Low-lying proton intruder state in 13B
via 4He(12Be,13Bγ) reaction

Shinsuke OTA

January, 2009

ABSTRACTThe proton-single-particle state and proton intruder state in the neutron-rich unstable
nucleus 13 B are studied by in-beam γ-ray spectroscopy with the proton transfer reaction (α, t)
in inverse kinematics. The experiment has been performed at RIPS beam line in RIKEN.
The secondary beam of 12 Be bombarded a liquid helium target with an incident energy of
50 MeV/nucleon. De-excitation γ rays were detected by a new detector array: GammaRay detector Array with Position and Energy sensitivity (GRAPE). Several γ lines wereobserved corresponding to the transition from four excited states directly to the ground state.
Angular differential cross sections have been deduced up to a scattering angle of 6 degrees in
the laboratory frame and analyzed with the DWBA-analysis method. Deduced transferredangular momentum gives a spin and parity assignment of 1/2+ to the 4.83-MeV excited state
and also possible assignments of the spins of the other excited states. The spectroscopic
strength for 4.83-MeV state was deduced as C 2 S = 0.20 ± 0.02(stat.) ± 0.1(syst.).
The excitation energy, spin and parity, and spectroscopic strength of the 4.83-MeV
excited state have been compared with a shell model calculation using SFO interaction,
which includes the effect of the tensor force. Any predicted 1/2+ state is not consistent with
our experimental results. The possible deformation of the nucleus is qualitatively examinedbased on the large deformation of the 12 Be core. The simple expression to the low-lying
1/2+ state at 4.83-MeV excitation energy is (π[220]1/2)1⊗12 Be. The deformation of thenucleus makes the [220]1/2 orbit lower and the whole system gains an energy of ∼8 MeV –including the change of the single particle energy and the surface energy in empirical mass
formula, which explains the existence of the proton intruder state in 13 B. The present study
suggests that nuclear deformation is important to the change of the proton shell structure
in neutron-rich unstable nuclei.

Contents
1 Introduction
1.1 Intruder state in light neutron-rich nuclei . . . . . . . . . . . . . . . . . . . .
1.2 Spectroscopic Studies via Transfer Reaction . . . . . . . . . . . . . . . . . .
1.3 Previous Works on 13 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
6
9

2 Experiment
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 RI Beam Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Projectile Fragment Separator: RIPS . . . . . . . . . . . . . . . . . .
2.2.2 Parameters of 12 Be Beam . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Detectors for Incident and Outgoing Particles . . . . . . . . . . . . . . . . .
2.3.1 Plastic Scintillators . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Parallel Plate Avalanche Counters . . . . . . . . . . . . . . . . . . . .
2.3.3 Plastic Scintillator Hodoscope . . . . . . . . . . . . . . . . . . . . . .
2.4 Detectors for γ Rays : GRAPE . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Structure of GRAPE . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Basic Concept for Extraction of γ-ray Interaction Position . . . . . .
2.4.3 Consideration on the positioning of GRAPE . . . . . . . . . . . . . .
2.4.4 Signal Processing for GRAPE . . . . . . . . . . . . . . . . . . . . . .
2.5 Secondary Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Signal Processing and Data Acquisition . . . . . . . . . . . . . . . . . . . . .
2.6.1 Signal Processing and Trigger Condition . . . . . . . . . . . . . . . .
2.6.2 Data Acquisition System . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.3 Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11
11
13
13
14
17
17
17
18
19
21
21
24
26
26
28
28
32
32

3 Data analysis
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Time calibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Walk correction for the plastic scintillators and Germanium detectors
3.3 Analysis of incident particle . . . . . . . . . . . . . . . . . . . . . . . . . . .

35
35
36
36
38

i

3.3.1 Identification of incident particles . . . . . . . . . . . . . . . . . . . .
3.3.2 Beam profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Target position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.4 Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Analysis of Outgoing Particles . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Identification of Atomic Number . . . . . . . . . . . . . . . . . . . . .
3.4.2 Identification of Mass Number . . . . . . . . . . . . . . . . . . . . . .
3.4.3 Position of Outgoing Particles . . . . . . . . . . . . . . . . . . . . . .
3.4.4 Velocity of Outgoing Particles . . . . . . . . . . . . . . . . . . . . . .
3.4.5 Detection efficiency of Hodoscope . . . . . . . . . . . . . . . . . . . .
Analysis of De-excitation γ Rays . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Energy Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.2 Timing Information . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.3 Extraction of γ-ray Interaction Position . . . . . . . . . . . . . . . . .
3.5.4 Doppler Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
γ-ray Yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Angular Distribution and DWBA Analysis . . . . . . . . . . . . . . . . . . .
Single folding potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38
38
43
44
45
45
45
47
48
48
51
52
55
56
57
60
63
65

4 Experimental Results
4.1 Energy Spectrum after Doppler Correction . . . . . . . . . . . . . . . . . . .
4.2 Cross Sections for Excited States in 13 B . . . . . . . . . . . . . . . . . . . . .
4.3 Angular Distributions for Excited States in 13 B . . . . . . . . . . . . . . . .
4.3.1 Spin and Parity Assignment . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Spectroscopic Strength . . . . . . . . . . . . . . . . . . . . . . . . . .

71
71
72
74
75
78

5 Discussion
5.1 Comparison with the Previous Works . . . . . . . . . . . . . . . . . . . . . .
5.2 Comparisons with Shell Model Calculation . . . . . . . . . . . . . . . . . . .
5.3 Property of 1/2+ Intruder State . . . . . . . . . . . . . . . . . . . . . . . . .

