U (r) =∫=

Φ∗c (r1 , r2 , ..., rA )A∑u(|r − r ′ |)δ(r ′ − ri )Φc (r1 , r2 , ..., rA )dr1 dr2 · · · drA dr ′i=1

ρ(r ′ )u(|r − r ′ |)dr ′ ,(5.2)

with ρ(r ′ ) is the point-nucleon density distribution of the 10 C ground state:′∫

ρ(r ) ≡

Φ∗c (r1 , r2 , ..., rA )A∑δ(r ′ − ri )Φc (r1 , r2 , ..., rA )dr1 dr2 · · · drA

i=1= ⟨Φc |A∑

δ(r ′ − ri )|Φc ⟩.

(5.3)

i=1

75

5.2 Determination of the range parameter
Since the spin parity of the ground state in 10 C is 0+ , only the transferred angularmomentum of ∆L = 0 is allowed, and thus the wave function and the density distributionof the ground state are treated as spherically symmetric hereafter.

10C

α

ri

r

Figure 5.1: Definition of the spacial coordinates in Eq. (5.2).

We parametrize the phenomenological α-N interaction as in Ref. [150]:
′ 2

2

′ 2

2

u(|r − r ′ |) = −V [1 + βρ2/3 (r′ )]e−|r−r | /αV−iW [1 + βρ2/3 (r′ )]e−|r−r | /αW .

(5.4)

The parameters V and W are the strengths of the real and imaginary interactions, respectively. β is the density dependence parameter of the eﬀective interaction. αV and αW are therange parameters of the real and imaginary interactions. These parameters are dependent on
the incident energy and nucleus. Therefore, they have to be determined phenomenologically.
Here, we note that the definition of αV and αW are changed from Ref. [150] so that they
have the dimension of the length.
In the present analysis, it is assumed that the range parameters of the real and imaginaryinteractions are the same (αV = αW = α), and the eﬀective interaction is density independent(β = 0). According to a previous study [143], the density dependent (β ̸= 0) and independentinteractions give almost the same cross sections for the 2+
1 states in the DWBA calculations
once the parameters of the eﬀective interaction are determined to reproduce the cross sectionsof the elastic scattering. Thus, we chose the density independent interaction for simplicity.

5.2 Determination of the range parameter
In an usual analysis as in Refs. [143, 151], the density distribution of the ground state is already determined from the electron scattering experiments, and only the eﬀective interaction
is optimized to reproduce the alpha elastic scattering. However, the density distribution of
the ground state in 10 C has not been obtained from the electron scattering, both the eﬀective
interaction and the density distribution must be determined at the same time. Unfortunately,
the eﬀective interaction and the density distribution are not fully decoupled in the calcula–76–

CHAPTER 5. DWBA ANALYSIS

Table 5.1: Eﬀective interaction parameters for the α+12 C scattering at 60 MeV/u.
Parameter

Value

α
V
W

2.13 fm
17.9 MeV
6.96 MeVtion of the cross section. Especially, the range parameter α in the eﬀective interaction andthe radius of the 10 C nucleus are strongly coupled. Therefore, these parameters can not be
determined uniquely.
In this analysis, the range parameter α was determined not from the present α+10 C
elastic scattering data, but from the previous α+12 C elastic scattering data at 60 MeV/u
measured in the Texas A&M K500 superconducting cyclotron facility [152].

The previous data was analyzed using the density independent eﬀective interaction described in Sec. 5.1. The point-nucleon density distribution of the ground state in 12 C was
calculated from the charge distribution obtained by the electron scattering experiment as
described in Appendix C.

The point-nucleon density distribution is a sum of the point-proton and point-neutron
distributions. In the case of 12 C, the point-neutron distribution was assumed to be the sameas the point-proton distribution because the charge symmetry is approximately established inlight nuclei. The eﬀective interaction parameters V , W , and α in Eq. (5.4) were determinedso as to minimize the chi-square (χ2 ) between the experimental and theoretical diﬀerential
cross section:∑( y exp − y cal )2
2
i
i
χ =
,
(5.5)
∆yi
i

where yiexp , yiexp , and ∆yi are the i-th experimental value of the cross section, cross section
calculated by the optical-model potential, and uncertainty of the cross section, respectively.
The optimization of the parameters were performed using the Nelder-Mead algorithm [153]
implemented in the SciPy library [154]. The obtained parameters are listed in Table 5.1.
The diﬀerential cross section calculated using the optimized parameters are compared withthe experimental data in Fig. 5.2. The optical-model calculation reasonably reproduces theexperimental cross section. We used the same range parameter α in analyzing the α+10 C
scattering data.
–77–

5.3 Analysis of α+10 C elastic scattering

104

C(α, α) 60 MeV/u

12

Exp.
Cal.

d σ /d Ω (mb/sr)

103
102
10
1
10−1
10−2
0

10

20

30

θ c.m. (deg)40

50

Figure 5.2: Diﬀerential cross section of the α+12 C elastic scattering at 60 MeV/u. The
black solid circle and the red solid line represent the experimental data from Ref. [152] andoptical single-folding model calculation, respectively.

5.3 Analysis of α+10C elastic scattering
The point-nucleon distribution of the ground state in 10 C was parametrized with the threeparameter Gaussian (3pG) function:
ρ(r) =

ρ0 (1 + wr2 /c2 ),1 + e(r2 −c2 )/z 2(5.6)

where c, z, and w are the parameters of the 3pG function. The normalization factor ρ0 isdetermined so that it satisfies the relation:∫
4π ρ(r)r2 dr = A,
(5.7)
where A = 10 is the mass number of 10 C. Note that while the point-nucleon density distribution of 12 C was deduced from the experimental results, the point-nucleon density distributionof 10 C was unknown.
The parameters V and W for the α-N eﬀective interaction and the 3pG parameters c,z, and w were optimized so as to reproduce the χ2 of the cross section of the α+10 C elastic
scattering, keeping the range parameter α at 2.13 fm which was taken from the α+12 C elastic
scattering analysis. The optimized parameters are listed in Table 5.2, and the calculated
cross section using these parameters is plotted by the solid line in Fig. 4.29. The reduced–78–

CHAPTER 5. DWBA ANALYSIS

Table 5.2: Optimized parameters for the α-N eﬀective interaction and the point-nucleon
distribution of the ground state in 10 C in the present analysis.
Interaction
α (fm)
2.13 (fixed)

V (MeV)
25.8+3.1−2.1

3pG
W (MeV)
17.0+2.7−2.0

c (fm)0.21

z (fm)
1.98

w−1.8 × 10−4

rms (fm)
2.6 ± 0.3

chi-square of the fitting χ2 /ν, where ν is the number of degrees of freedom, is 4.98/5, whichmeans that the optical-model calculation reasonably reproduce the experimental data.
The standard uncertainties of V and W in Table 5.2 were estimated by allowing toincrease the χ2 value from the minimized value in the range
χ2 ≤ χ2min + ∆χ2 .

(5.8)

When one parameter was varied, the other parameters were determined to minimize theχ2 value. Then, ∆χ2 obeys the χ2 probability distribution for 1 degree of freedom. Thus,
when ∆χ2 is equal to 1, the variation of the parameter corresponds to the uncertainty at aconfidence level of 68.3%. Figure 5.3 shows the χ2 value as functions of the strengths of theeﬀective α-N interaction: (a) V and (b) W . The χ2 lines at χ2min + 1 are indicated by the
red solid lines. The uncertainties of V and W in Table 5.2 were obtained from the ranges
where χ2 ≤ χ2min + 1.

35

18 (b)

(a)

16

30

14

χ2

χ2

25
20

12
10

15

8
10

χ2 = χ2min + 1

6

35

4
10

χ2 = χ2min + 1

5
20

25

30

40

V (MeV)

12 14

16 18

20

22 24

26

W (MeV)

Figure 5.3: χ2 values as functions of interaction strengths: (a) V and (b) W .

The obtained point-nucleon distribution of the ground state in 10 C is plotted in Fig.
5.4. The vertical axes of the left and right panels represent ρ(r) and r2 ρ(r), respectively.
The point-nucleon distribution expressed by the best-fit 3pG function is plotted by the blue–79–

5.3 Analysis of α+10 C elastic scatteringsolid line associated with its error band. The error band was estimated as an envelopewith ∆χ2 ≤ 3.5 when the three parameters c, z, and w in the 3pG function were randomly
varied. ∆χ2 for multi parameters obeys the χ2 distribution of ν degrees of freedom, whereν is the number of free parameters [155]. At a confidence level of 68.3%, ∆χ2 for the threeparameters c, z, and w in the 3pG function is 3.5. When the three parameters were varied,
V and W were optimized to minimize χ2 . The uncertainty of ρ(r) is large at small r because
the present measurement is limited at forward angles in the c.m. frame.

0.25

10

C point nucleon distribution

0.35

r 2ρ (r ) (fm-1)

0.2

ρ (r ) (fm-3)

0.4

3pG
AMD

0.15

0.1

0.3
0.25
0.2
0.15
0.1

0.05

0.05
0
0

1

2

3

4

5

6

7

0
0

8

r (fm)

1

2

3

4

5

6

7

8

r (fm)

Figure 5.4: Point-nucleon distribution of the ground state in 10 C. Left panel: ρ(r) as afunction of r. Right panel: r2 ρ(r) as a function of r. The density distribution obtained fromthe present work is shown by the blue solid line associated with the error band. The black
dashed lines represent the density distribution calculated by AMD [156].

The obtained point-nucleon density distribution was compared with the theoretical prediction by the antisymmetrized molecular dynamics (AMD) model [156] which is plottedby the black dashed lines in Fig .5.4. The present result is consistent with the AMD calculation within the error band. The root-mean-square (RMS) radius of the point-nucleon
distribution, defined as∫ 4√r ρ(r)dr⟨r2 ⟩ = ∫ 2,(5.9)
r ρ(r)dris compared between the present result, previous experiments obtained by the proton elastic
scattering [94] and the interaction cross section [157], and the AMD calculation [156] inTable 5.3. The present rms value is consistent with those from the proton elastic scatteringexperiment [94] and the AMD calculation, but slightly larger than that from the interaction
cross section measurement [157].
–80–

CHAPTER 5. DWBA ANALYSIS

Table 5.3: Comparison of the RMS radii of the ground state in 10 C obtained from thepresent and previous works.
Present
Proton elastic scattering [94]Interaction cross section [157]
AMD calculation [156]2.6 ± 0.3 fm
2.42 ± 0.10 fm
2.27 ± 0.03 fm
2.52 fm

5.4 Analysis of α+10C inelastic scattering to the 2+
1state
In order to calculate the diﬀerential cross section of alpha inelastic scatterings using DWBA,
we need to determine the transition potential δU (r), which is defined as∫δU (r) =A∑Φ∗c′ (r1 , r2 , ..., rA )u(|r − ri |)Φc (r1 , r2 , ..., rA )dr1 dr2 · · · drA .

(5.10)

i=1

where Φ∗c′ (r1 , r2 , ..., rA ) is the wave function of the excited state. The eﬀective interaction
parameters are the same as those determined in Sec. 5.3. Using a delta function, one obtains∫
δU (r) =∫=Φ∗c′ (r1 , r2 , ..., rA )A∑u(|r − r ′ |)δ(r ′ − ri )Φc (r1 , r2 , ..., rA )dr1 ddr2 · · · drA dr ′i=1

δρ(r)u(|r − r ′ |)dr ′ .

(5.11)

δρ(r) is the transition density between the ground and excited states which is defined as∫δρ(r) ≡

Φ∗c′ (ξ)

A∑

δ(r − ri )Φc (ξ)dξ = ⟨Φc′ |

i=1

A
∑

δ(r − ri )|Φc ⟩.

(5.12)

i=1

The transition density expresses the overlap of the wave functions between the initial andfinal states. It can be expanded with the sum of diﬀerent multipoles as [158]δρ(r) =∑
λ

δρ(λ) (r)∑∗Yλµ(r̂)⟨Jc Mc λµ|Jc′ Mc′ ⟩,(5.13)µ

where Jc,c′ and Mc,c′ represent the angular momenta and magnetic quantum numbers. The
Clebsch-Gordan coeﬃcients have the non-zero values only when λ and µ take |Jc′ − Jc | ≤+λ ≤ |Jc′ + Jc | and µ = Mc′ − Mc . For the transition from the 0+
1 state to the 21 state, only
the λ = 2 terms have non-zero values.
In the present analysis, the macroscopic model [159] was used. The macroscopic model
treats the nuclear excitation modes as small-amplitude collective oscillations of a liquid drop–81–

5.4 Analysis of α+10 C inelastic scattering to the 2+
1 stateabout its equilibrium shape. Using this model, the quadrupole transition (∆L = 2) densityfrom the ground state to the 2+
1 state is given as(2)δρn(p) (r) = −δn(p)

d
ρ
(r),
dr n(p)(5.14)

where δn(p) is the deformation length for neutron (proton), and ρn(p) (r) is the point density
distribution of neutron (proton) of the ground state in 10 C. Assuming that the neutron andproton distributions have the same shape, one can obtain the point proton and neutron
distribution:ρp (r) =ρn (r) =

Z
ρ(r)
A
N
ρ(r).
A

(5.15)(5.16)The density distribution ρ(r) is expressed by Eq. (5.6) with the parameters in Table 5.2.
The quadrupole transition matrix element of neutron (proton) was calculated from thetransition density with∫
Mn(p) =

(2)

r4 δρn(p) (r)dr.

(5.17)+The reduced electromagnetic transition rate B(E2; 0+
1 → 21 ) is related to the proton transition matrix element Mp by+
2
2
B(E2; 0+1 → 21 ) = e |Mp | .

(5.18)

Here we note that since a proton is not a point-like particle strictly, its charge form factor+
should be taken into account when Mp is compared with B(E2; 0+
1 → 21 ) in principle.
However, the diﬀerence of Mp calculated directly from the point-proton distribution and that
calculated from the charge distribution obtained by applying Eq. (C.3) to the 3pG pointproton distribution is only 1.9%. Since this diﬀerence is much smaller than the uncertainties
of Mn described below, we calculated Mp directly from the point-proton distribution forsimplicity. The proton deformation length δp was determined so as to reproduce the known+
10 C reported in Ref. [117]:
B(E2; 0+1 → 21 ) value in+
2
4
B(E2; 0+1 → 21 ) = 44.5 ± 1.5 e fm .

(5.19)It should be noted that the neutron deformation length δn is the last remaining free parameterto calculate the diﬀerential cross section of the alpha inelastic scattering using the DWBA
and the transition potential in Eq. (5.11).
δn was determined to be 2.4 fm by fitting the calculated cross sections to the experimental
data as shown by the dashed line in Fig. 4.29. The reduced χ2 of the fitting was χ2 /ν =10.9/7. Using Eq. (5.14), the transition densities from the ground state to the 2+
1 statewere obtained as shown in Fig. 5.5. By integrating the neutron transition density with Eq.
(5.17), we obtained the neutron transition matrix element from the ground state to the 2+
1–82–

CHAPTER 5. DWBA ANALYSIS
state as Mn = 6.9 fm2 .
0.06

10

C 0+1 → 2+1
ProtonNeutron

0.04
0.03

(2)

δ ρ (r) (fm-3)

0.05

0.02
0.01
0
0

1

2

3

4

5

6

7

8

r (fm)

Figure 5.5: Proton and neutron quadrupole transition densities from the ground state to
10 C. The proton transition density is plotted by the red solid line, while thethe 2+
1 state inneutron transition density is plotted by the blue dashed line.