81
81
83
83

6 Conclusion

87

A Data Table of Angular Differential Cross Section

97

B Comparison between analog and digital sum of GRAPE

99

C Tailing function of Full Energy Peak in Germanium Detector

101

3.4

3.5

3.6
3.7
3.8

List of Tables
1.1

Comparison of matching condition in transfer reactions . . . . . . . . . . . .

8

2.1
2.2

Parameter of RIPS for secondary 12 Be beam . . . . . . . . . . . . . . . . . .
List of the data set for the analysis . . . . . . . . . . . . . . . . . . . . . . .

17
33

3.1
3.2
3.3
3.4
3.5

The detection efficiencies of PPAC’s. . . . . . . . . . . . . . . . . . . . . . .
The list of time resolutions of plastic scintillators and anodes in PPAC’s. . .
The list of timing and position resolutions of cathodes in PPAC’s. . . . . . .
Global optical potential parameters . . . . . . . . . . . . . . . . . . . . . . .
Parameters of single folding potentials . . . . . . . . . . . . . . . . . . . . .

41
42
42
65
69

4.1
4.2

List of the cross sections for the excited states in 13 B via the proton transfer
reaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Result of DWBA analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74
75

5.1

List of the relative populations for the present and previous works . . . . . .

82

A.1 Data table of angular differential cross section for 4.83-MeV excited state. . .

97

iii

List of Figures
1.1
1.2
1.3
1.4
1.5
1.6

Nilsson diagram for N(Z) ≤ 50 . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic picture of the monopole interaction . . . . . . . . . . . . . . . . .
Neutron energy in Weakly bound system . . . . . . . . . . . . . . . . . . . .
Effective single particle energies in 12 Be and 13 B . . . . . . . . . . . . . . . .
Diagram of the transfer process . . . . . . . . . . . . . . . . . . . . . . . . .
Level structure in 13 B deduced from experimental studies . . . . . . . . . . .

3
4
4
5
7
10

2.1 Schematic view of the experimental setup . . . . . . . . . . . . . . . . . . . .
2.2 Layout of RARF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Layout of RIPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Exploded view of PPAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Schematic view of plastic scintillator hodoscope . . . . . . . . . . . . . . . .
2.6 Schematic view of segmented planar Ge detector . . . . . . . . . . . . . . . .
2.7 Schematic pulse shape of GRAPE . . . . . . . . . . . . . . . . . . . . . . . .
2.8 Simulated correlation of zero-cross timings . . . . . . . . . . . . . . . . . . .
2.9 Positioning of Ge detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10 Electronic circuits for signal processing of GRAPE . . . . . . . . . . . . . . .
2.11 Schematic view of the liquid target system . . . . . . . . . . . . . . . . . . .
2.12 Electronic circuit diagram for beam counter . . . . . . . . . . . . . . . . . .
2.13 Electronic circuit diagram for PPAC . . . . . . . . . . . . . . . . . . . . . .
2.14 Electronic circuit diagram for hodoscope . . . . . . . . . . . . . . . . . . . .

12
15
16
18
20
22
23
24
25
27
29
30
30
31

3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9

37
39
40
43
44
46
47
49
50

Example of time calibration . . . . . . . . . . . . . . . . . . . . . . . . . . .
Scatter plot for identification of incident particles . . . . . . . . . . . . . . .
Single hit event selection using PPAC . . . . . . . . . . . . . . . . . . . . . .
Beam profile at secondary target . . . . . . . . . . . . . . . . . . . . . . . .
Position of secondary target . . . . . . . . . . . . . . . . . . . . . . . . . . .
Identification of atomic number of outgoing particle . . . . . . . . . . . . . .
Identification of mass number for boron isotopes . . . . . . . . . . . . . . . .
Position calibration for plastic scintillator hodoscope . . . . . . . . . . . . .
Selection of two-body reaction . . . . . . . . . . . . . . . . . . . . . . . . . .

v

3.10 Acceptance of hodoscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.11 Energy deviation of GRAPE . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.12 Energy resolution and efficiency of GRAPE . . . . . . . . . . . . . . . . . .
3.13 Timing distribution of γ-ray detectors . . . . . . . . . . . . . . . . . . . . . .
3.14 Correlation between Tsum and Tseg . . . . . . . . . . . . . . . . . . . . . . . .
3.15 Distance between simulation and experimental data in the correlation between
Tsum and Tseg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.16 Performance of Doppler-shift correction with the GRAPE . . . . . . . . . . .
3.17 Comparison between the experimentally obtained and simulated response
functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.18 Experimental and simulated efficiencies of the GRAPE . . . . . . . . . . . .
3.19 DWBA prediction with energy dependent optical potential . . . . . . . . . .
3.20 Single folding potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.21 Differential cross section of inelastic scattering on 12 Be . . . . . . . . . . . .

51
54
55
56
57

61
63
65
67
68

4.1
4.2
4.3
4.4
4.5

Doppler corrected γ-ray energy spectrum of 13 B . . . . . . . . . . . . . . . .
Doppler corrected γ-ray energy spectra of 10,11,12 B . . . . . . . . . . . . . . .
Differential cross section of each excited state in 13 B . . . . . . . . . . . . . .
Angular distribution for 4.83-MeV excited state . . . . . . . . . . . . . . . .
Search for the best fit parameters of single folding postential . . . . . . . . .

72
73
76
77
79

5.1
5.2

Experimental and predicted level scheme in 13 B . . . . . . . . . . . . . . . .
Proton orbitals in deformed potential . . . . . . . . . . . . . . . . . . . . . .