The uncertainty of Mn associated with the fitting procedure was estimated by changing
Mn values in the range that satisfies Eq. (5.8). In the present fitting, since the free parameteris only Mn or δn , ∆χ2 is equal to 1. Figure 5.6 shows the χ2 value calculated by changing
Mn . The red horizontal line is drawn at χ2 = χ2min + 1. The resulting uncertainty of Mn wasdetermined to be ±0.4 fm2 from the range where χ2 ≤ χ2min + 1.

60
50

χ2

40
30
20
χ2 = χ2 + 1
min

10
4

5

6

7

8

9

10

M n (fm2)
10 C as a function of M .
Figure 5.6: χ2 value of the cross section for the 2+n
1 state in

The uncertainties of the eﬀective interaction and the 3pG parameters also result in an
additional uncertainty in Mn . This uncertainty was estimated to be ±0.6 fm2 by taking–83–5.4 Analysis of α+10 C inelastic scattering to the 2+
1 statediﬀerent parameter sets of V , W , c, z, and w randomly generated in the range that satisfies
χ2 ≤ χ2min + 1 for the elastic scattering data and deduced the Mn values using these parameters. Adding this uncertainty to that obtained in the previous paragraph in quadrature, we√
obtained the total uncertainty coming from the fitting procedure as (0.4)2 + (0.6)2 = 0.7
fm2 .
In Ref. [143], the diﬀerential cross sections of the alpha inelastic scattering exciting
the 2+ states in stable self-conjugate even-even nuclei from 12 C to 40 Ca were systematicallymeasured at Eα = 386 MeV. The authors calculated the diﬀerential cross sections using
a single-folding model similar to the present method. They obtained δp from the know+
B(E2; 0+1 → 21 ) values, assuming Mn = Mp because the proton and neutron numbers are
the same. The ratio R of the experimental cross sections to the calculated cross sections areshown in Fig. 5.7, which is taken from Fig. 15 in Ref. [143]. The fluctuation of R aroundthe unity is 17% in the standard deviation. This fluctuation corresponds to the uncertaintyof the single-folding model in the DWBA calculation. Therefore, we adopt ±17% (±1.2
fm2 ) as a systematic uncertainty associated to the error in the DWBA analysis with thesingle-folding model. Other systematic uncertainties, i.e., uncertainties of the target density(0.1%) and the number of beam particles (3%) are negligibly small compared to that of the
DWBA analysis.
+
10 C
Finally, we obtained the Mn value and its uncertainties in the 0+
1 → 21 transition inas
Mn = 6.9 ± 0.7 (fit) ± 1.2 (sys) fm2 .
(5.20)

Figure 5.7: Ratio R = σexp /σcal of the measured cross sections to the calculated crosssections using the single-folding model. The figure is taken from Fig. 15 in Ref. [143].

–84–

6 DISCUSSION
6.1 Comparison with the previous results
If one assumes the charge symmetry in the 10 C-10 Be system, the neutron transition matrix
element Mn in 10 C should be equal to the proton transition matrix element Mp in 10 Be. The
Mp value in 10 Be was measured as Mp = 6.78 ± 0.11 fm2 using the Doppler shift attenuation
method [160]. This value is actually close to the present Mn value in 10 C, and thus the
charge symmetry in the A = 10 system is almost conserved.
On the other hand, the previous proton inelastic scattering measurement reported adiﬀerent Mn value in 10 C of Mn = 5.51 ± 1.09 fm2 [94], which is significantly smaller thanthe Mp value in 10 Be. The authors claimed that the mirror symmetry was not fulfilled in
the 10 C-10 Be system, which is attributed to the Coulomb eﬀect. The discrepancy between
the present and previous results is possibly because the authors in Ref. [94] used the old+2
4
B(E2; 0+1 → 21 ) value of 61.5 ± 10 e fm from Ref. [114]. This value is larger than the
new value of 44.0 ± 1.5 e2 fm4 [117] which was reported after the proton inelastic scattering+
measurement. This larger B(E2; 0+
1 → 21 ) value might lead to a smaller value of Mn+
in the previous analysis. If we use the old B(E2; 0+
1 → 21 ) value to analyze the present
cross section of the alpha inelastic scattering, we obtained the smaller Mn value in 10 C as
Mn = 5.7 ± 0.6 (fit) ± 1.0 (sys) fm2 which is close to the previous result.

6.2 Comparison with theoretical calculations
As mentioned in Sec. 1.6, the ratio of the neutron and proton transition matrix elementsfrom the ground states to the 2+
1 states Mn /Mp (single ratio) is a useful probe to investigate
nuclear shell structures. The single ratio in 10 C deduced from the present measurement is
Mn /Mp = 1.05 ± 0.11 (fit) ± 0.17 (sys),(6.1)

which is close to unity.
The present results of the transition matrix elements and the single ratio are comparedwith the results of the proton inelastic scattering [94] and various theoretical calculations inTable 6.1. In the 2p+2α four-body cluster-model calculation [161], the fully antisymmetrized
10-nucleon wave functions were built in a microscopic 2p + 2α configuration space using theMinnesota potential [162]. The no-core shell model calculation [163] was conducted using the
CD-Bonn N -N potential [164] in the configuration space up to 8h̄Ω with the harmonic oscillator frequency of h̄Ω = 14 MeV. The shell-model calculation [165] was carried out within85

6.3 Z = 6 shell closure

Table 6.1: Comparison of the transition matrix elements of proton and neutron from the
10 C. Single ratios of the transition matrix elements M /M
ground state to the 2+np
1 state inare also listed.
Present(p, p′ ) [94]Cluster [161]No-core shell model [163]Shell model [165]Monte Carlo shell model [167]
AMD [169]

Mp (fm2 )6.63 ± 0.11 [117]7.84 ± 0.64 [114]
5.5
5.3
3.3
6.8
5.3

Mn (fm2 )6.9 ± 0.7 ± 1.2
5.51 ± 1.09
4.4
5.7
4.3
6.8
6.9

Mn /Mp1.05 ± 0.11 ± 0.17
0.70 ± 0.08
0.8
1.1
1.3
1.0
1.3

the p shell using the Cohen and Kurath (8-16)2BME interaction [166]. The shell-model transition matrix elements were calculated using single-particle wave functions in the harmonic
oscillator potential with b = 1.64 fm and eﬀective charges of ep = 1.3e and en = 0.5e. In theMonte Carlo shell-model calculation [167], the unitary-correlation-operator-method potential
based on the chiral next-to-next-to-next-to-leading-order two nucleon interaction [168] withthe bare charges (ep = e and en = 0) was used to calculate the transition matrix elements.
The Mn value in 10 C by the Monte Carlo shell-model calculation was taken from the Mp
value calculated in the mirror nucleus 10 Be assuming the charge symmetry. In the AMD
calculation [169], the wave function was given by a single Slater determinant of Gaussianfunctions with the width parameter of ν = 0.15 fm−2 .
The theoretical calculations tend to underestimate the transition matrix elements Mp
and Mn except the Monte Carlo shell-model calculation and the Mn value calculated by
the AMD model. Especially, the Mp and Mn values predicted by the shell-model calculation are considerably smaller than the experimental values. However, all of the theoretical
calculations agree with the Mn /Mp values close to unity.

6.3 Z = 6 shell closure
The Mn /Mp ratio in 10 C is almost unity, whereas a large Mn /Mp ratio of 3.2 ± 0.7 wasreported in 16 C [101]. This suggests that the quadrupole transition in 10 C is less neutron
dominant than 16 C and the eﬀect of the Z = 6 shell closure proposed in the neutron-richcarbon isotopes [77] is less evident in proton-rich 10 C.
If one assumes that the deformation lengths of the proton and neutron distributions are
the same, the Mn /Mp ratio is expected to be N/Z [see Eq. (1.9)]. Taking this assumption,
the double ratio (Mn /Mp )/(N/Z) has been widely used to discuss the nuclear shell structures[85]. When the neutron (proton) shell is closed, the double ratio becomes smaller (larger)
than unity because the transition of the neutron (proton) is remarkably suppressed. The
experimental double ratios in various single or double closed nuclei are shown in Fig. 6.1.
–86–

CHAPTER 6. DISCUSSION
The plotted values are the weighted averages of the experimental values derived by various
hadronic probes compiled in Ref. [84]. The double ratios in proton-closed nuclei plottedby the circles are systematically larger than unity, whereas those in neutron-closed nuclei
plotted by the triangles are smaller than unity.

2.5

Double closed(Mn / Mp ) / (N / Z )2

Proton closed
1.5

1

0.5

Ca
Pb

48

208

Ti
Cr
54
Fe
88
Sr
90
Zr
92
Mo
138
Ba
140
Ce
144
Sm

50

52

18

42

0

O
Ca
44
Ca
58
Ni
60
Ni
62
Ni
64
Ni
112
Sn
116
Sn
118
Sn
120
Sn
122
Sn
124
Sn

Neutron closed

Figure 6.1: Double ratios in proton-closed (circles), neutron-closed (triangles), and doubleclosed (diamonds) nuclei. The plotted values are the weighted averages of the experimental
values compiled in Ref. [84].
The double ratios in the even-even Tz = −1 (Z = N + 2) nuclei are plotted in Fig. 6.2.
+
The Mp values were calculated using Eq. (5.18) and the B(E2; 0+
1 → 21 ) values were takenfrom Refs. [170, 171]. The Mn values in 18 Ne–46 Cr were obtained by assuming the charge
symmetry:
Mn (N, Z) = Mp (Z, N ).
(6.2)
As seen in the figure, the double ratios are close to unity when both the proton and neutron
shells are open (open squares). On the other hand, the double ratios deviate from unity when
the proton or neutron shell is closed. In case of 18 Ne and 42 Ti (open triangles), neutrons tend
to contribute less to the excitation than protons since the neutron shells are closed at the
1p1/2 and 1d3/2 orbitals, respectively. In case of 38 Ca (open circle), the proton contribution
is reduced because the proton shell is closed at the 1d3/2 orbital.
The double ratio in 10 C obtained from the present work is plotted by the solid circle.
The fitting and systematic uncertainties are indicated by the solid and dashed vertical lines,
respectively. It should be noted that the double ratio in 10 C was deduced in the present work–87–

6.3 Z = 6 shell closure
without relying on the charge symmetry. Although we concluded that the Z = 6 shell closure
or magicity is less evident in 10 C, it was found that the double ratio in 10 C is significantly
larger than unity as those in the proton-closed nuclei.
(Mn /Mp )/(N/Z) = 1.57 ± 0.17 (fit) ± 0.27 (sys).

1.8

(6.3)

This work(Mn / Mp ) / (N / Z )

1.6
1.4
1.2
1
0.8
0.6
0.4
10

C

14

O

18

Ne 22Mg 26Si

30

S

34

Ar

38

Ca

42

Ti

46

Cr

Figure 6.2: Double ratios (Mn /Mp )/(N/Z) for the quadrupole transitions from the ground
10 C determinedstates to the 2+
1 states in the even-even Tz = −1 nuclei. The double ratio inin the present measurement is plotted by the solid circle, and the fitting and systematic
uncertainties are presented by the vertical solid and dashed lines, respectively. The open
squares are the double ratios in open shell nuclei. The open circle and triangles represent
proton and neutron closed nuclei, respectively.

The AMD calculation [172] explains the reason of this contradiction. The deformationsin the ground states in 16 C and 20 C predicted by the AMD calculation are illustrated in Fig.
6.3. The AMD calculation suggests both the proton and neutron distributions exhibit the
oblate deformation in 20 C as illustrated in the right side. Such a same deformation in protonand neutron distributions is a usual trend in stable nuclei. On the other hand, in case of 16 C,
the deformation of the proton and neutron distributions are decoupled, that is, the oblate
proton and prolate neutron deformations whose symmetric axes are perpendicular to eachother. Therefore, the assumption to use the double ratio, that the deformation lengths of the
proton and neutron distributions are the same, is no longer valid in this nucleus. The AMD
calculation suggests that the proton and neutron deformations in 10 C are also decoupled like
16 C [173]. Therefore, the large double ratio observed in 10 C does not necessarily suggests
the Z = 6 shell closure in the proton-rich carbon isotope.
The present conclusion that the Z = 6 shell closure is less evident in 10 C is consistent–88–CHAPTER 6. DISCUSSIONzz

Z
neutronneutronprotonproton

y

x

y

Z x

(b) 20 C(a) 16 C

Figure 6.3: Deformations of the proton and neutron distributions in carbon isotopes in the
AMD calculation. (a): Oblate proton and prolate neutron deformations in 16 C. (b): Oblate
proton and oblate neutron deformations in 20 C. The principal axes for the ground-state
K π = 0+ band are denoted as Z. The figure is taken from Fig. 4 in Ref. [172].

+with the known B(E2; 0+
1 → 21 ) values in carbon isotopes. Since the quadrupole transition is described as rearrangement of the particle-hole configuration in the valence shell,+
the B(E2; 0+
1 → 21 ) value tends to become small for proton-closed nuclei. The measured+
16 C [93, 100, 101, 105], 18 C [101, 106], 20 C [104] are only 1.1–2.3
B(E2; 0+1 → 21 ) values in
W.u.. These values are comparable to the proton-closed oxygen isotopes, i.e., 3.1 W.u. in
18 O [174], and thus the Z = 6 shell closure is evident in above neutron-rich carbon isotopes.
+
10 C is 6.9 ± 0.2 W.u. [117] and not small
On the other hand, the B(E2; 0+
1 → 21 ) value incompared to those in the neutron-rich carbon isotopes. This fact also supports the present
conclusion that the Z = 6 shell closure is less evident in the proton-rich side.

6.4 Future perspectives
6.4.1 Upgrade of the MAIKo detector
The present work demonstrated that the MAIKo active target is successful in performing themissing-mass spectroscopy to investigate the nuclear structures of unstable nuclei. It shallbe utilized in various incoming RI beam experiments. However, it is expected that the beam
intensity and the statistics will become smaller when we are to explore with the unstable
nuclei beam far from the stability region. The active target method can compensate thereduction of the beam intensity by extending the detector size along the beam axis which
increases the target thickness.
We are planning to enlarge the sensitive volume of the MAIKo TPC from 100 × 100 × 140–89–

6.4 Future perspectives
mm2 to 300 × 300 × 300 mm2 . This upgrade will increases the target thickness by a factorof three. Moreover, the energy acceptance of the recoil alpha particles which stop inside the
TPC sensitive volume will become three times larger than that of the present TPC. This cansignificantly reduce the beam time because the measurements for low-energy recoil particles
at forward c.m. angles and for high-energy recoil particles at relatively backward c.m. angles
were carried out separately in the present work. The statistics will become approximately afactor of nine.
In addition to the enlargement of the TPC, the configuration of the readout strip will bechanged from the two-axial X–Y structure to the three-axial X–U –V structure as shown in
Fig. 6.4. In the present two-axial strips, only the vertical angle of the beam particles canbe determined since the X strips are parallel to the beam axis. We installed the additional
MWDC detectors to monitor the horizontal angle, but this makes the analysis procedurecomplicated if we are to determine the angle of the incident beam which is scattered in the
TPC. In the upgraded MAIKo TPC, we are considering to arrange the readout strips at 60
degrees. Although the X strips are still parallel to the beam axis, the three dimensional
angle of the beam particles can be determined from the U and V strips.

V-s
t

U-

Y-striprip

X-strip

ipstr

X-strip

Figure 6.4: Structures of the readout strips. Left: Two-axial strips utilized in the present
MAIKo TPC. Right: Three-axial strips planed in the upgraded TPC.

6.4.2 Search for alpha cluster states
As described in Sec. 1.4.2.2, the MAIKo active target is a suited device to search for alphacluster states in unstable nuclei because it enables the measurement at forward c.m. angles
where the cross sections for exciting spatially well-developed cluster states become large. Asa next physics experiment with MAIKo, we aim to examine the alpha cluster states in 10 Be
and 10 C.
A recent theoretical calculation with the generalized two-center cluster model (GTCM)
describes the cluster structures in the 10 Be-10 C mirror system and suggests that the diﬀer–90–

CHAPTER 6. DISCUSSION
ences of the cluster configurations are reflected in monopole transition strengths and energy
shifts.
Figure 6.5 schematically shows the alpha cluster structures in 10 C predicted by GTCM
[175]. The 0+
1 state is considered to be a 2α + 2p molecular state in which two valence
protons occupy the π-molecular orbital formed by two alpha particles. The 0+
2 state isanother 2α+2p molecular structure in which the valence protons occupy the σ-molecular+
6
orbital. On the other hand, the 0+
3 and 04 states are predicted to have α+ Be structures inwhich the relative motion between the two alpha clusters are dominated by the s wave andd wave, respectively.
The isoscalar monopole transition strengths B(E0) from the ground state to excitedstates are enhanced when the relative motion of the two alpha clusters in the ground stateis excited. According to the GTCM calculation, the 0+
4 state is excited by stimulating therelative motion of the two alpha clusters in the ground state. This transition is denoted ascluster excitation in the figure. The 0+
2 state is excited by rearranging the valence protons
which is denoted as molecular orbital (MO) excitation. The 0+
3 state is excited by the cluster++
excitation of the 02 state. The 04 state is predicted to have the largest B(E0) value of 57+
fm4 among the three excited states while the 0+
2 and 03 states have about half B(E0) valuesof 28 fm4 and 23 fm4 , respectively. Therefore, the B(E0) values provide information to
distinguish these diﬀerent alpha cluster configurations.

α + 6Be(2+)

6

2+

04+

B(E0) = 57 fm4B(E0) = 23 fm4

03+

0+

0

α + 6Beg.s.

02+

Clu
ste
rE
x.

2
Ex
ster
Clu
.

-2
-4

MO

Ex.

Energy ( MeV )4

α + 6Be(0+)(π-)2B(E0) = 28 fm4 (σ+)201+-6Figure 6.5: Alpha cluster structures in 10 C predicted by the GTCM calculation [175].

–91–

6.4 Future perspectives
The energy shifts between the isobaric analog states are expected to be a new observableto clarify the alpha cluster structures [176–178]. Due to the Coulomb interaction, the analog
states in the proton-rich nuclei usually shift to higher energies than those in the neutronrich mirror nuclei. It is naively expected that these energy shifts become small as the radiiof the analog states increase. Figure 6.6 shows the level structure of the 0+ states in the
10 Be-10 C system predicted by the GTCM calculation [176]. The energy shift of the 0+ states1
is predicted to be 2.3 MeV, which is the largest energy shift among the four 0+ states.
This large energy shift is because of the spatially compact structure due to the two valence
nucleons in the π-molecular orbital. The energy shift of the 0+
2 states is predicted to be+
1.3 MeV which is smaller than that between the 01 states due to the loosely bound σ+
66molecular orbital. Because the 0+
3 and 04 states have spatially well-developed α+ He/ Be
cluster structures, the energy shifts of these states are even smaller: 0.6 MeV and 0.8 MeV,+
respectively. The smaller energy shift of the 0+
3 states than the 04 states is because therelative angular momenta between the two clusters in the 0+
3 states are zero. They are
not confined in the centrifugal potential which makes them spatially more expanded thanthe 0+
4 states. These diﬀerences in the energy shift are the the analogy of the well-known
Thomas-Ehrman shift originally discussed in the single particle states in the 13 C-13 N system[179, 180].
As discussed above, a systematic measurement of the energies and the isoscalar monopoletransition strengths in the 10 Be-10 C system are desired to investigate the cluster structures.
10 Be was experimentally found at E = 6.18 MeV [113]. However, the 0+
The 0+x
3
2 state in
10 Be. On the other hand, in 10 C, no 0+ excitedand 0+
states
have
never
been
found
in
4
state has been found. We obtained the excitation-energy spectra in 10 C above the particle
decay threshold from the present measurement as shown in Fig. 4.24. In order to deducethe monopole transition strengths from the present measurement, the angular distribution of
the diﬀerential cross section at each excitation-energy should be determined. However, thepresent statistics is not suﬃcient to reliably determine the diﬀerential cross section at each
excitation energy and at each c.m. angle. We will carry out further measurements both on
10 Be and 10 C after upgrading the MAIKo active target to achieve about nine times higher
statistics which ensures us to obtain the angular distribution of the diﬀerential cross section
at each excitation-energy.

–92–CHAPTER 6. DISCUSSION

Figure 6.6: Level structure of the 0+ states in the 10 Be (left)-10 C (right) mirror system.
The figure was taken from Fig. 1 in Ref. [176].

–93–

7 SUMMARY
The missing-mass spectroscopy is a promising method to investigate the structures in unstable nuclei. However, the measurement at forward scattering angles in the c.m. system istechnically diﬃcult because the energies of the recoil particles are extremely small. In orderto realize the detection of such low-energy recoil particle, we developed a new active target
system named MAIKo [41]. MAIKo is based on a TPC in which the detector gas is used as atarget gas. Since the scatterings occur inside the sensitive volume of the TPC, the detection
threshold for the recoil particles are lowered. The size of the MAIKo TPC is 100 × 100 × 140
mm3 . We introduced the µ-PIC [42] for the amplification and detection of the drift electrons
and achieved the finest readout pitch among the existing active target systems.
As the first physics experiment using MAIKo, we performed the alpha elastic and inelastic
scatterings oﬀ 10 C at 68 MeV/u at RCNP, Osaka University [76]. The aim of the experimentis to deduce neutron transition matrix element Mn from the ground state to the 2+
1 state at10
Ex = 3.35 MeV in C and discuss the Z = 6 magicity in the proton-rich carbon isotope.
An almost pure 10 C secondary beam with an intensity of 70 kcps was produced from a 12 C
primary beam at 96 MeV/u. The missing-mass spectroscopy with MAIKo was performed todetermine the excitation energy in 10 C. MAIKo was operated with the He(96%)+CO2 (4%)
gas mixture at 500 and 1000 hPa. We successfully detected low-energy recoil alpha particles
down to Eα = 0.5 MeV with MAIKo as designed, which corresponds to the momentum
transfer down to q = 0.4 fm−1 .
The trajectories of the recoil alpha particles were reconstructed with the Hough transformation. Only the α+10 C events were selected from other background events by the trackreconstruction algorithm and the particle identification. The parameters in the track reconstruction procedures were optimized from the eye-scanned data samples. The excitationenergy resolution was about 1 MeV in sigma, which was good enough to distinguish the first
10 C. A Monte Carlo simulation
excited 2+
1 state at Ex = 3.35 MeV from the ground state in
was performed to estimate the detection and track reconstruction eﬃciencies. The diﬀerential cross sections of the α+10 C elastic scattering and the inelastic scattering exciting the 2+
1◦◦state were determined at 4.5 < θc.m. < 15 .
The diﬀerential cross section of the α+12 C elastic scattering was also measured using a
primary 12 C beam at 94 MeV/u. The measured cross section was compared with the previous
data measured under the normal kinematic condition using a 4 He beam at 96 MeV/u [143].
We confirmed that both results are consistent quantitatively, demonstrating the reliabilityof the present measurement with MAIKo.
From the analysis of the cross section of the α+10 C elastic scattering, the phenomenological α-N eﬀective interaction and the point-nucleon distribution of the ground state in95

10 C were determined. The rms radius of 2.6 ± 0.3 fm in 10 C is consistent with the theoreticalprediction by the AMD calculation [156] and the experimental result of the previous proton
elastic scattering [94], but slightly larger than that deduced from the interaction cross section[157].
By comparing the measured cross section of the α+10 C inelastic scattering exciting the2+
1 state with the DWBA calculation, the neutron transition matrix element of Mn = 6.9 ±
0.7 (fit) ± 1.2 (sys) was obtained. The present experiment is the first attempt to determine
the neutron transition matrix element in unstable nuclei from the alpha inelastic scattering.
The alpha inelastic scattering enables us to deduce the neutron transition matrix element
without model ambiguity better than the inelastic proton scattering which has been widely
applied to the RI beam experiments.
Because the obtained Mn value in 10 C is close to the Mp value in the mirror nucleus
10 Be, we concluded that the isospin symmetry in the A = 10 system is almost conserved.
The ratio of the neutron transition matrix element to the proton transition matrix element
was determined as Mn /Mp = 1.05 ± 0.11 (fit) ± 0.17 (sys). The ratio close to unity suggeststhat the recently proposed Z = 6 shell closure is less evident in proton-rich 10 C nucleus than
the neutron-rich side. The fact that Mn /Mp larger than N/Z = 2/3 can be attributed to thedecoupling of the proton and neutron distributions predicted by the AMD calculation [172].
After the long-standing development since 2011, the first physics experiment using MAIKo
has been successfully completed. The MAIKo active target provides a breakthrough to realize the measurement of the alpha inelastic scattering at low-momentum transfer region
under the inverse kinematic condition. MAIKo is now under an upgrade program to explore
more exotic side and it shall be utilized in many incoming RI beam experiments in the near
future.

–96–

Acknowledgement
The present thesis would not have been completed without the dedicated and kind supports
from many people. I would like to take this opportunity to express my sincere thanks tothem.
First and foremost, I would like to express my great appreciation to my supervisor,
Prof. Takahiro Kawabata. He provided me the opportunity to undertake the interesting
and challenging project to develop a new active target system from almost scratch. Since I
was an undergraduate student, he has been continuously encouraged and supported for theplanning and carrying out the experiments, the data analysis, and writing the thesis evenafter he moved from Kyoto University to Osaka University. Moreover, he taught me themost important thing, to enjoy research and life. It was my privilege to be his student.
I am really grateful to Prof. Hooi Jin Ong at RCNP. He gave me invaluable advice on
the experiment at RCNP and discussions on the obtained physics results. He also kindly
helped me for writing the articles.
I gratefully acknowledge the present and past members of our research group: Dr. Satoshi
Adachi, Mr. Naoki Yokota, Mr. Tomosuke Kadoya, Mr. Hidetomo Watanabe, Ms. Miho
Tsumura, Mr. Tatsuo Baba, Mr. Motoki Murata, Mr. Yuki Ishii, Mr. Takahiro Morimoto,Ms. Akane Sakaue, Mr. Kento Inaba, Mr. Yu Takahashi, Mr. Tomoya Takeda, Mr. Yuki
Fujikawa, Mr. Takanobu Doi, Mr. Shintaro Okamoto, and Mr. Kohsuke Sakanashi. We
spent quite a long time on many experiments through which I felt a genuine camaraderie.
They made great contributions to the development of MAIKo and the present measurement.
I also enjoyed daily conversation with them. The long time with them has been matured mypersonality.
The present E463 experiment was successfully carried out thanks to the hard and professional works by the collaborators: Dr. Yassid Ayyad, Prof. Michele Sferrazza, Prof. Isao
Tanihata, and Dr. Dinh Trong Tran.
I would like to express my gratitude to Prof. Nori Aoi, Prof. Yoshiko Kanada-En’yo,
Prof. Kazuyuki Ogata, and Prof. Harutaka Sakaguchi for the fruitful discussions about thetheoretical interpretation of the results. My special thanks go to Prof. Makoto Ito who
motivated me to start the present experiment showing me the interesting clustering aspectsof 10 C. I am grateful to Prof. Tomokazu Suzuki, Dr. Takashi Hashimoto and Dr. Junki
Tanaka for helping the development of MAIKo.
The MAIKo active target was successfully developed owing to the supports from theCosmic Ray group at Kyoto University. I would like to express my great appreciation toProf. Toru Tanimori, Prof. Hidetoshi Kubo, Prof. Atsushi Takada, Dr. Shigeto Kabuki,
Dr. Tatsuya Sawano, Dr. Yoshitaka Mizumura, Dr. Tetsuya Mizumoto, and Dr. Yoshihiro
Matsuoka.
97

The low-pressure MWDCs which were used in the present experiment were newly developed with the help from Prof. Yohei Matsuda, Prof. Satoru Terashima, Prof. Juzo Zenihiro.
I am also grateful to Kaizu Works company, Liftec company, Swagelok company, MKS company, Seiwa Sangyo company, Horiba company, and Horiguchi company for the kind help
and advice in making the experimental equipment used in the present measurement.
I am truly grateful to the staﬀs in RCNP for their supports. Especially, Prof. Atsushi
Tamii, Prof. Nobuyuki Kobayashi, Mr. Shinji Ano, Prof. Tomokazu Suzuki, Mr. Michio
Uraki, Dr. Yusuke Yasuda, and Dr. Hidetomo Yoshida for technical supports on experiments.
The experiment was stably performed thanks to the cyclotron crews. I deeply appreciate the
staﬀs at the RCNP oﬃce for providing the comfortable guest house during the experiments.
I would like to express my great gratitude to the staﬀs at Experimental Nuclear andHadronic Physics Laboratory (NH) at Kyoto University, Prof. Tomofumi Nagae, Prof.
Megumi Naruki, Prof. Juzo Zenihiro, Prof. Tetsuya Murakami, Prof. Hiroyuki Fujioka,
Prof. Masayuki Niiyama, and Prof. Toshiyuki Gogami for the encouragements and suggestions throughout the time in the university. I would like to also acknowledge all of themembers in the NH group and oﬃce staﬀs at Department of Physics. I enjoyed discussions,research activities, recreations, BBQ, and drinkings with them.
After six year in Kyoto University, I started to work at RCNP. I am grateful to all of thestaﬀs and students in NP-1A group for discussions and supports.
I appreciate the financial supports of Grant-in-Aid for JSPS Research Fellow JP14J00949.
I am indebted to all of my friends in my university life for the emotional supports. I wasreally happy with them.
Finally but not least, I would like to express my sincere appreciation to my parents,
sister, and grand parents. Without their kind supports and understandings, it would havebeen impossible to reach this goal.

–98–

Appendix A
Hough transformation
The Hough transform algorithm was originally developed by Hough for the analysis of the
bubble chamber pictures [181] and later modified by Duda and Hart [182]. It enables toextract features in a picture such as lines or circles without knowing how many features exist
in the picture. This algorithm is now widely applied in image recognition because of its highimmunity to random noises compared to other algorithms like the least-square method.
The anode and cathode images obtained by MAIKo contain 256 × 1024 black-and-white
pixels. When coordinates of the i-th black pixel in an image are expressed as (xi , yi ) in the
track space, this pixel is transformed to a curve in the (θ, r) parameter space (Hough space)given by Eq. (A.1) in the Hough transformation.
r = xi cos θ + yi sin θ.