84
85

58
59

B.1 Comparison between analog and digital sum . . . . . . . . . . . . . . . . . . 100
C.1 Modification for tailing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Chapter 1
Introduction
1.1

Intruder state in light neutron-rich nuclei

A picture of the shell structure was originally applied to the atomic nuclei so as to explain
the discrepancies in the systematics of the binding energies and relative abundances in 1934
by Elsasser [1]. The gross systematics itself was well explained by the liquid drop model;
however, the discrepancies were seen when the proton number (Z) or neutron number (N)is 2, 8, 20, 28, 50, 82, or 126, which are now called ‘magic number’. The magic number
corresponds to the number of filled nucleons of each type up to the closed shell, in contrast
to the number of electrons in atomic closed shell. In order to reproduce the magic number,
Mayer [2, 3, 4] and Jensen [5] introduced spin-orbit force into the nuclear shell structure in
1949. The shell model including the effect of spin-orbit force well represents not only theexistence of the magic number but also the spins of the ground states, especially of odd-massnuclei. The spin of a odd-mass nucleus is characterized by the last unpaired one nucleon
since identical nucleons in a same orbit are paired to form the spin and parity of 0+ .
Nuclear deformation was introduced into shell structure [6] in 1953. In this picture
the nuclear potential in an atomic nucleus is deformed and the resultant shell structure isdifferent from the spherical one. This picture described the large quadruple moments andexcitation energies in rare-metal isotopes. Figure 1.1 shows a Nilsson diagram for up toaround N(Z) ≤ 50. At the deformation parameter ǫ2 = 0, i.e. spherical limit, the magic
numbers are seen. The positive and negative ǫ2 values correspond to prolate and oblate
deformations, respectively. When ǫ2 value is non-zero, the degeneracies of the magnetic
sub-states are resolved, and some orbits are pushed up while some others are pulled down.
This picture of spherical and deformed shell structure explains the properties of theground and low-lying excited states in β-stable nuclei very well. However, the situation
changed after the exploration of the properties of unstable nuclei far from the β-stabilityline. One of the most striking phenomena is the disappearance of the nuclear magic number
in light neutron-rich nuclei. The magic number N = 8 disappears in light neutron-rich

1

2

CHAPTER 1. INTRODUCTION+
12
unstable nuclei. The low excitation energies of low-lying 2+
Be
1 [7] and 02 [8, 9] states insuggest a 2~ω configuration in these states. The large deformation of the ground state in
12
Be was also indicated by an AMD calculation [10]. Around the N = 20 region, an
inversion of the orbits in sd-shell appears in many neutron-rich unstable nuclei in the socalled ‘island of inversion’ region [11, 12]. These disappearances of the magic numbers are
believed to originate from the change of the single-particle orbits in a nuclear potential. In
this situation, the low-lying excited states having many ~ω excitations (or configurations)appear and can account for the disappearance of the magic number.

An attractive phenomena associated with this change of the single-particle orbits is theexistence of intruder states. Intruder states are characterized by the last nucleon in the
intruder orbit, which comes down from upper shell. Since the parities of neighboring shells
are opposite, the parities of the intruder states are atypical. The existence of intruder statesin light neutron-rich unstable nuclei is often considered to be evidence for one or more ~ω
configurations in the low-lying states around the psd shell. The ground state of 11 Be is 1/2+
which is lower in energy by 0.3 MeV than the 1/2− state [13]. In 12 Be, there is a 1− intruder
state at 2.7-MeV excitation energy [14]. The energies of these low-lying, abnormal parity
states indicate 1~ω configurations.
Three theoretical interpretations have been proposed for these one or more ~ω configurations in the low-lying states of neutron-rich unstable nuclei: (1) the monopole interactionof the tensor force [15, 16]; (2) the loosely bound nature of some orbitals [17]; (3) nuclear
deformation [9, 18, 19]. In Ref. [15, 16], the effective interaction was determined so that themodel reproduces the energy levels in light neutron-rich nuclei including intruder states, andthe importance of monopole interaction due to the tensor force was pointed out. A schematic
picture of monopole interaction is illustrated in Fig. 1.2. In this figure the monopole interaction is produced by the tensor force between a proton in j>,< = l ± 1/2 orbit and a neutron′′
is attractive; onin j>,<= l′ ± 1/2 orbit. The tensor force between the j>(<) and j<(>)′
the other hand, the one between j>(<) and j>(<) is repulsive. The existence of less protons
changes the energy of neutron orbital, and the neutron shell gap becomes small. Reference[17] discusses the fact that the 2s1/2 orbital gains its energy relative to the other orbitals
in the psd shell due to its loosely bound nature. Figure 1.1 shows an example of the effectof this weakly bound nature. Near the zero binding energy, the 2s1/2 orbital becomes close
to the 1p1/2 orbital and the gap between them becomes small. The non-zero ~ω configurations can also be intuitively explained by the deformed mean field picture. As seen in
the Nilsson diagram (Fig. 1.1), the [220]1/2 orbit becomes rapidly lower, and the [101]1/2
becomes slightly higher, i.e. the gap becomes smaller with increasing deformation. Since
these models provide the corresponding effects for the change of the neutron shell structure
in neutron-rich unstable nuclei, such as 11,12 Be from the different point of view, the relative
importance of three theory has not been clarified.
These effects are expected to change for the proton shell in light neutron-rich nuclei.

1.1. INTRUDER STATE IN LIGHT NEUTRON-RICH NUCLEI

Figure 1.1: Nilsson diagram for N (Z) ≤ 50 with the deformation parameter ǫ2 .
Spherical nuclei are described by ǫ2 = 0, and prolate and oblate deformation are
described by ǫ2 > 0 and ǫ2 < 0, respectively. At ǫ2 = 0 the magic numbers appear.

3

4

CHAPTER 1. INTRODUCTION

Figure 1.2: Schematic picture of the monopole interaction produced by the tensor′
force between a proton in j>,< = l ± 1/2 and a neutron in j>,<= l′ ± 1/2, taken from
Ref. [16].

Figure 1.3: The energy of the neutron orbital in the Woods-Saxon potential, takenfrom Ref.[17]. The x-axis expresses the strength of the potential. When the neutron is
weakly bound, the s1/2 orbital becomes near to the p1/2 orbital.

1.1. INTRUDER STATE IN LIGHT NEUTRON-RICH NUCLEI

10

5

10
3

5

3

5
4

0
-5

3

5

4

1

0

5

4

3

5
4

-5

5
1

2

-10

-10
1

-15
-20

2
1

-15
2

-20

(a) 12 Be(g.s)2(a) 13 B(g.s)

Figure 1.4: Effective single particle energies in the ground states of 12 Be (a) and
13 B (b), calculated using SFO interaction [15], are shown. Open circles indicate thenucleon in single particle orbitals. The numbers beside single particle orbitals indicate
the orbitals in order of p1/2 , p3/2 , d3/2 , d5/2 , and s1/2 . Due to the tensor force, one
more proton in the orbital 1 (p3/2 ) changes the neutron shell gap between the orbital
2 (p1/2 ) and 5 (s3/2 ).

The effect due to the weakly bound nature is expected to be negligible because protons inneutron-rich nuclei are generally bound in a deep nuclear potential. It is also expected thatthe effect of the monopole interaction due to the tensor force is small. Figure 1.4 illustrates
the occupation numbers in single particle orbitals and effective single particle energies in
psd shell, for the cases of 12 Be and 13 B. The number of open circles indicates the occupation
number in the corresponding single particle orbit. The effective single particle energies arecalculated using the single particle energies and monopole interaction in SFO interaction(“present′ ” interaction in Ref. [15]). Comparing the single particle energies in their groundstates, less occupation of proton p3/2 orbit in 12 Be makes the gap between the p1/2 and s1/2
orbits in the neutron shell small, while the proton shell structures are almost identical in both
nuclei. Nuclear deformation, however, is still presumed to affect the proton shell structure
when the distributions of not only neutrons but also protons deform, as seen in the Nilsson
diagram. Proton intruder states in light, neutron-rich nuclei are, therefore, signatures forthe importance of deformation as the effect for the change of the single-particle orbital, i.e.
the change of the shell structure.

6

1.2

CHAPTER 1. INTRODUCTION

Spectroscopic Studies via Transfer Reaction

In 1950, the direct transfer reactions such as the (d, p), (d, n), (d,3 He), (α, t) and (α,3 He)
reactions, opened up spectroscopic studies of the nuclear shell structure. The first spectroscopic study was performed by Butler for the ground and first excited state of 17 O nucleus
based on the analysis of the angular distribution of (d, p) reaction [20, 21]. After the suggestion by Bethe and Butler [22], many stripping reactions on various nuclei were measured
and the property of levels were studied [23, 24, 25, 26, 27]. Although the method of Butler
was effective for the light nuclei and could predict the relative cross section, this method
overestimated the absolute value of reaction cross section. In Ref. [28], the importance ofthe distortion of the outgoing wave was pointed out, which Butler’s method neglected. At
about the same time, the optical model was discussed, and the refraction and absorption
in interior of nuclei were taken into account for the reaction cross section [29]. Analysis
with distorted wave Born approximation (DWBA) has enabled us quantitative discussions
on differential cross sections of direct reactions. A factorization of spectroscopic factor has
also been introduced as a basic quantity to compare experimental data with predictions of
nuclear models [30].
Spectroscopic studies of the shell structure via direct reactions with unstable nuclei has
been opened by the one-nucleon knockout reaction [31] using a fast radioactive beam. In thisreaction the ground state property of the incident nucleus was deduced from the longitudinal momentum distribution. From many measurements of this reactions with neutron-richnuclei, the systematics of the ground state properties were revealed. The transfer reactionsof (d, n) reaction on unstable nuclei were performed with deuteron in inverse kinematics[32, 33]. In these (d, n) reactions at incident energies higher than so called intermediate energy of a few tens MeV/nucleon, the reaction cross sections rapidly decrease and the kinetic
energies of the recoiled particles were small in inverse kinematics. However, an applicationof α-induced transfer reaction on neutron-rich, unstable nuclei has been introduced recently[34]. The kinematic matching conditions become loose in this reaction even at intermediate
incident energy region due to large negative Q-value of the reaction and to the fact thata proton in α particle is deeply bound and has a high-momentum component. This results in the moderate decreasing of the reaction cross section of the (α, t) reaction even atintermediate energies.
The difference in the energy dependence of the reaction cross sections between the (α, t)and (d, n) reactions can be explained by an estimation of the kinematic matching conditions[35, 36]. The conditions are the conservation of the linear and angular momenta, and therequirement that the transferred nucleon is in the reaction plane. The notations commonlyused are listed below:

1.2. SPECTROSCOPIC STUDIES VIA TRANSFER REACTION

7

Figure 1.5: Schematic diagram of the transfer process. For notations, see the text.
(Quoted from Ref. [36].)