(A.1)

In the track space as shown in Fig. A.1(a), the red, blue, and green points at (xi , yi ),(xi+1 , yi+1 ), and (xi+2 , yi+2 ) are transformed into the red, blue, and green curves in the
Hough space as shown in Fig. A.1(b), respectively. The j-th pixel at (θj , rj ) in the Hough
space corresponds to a straight line in the track space as:
y=−

rj
x
+
.
tan θjsin θj(A.2)

As shown in Fig. A.1(a), r corresponds to the distance from the origin to the straight line,and θ corresponds to the angle between the x axis and the perpendicular line from the origin.
If several pixels in the track space lie on a straight line, their transformed curves intersect
at one point in the Hough space, which corresponds to the original straight line in the track
space.
In the present analysis, the transformed curves were discretized into bins and these bins
were booked into a two-dimensional histogram in the Hough space. The number of the binsof the histogram in the Hough space was 180 × 256. The trajectories with sizable length
make clear peaks in the Hough histogram while the small spots caused by noises do not. By
finding the bin with the maximum entry in the Hough histogram, which is easy to implement
in a computer program, the longest trajectory in the image was extracted.
After the first trajectory was extracted, the black pixels near the first line were eliminatedfrom the image. The remaining pixels were again transformed into the Hough space. The
second trajectory was determined by the highest bin in the second Hough histogram. The
Hough transformation and the elimination were repeated until the number of entry in the99highest bin in the Hough histogram became lower than a certain threshold. With thismethod, multiple trajectories were successfully extracted from the images.
Here it should be noted that the smaller bin size in the Hough histogram does notnecessarily lead to an accurate determination of the trajectories. If we adopt the smaller
bin size, the number of the entries in the highest bin in the Hough histogram becomes
smaller, thus this makes the analysis susceptible to noises and diﬃcult to determine thehighest bin. In order to achieve the better accuracy in determining the trajectories, we
fitted another straight line to the black pixels near the straight line extracted by the Houghtransformation.

rj(b)Hough space(x i, yi)(x i+1, yi+1)(x i+2, yi+2)
rj
y = -x +tanθ j sinθ j(θ j , rj )ry(a)Track space

θj
θ

x

Figure A.1: Example of the Hough transformation. The the red, blue, and green points
at (xi , yi ), (xi+1 , yi+1 ), and (xi+2 , yi+2 ) in the left panel (a) are transformed into the red,
blue, and green curves in the right panel (b) according to Eq. (A.1), respectively.

–100–Appendix B
Relativistic kinematics
In this appendix, the formulae to reconstruct the excitation energy and center of mass angle
in the inverse kinematic condition are presented.
The excitation energy of the incident particle Ex is calculated using the relativistic kinematics as√
Ex = s + t + u − m21 − m22 − m23 − m4 ,(B.1)
where m1 and m4 are the rest mass of the incident particle and m2 and m3 are the rest massof the target particle. s, t, and u are the Mandelstam variables defined ass = m21 + m22 + 2m2 E1 ,t =u =m21 + m23 + 2(p1 p3 cos θ3 − E1 E3 ),m22 + m23 − 2m2 E3 .

(B.2)
(B.3)
(B.4)

Here, θ3 is the angle of the recoil particle in the laboratory frame. E1 , E3 are the energies
while p1 and p3 are the momenta given as
E1 = m1 + K1 ,(B.5)

E3 = m3 + K3 ,√p1 =E12 − m21 ,√p3 =E32 − m23 ,

(B.6)
(B.7)
(B.8)

where K1 and K3 are the kinetic energies of the incident and recoil particles, respectively.
The center of the mass scattering angle θc.m. is calculated from several parameters below.
W =√m21 + m22 + 2m1 E1 ,(B.9)

is the total energy in the c.m. frame.
p1,c.m. = p2,c.m. =p3,c.m. = p4,c.m. =1 √ 2[W − (m1 + m2 )2 ][W 2 − (m1 − m2 )2 ],2W
1 √ 2[W − (m3 + m4 )2 ][W 2 − (m3 − m4 )2 ],2W
101(B.10)(B.11)

are the momenta in the c.m. frame.
Ei,c.m.

√=m2i + p2i,c.m. ,γi,c.m. =βi,c.m. =Ei,c.m.
,mipi,c.m.
,Ei,c.m.

(B.12)(B.13)(B.14)

are the energy, gamma factor, and velocity in the c.m. frame, respectively. Finally, θc.m. is
calculated as
θc.m. = arccos(λ + τ ),(B.15)
where
β2,c.m. (γi,c.m. tan θ3 )2
λ = −,β3,c.m. 1 + (γi,c.m. tan θ3 )2√[1 − (β2,c.m. /β3,c.m. )2 ]γ2,c.m. tan θ3 + 1τ =.
1 + (γi,c.m. tan θ3 )2–102–(B.16)(B.17)

Appendix C
Point proton density distribution of
the ground state in 12C
In this appendix, the procedure to obtain the point proton density distribution of the groundstate in 12 C from the charge distribution is described.
The charge distribution ρc0 given in Ref. [183] was parametrized with the Gauusians(SOG) function as follows:ρc0 (r) =

∑
i

with

{
Ai

) ]
[ (
) ]}
[ (
r + Ri 2
r − Ri 2+ exp −
,
exp −
γ
γ

[
(
)]
2R2 −1
Ai = ZeQi 2π 2/3 γ 3 1 + 2i
.
γ

(C.1)

(C.2)

The parameters Ri , Qi , and γ in Eqs. (C.1) and (C.2) were determined so as to reproduce
the cross section of the elastic electron scattering. We took the values from Ref. [183] aslisted in Table C.1.
The point-proton distribution ρp0 (r) was calculated by unfolding the charge distributionof Eq. (C.1) with the proton charge form factor [184] since the charge distribution ρc0 (r) is
given by folding ρp0 (r) with the charge distribution in a proton ρcp (r) as follows:∫ρc0 (r) =ρp0 (r′ )ρcp (r − r′ )dr′ .

(C.3)

Here, we assumed that ρp0 (r) and ρcp (r) are spherically symmetric. The Fourier transformationof this equation is expressed by
F c (q) = F p (q)Gp (q),(C.4)

where q is the momentum transfer and F c (q), F p (q), and Gp (q) represent the Fourier transformations of ρc0 (r), ρp0 (r), and ρcp (r), respectively. F c (q) was calculated from the charge
density in Eq. (C.1) by∫
4πc
F (q) =
ρc0 (r) sin(qr)dr,(C.5)
q
and Gp (q), known as Sachs electric form factor of proton [185], was empirically parametrized103

Table C.1:
Ref. [183].

Parameters of SOG for the charge distribution of the 12 C ground state from

Parameter

Value

i

Ri (fm)

Qi

1
2
3
4
5
6
7
8
9
10
11

0.0
0.4
1.0
1.3
1.7
2.3
2.7
3.5
4.3
5.4
6.7

0.016690
0.050325
0.128621
0.180515
0.219097
0.278416
0.058779
0.057817
0.007739
0.002001
0.000007γ (fm)

0.98

in Ref. [186] as:
Gp (q) =

0.312
1.312
0.709
0.085
+
−
+
.
2
2
2
1 + q /6.0 1 + q /15.02 1 + q /44.08 1 + q 2 /154.2(C.6)

Therefore, Fp (q) was calculated from Eqs. (C.4), (C.5) and (C.6). Finally, the point-proton
distribution ρp0 (r) was obtained by the inverse Fourier transformation of Fp (q) as∫
ρp0 (r) =F c (q)sin(rq)dq.
Gp (q)The obtained charge and point-proton density distribution are compared in Fig. C.1.

–104–

(C.7)

APPENDIX C. POINT PROTON DENSITY DISTRIBUTION OF THE
GROUND STATE IN 12 C

12C ground state distribution

ρ (fm-3)

0.08

ChargePoint-proton

0.06
0.04
0.02
0
0

1

2

3

4

5

6

r (fm)

Figure C.1: Density distribution of the 12 C ground state. The charge distribution given by
the SOG function is plotted by the red solid line, while the unfolded point-proton distribution
is plotted by the blue dashed line.

–105–

Appendix D
Data tables of cross sections
Table D.1: Data table for α+10 C elastic scattering cross section.
θc.m. (deg)dσ/dΩ (mb/sr)

4.25
4.75
5.94
6.31
6.69
7.06
9.50
11.0
13.0
15.0

(1.35 ± 0.22) × 104(9.01 ± 1.44) × 103(4.06 ± 0.66) × 103(2.90 ± 0.47) × 103(2.07 ± 0.33) × 103(1.36 ± 0.22) × 103(3.04 ± 0.49) × 102(3.50 ± 0.56) × 102(2.48 ± 0.40) × 102(1.10 ± 0.18) × 102

Table D.2:
Data table for α+10 C inelastic scattering cross section to the 2+
1 state at
Ex = 3.35 MeV.
θc.m. (deg)dσ/dΩ (mb/sr)

5.94
6.31
6.69
7.06
9.50
11.0
13.0
15.0

(1.50 ± 0.29) × 102(1.30 ± 0.25) × 102(1.03 ± 0.20) × 103(1.30 ± 0.25) × 103(6.65 ± 1.14) × 101(3.27 ± 0.56) × 101(1.76 ± 0.31) × 101(2.33 ± 0.40) × 101

107

References[1] T. Kubo, M. Ishihara, N. Inabe, H. Kumagai, I. Tanihata, K. Yoshida, T. Nakamura,
H. Okuno, S. Shimoura, and K. Asahi, Nucl. Instrum. Methods Phys. Res. B 70, 309(1992).
[2] Y. Yano, Nucl. Instrum. Methods Phys. Res. B 261, 1009 (2007).
[3] M. Thoennessen, Nucl. Phys. A 834, 688 (2010).
[4] P. Spiller and G. Franchetti, Nucl. Instrum. Methods Phys. Res. A 561, 305 (2006).
[5] S. Gales, Nucl. Phys. A 834, 717 (2010).
[6] H. Moeini, S. Ilieva, F. Aksouh, K. Boretzky, a. Chatillon, a. Corsi, P. Egelhof,
H. Emling, G. Ickert, J. Jourdan, N. Kalantar Nayestanaki, D. Kiselev, O. Kiselev,
C. Kozhuharov, T. Le Bleis, X. Le, Y. Litvinov, K. Mahata, J. Meier, F. Nolden,
S. Paschalis, U. Popp, H. Simon, M. Steck, T. Stöhlker, H. Weick, D. Werthmüller,
and a. Zalite, Nucl. Instrum. Methods Phys. Res. A 634, 77 (2011).
[7] T. Kröll, M. Von Schmid, J. C. Zamora, S. Bagchi, S. Bönig, M. Csatlós, I. Dillmann, C. Dimopoulou, P. Egelhof, V. Eremin, T. Furuno, H. Geissel, R. Gernhäuser,
M. N. Harakeh, A. L. Hartig, S. Ilieva, N. Kalantar-Nayestanaki, O. Kiselev, H. Kollmus, C. Kozhuharov, A. Krasznahorkay, M. Kuilman, S. Litvinov, Y. A. Litvinov,
M. Mahjour-Shafiei, M. Mutterer, D. Nagae, M. A. Najafi, C. Nociforo, F. Nolden,
U. Popp, C. Rigollet, S. Roy, C. Scheidenberger, M. Steck, B. Streicher, L. Stuhl,
M. Thürauf, T. Uesaka, H. Weick, J. S. Winfield, D. Winters, P. J. Woods, T. Yamaguchi, K. Yue, and J. Zenihiro, Jour. Phys.: Conf. Ser. 724, 012026 (2016).
[8] B. Franzke, Nucl. Instrum. Methods Phys. Res. B 24, 18 (1987).
[9] H. Reich, W. Bourgeois, B. Franzke, a. Kritzer, and V. Varentsov, Nucl. Phys. A 626,
417 (1997).
[10] B. Streicher, P. Egelhof, S. Ilieva, N. Kalantar-Nayestanaki, H. Kollmus, T. Kröll,
M. Mutterer, M. von Schmid, and M. Träger, Nucl. Instrum. Methods Phys. Res. A
654, 604 (2011).
[11] M. Steck, P. Beller, K. Beckert, B. Franzke, and F. Nolden, Nucl. Instrum. Methods
Phys. Res. A 532, 357 (2004).
[12] M. von Schmid, S. Bagchi, S. Bönig, M. Csatlós, I. Dillmann, C. Dimopoulou, P. Egelhof, V. Eremin, T. Furuno, H. Geissel, R. Gernhäuser, M. N. Harakeh, A.-L. Hartig,
S. Ilieva, N. Kalantar-Nayestanaki, O. Kiselev, H. Kollmus, C. Kozhuharov, A. Krasznahorkay, T. Kröll, M. Kuilman, S. Litvinov, Y. A. Litvinov, M. Mahjour-Shafiei,109

M. Mutterer, D. Nagae, M. A. Najafi, C. Nociforo, F. Nolden, U. Popp, C. Rigollet,
S. Roy, C. Scheidenberger, M. Steck, B. Streicher, L. Stuhl, M. Thürauf, T. Uesaka,
H. Weick, J. S. Winfield, D. Winters, P. J. Woods, T. Yamaguchi, K. Yue, J. C.
Zamora, and J. Zenihiro, Physica Scripta T166, 014005 (2015).
[13] J. C. Zamora, T. Aumann, S. Bagchi, S. Bönig, M. Csatlós, I. Dillmann,
C. Dimopoulou, P. Egelhof, V. Eremin, T. Furuno, H. Geissel, R. Gernhäuser,
M. Harakeh, A.-L. Hartig, S. Ilieva, N. Kalantar-Nayestanaki, O. Kiselev, H. Kollmus, C. Kozhuharov, A. Krasznahorkay, T. Kröll, M. Kuilman, S. Litvinov, Y. Litvinov, M. Mahjour-Shafiei, M. Mutterer, D. Nagae, M. Najafi, C. Nociforo, F. Nolden,
U. Popp, C. Rigollet, S. Roy, C. Scheidenberger, M. von Schmid, M. Steck, B. Streicher, L. Stuhl, M. Thürauf, T. Uesaka, H. Weick, J. Winfield, D. Winters, P. Woods,
T. Yamaguchi, K. Yue, and J. Zenihiro, Phys. Lett. B 763, 16 (2016).
[14] J. C. Zamora, T. Aumann, S. Bagchi, S. Bönig, M. Csatlós, I. Dillmann, C. Dimopoulou, P. Egelhof, V. Eremin, T. Furuno, H. Geissel, R. Gernhäuser, M. N.
Harakeh, A. L. Hartig, S. Ilieva, N. Kalantar-Nayestanaki, O. Kiselev, H. Kollmus,
C. Kozhuharov, A. Krasznahorkay, T. Kröll, M. Kuilman, S. Litvinov, Y. A. Litvinov,
M. Mahjour-Shafiei, M. Mutterer, D. Nagae, M. A. Najafi, C. Nociforo, F. Nolden,
U. Popp, C. Rigollet, S. Roy, C. Scheidenberger, M. Von Schmid, M. Steck, B. Streicher, L. Stuhl, M. Thürauf, T. Uesaka, H. Weick, J. S. Winfield, D. Winters, P. J.
Woods, T. Yamaguchi, K. Yue, and J. Zenihiro, Phys. Rev. C 96, 034617 (2017).
[15] J. Glorius, C. Langer, Z. Slavkovská, L. Bott, C. Brandau, B. Brückner, K. Blaum,
X. Chen, S. Dababneh, T. Davinson, P. Erbacher, S. Fiebiger, T. Gaßner, K. Göbel,
M. Groothuis, A. Gumberidze, G. Gyürky, M. Heil, R. Hess, R. Hensch, P. Hillmann,
P.-M. Hillenbrand, O. Hinrichs, B. Jurado, T. Kausch, A. Khodaparast, T. Kisselbach,
N. Klapper, C. Kozhuharov, D. Kurtulgil, G. Lane, C. Lederer-Woods, M. Lestinsky,
S. Litvinov, Y. A. Litvinov, B. Löher, F. Nolden, N. Petridis, U. Popp, T. Rauscher,
M. Reed, R. Reifarth, M. S. Sanjari, D. Savran, H. Simon, U. Spillmann, M. Steck,
T. Stöhlker, J. Stumm, A. Surzhykov, T. Szücs, T. T. Nguyen, A. Taremi Zadeh,
B. Thomas, S. Y. Torilov, H. Törnqvist, M. Träger, C. Trageser, S. Trotsenko, L. Varga,
M. Volknandt, H. Weick, M. Weigand, C. Wolf, P. J. Woods, and Y. M. Xing, Phys.
Rev. Lett. 122, 092701 (2019).
[16] G. A. Korolev, A. A. Vorobyov, and Y. K. Zalite, Nucl. Instrum. Methods 97, 323(1971).
[17] S. Ilieva, F. Aksouh, G. D. Alkhazov, L. Chulkov, A. V. Dobrovolsky, P. Egelhof,
H. Geissel, M. Gorska, A. Inglessi, R. Kanungo, A. V. Khanzadeev, O. A. Kiselev,
G. A. Korolev, X. C. Le, Y. A. Litvinov, C. Nociforo, D. M. Seliverstov, L. O. Sergeev,
H. Simon, V. A. Volkov, A. A. Vorobyov, H. Weick, V. I. Yatsoura, and A. A. Zhdanov,Nucl. Phys. A 875, 8 (2012).
[18] S. Neumaier, G. Alkhazov, M. Andronenko, A. Dobrovolsky, P. Egelhof, G. Gavrilov,
H. Geissel, H. Irnich, A. Khanzadeev, G. Korolev, A. Lobodenko, G. Münzenberg,–110–