R1(2) Radius of the nucleus 1(2)l1(2) Orbital angular momentum which the transferred nucleon has in the nucleus
1(2)λ1(2) Magnetic sub-state of orbital angular momentum which the transferred nucleon has in the nucleus 1(2)m Mass of the transferred nucleon
v Relative velocity of the two nuclei in the region of transfer.
The schematic diagram of the transfer process is shown Fig. 1.5. A nucleon is transferredfrom nucleus 1 to nucleus 2.
Conservation of linear momentum: The momentum of a transferred nucleon should bealmost conserved along y-axis and this conservation law is expressed as:∆k = k0 −

λ2
λ1
−
≈ 0,
R1 R2

(1.1)

where k0 is a relative wave number mv/~. For the more realistic case, a transition probability
becomes large when ∆k0 ≤ δk = 2π/(R1 + R2 ).

8

CHAPTER 1. INTRODUCTION
Reaction
12
Be(d, n)13 B
12Be(α, t)13 B

Jπ
1/2+
5/2+
1/2+
5/2+

Q (MeV)
9.579
9.579
-8.01
-8.01

l2
0
2
0
2

λ2
0
2
0
2

∆k/δk
1.2
0.4
1.1
0.6

∆L/δL
0.1
1.7
0.7
0.2

Table 1.1: Matching conditions estimated for the proton transfer reactions
12 Be(d, n)13 B and 12 Be(α, t)13 B at 50 MeV/nucleon. For detail see the text.

Conservation of angular momentum: The angular momentum of the two interacting
nuclei should be conserved as below,
1
Q∆L = λ2 − λ1 + k0 (R1 − R2 ) + (R1 + R2 ),
2
~v

(1.2)

where Q is the difference in the binding energies of the incident and outgoing particles,
called the Q-value. The first term represents the difference of the angular momenta ofthe transferred nucleon before and after transfer. The other terms represent the change ofthe relative angular momentum due to the change of reduced mass, velocity of transferred
nucleon, andpdistance (or R1 + R2 ) between the two nuclei. For a more realistic case,∆L ≤ δL = γ1 (R1 + R2 ), where γ1 is related to the binding energy ǫ1 of the initial stateby γ1 = 2mǫ1 /~2 .
Requirement for location of the transferred nucleon: The transition probability islarge when the transferred nucleon is near the reaction plane. This is described by,
l1 + λ1 =even,

l2 + λ2 =

even,

(1.3)

where l1,2 is the orbital angular momentum of the transferred nucleon in the nucleus 1(2).
We estimated the conditions of 12 Be(α, t) and 12 Be(d, p) reactions, listed in Table 1.1.
The incident energy is 50 MeV/nucleon, which corresponds to v/c ∼ 0.315 for both thereactions. The nuclear radii for this estimation
are 2.116 fm for d [37], 1.671 fm for α [37],√3
13
and 2.939 fm for B calculated from 1.25× A, respectively. Since the criteria for linear andangular momenta matching are given by ∆k/δk ≤ 1 and ∆L/δL ≤ 1, respectively, the (α, t)reaction has better kinematics condition than the (d, n) reaction for populating proton-single
particle states. The better kinematical conditions of the (α, t) reaction are caused by thelarge negative Q-value of the reaction and the fact that a proton in an α particle is deeply
bound and has a high momentum component as stated previously. Therefore, we havechosen the (α, t) transfer reaction for the in-beam γ-ray spectroscopy in inverse kinematics
to investigate proton single particle states.

1.3. PREVIOUS WORKS ON 13 B

1.3

9Previous Works on 13B

The light, neutron-rich unstable nucleus 13 B has been experimentally studied through various
measurement such as β-decay measurements [38, 39], 7 Li-induced reactions [40, 41], twoneutron transfer reaction of (t,p) reactions [42, 43], neutron knockout reaction [44], β-delayed
neutron measurement [45, 46], measurements of nuclear moments [47, 48, 49], multi-nucleon
transfer reaction [50] and lifetime measurements of the excited states [51]. The ground state
properties such as spin and parity were deduced from the measurement of β-decay of 13 B. A
spin and parity of 3/2− has been assigned to the ground state. The nuclear moments of theground state, such as dipole and quadruple moment, were measured precisely in this decadeand were compared with the result of a spherical shell model calculation [52]. The shellmodel calculation reproduces the measured nuclear moments and indicates a spherical shapefor the ground state. The wave function of the ground state is reported in Ref. [53], where
the spectroscopic factor of proton stripping reaction on 12 Be to the ground state of the 13 B
nucleus and the configuration such as (νsd)2 and (ν1p)2 in the ground state are evaluatedcombining the measurement of delayed neutron emitting β decay of 14 B. It is found thatthe 13 B nucleus has a 20% admixture of (νsd)2 in its ground state although the number ofneutron in 13 B is a magic number (N = 8).
There are six known excited states below the neutron threshold of 4.87 MeV in 13 B foundthrough the (t, p) reactions and Lithium-induced reactions. Although the parities of several
excited states were determined by analysing the angular distribution of (t, p) reactions[42, 43],
the spins have not been uniquely determined. One of the difficulty in the spin and parity
assignment to the excited states in 13 B is the lack of bound 13 Be nuclei. Then the β decay of
13
Be to 13 B cannot be applied. In spite of this difficulty the information of the excited states
have been extracted and the possible spins have been suggested through the experimental
studies such as lifetime measurement of the excited states [51], spectroscopic study using
knockout reaction [44], and measurement of multi-nucleon transfer reaction [50]. However,the spin of any excited state has been uniquely determined.
The aim of the present study is, by assigning spins to such excited states, to investigate
the proton single particle state, and especially intruder state in the neutron-rich nuclei,
and subsequently discuss the effects in three theoretical models on the change of the shellstructure described in Sec. 1.1. For this purpose the in-beam γ-ray spectroscopy has been
performed using the proton transfer reaction onto the largely deformed, neutron-rich nuclei
12Be.
Chapter 2 describes the experimental setup including the details of the accelerator facility, beam monitors and counters, detector for reaction products and a newly developed
γ-ray detector array: GRAPE. Chapter 3 describes the analysis procedure: the particle
identification of incident and outgoing particles; the reconstruction of the γ-ray energies inparticle-rest frame; and the deduction of the populations and differential cross sections by