M. Mutterer, W. Schwab, D. Seliverstov, T. Suzuki, N. Timofeev, A. Vorobyov, andV. Yatsoura, Nucl. Phys. A 712, 247 (2002).
[19] T. Hashimoto, H. Ishiyama, T. Ishikawa, T. Kawamura, K. Nakai, Y. Watanabe,
H. Miyatake, M. Tanaka, Y. Fuchi, N. Yoshikawa, S. Jeong, I. Katayama, T. Nomura,
T. Furukawa, S. Mitsuoka, K. Nishio, M. Matsuda, H. Ikezoe, T. Fukuda, S. Das,
P. Saha, Y. Mizoi, T. Komatsubara, M. Yamaguchi, and Y. Tagishi, Nucl. Instrum.
Methods Phys. Res. A 556, 339 (2006).
[20] C. Demonchy, M. Caamaño, H. Wang, W. Mittig, P. Roussel-Chomaz, H. Savajols,
M. Chartier, D. Cortina-Gil, A. Fomichev, G. Frémont, P. Gangnant, A. Gillibert,
L. Giot, M. Golovkov, B. Jurado, J. Libin, A. Obertelli, E. Pollaco, A. Rodin, C. Spitaels, S. Stepantsov, G. Ter-Akopian, and R. Wolski, Nucl. Instrum. Methods Phys.
Res. A 583, 341 (2007).
[21] D. Suzuki, M. Ford, D. Bazin, W. Mittig, W. G. Lynch, T. Ahn, S. Aune, E. Galyaev,
A. Fritsch, J. Gilbert, F. Montes, A. Shore, J. Kolata, J. Browne, A. Howard,
A. Roberts, and X. Tang, Nucl. Instrum. Methods Phys. Res. A 691, 39 (2012).
[22] S. Ota, H. Tokieda, C. S. Lee, and Y. N. Watanabe, Jour. Rad. Nucl. Chem. 305, 907(2015).
[23] J. Bradt, D. Bazin, F. Abu-Nimeh, T. Ahn, Y. Ayyad, S. B. Novo, L. Carpenter,
M. Cortesi, M. Kuchera, W. Lynch, W. Mittig, S. Rost, N. Watwood, and J. Yurkon,
Nucl. Instrum. Methods Phys. Res. A 875, 65 (2017).
[24] T. Roger, J. Pancin, G. Grinyer, B. Mauss, A. Laﬀoley, P. Rosier, H. Alvarez-Pol,
M. Babo, B. Blank, M. Caamaño, S. Ceruti, J. Daemen, S. Damoy, B. Duclos,
B. Fernández-Domı́nguez, F. Flavigny, J. Giovinazzo, T. Goigoux, J. Henares, P. Konczykowski, T. Marchi, G. Lebertre, N. Lecesne, L. Legeard, C. Maugeais, G. Minier,
B. Osmond, J. Pedroza, J. Pibernat, O. Poleshchuk, E. Pollacco, R. Raabe, B. Raine,
F. Renzi, F. Saillant, P. Sénécal, P. Sizun, D. Suzuki, J. Swartz, C. Wouters, G. Wittwer, and J. Yang, Nucl. Instrum. Methods Phys. Res. A 895, 126 (2018).
[25] B. Mauss, P. Morfouace, T. Roger, J. Pancin, G. Grinyer, J. Giovinazzo, V. Alcindor, H. Álvarez-Pol, A. Arokiaraj, M. Babo, B. Bastin, C. Borcea, M. Caamaño,S. Ceruti, B. Fernández-Domı́nguez, E. Foulon-Moret, P. Gangnant, S. Giraud, A. Laffoley, G. Mantovani, T. Marchi, B. Monteagudo, J. Pibernat, O. Poleshchuk, R. Raabe,
J. Refsgaard, A. Revel, F. Saillant, M. Stanoiu, G. Wittwer, and J. Yang, Nucl. Instrum. Methods Phys. Res. A 940, 498 (2019).
[26] E. Koshchiy, G. V. Rogachev, E. Pollacco, S. Ahn, E. Uberseder, J. Hooker, J. Bishop,
E. Aboud, M. Barbui, V. Z. Goldberg, C. Hunt, H. Jayatissa, C. Magana, R. O’Dwyer,
B. T. Roeder, A. Saastamoinen, and S. Upadhyayula, arXiv, 1906.07845 (2019).
[27] I. Tanihata, M. Alcorta, D. Bandyopadhyay, R. Bieri, L. Buchmann, B. Davids,
N. Galinski, D. Howell, W. Mills, S. Mythili, R. Openshaw, E. Padilla-Rodal,–111–

G. Ruprecht, G. Sheﬀer, A. C. Shotter, M. Trinczek, P. Walden, H. Savajpls, T. Roger,
M. Caamaño, W. Mittig, P. Roussel-Chomaz, R. Kanungo, A. Gallant, M. Notani,
G. Savard, and I. J. Thompson, Phys. Rev. Lett. 100, 192502 (2008).
[28] M. Vandebrouck, J. Gibelin, E. Khan, N. L. Achouri, H. Baba, D. Beaumel, Y. Blumenfeld, M. Caamaño, L. Càceres, G. Colò, F. Delaunay, B. Fernandez-Dominguez,
U. Garg, G. F. Grinyer, M. N. Harakeh, N. Kalantar-Nayestanaki, N. Keeley, W. Mittig, J. Pancin, R. Raabe, T. Roger, P. Roussel-Chomaz, H. Savajols, O. Sorlin,
C. Stodel, D. Suzuki, and J. C. Thomas, Phys. Rev. Lett. 113, 032504 (2014).
[29] M. Vandebrouck, J. Gibelin, E. Khan, N. L. Achouri, H. Baba, D. Beaumel, Y. Blumenfeld, M. Caamaño, L. Càceres, G. Colò, F. Delaunay, B. Fernandez-Dominguez,
U. Garg, G. F. Grinyer, M. N. Harakeh, N. Kalantar-Nayestanaki, N. Keeley, W. Mittig, J. Pancin, R. Raabe, T. Roger, P. Roussel-Chomaz, H. Savajols, O. Sorlin,
C. Stodel, D. Suzuki, and J. C. Thomas, Phys. Rev. C 92, 024316 (2015).
[30] S. Bagchi, J. Gibelin, M. Harakeh, N. Kalantar-Nayestanaki, N. Achouri, H. Akimune,
B. Bastin, K. Boretzky, H. Bouzomita, M. Caamaão, L. Càceres, S. Damoy, F. Delaunay, B. Fernàndez-Domı́nguez, M. Fujiwara, U. Garg, G. Grinyer, O. Kamalou,
E. Khan, A. Krasznahorkay, G. Lhoutellier, J. Libin, S. Lukyanov, K. Mazurek, M. Najafi, J. Pancin, Y. Penionzhkevich, L. Perrot, R. Raabe, C. Rigollet, T. Roger, S. Sambi,
H. Savajols, M. Senoville, C. Stodel, L. Suen, J. Thomas, M. Vandebrouck, and J. V.
de Walle, Phys. Lett. B 751, 371 (2015).
[31] C. Rodrı́guez-Tajes, F. Farget, L. Acosta, H. Alvarez-Pol, M. Babo, F. Boulay,
M. Caamaño, S. Damoy, B. Fernàndez-Domı́nguez, D. Galaviz, G. Grinyer, J. Grinyer,
M. Harakeh, P. Konczykowski, I. Martel, J. Pancin, G. Randisi, F. Renzi, T. Roger,
A. Sánchez-Benı́tez, P. Teubig, and M. Vandebrouck, Nucl. Phys. A 958, 246 (2017).
[32] D. Suzuki, A. Shore, W. Mittig, J. J. Kolata, D. Bazin, M. Ford, T. Ahn, F. D.
Becchetti, S. Beceiro Novo, D. Ben Ali, B. Bucher, J. Browne, X. Fang, M. Febbraro,
A. Fritsch, E. Galyaev, A. M. Howard, N. Keeley, W. G. Lynch, M. Ojaruega, A. L.
Roberts, and X. D. Tang, Phys. Rev. C 87, 054301 (2013).
[33] A. Fritsch, S. Beceiro-Novo, D. Suzuki, W. Mittig, J. J. Kolata, T. Ahn, D. Bazin, F. D.
Becchetti, B. Bucher, Z. Chajecki, X. Fang, M. Febbraro, A. M. Howard, Y. KanadaEn’yo, W. G. Lynch, A. J. Mitchell, M. Ojaruega, A. M. Rogers, A. Shore, T. Suhara,
X. D. Tang, R. Torres-Isea, and H. Wang, Phys. Rev. C 93, 014321 (2016).
[34] J. Bradt, Y. Ayyad, D. Bazin, W. Mittig, T. Ahn, S. B. Novo, B. Brown, L. Carpenter, M. Cortesi, M. Kuchera, W. Lynch, S. Rost, N. Watwood, J. Yurkon, J. Barney,
U. Datta, J. Estee, A. Gillibert, J. Manfredi, P. Morfouace, D. Pérez-Loureiro, E. Pollacco, J. Sammut, and S. Sweany, Phys. Lett. B 778, 155 (2018).
[35] Y. Ayyad, B. Olaizola, W. Mittig, G. Potel, V. Zelevinsky, M. Horoi, S. BeceiroNovo, M. Alcorta, C. Andreoiu, T. Ahn, M. Anholm, L. Atar, A. Babu, D. Bazin,–112–

N. Bernier, S. S. Bhattacharjee, M. Bowry, R. Caballero-Folch, M. Cortesi, C. Dalitz,
E. Dunling, A. B. Garnsworthy, M. Holl, B. Kootte, K. G. Leach, J. S. Randhawa,
Y. Saito, C. Santamaria, P. Šiurytė, C. E. Svensson, R. Umashankar, N. Watwood,and D. Yates, Phys. Rev. Lett. 123, 082501 (2019).
[36] J. Hooker, G. V. Rogachev, E. Koshchiy, S. Ahn, M. Barbui, V. Z. Goldberg, C. Hunt,
H. Jayatissa, E. C. Pollacco, B. T. Roeder, A. Saastamoinen, and S. Upadhyayula,
Phys. Rev. C 100, 054618 (2019).
[37] J. Pancin, J. L. Pedroza, J. Pibernat, E. Pollacco, A. Rebii, T. Roger, and P. Sizun,
Nucl. Instrum. Methods Phys. Res. A 840, 15 (2016).
[38] S. Beceiro-Novo, T. Ahn, D. Bazin, and W. Mittig, Prog. Part. Nucl. Phys. 84, 124(2015).
[39] Y. Ayyad, D. Bazin, S. Beceiro-Novo, M. Cortesi, and W. Mittig, Eur. Phys. J. A 54,
181 (2018).
[40] E. Koshchiy, G. Rogachev, E. Pollacco, S. Ahn, E. Uberseder, J. Hooker, J. Bishop,
E. Aboud, M. Barbui, V. Goldberg, C. Hunt, H. Jayatissa, C. Magana, R. Dwyer,
B. Roeder, A. Saastamoinen, and S. Upadhyayula, Nucl. Instrum. Methods Phys. Res.
A 957, 163398 (2020).
[41] T. Furuno, T. Kawabata, H. J. Ong, S. Adachi, Y. Ayyad, T. Baba, Y. Fujikawa,
T. Hashimoto, K. Inaba, Y. Ishii, S. Kabuki, H. Kubo, Y. Matsuda, Y. Matsuoka,
T. Mizumoto, T. Morimoto, M. Murata, T. Sawano, T. Suzuki, A. Takada, J. Tanaka,
I. Tanihata, T. Tanimori, D. T. Tran, M. Tsumura, and H. D. Watanabe, Nucl. Instrum. Methods Phys. Res. A 908, 215 (2018).
[42] A. Ochi, T. Nagayoshi, S. Koishi, T. Tanimori, T. Nagae, and M. Nakamura, Nucl.
Instrum. Methods Phys. Res. A 471, 264 (2001).
[43] H. D. Watanabe, Master thesis, Kyoto University (2013).
[44] T. Furuno, Master thesis, Kyoto University (2014).
[45] T. Furuno, T. Kawabata, S. Adachi, Y. Ayyad, T. Baba, T. Hashimoto, Y. Ishii,
S. Kabuki, H. Kubo, Y. Matsuda, Y. Matsuoka, T. Mizumoto, M. Murata, H. J. Ong,
T. Sawano, A. Takada, J. Tanaka, I. Tanihata, T. Tanimori, M. Tsumura, and H. D.
Watanabe, Jour. Phys.: Conf. Ser. 569, 012042 (2014).
[46] M. Murata, Master thesis, Kyoto University (2014).
[47] T. Furuno, T. Kawabata, M. Murata, S. Adachi, Y. Ayyad, H. Baba, T. Baba, M. Cavallaro, T. Hashimoto, Y. Ishii, R. Kobayakawa, Y. Matsuda, Y. Matsuoka, T. Morimoto, T. Nanamura, S. Beceilo-Novo, H. J. Ong, A. Sakaue, J. Tanaka, I. Tanihata,
D. T. Trong, M. Tsumura, and H. D. Watanabe, Jour. Phys.: Conf. Ser. 863, 012076(2017).
–113–