10

CHAPTER 1. INTRODUCTION

Ex (MeV)

1111
0000
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111

Figure 1.6: The level structure in 13 B deduced from experimental studies. The excitedstates below 5-MeV excitation energy are drawn with adopted spin and parities.

the decomposition of the γ-ray energy spectrum. The DWBA analysis of the angular differential cross sections also is described. The experimental results are described in Chapter 4.
The γ-ray energy spectrum coincident with 13 B is shown with its energy decomposition. The
deduced populations and angular differential cross sections with the result of DWBA analysis are also described. Discussion of the spin and parity assignment, and the property ofsingle-particle state and intruder state are described in Chapter 5. Conclusion of the thesis
are given in Chapter 6.

Chapter 2
Experiment2.1

Overview

We performed an experiment of in-beam γ-ray spectroscopy of 13 B. The main object of thisexperiment is to clarify the property of the proton shell structure in N = 8, light, neutronrich, unstable nucleus 13 B via the proton transfer reaction using a 12 Be beam with a liquid
helium target. In order to identify the excited states and to assign the spin and parity, thede-excitation γ-ray and the angular distribution of 13 B were measured in coincidence.
The experiment was performed at RIKEN Accelerator Research Facility (RARF). A
schematic view of the experimental setup is shown in Fig. 2.1. A 12 Be (secondary) beam
was produced by using the fragmentation reaction of a 18 O (primary) beam and separatedusing RIKEN Projectile Fragment Separator (RIPS). The incident particles were identified
on an event-by-event basis using the measured time-of-flight and the energy deposition. The
time-of-flight, over a path length of 5.4 m, and the energy deposition were measured using
two plastic scintillators located at the last two foci of the RIPS. The intensity and the purityof the 12 Be beam were typically 2 × 105 counts per second and 90%, respectively.
The 50 AMeV 12 Be beam bombarded a secondary target of liquid helium[54] located at
the final focus of RIPS. A liquid helium target was chosen because of its statistical advantage(see Sec. 1.2) in terms of the experimental yields. The helium was condensed by a cryogenic
refrigerator and kept below 4 K during the experiment. A target thickness of 143±5 mg/cm2
was estimated from the velocity difference between outgoing particles measured with andwithout the liquid helium. The position and direction of the incident particles on the secondary target were deduced from the trajectory information of two parallel plate avalanche
counters [55] located 30-cm apart from each other before the final focus. The outgoing particles were detected by a plastic scintillator hodoscope 3.5-m downstream from the secondary
target with a 1 × 1 m2 active area and an angular acceptance of up to 8 degrees in the laboratory frame. The plastic scintillator hodoscope consisted of 5-mm thick ∆E and 60-mm
thick E layers. The ∆E layer was divided into 13 plastic scintillator bars vertically and the

11

12

CHAPTER 2. EXPERIMENT
GRAPE

liquid helium target
hodoscope

B

13

Be

12

PPAC

F3Pl

vacuum

NaI

1.33 m3.54 mFigure 2.1: A schematic view of the experimental setup in the experimental area. The
secondary beam of 12 Be comes from the left. Two PPACs are located before the last
focus F3 in order to monitor the position and direction of the beam particles. A plastic
scintillation counter is located to measure the time-of-flight of the incident particles
and outgoing particles in combination with a plastic scintillation counter located at thesecond focus and a plastic scintillator hodoscope located at the end of the beam line.
A liquid helium target is set on the beam axis. GRAPE was located at 140◦ respectedto the beam axis. The outgoing particles stops in the plastic scintillator hodoscope.
The inside of beam pipe is kept vacuum.

E layer into 16 plastic scintillator bars horizontally. The outgoing particles were identified
on an event-by-event basis using the measured time-of-flight and the energy deposited inthe ∆E and E layers. The mass distribution for Z = 5 particles was determined to be∼0.25 atomic mass unit (one standard deviation). The time-of-flight between the secondary
target and the hodoscope was deduced from the time information of the plastic scintillators
located upstream from the secondary target and the plastic scintillator hodoscope. Position
information for the outgoing particles was deduced from the time difference between the
output signals from the two photomultiplier tubes attached to both ends of each scintillator
bar and was used to determine the scattering angle. The angular resolution of the scattering
angle in the laboratory frame was 0.5-degrees (one standard deviation).
The de-excitation γ rays were detected by an array of germanium detectors: GammaRay detector Array with Position and Energy sensitivities (GRAPE)[56]. This consisted of 6
germanium detectors located at 140◦ with respect to the beam axis. Each detector contains
two cylindrical crystals 6 cm in diameter and 2 cm thick, with a common anode between
them. The cathode attached to each crystal is segmented into a 3 × 3 matrix. GRAPE
provides position information of the γ-ray interaction point, which is extracted from a pulse
shape analysis of the signal from the cathode [56, 57]. The intrinsic energy resolution and2.2. RI BEAM PRODUCTION

13

the full energy peak efficiency were typically 2.7 keV (FWHM) and 0.4%, respectively, for1332-keV γ rays from a standard 60 Co source. The energy resolution after correcting for the
Doppler shift was deduced to be 1.3% (FWHM) for 2.1-MeV γ rays, corresponding to the
decay of the first 2+ state of 12 Be moving with 30% the velocity of light. The excited statesof 13 B populated in the reaction were identified by measuring the energy of the de-excitation
γ rays.
The trigger for the data acquisition required that all the signals from the plastic scintillation counter at F3, the hodoscope and the GRAPE were coincident.
In Sec. 2.2, the production of the secondary beam is descried with a summary of RIPS
beam line. The details of the detector systems such as the counter and tracker for theincident particles, the plastic scintillator array for the outgoing particles, and the γ-ray
detector array are described in Sec. 2.3, 2.3.3, and 2.4, respectively. The secondary target
is described in Sec. 2.5. The electronic circuit, data acquisition system and analyzed data
set are summarized in Sec. 2.6.