[48] T. Morimoto, Master thesis, Kyoto University (2017).
[49] K. Inaba, Master thesis, Kyoto University (2018).
[50] M. G. Mayer, Phys. Rev. 75, 1969 (1949).
[51] O. Haxel, J. H. D. Jensen, and H. E. Suess, Phys. Rev. 75, 1766 (1949).
[52] A. Navin, D. W. Anthony, T. Aumann, T. Baumann, D. Bazin, Y. Blumenfeld, B. A.
Brown, T. Glasmacher, P. G. Hansen, R. W. Ibbotson, P. A. Lofy, V. Maddalena,
K. Miller, T. Nakamura, B. V. Pritychenko, B. M. Sherrill, E. Spears, M. Steiner,
J. A. Tostevin, J. Yurkon, and A. Wagner, Phys. Rev. Lett. 85, 266 (2000).
[53] H. Iwasaki, T. Motobayashi, H. Akiyoshi, Y. Ando, N. Fukuda, H. Fujiwara, Z. Fülöp,
K. I. Hahn, Y. Higurashi, M. Hirai, I. Hisanaga, N. Iwasa, T. Kijima, T. Minemura,
T. Nakamura, M. Notani, S. Ozawa, H. Sakurai, S. Shimoura, S. Takeuchi, T. Teranishi,
Y. Yanagisawa, and M. Ishihara, Phys. Lett. B 481, 7 (2000).
[54] H. Iwasaki, T. Motobayashi, H. Akiyoshi, Y. Ando, N. Fukuda, H. Fujiwara, Z. Fülöp,
K. Hahn, Y. Higurashi, M. Hirai, I. Hisanaga, N. Iwasa, T. Kijima, A. Mengoni,
T. Minemura, T. Nakamura, M. Notani, S. Ozawa, H. Sagawa, H. Sakurai, S. Shimoura,
S. Takeuchi, T. Teranishi, Y. Yanagisawa, and M. Ishihara, Phys. Lett. B 491, 8 (2000).
[55] S. Shimoura, A. Saito, T. Minemura, Y. Matsuyama, H. Baba, H. Akiyoshi, N. Aoi,
T. Gomi, Y. Higurashi, K. Ieki, N. Imai, N. Iwasa, H. Iwasaki, S. Kanno, S. Kubono,
M. Kunibu, S. Michimasa, T. Motobayashi, T. Nakamura, H. Sakurai, M. Serata,
E. Takeshita, S. Takeuchi, T. Teranishi, K. Ue, K. Yamada, Y. Yanagisawa, M. Ishihara, and N. Itagaki, Phys. Lett. B 560, 31 (2003).
[56] C. Thibault, R. Klapisch, C. Rigaud, A. M. Poskanzer, R. Prieels, L. Lessard, and
W. Reisdorf, Phys. Rev. C 12, 644 (1975).
[57] T. Motobayashi, Y. Ikeda, K. Ieki, M. Inoue, N. Iwasa, T. Kikuchi, M. Kurokawa,
S. Moriya, S. Ogawa, H. Murakami, S. Shimoura, Y. Yanagisawa, T. Nakamura,
Y. Watanabe, M. Ishihara, T. Teranishi, H. Okuno, and R. Casten, Phys. Lett. B
346, 9 (1995).
[58] B. Bastin, S. Grévy, D. Sohler, O. Sorlin, Z. Dombrádi, N. L. Achouri, J. C. Angélique,F. Azaiez, D. Baiborodin, R. Borcea, C. Bourgeois, A. Buta, A. Bürger, R. Chapman,
J. C. Dalouzy, Z. Dlouhy, A. Drouard, Z. Elekes, S. Franchoo, S. Iacob, B. Laurent,
M. Lazar, X. Liang, E. Liénard, J. Mrazek, L. Nalpas, F. Negoita, N. A. Orr, Y. Penionzhkevich, Z. Podolyák, F. Pougheon, P. Roussel-Chomaz, M. G. Saint-Laurent,
M. Stanoiu, I. Stefan, F. Nowacki, and A. Poves, Phys. Rev. Lett. 99, 022503 (2007).
[59] S. Takeuchi, M. Matsushita, N. Aoi, P. Doornenbal, K. Li, T. Motobayashi, H. Scheit,
D. Steppenbeck, H. Wang, H. Baba, D. Bazin, L. Càceres, H. Crawford, P. Fallon,
R. Gernhäuser, J. Gibelin, S. Go, S. Grévy, C. Hinke, C. R. Hoﬀman, R. Hughes,
E. Ideguchi, D. Jenkins, N. Kobayashi, Y. Kondo, R. Krücken, T. Le Bleis, J. Lee,–114–

G. Lee, A. Matta, S. Michimasa, T. Nakamura, S. Ota, M. Petri, T. Sako, H. Sakurai, S. Shimoura, K. Steiger, K. Takahashi, M. Takechi, Y. Togano, R. Winkler, andK. Yoneda, Phys. Rev. Lett. 109, 182501 (2012).
[60] A. Ozawa, T. Kobayashi, T. Suzuki, K. Yoshida, and I. Tanihata, Phys. Rev. Lett. 84,
5493 (2000).
[61] J. Prisciandaro, P. Mantica, B. Brown, D. Anthony, M. Cooper, A. Garcia, D. Groh,
A. Komives, W. Kumarasiri, P. Lofy, A. Oros-Peusquens, S. Tabor, and M. Wiedeking,Phys. Lett. B 510, 17 (2001).
[62] R. Janssens, B. Fornal, P. Mantica, B. Brown, R. Broda, P. Bhattacharyya, M. Carpenter, M. Cinausero, P. Daly, A. Davies, T. Glasmacher, Z. Grabowski, D. Groh,
M. Honma, F. Kondev, W. Królas, T. Lauritsen, S. Liddick, S. Lunardi, N. Marginean,
T. Mizusaki, D. Morrissey, A. Morton, W. Mueller, T. Otsuka, T. Pawlat, D. Seweryniak, H. Schatz, A. Stolz, S. Tabor, C. Ur, G. Viesti, I. Wiedenhöver, and J. Wrzesiński,
Phys. Lett. B 546, 55 (2002).
[63] A. Gade, R. V. F. Janssens, D. Bazin, R. Broda, B. A. Brown, C. M. Campbell,
M. P. Carpenter, J. M. Cook, A. N. Deacon, D.-C. Dinca, B. Fornal, S. J. Freeman,
T. Glasmacher, P. G. Hansen, B. P. Kay, P. F. Mantica, W. F. Mueller, J. R. Terry,J. A. Tostevin, and S. Zhu, Phys. Rev. C 74, 021302 (2006).
[64] F. Wienholtz, D. Beck, K. Blaum, C. Borgmann, M. Breitenfeldt, R. B. Cakirli,
S. George, F. Herfurth, J. D. Holt, M. Kowalska, S. Kreim, D. Lunney, V. Manea,
J. Menéndez, D. Neidherr, M. Rosenbusch, L. Schweikhard, A. Schwenk, J. Simonis,
J. Stanja, R. N. Wolf, and K. Zuber, Nature 498, 346 (2013).
[65] M. Rosenbusch, P. Ascher, D. Atanasov, C. Barbieri, D. Beck, K. Blaum, C. Borgmann,
M. Breitenfeldt, R. B. Cakirli, A. Cipollone, S. George, F. Herfurth, M. Kowalska,
S. Kreim, D. Lunney, V. Manea, P. Navrátil, D. Neidherr, L. Schweikhard, V. Somà,
J. Stanja, F. Wienholtz, R. N. Wolf, and K. Zuber, Phys. Rev. Lett. 114, 202501(2015).
[66] D. Steppenbeck, S. Takeuchi, N. Aoi, P. Doornenbal, M. Matsushita, H. Wang, Y. Utsuno, H. Baba, S. Go, J. Lee, K. Matsui, S. Michimasa, T. Motobayashi, D. Nishimura,
T. Otsuka, H. Sakurai, Y. Shiga, N. Shimizu, P.-A. Söderström, T. Sumikama, R. Taniuchi, J. J. Valiente-Dobón, and K. Yoneda, Phys. Rev. Lett. 114, 252501 (2015).
[67] D. Steppenbeck, S. Takeuchi, N. Aoi, P. Doornenbal, M. Matsushita, H. Wang,
H. Baba, N. Fukuda, S. Go, M. Honma, J. Lee, K. Matsui, S. Michimasa, T. Motobayashi, D. Nishimura, T. Otsuka, H. Sakurai, Y. Shiga, P. A. Söderström,
T. Sumikama, H. Suzuki, R. Taniuchi, Y. Utsuno, J. J. Valiente-Dobón, and K. Yoneda,
Nature 502, 207 (2013).
[68] S. Michimasa, M. Kobayashi, Y. Kiyokawa, S. Ota, D. S. Ahn, H. Baba, G. P. A. Berg,
M. Dozono, N. Fukuda, T. Furuno, E. Ideguchi, N. Inabe, T. Kawabata, S. Kawase,–115–

K. Kisamori, K. Kobayashi, T. Kubo, Y. Kubota, C. S. Lee, M. Matsushita, H. Miya,
A. Mizukami, H. Nagakura, D. Nishimura, H. Oikawa, H. Sakai, Y. Shimizu, A. Stolz,
H. Suzuki, M. Takaki, H. Takeda, S. Takeuchi, H. Tokieda, T. Uesaka, K. Yako, Y. Yamaguchi, Y. Yanagisawa, R. Yokoyama, K. Yoshida, and S. Shimoura, Phys. Rev. Lett.
121, 022506 (2018).
[69] S. Chen, J. Lee, P. Doornenbal, A. Obertelli, C. Barbieri, Y. Chazono, P. Navrátil,
K. Ogata, T. Otsuka, F. Raimondi, V. Somà, Y. Utsuno, K. Yoshida, H. Baba,
F. Browne, D. Calvet, F. Château, N. Chiga, A. Corsi, M. L. Cortés, A. Delbart, J.-M.
Gheller, A. Giganon, A. Gillibert, C. Hilaire, T. Isobe, J. Kahlbow, T. Kobayashi,
Y. Kubota, V. Lapoux, H. N. Liu, T. Motobayashi, I. Murray, H. Otsu, V. Panin,
N. Paul, W. Rodriguez, H. Sakurai, M. Sasano, D. Steppenbeck, L. Stuhl, Y. L. Sun,
Y. Togano, T. Uesaka, K. Wimmer, K. Yoneda, N. Achouri, O. Aktas, T. Aumann,
L. X. Chung, F. Flavigny, S. Franchoo, I. Gašparić, R.-B. Gerst, J. Gibelin, K. I. Hahn,
D. Kim, T. Koiwai, Y. Kondo, P. Koseoglou, C. Lehr, B. D. Linh, T. Lokotko, M. MacCormick, K. Moschner, T. Nakamura, S. Y. Park, D. Rossi, E. Sahin, D. Sohler, P.-A.
Söderström, S. Takeuchi, H. Törnqvist, V. Vaquero, V. Wagner, S. Wang, V. Werner,
X. Xu, H. Yamada, D. Yan, Z. Yang, M. Yasuda, and L. Zanetti, Phys. Rev. Lett.
123, 142501 (2019).
[70] K. Ikeda, N. Takigawa, and H. Horiuchi, Prog. Theor. Phys. Suppl E68, 464 (1968).
[71] F. Hoyle, Astrophys. J. Suppl. Ser. 1, 121 (1954).
[72] Y. Kanada-En’yo, H. Horiuchi, and A. Doté, Phys. Rev. C 60, 064304 (1999).
[73] Y. Kanada-En’yo, M. Kimura, and A. Ono, Prog. Theor. Exp. Phys. 2012, 01A202(2012).
[74] T. Kawabata, H. Akimune, H. Fujita, Y. Fujita, M. Fujiwara, K. Hara, K. Hatanaka,
M. Itoh, Y. Kanada-En’yo, S. Kishi, K. Nakanishi, H. Sakaguchi, Y. Shimbara,
A. Tamii, S. Terashima, M. Uchida, T. Wakasa, Y. Yasuda, H. Yoshida, and M. Yosoi,
Phys. Lett. B 646, 6 (2007).
[75] T. Yamada, Y. Funaki, H. Horiuchi, K. Ikeda, and A. Tohsaki, Prog. Theor. Phys.
120, 1139 (2008).
[76] T. Furuno, T. Kawabata, S. Adachi, Y. Ayyad, Y. Kanada-En’yo, Y. Fujikawa, K. Inaba, M. Murata, H. J. Ong, M. Sferrazza, Y. Takahashi, T. Takeda, I. Tanihata, D. T.
Tran, and M. Tsumura, Phys. Rev. C 100, 054322 (2019).
[77] D. T. Tran, H. J. Ong, G. Hagen, T. D. Morris, N. Aoi, T. Suzuki, Y. Kanada-En’yo,
L. S. Geng, S. Terashima, I. Tanihata, T. T. Nguyen, Y. Ayyad, P. Y. Chan, M. Fukuda,
H. Geissel, M. N. Harakeh, T. Hashimoto, T. H. Hoang, E. Ideguchi, A. Inoue, G. R.
Jansen, R. Kanungo, T. Kawabata, L. H. Khiem, W. P. Lin, K. Matsuta, M. Mihara,
S. Momota, D. Nagae, N. D. Nguyen, D. Nishimura, T. Otsuka, A. Ozawa, P. P. Ren,
H. Sakaguchi, C. Scheidenberger, J. Tanaka, M. Takechi, R. Wada, and T. Yamamoto,
Nature Communications 9, 1594 (2018).
–116–

[78] H. R. Collard, L. R. B. Elton, and R. Hofstadter, Nuclear radii 2 (Springer 1967).
[79] I. Angeli and M. Csatlós, Nucl. Phys. A 288, 480 (1977).
[80] I. Angeli and K. P. Marinova, J. Phys. G 42, 055108 (2015).
[81] A. Bohr and B. R. Mottelson, NUCLEAR STRUCTURE VOL I (W. A. BENJAMIN,
INC. 1971).
[82] E. P. Wigner and E. Feenberg, Rep. Prog. Phys. 8, 274 (1941).
[83] M. G. Mayer, The shell model, Novel Lectures pp. 20-37, (1963).
[84] A. Bernstein, V. Brown, and V. Madsen, Phys. Lett. B 103, 255 (1981).
[85] A. M. Bernstein, V. R. Brown, and V. A. Madsen, Comments Nucl. Part. Phys. 11,
203 (1983).
[86] M. A. Kennedy, P. D. Cottle, and K. W. Kemper, Phys. Rev. C 46, 1811 (1992).
[87] L. A. Riley, J. K. Jewell, P. D. Cottle, T. Glasmacher, K. W. Kemper, N. Alamanos,
Y. Blumenfeld, J. A. Carr, M. J. Chromik, R. W. Ibbotson, F. Maréchal, W. E.
Ormand, F. Petrovich, H. Scheit, and T. Suomijärvi, Phys. Rev. Lett. 82, 4196 (1999).
[88] J. K. Jewell, L. A. Riley, P. D. Cottle, K. W. Kemper, T. Glasmacher, R. W. Ibbotson, H. Scheit, M. Chromik, Y. Blumenfeld, S. E. Hirzebruch, F. Maréchal, andT. Suomijärvi, Phys. Lett. B 454, 181 (1999).
[89] H. Iwasaki, T. Motobayashi, H. Akiyoshi, Y. Ando, N. Fukuda, H. Fujiwara, Z. Fülöp,
K. I., Y. Higurashi, M. Hirai, I. Hisanaga, N. Iwasa, T. Kijima, T. Minemura, T. Nakamura, M. Notani, S. Ozawa, H. Sakurai, S. Shimoura, S. Takeuchi, T. Teranishi,
Y. Yanagisawa, and M. Ishihara, Phys. Lett. B 481, 7 (2000).
[90] E. Khan, T. Suomijärvi, Y. Blumenfeld, N. Van Giai, N. Alamanos, F. Auger,
E. Bauge, D. Beaumel, J. P. Delaroche, P. Delbourgo-Salvador, A. Drouart, S. Fortier,N. Frascaria, A. Gillibert, M. Girod, C. Jouanne, K. W. Kemper, A. Lagoyannis, V. Lapoux, A. Lépine-Szily, I. Lhenry, J. Libert, F. Maréchal, J. M. Maison,
A. Musumarra, S. Ottini-Hustache, P. Piattelli, S. Pita, E. C. Pollacco, P. RousselChomaz, D. Santonocito, J. E. Sauvestre, J. A. Scarpaci, and T. Zerguerras, Nucl.
Phys. A 694, 103 (2001).
[91] P. D. Cottle, Z. Hu, B. V. Pritychenko, J. A. Church, M. Fauerbach, T. Glasmacher,
R. W. Ibbotson, K. W. Kemper, L. A. Riley, H. Scheit, and M. Steiner, Phys. Rev.
Lett. 88, 172502 (2002).
[92] Z. Elekes, Z. Dombrádi, A. Krasznahorkay, H. Baba, M. Csatlós, L. Csige, N. Fukuda,
Z. Fülöp, Z. Gácsi, J. Gulyás, N. Iwasa, H. Kinugawa, S. Kubono, M. Kurokawa,
X. Liu, S. Michimasa, T. Minemura, T. Motobayashi, A. Ozawa, A. Saito, S. Shimoura,
S. Takeuchi, I. Tanihata, P. Thirolf, Y. Yanagisawa, and K. Yoshida, Phys. Lett. B
586, 34 (2004).
–117–