2.2

RI Beam Production

2.2.1

Projectile Fragment Separator: RIPS

The layout of the RIKEN Accelerator Research Facility is shown in Fig. 2.2. To produce
an RI beam, a heavy-ion primary beam is accelerated by the ring cyclotron K = 540 up to
several tens to 135 MeV/nucleon and is transported to a production target in the D room. A
layout of RIPS [58] is shown in Fig. 2.3. The RIPS beam line consists of two dipole magnets
D1-D2, twelve quadrupole magnets Q1-Q12, and four sextupole magnets SX1-SX4. There
are four focal point interest, denoted F0, F1, F2, and F3, respectively. The production target
is mounted at F0 in the D room. A dispersive focal plane F1 and a achromatic focus F2 areused for the isotope separation, while the final focal plane F3 is used as the target position
for the experiment. The experimental devices of the F3 chamber are installed for measuring
the secondary beam. The secondary target to populate reaction products is usually placedaround F3.
The isotope separation of reaction fragments is achieved as detailed presently. Various
nuclei are produced in projectile fragmentation reactions of a primary beam on a production
target at F0. The reaction fragments are emitted in a small forward cone with almost thesame velocity as the projectile beam. The magnetic rigidity, Bρ ∝ p/Q, of the secondary
beams can be selected by a slit at the momentum dispersive focal plane F1, where the pand Q stand for the momentum and the charge state of fragment, respectively. In the caseof light nuclei (A < 80), the fragments are fully stripped so that the charge of the fragment
equals the atomic number, Q = Z. In the stage from F0 to F1, isotopes are thus selectedroughly by their A/Z values, since the fragments have almost the same velocity. In order

14

CHAPTER 2. EXPERIMENT

to improve the purity of the secondary beam of interest, a wedge-shaped energy degrader,made of aluminum, is installed at F1. After passing through the degrader, the magnetic
rigidities of the beam particles are modified due to the energy loss, which depends on Z andthe velocity of the isotopes. The modified rigidity is analyzed by the D2 magnet and selectedby a slit at the first achromatic focus F2. In the next stage from F1 to F2, the beams are
selected by a different function of A and Z. This function is approximately proportional toA2.5 /Z 1.5 of the isotopes. The isotope separation at RIPS is thus achieved by combining the
1.5
A/Z selection at F1 and A2.5/Z selection at F2. Finally, the selected isotope and cocktail
beam are transported to the final achromatic focal plane F3, where a detector setup is placed.

2.2.2

Parameters of 12Be BeamA secondary 12 Be beam was produced by a projectile fragmentation of a 100 AMeV 18 O
beam on a beryllium primary target with a thickness of 1.85 g/cm2 , and separated by using
RIPS. The primary beam of 18 O was chosen by the intensity of the primary beam itself(available up to 1 pµA) and the one of the secondary beam. The 18 O nucleus is empiricallyknown to be good for the production of the light neutron-rich unstable nuclei below Z = 8.
The primary target of 9 Be was also known to be good for the production of the neutron-rich
unstable nuclei.
The rate of 2 × 105 Hz was reached for the secondary beam. Although the more intense
beam was available, higher intensity induces too many pile up events. The momentum
window ∆p/p was set to ± 1% by using a slit at the first momentum dispersive focal plane,
which was narrow enough to discriminate the 12 Be from contaminant particles by setting thetime window on the time distribution (see section 3.3.1). At the first achromatic focus (thesecond focus), the window from −10mm to 0mm was set to separate the 12 Be. The energy ofthe secondary beam was calculated from the Bρ(D2) value, however, it needed a correction
since the window was asymmetric, i.e., the extracted beam didn’t traverse the center of F2,
since the beam axis was slightly mismatched to the optical axis in the present experiment.
The correction factor was calculated from the momentum dispersion at the F2 of 37mm/%.
The energy was higher by 0.135% than that obtained from the BρD2 value. The energy
was 49.90 AMeV after the last focus F3 with consideration for the energy deposited in thematerial between F2 and F3. The purity of 12 Be was ∼90%. The detailed parameters of
RIPS optimized for the production of 12 Be are listed in Table 2.1.

2.2. RI BEAM PRODUCTION

Figure 2.2: (a) Layout of the RIKEN Accelerator Facility. A primary beam is accelerated by AVF cyclotron and RIKEN ring cyclotron (RRC). (b) A enlargement of
RIPS in the figure (a).

15

16

CHAPTER 2. EXPERIMENT

Figure 2.3: Layout of the RIKEN Projectile Fragment Separator, RIPS. The primary
projectile fragmentation reaction occurred at F0. Projectile-like fragments are analyzed
and selected in two stages of F0-F1 and F2-F3 to produce and select radioactive beam
particles of desired nuclei. The final focus of F3 is provided for the experimental setup.