[93] N. Imai, H. J. Ong, N. Aoi, H. Sakurai, K. Demichi, H. Kawasaki, H. Baba,
Z. Dombrádi, Z. Elekes, N. Fukuda, Z. Fülöp, A. Gelberg, T. Gomi, H. Hasegawa,K. Ishikawa, H. Iwasaki, E. Kaneko, S. Kanno, T. Kishida, Y. Kondo, T. Kubo, K. Kurita, S. Michimasa, T. Minemura, M. Miura, T. Motobayashi, T. Nakamura, M. Notani, T. K. Onishi, A. Saito, S. Shimoura, T. Sugimoto, M. K. Suzuki, E. Takeshita,
S. Takeuchi, M. Tamaki, K. Yamada, K. Yoneda, H. Watanabe, and M. Ishihara, Phys.
Rev. Lett. 92, 062501 (2004).
[94] C. Jouanne, V. Lapoux, F. Auger, N. Alamanos, A. Drouart, A. Gillibert, G. Lobo,
A. Musumarra, L. Nalpas, E. Pollacco, J.-L. Sida, M. Trotta, Y. Blumenfeld, E. Khan,
T. Suomijärvi, T. Zerguerras, P. Roussel-Chomaz, H. Savajols, A. Lagoyannis, and
A. Pakou, Phys. Rev. C 72, 014308 (2005).
[95] L. A. Riley, M. A. Abdelqader, D. Bazin, M. J. Bojazi, B. A. Brown, C. M. Campbell,
J. A. Church, P. D. Cottle, D. C. Dinca, J. Enders, A. Gade, T. Glasmacher, M. Honma,
S. Horibe, Z. Hu, K. W. Kemper, W. F. Mueller, H. Olliver, T. Otsuka, B. C. Perry,
B. T. Roeder, B. M. Sherrill, T. P. Spencer, and J. R. Terry, Phys. Rev. C 72, 024311(2005).
[96] K. Yamada, T. Motobayashi, N. Aoi, H. Baba, K. Demichi, Z. Elekes, J. Gibelin,
T. Gomi, H. Hasegawa, N. Imai, H. Iwasaki, S. Kanno, T. Kubo, K. Kurita, Y. Matsuyama, S. Michimasa, T. Minemura, M. Notani, T. Onishi K., H.J. Ong, S. Ota,
A. Ozawa, A. Saito, H. Sakurai, S. Shimoura, E. Takeshita, S. Takeuchi, M. Tamaki,
Y. Togano, Y. Yanagisawa, K. Yoneda, and I. Tanihata, Eur. Phys. J. A 25, 409 (2005).
[97] H. J. Ong, N. Imai, N. Aoi, H. Sakurai, Z. Dombrádi, A. Saito, Z. Elekes, H. Baba,
K. Demichi, Z. S. Fülöp, J. Gibelin, T. Gomi, H. Hasegawa, M. Ishihara, H. Iwasaki,
S. Kanno, S. Kawai, T. Kubo, K. Kurita, Y. U. Matsuyama, S. Michimasa, T. Minemura, T. Motobayashi, M. Notani, S. Ota, H. K. Sakai, S. Shimoura, E. Takeshita,
S. Takeuchi, M. Tamaki, Y. Togano, K. Yamada, Y. Yanagisawa, and K. Yoneda, Phys.
Rev. C 73, 024610 (2006).
[98] E. Becheva, Y. Blumenfeld, E. Khan, D. Beaumel, J. M. Daugas, F. Delaunay, C.-E.
Demonchy, A. Drouart, M. Fallot, A. Gillibert, L. Giot, M. Grasso, N. Keeley, K. W.
Kemper, D. T. Khoa, V. Lapoux, V. Lima, A. Musumarra, L. Nalpas, E. C. Pollacco,
O. Roig, P. Roussel-Chomaz, J. E. Sauvestre, J. A. Scarpaci, F. Skaza, and H. S. Than,
Phys. Rev. Lett. 96, 012501 (2006).
[99] C. M. Campbell, N. Aoi, D. Bazin, M. D. Bowen, B. A. Brown, J. M. Cook, D. C.
Dinca, A. Gade, T. Glasmacher, M. Horoi, S. Kanno, T. Motobayashi, L. A. Riley,
H. Sagawa, H. Sakurai, K. Starosta, H. Suzuki, S. Takeuchi, J. R. Terry, K. Yoneda,
and H. Zwahlen, Phys. Lett. B 652, 169 (2007).
[100] M. Wiedeking, P. Fallon, A. O. Macchiavelli, J. Gibelin, M. S. Basunia, R. M. Clark,
M. Cromaz, M.-A. Deleplanque, S. Gros, H. B. Jeppesen, P. T. Lake, I.-Y. Lee, L. G.
Moretto, J. Pavan, L. Phair, E. Rodriguez-Vietiez, L. A. Bernstein, D. L. Bleuel, J. T.
–118–

Burke, S. R. Lesher, B. F. Lyles, and N. D. Scielzo, Phys. Rev. Lett. 100, 152501(2008).
[101] H. J. Ong, N. Imai, D. Suzuki, H. Iwasaki, H. Sakurai, T. K. Onishi, M. K. Suzuki,
S. Ota, S. Takeuchi, T. Nakao, Y. Togano, Y. Kondo, N. Aoi, H. Baba, S. Bishop,
Y. Ichikawa, M. Ishihara, T. Kubo, K. Kurita, T. Motobayashi, T. Nakamura, T. Okumura, and Y. Yanagisawa, Phys. Rev. C 78, 014308 (2008).
[102] N. Iwasa, T. Motobayashi, S. Bishop, Z. Elekes, J. Gibelin, M. Hosoi, K. Ieki,
K. Ishikawa, H. Iwasaki, S. Kawai, S. Kubono, K. Kurita, M. Kurokawa, N. Matsui,
T. Minemura, H. Morikawa, T. Nakamura, M. Niikura, M. Notani, S. Ota, A. Saito,
H. Sakurai, S. Shimoura, K. Sugawara, T. Sugimoto, H. Suzuki, T. Suzuki, I. Tanihata,
E. Takeshita, T. Teranishi, Y. Togano, K. Yamada, K. Yamaguchi, and Y. Yanagisawa,Phys. Rev. C 78, 024306 (2008).
[103] Z. Elekes, Z. Dombrádi, T. Aiba, N. Aoi, H. Baba, D. Bemmerer, B. A. Brown, T. Furumoto, Z. Fülöp, N. Iwasa, Á. Kiss, T. Kobayashi, Y. Kondo, T. Motobayashi,
T. Nakabayashi, T. Nannichi, Y. Sakuragi, H. Sakurai, D. Sohler, M. Takashina,
S. Takeuchi, K. Tanaka, Y. Togano, K. Yamada, M. Yamaguchi, and K. Yoneda,
Phys. Rev. C 79, 011302 (2009).
[104] M. Petri, P. Fallon, A. O. Macchiavelli, S. Paschalis, K. Starosta, T. Baugher, D. Bazin,
L. Cartegni, R. M. Clark, H. L. Crawford, M. Cromaz, A. Dewald, A. Gade, G. F.
Grinyer, S. Gros, M. Hackstein, H. B. Jeppesen, I. Y. Lee, S. McDaniel, D. Miller,
M. M. Rajabali, A. Ratkiewicz, W. Rother, P. Voss, K. A. Walsh, D. Weisshaar,
M. Wiedeking, and B. A. Brown, Phys. Rev. Lett. 107, 102501 (2011).
[105] M. Petri, S. Paschalis, R. M. Clark, P. Fallon, A. O. Macchiavelli, K. Starosta,
T. Baugher, D. Bazin, L. Cartegni, H. L. Crawford, M. Cromaz, U. Datta Pramanik, G. de Angelis, A. Dewald, A. Gade, G. F. Grinyer, S. Gros, M. Hackstein,
H. B. Jeppesen, I. Y. Lee, S. McDaniel, D. Miller, M. M. Rajabali, A. Ratkiewicz,
W. Rother, P. Voss, K. A. Walsh, D. Weisshaar, M. Wiedeking, B. A. Brown,
C. Forssén, P. Navrátil, and R. Roth, Phys. Rev. C 86, 044329 (2012).
[106] P. Voss, T. Baugher, D. Bazin, R. M. Clark, H. L. Crawford, A. Dewald, P. Fallon, A. Gade, G. F. Grinyer, H. Iwasaki, A. O. Macchiavelli, S. McDaniel, D. Miller,
M. Petri, A. Ratkiewicz, W. Rother, K. Starosta, K. A. Walsh, D. Weisshaar,
C. Forssén, R. Roth, and P. Navrátil, Phys. Rev. C 86, 011303 (2012).
[107] Y. Togano, Y. Yamada, N. Iwasa, K. Yamada, T. Motobayashi, N. Aoi, H. Baba,
S. Bishop, X. Cai, P. Doornenbal, D. Fang, T. Furukawa, K. Ieki, T. Kawabata,S. Kanno, N. Kobayashi, Y. Kondo, T. Kuboki, N. Kume, K. Kurita, M. Kurokawa,
Y. G. Ma, Y. Matsuo, H. Murakami, M. Matsushita, T. Nakamura, K. Okada, S. Ota,
Y. Satou, S. Shimoura, R. Shioda, K. N. Tanaka, S. Takeuchi, W. Tian, H. Wang,
J. Wang, and K. Yoneda, Phys. Rev. Lett. 108, 222501 (2012).
–119–[108] S. Michimasa, Y. Yanagisawa, K. Inafuku, N. Aoi, Z. Elekes, Z. Fülöp, Y. Ichikawa,
N. Iwasa, K. Kurita, M. Kurokawa, T. Machida, T. Motobayashi, T. Nakamura,
T. Nakabayashi, M. Notani, H. J. Ong, T. K. Onishi, H. Otsu, H. Sakurai, M. Shinohara, T. Sumikama, S. Takeuchi, K. Tanaka, Y. Togano, K. Yamada, M. Yamaguchi,
and K. Yoneda, Phys. Rev. C 89, 054307 (2014).
[109] L. A. Riley, M. L. Agiorgousis, T. R. Baugher, D. Bazin, M. Bowry, P. D. Cottle, F. G.
DeVone, A. Gade, M. T. Glowacki, K. W. Kemper, E. Lunderberg, D. M. McPherson,
S. Noji, F. Recchia, B. V. Sadler, M. Scott, D. Weisshaar, and R. G. T. Zegers, Phys.
Rev. C 90, 011305 (2014).
[110] A. Corsi, S. Boissinot, A. Obertelli, P. Doornenbal, M. Dupuis, F. Lechaftois, M. Matsushita, S. Péru, S. Takeuchi, H. Wang, N. Aoi, H. Baba, P. Bednarczyk, M. Ciemala,A. Gillibert, T. Isobe, A. Jungclaus, V. Lapoux, J. Lee, M. Martini, K. Matsui, T. Motobayashi, D. Nishimura, S. Ota, E. Pollacco, H. Sakurai, C. Santamaria, Y. Shiga,
D. Sohler, D. Steppenbeck, and R. Taniuchi, Phys. Lett. B 743, 451 (2015).
[111] M. L. Cortés, P. Doornenbal, M. Dupuis, S. M. Lenzi, F. Nowacki, A. Obertelli, S. Péru,
N. Pietralla, V. Werner, K. Wimmer, G. Authelet, H. Baba, D. Calvet, F. Château,
A. Corsi, A. Delbart, J.-M. Gheller, A. Gillibert, T. Isobe, V. Lapoux, C. Louchart,
M. Matsushita, S. Momiyama, T. Motobayashi, M. Niikura, H. Otsu, C. Péron,
A. Peyaud, E. C. Pollacco, J.-Y. Roussé, H. Sakurai, C. Santamaria, M. Sasano,Y. Shiga, S. Takeuchi, R. Taniuchi, T. Uesaka, H. Wang, K. Yoneda, F. Browne,
L. X. Chung, Z. Dombradi, S. Franchoo, F. Giacoppo, A. Gottardo, K. HadynskaKlek, Z. Korkulu, S. Koyama, Y. Kubota, J. Lee, M. Lettmann, R. Lozeva, K. Matsui,
T. Miyazaki, S. Nishimura, L. Olivier, S. Ota, Z. Patel, E. Sahin, C. M. Shand, P.-A.
Söderström, I. Stefan, D. Steppenbeck, T. Sumikama, D. Suzuki, Z. Vajta, J. Wu, andZ. Xu, Phys. Rev. C 97, 044315 (2018).
[112] L. A. Riley, D. Bazin, J. Belarge, P. C. Bender, B. A. Brown, P. D. Cottle, B. Elman,
A. Gade, S. D. Gregory, E. B. Haldeman, K. W. Kemper, B. R. Klybor, M. A. Liggett,
S. Lipschutz, B. Longfellow, E. Lunderberg, T. Mijatovic, J. Pereira, L. M. Skiles,
R. Titus, A. Volya, D. Weisshaar, J. C. Zamora, and R. G. T. Zegers, Phys. Rev. C
100, 044312 (2019).
[113] D. Tilley, J. Kelley, J. Godwin, D. Millener, J. Purcell, C. Sheu, and H. Weller, Nucl.
Phys. A 745, 155 (2004).
[114] T. R. Fisher, S. S. Hanna, D. C. Healey, and P. Paul, Phys. Rev. 176, 1130 (1968).
[115] Y. Blumenfeld, F. Auger, J. E. Sauvestre, F. Maréchal, S. Ottini, N. Alamanos, A. Barbier, D. Beaumel, B. Bonnereau, D. Charlet, J. F. Clavelin, P. Courtat, P. DelbourgoSalvador, R. Douet, M. Engrand, T. Ethvignot, A. Gillibert, E. Khan, V. Lapoux,
A. Lagoyannis, L. Lavergne, S. Lebon, P. Lelong, A. Lesage, V. Le Ven, I. Lhenry,J. M. Martin, A. Musumarra, S. Pita, L. Petizon, E. Pollacco, J. Pouthas, A. Richard,
D. Rougier, D. Santonocito, J. A. Scarpaci, J. L. Sida, C. Soulet, J. S. Stutzmann,–120–

T. Suomijärvi, M. Szmigiel, P. Volkov, and G. Voltolini, Nucl. Instrum. Methods Phys.
Res. A 421, 471 (1999).
[116] J.-P. Jeukenne, A. Lejeune, and C. Mahaux, Phys. Rev. C 16, 80 (1977).
[117] E. A. McCutchan, C. J. Lister, S. C. Pieper, R. B. Wiringa, D. Seweryniak, J. P.
Greene, P. F. Bertone, M. P. Carpenter, C. J. Chiara, G. Gürdal, C. R. Hoﬀman,
R. V. F. Janssens, T. L. Khoo, T. Lauritsen, and S. Zhu, Phys. Rev. C 86, 014312(2012).
[118] R.
Veenhof,Garfield
http://garfield.web.cern.ch/garfield/.

simulationofgaseousdetectors,[119] S. Mukhopadhyay and N. Majumdar, Engineering Analysis with Boundary Elements
33, 105 (2009).
[120] A. Ochi, T. Nagayoshi, S. Koishi, T. Tanimori, T. Nagae, and M. Nakamura, Nucl.
Instrum. Methods Phys. Res. A 471, 264 (2001).
[121] Y. Mizumura, T. Tanimori, H. Kubo, A. Takada, J. D. Parker, T. Mizumoto, S. Sonoda, D. Tomono, T. Sawano, K. Nakamura, Y. Matsuoka, S. Komura, S. Nakamura,
M. Oda, K. Miuchi, S. Kabuki, Y. Kishimoto, S. Kurosawa, and S. Iwaki, Journal of
Instrumentation 9, C05045 (2014).
[122] D. Tomono, T. Mizumoto, A. Takada, S. Komura, Y. Matsuoka, Y. Mizumura, M. Oda,
and T. Tanimori, Scientific Reports 7, 41972 (2017).
[123] T. Tanimori, Y. Mizumura, A. Takada, S. Miyamoto, T. Takemura, T. Kishimoto, S. Komura, H. Kubo, S. Kurosawa, Y. Matsuoka, K. Miuchi, T. Mizumoto,Y. Nakamasu, K. Nakamura, J. D. Parker, T. Sawano, S. Sonoda, D. Tomono, andK. Yoshikawa, Scientific Reports 7, 41511 (2017).
[124] S. Komura, A. Takada, Y. Mizumura, S. Miyamoto, T. Takemura, T. Kishimoto,H. Kubo, S. Kurosawa, Y. Matsuoka, K. Miuchi, T. Mizumoto, Y. Nakamatsu,
K. Nakamura, M. Oda, J. D. Parker, T. Sawano, S. Sonoda, T. Tanimori, D. Tomono,and K. Yoshikawa, The Astrophysical Journal 839, 41 (2017).
[125] K. Miuchi, H. Nishimura, K. Hattori, N. Higashi, C. Ida, S. Iwaki, S. Kabuki,
H. Kubo, S. Kurosawa, K. Nakamura, J. Parker, T. Sawano, M. Takahashi, T. Tanimori, K. Taniue, K. Ueno, H. Sekiya, A. Takeda, K. Tsuchiya, and A. Takada, Phys.
Lett. B 686, 11 (2010).
[126] K. Nakamura, K. Miuchi, T. Tanimori, H. Kubo, A. Takada, J. Parker, T. Mizumoto,Y. Mizumura, H. Nishimura, H. Sekiya, A. Takeda, T. Sawano, Y. Matsuoka, S. Komura, Y. Yamaguchi, and T. Hashimoto, Progress of Theoretical and Experimental
Physics 2015, 043F01 (2015).
–121–

[127] J. Parker, K. Hattori, H. Fujioka, M. Harada, S. Iwaki, S. Kabuki, Y. Kishimoto,H. Kubo, S. Kurosawa, K. Miuchi, T. Nagae, H. Nishimura, T. Oku, T. Sawano,
T. Shinohara, J. Suzuki, A. Takada, T. Tanimori, and K. Ueno, Nucl. Instrum. Methods
Phys. Res. A 697, 23 (2013).
[128] J. Parker, M. Harada, K. Hattori, S. Iwaki, S. Kabuki, Y. Kishimoto, H. Kubo, S. Kurosawa, Y. Matsuoka, K. Miuchi, T. Mizumoto, H. Nishimura, T. Oku, T. Sawano,
T. Shinohara, J. Suzuki, A. Takada, T. Tanimori, and K. Ueno, Nucl. Instrum. Methods Phys. Res. A 726, 155 (2013).
[129] C. Altunbas, M. Capéans, K. Dehmelt, J. Ehlers, J. Friedrich, I. Konorov, a. Gandi,
S. Kappler, B. Ketzer, R. De Oliveira, S. Paul, a. Placci, L. Ropelewski, F. Sauli,
F. Simon, and M. Van Stenis, Nucl. Instrum. Methods Phys. Res. A 490, 177 (2002).
[130] F. Sauli, Nucl. Instrum. Methods Phys. Res. A 386, 531 (1997).
[131] T. Mizumoto, Y. Matsuoka, Y. Mizumura, T. Tanimori, H. Kubo, A. Takada,
S. Iwaki, T. Sawano, K. Nakamura, S. Komura, S. Nakamura, T. Kishimoto, M. Oda,
S. Miyamoto, T. Takemura, J. Parker, D. Tomono, S. Sonoda, K. Miuchi, and S. Kurosawa, Nucl. Instrum. Methods Phys. Res. A 800, 40 (2015).
[132] H. Baba, T. Ichihara, T. Ohnishi, S. Takeuchi, K. Yoshida, Y. Watanabe, S. Ota, andS. Shimoura, Nucl. Instrum. Methods Phys. Res. A 616, 65 (2010).
[133] T. Shimoda, H. Miyatake, and S. Morinobu, Nucl. Instrum. Methods Phys. Res. B 70,
320 (1992).
[134] S. Mitsuoka, T. Shimoda, H. Miyatake, Y. Mizoi, H. Kobayashi, M. Sasaki, T. Shirakura, N. Takahashi, T. Murakami, and S. Morinobu, Nucl. Instrum. Methods Phys.
Res. A 372, 489 (1996).
[135] H. J. Ong, AIP Conf. Proc. 1588, 146 (2014).
[136] H. Nishibata, R. Leguillon, A. Odahara, T. Shimoda, C. M. Petrache, Y. Ito,
J. Takatsu, K. Tajiri, N. Hamatani, R. Yokoyama, E. Ideguchi, H. Watanabe, Y. Wakabayashi, K. Yoshinaga, T. Suzuki, S. Nishimura, D. Beaumel, G. Lehaut, D. Guinet,
P. Desesquelles, D. Curien, K. Higashiyama, and N. Yoshinaga, Phys. Rev. C 91,
054305 (2015).
[137] Y. L. Sun, J. Lee, Y. L. Ye, A. Obertelli, Z. H. Li, N. Aoi, H. J. Ong, Y. Ayyad,
C. A. Bertulani, J. Chen, A. Corsi, F. Cappuzzello, M. Cavallaro, T. Furono, Y. C.
Ge, T. Hashimoto, E. Ideguchi, T. Kawabata, J. L. Lou, Q. T. Li, G. Lorusso, F. Lu,
H. N. Liu, S. Nishimura, H. Suzuki, J. Tanaka, M. Tanaka, D. T. Tran, M. B. Tsang,
J. Wu, Z. Y. Xu, and T. Yamamoto, Phys. Rev. C 93, 044607 (2016).
[138] J. Chen, J. L. Lou, Y. L. Ye, J. Rangel, A. M. Moro, D. Y. Pang, Z. H. Li, Y. C.
Ge, Q. T. Li, J. Li, W. Jiang, Y. L. Sun, H. L. Zang, Y. Zhang, N. Aoi, E. Ideguchi,
H. J. Ong, J. Lee, J. Wu, H. N. Liu, C. Wen, Y. Ayyad, K. Hatanaka, T. D. Tran,–122–

T. Yamamoto, M. Tanaka, T. Suzuki, and T. T. Nguyen, Phys. Rev. C 94, 064620(2016).
[139] J. Chen, J. L. Lou, Y. L. Ye, Z. H. Li, Y. C. Ge, Q. T. Li, J. Li, W. Jiang, Y. L.
Sun, H. L. Zang, N. Aoi, E. Ideguchi, H. J. Ong, Y. Ayyad, K. Hatanaka, D. T. Tran,
T. Yamamoto, M. Tanaka, T. Suzuki, N. T. Tho, J. Rangel, A. M. Moro, D. Y. Pang,
J. Lee, J. Wu, H. N. Liu, and C. Wen, Phys. Rev. C 93, 034623 (2016).
[140] D. T. Tran, H. J. Ong, T. T. Nguyen, I. Tanihata, N. Aoi, Y. Ayyad, P. Y. Chan,
M. Fukuda, T. Hashimoto, T. H. Hoang, E. Ideguchi, A. Inoue, T. Kawabata, L. H.
Khiem, W. P. Lin, K. Matsuta, M. Mihara, S. Momota, D. Nagae, N. D. Nguyen,
D. Nishimura, A. Ozawa, P. P. Ren, H. Sakaguchi, J. Tanaka, M. Takechi, S. Terashima,
R. Wada, and T. Yamamoto, Phys. Rev. C 94, 064604 (2016).
[141] H. Kumagai, A. Ozawa, N. Fukuda, K. Summerer, and I. Tanihata, Nucl. Instrum.
Methods Phys. Res. A 470, 562 (2001).
[142] Y. Matsuda, J. Zenihiro, W. Chao, H. Sakaguchi, J. Tanaka, M. Tsumura, and
N. Nakatsuka, RIKEN Accel. Prog. Rep. 48, 211 (2015).
[143] S. Adachi, T. Kawabata, K. Minomo, T. Kadoya, N. Yokota, H. Akimune, T. Baba,
H. Fujimura, M. Fujiwara, Y. Funaki, T. Furuno, T. Hashimoto, K. Hatanaka, K. Inaba, Y. Ishii, M. Itoh, C. Iwamoto, K. Kawase, Y. Maeda, H. Matsubara, Y. Matsuda,
H. Matsuno, T. Morimoto, H. Morita, M. Murata, T. Nanamura, I. Ou, S. Sakaguchi,
Y. Sasamoto, R. Sawada, Y. Shimizu, K. Suda, A. Tamii, Y. Tameshige, M. Tsumura,
M. Uchida, T. Uesaka, H. P. Yoshida, and S. Yoshida, Phys. Rev. C 97, 014601 (2018).
[144] J. Ziegler, M. Ziegler, and J. Biersack, Nucl. Instrum. Methods Phys. Res. B 268, 1818
(2010).
[145] S. Biagi, Nucl. Instrum. Methods Phys. Res. A 421, 234 (1999).
[146] https://garfieldpp.web.cern.ch/garfieldpp/.
[147] A. Takada, T. Tanimori, H. Kubo, J. D. Parker, T. Mizumoto, Y. Mizumura, S. Iwaki,
T. Sawano, K. Nakamura, K. Taniue, N. Higashi, Y. Matsuoka, S. Komura, Y. Sato,S. Namamura, M. Oda, S. Sonoda, D. Tomono, K. Miuchi, S. Kabuki, Y. Kishimoto,and S. Kurosawa, Journal of Instrumentation 8, C10023 (2013).
[148] M. Fujiwara, H. Akimune, I. Daito, H. Fujimura, Y. Fujita, K. Hatanaka, H. Ikegami,
I. Katayama, K. Nagayama, N. Matsuoka, S. Morinobu, T. Noro, M. Yoshimura,
H. Sakaguchi, Y. Sakemi, A. Tamii, and M. Yosoi, Nucl. Instrum. Methods Phys.
Res. A 422, 484 (1999).
[149] J. Raynal, Computer program: ECIS-95, NEA-0850/19.
[150] G. R. Satchler and D. T. Khoa, Phys. Rev. C 55, 285 (1997).
–123–

[151] M. Uchida, H. Sakaguchi, M. Itoh, M. Yosoi, T. Kawabata, H. Takeda, Y. Yasuda,
T. Murakami, T. Ishikawa, T. Taki, N. Tsukahara, S. Terashima, U. Garg, M. Hedden,
B. Kharraja, M. Koss, B. K. Nayak, S. Zhu, M. Fujiwara, H. Fujimura, K. Hara,
E. Obayashi, H. P. Yoshida, H. Akimune, M. N. Harakeh, and M. Volkerts, Phys. Lett.
B 557, 12 (2003).
[152] J. Bency, Y. Tokimoto, Y. W. Lui, H. L. Clark, X. Chen, and D. H. Youngblood, Phys.
Rev. C 68, 014305 (2003).
[153] J. A. Nelder and R. Mead, The Computer Journal 7, 308 (1965).
[154] E. Jones, T. Oliphant, P. Peterson et al., SciPy: Open source scientific tools for Python(2001–).
[155] P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the
Physical Sciences (McGraw-Hill 2003).
[156] Y. Kanada-En’yo, Phys. Rev. C 91, 014315 (2015).
[157] A. Ozawa, I. Tanihata, T. Kobayashi, Y. Sugahara, O. Yamakawa, K. Omata, K. Sugimoto, D. Olson, W. Christie, and H. Wieman, Nucl. Phys. A 608, 63 (1996).
[158] M. Kamimura, Nucl. Phys. A 351, 456 (1981).
[159] G. Satchler, Nucl. Phys. A 472, 215 (1987).
[160] E. A. McCutchan, C. J. Lister, R. B. Wiringa, S. C. Pieper, D. Seweryniak, J. P.
Greene, M. P. Carpenter, C. J. Chiara, R. V. F. Janssens, T. L. Khoo, T. Lauritsen,
I. Stefanescu, and S. Zhu, Phys. Rev. Lett. 103, 192501 (2009).
[161] Y. Ogawa, K. Arai, Y. Suzuki, and K. Varga, Nucl. Phys. A 673, 122 (2000).
[162] D. Thompson, M. Lemere, and Y. Tang, Nucl. Phys. A 286, 53 (1977).
[163] E. Caurier, P. Navrátil, W. E. Ormand, and J. P. Vary, Phys. Rev. C 66, 024314(2002).
[164] R. Machleidt, F. Sammarruca, and Y. Song, Phys. Rev. C 53, R1483 (1996).
[165] H. Sagawa, X. R. Zhou, X. Z. Zhang, and T. Suzuki, Phys. Rev. C 70, 054316 (2004).
[166] S. Cohen and D. Kurath, Nucl. Phys. 73, 1 (1965).
[167] L. Liu, T. Otsuka, N. Shimizu, Y. Utsuno, and R. Roth, Phys. Rev. C 86, 014302(2012).
[168] R. Machleidt and D. R. Entem, Phys. Rep. 503, 1 (2011).
[169] Y. Kanada-En’yo, Phys. Rev. C 84, 024317 (2011).
[170] I. Angeli and K. P. Marinova, At. Data and Nucl. Data Tab. 99, 69 (2013).
–124–

[171] J. Henderson, G. Hackman, P. Ruotsalainen, S. R. Stroberg, K. D. Launey, J. D. Holt,
F. A. Ali, N. Bernier, M. A. Bentley, M. Bowry, R. Caballero-Folch, L. J. Evitts,
R. Frederick, A. B. Garnsworthy, P. E. Garrett, B. Jigmeddorj, A. I. Kilic, J. Lassen,
J. Measures, D. Muecher, B. Olaizola, E. O’Sullivan, O. Paetkau, J. Park, J. Smallcombe, C. E. Svensson, R. Wadsworth, and C. Y. Wu, Phys. Lett. B 782, 468 (2018).
[172] Y. Kanada-En’yo, Phys. Rev. C 71, 014310 (2005).
[173] Y. Kanada-En’yo and H. Horiuchi, Phys. Rev. C 55, 2860 (1997).
[174] B. Pritychenko, M. Birch, B. Singh, and M. Horoi, At. Data and Nucl. Data Tab. 107,
1 (2016).
[175] M. Ito, Private communication.
[176] M. Ito, EPJ Web Conf. 117, 06014 (2016).
[177] M. Nakao, H. Umehara, S. Ebata, and M. Ito, Phys. Rev. C 98, 054318 (2018).
[178] T. Baba and M. Kimura, Phys. Rev. C 99, 021303 (2019).
[179] J. B. Ehrman, Phys. Rev. 81, 412 (1951).
[180] R. G. Thomas, Phys. Rev. 88, 1109 (1952).
[181] P. Hough, 2nd Int. Conf. High-Energy Acc. Inst. pp. 554–558 (1959).
[182] R. O. Duda and P. E. Hart, Commun. ACM 15, 11 (1972).
[183] H. D. Vries, C. D. Jager, and C. D. Vries, At. Data and Nucl. Data Tab. 36, 495(1987).
[184] J. J. Kelly, Phys. Rev. C 70, 068202 (2004).
[185] R. G. Sachs, Phys. Rev. 126, 2256 (1962).
[186] S. Adachi, Ph.D. thesis, Kyoto University (2018).

–125–