2.3. DETECTORS FOR INCIDENT AND OUTGOING PARTICLES
Primary Beam
18
O

Energy100 AMeV

Intensity
100 pnA

Primary Target
9
Be

Thickness
1.85 g/cm2

F1 Al Degrader
1099 mg/cm2

D1 Field
3.525 Tm

D2 Field
3.150 Tm

F1 Slit±1 %Energy@F3
50 A MeV

Intensity
200 kcps

F2 Slit-10mm +0mm
Secondary Beam
12
Be

17

Table 2.1: Parameters of RIPS to product secondary 12 Be beam via the projectile
fragmentation reaction. See the text for details.

2.3

Detectors for Incident and Outgoing Particles

2.3.1

Plastic Scintillators

Two plastic scintillators 0.5 mm-thick with 7 × 7-cm2 active are were placed at two foci, F2
and F3, of RIPS (F2PL and F3PL, respectively) in order to deduce the velocity of the incident
particles and to identify incident particles on event-by-event basis through the measurementof energy deposition (∆E) and time-of-flight (TOF). The flight length between F2PL and
F3PL were 5.4 m. Two 2-inch photomultiplier tubes (PMT’s) were attached to the both sideof each plastic scintillator in order to reduce the ambiguity of timing and ∆E depending on
the hit position of particles. The timing information were obtained as the arithmetic mean of
two timing signals from the two PMTs. The ∆E information were obtained as the geometricmean of two pulse height signals from the two PMTs. The time difference between the F2PL
and F3PL was used as the TOF information for the incident beam velocity and the particle
identification. The 12 Be beam was identified and discriminated from contaminants by using
the ∆E-TOF method.

2.3.2

Parallel Plate Avalanche Counters

Two delay-line type Parallel Plate Avalanche Counters (PPACs) [55] were placed at 185and 155-cm upstream of the secondary target in order to monitor the position and directionof secondary beam. The expanded view of a PPAC is shown in Fig. 2.4. The active areaof each PPAC was 100 × 100 mm2 , and the effective length of the delay line was 100 mm.
The PPAC was filled typically 8.6-Torr C3 F8 gas and supplied +910-V voltage to the anode

18

CHAPTER 2. EXPERIMENT

Figure 2.4: Exploded view of the delay line parallel plate avalanche counter(PPAC),
taken figure from Ref. [55].

plane. The high voltage was optimized to maximize the detection efficiency. The hit position
on PPAC was obtained from the time difference between two timings of the signals from theboth ends of the delay line. The typical transition time of the signal was 0.8 ns/mm and thesum of transition time of two signals was to be a constant, e.g., 80 ns for the effective lengthof the delay line. This condition was used to reject multi-hit events. The position of theincident particles on the secondary target was determined by extrapolating the hit positions
obtained the two PPACs. The typical position resolution on the PPAC was 0.5 mm (FWHM),
and the resolution of the extrapolated position on the secondary target was 4.3 mm. The
corresponding angular resolution was 3.4 mrad. The details of the resolutions are describedin section 3.3.2.

2.3.3

Plastic Scintillator Hodoscope

A plastic scintillator hodoscope was placed at the end of beam line in order to identify
the outgoing particle and to determine their velocity and scattering angle. The hodoscope2.4. DETECTORS FOR γ RAYS : GRAPE

19

was installed in the vacuum chamber in order to decrease the energy losses and multiple
scatterings of outgoing charged particles along the flight path. Figure 2.5 shows a schematic
view of the hodoscope. The hodoscope itself consisted of three layers, ∆E, E, and E ′ .
However, the first two of them were used since the outgoing particles deposited all the
kinematic energy and stopped in the E layer. The used layers were 5 mm-thick ∆E and
60 mm-thick E layers. The hodoscope had 1 × 1 m2 active area and was located 3.5m downstream of the secondary target with its symmetry axis along the beam line. The∆E layer was divided horizontally into 13, and the E layer was divided vertically into 16
scintillator bars, respectively. The widths of the scintillator bars at the central part in eachlayer were narrower to improve the balance of the counting rate among the scintillator bars,
and they were 40 mm and 38 mm for the ∆E and E layers, respectively. The widths at
the other part were 100 mm and 75 mm, respectively. The timing of the particle arrival
was deduced from the sum of two timings of the PMT signals without the ambiguity due
to the hit position. The velocity after the secondary target was deduced in combination
with the F2PL and F3PL. The energy deposited was deduced from the multiplication of
the charge integration of the PMT signals without the ambiguity due to the hit position.
The hit position along the vertical and horizon axis were deduced from the time difference
between the two timings of the PMT signals in the ∆E and E layers, respectively. The time
resolutions were typically 121 ps and 106 ps and the position resolutions were 19.5 mm and17.1 mm for ∆E and E layers, respectively. The angular resolution was 0.5 degrees to thesecondary target in laboratory frame and the corresponding one in center-of-mass frame was
2.4 degrees.

2.4

Detectors for γ Rays : GRAPE

An array of germanium detectors, called “GRAPE” (Gamma-Ray detector Array with
Position and Energy sensitivity), has been developed aiming at the precise determination ofthe γ-ray interaction position and the resulting precise Doppler-shift correction for the γ rays
emitted from fast moving (β ∼ 0.3) nuclei. The position resolution after pulse shape analysis
was obtained to be less than 3 mm while the thickness of the germanium crystal was 2 cm.
The energy resolution after Doppler correction was 1.3% (FWHM) for the γ ray emitted
from the fast moving 12 Be, so far. In this section, the structure of the GRAPE (in section
2.4.1), the principle of the pulse shape analysis (in section 2.4.2), and the consideration of
the location (in section 2.4.3) were described. The electronic circuits for the GRAPE are
described in section 2.4.4

20

CHAPTER 2. EXPERIMENT

top view

E

E’

dE

PMT
Light guide1m

front viewside view

Figure 2.5: Schematic view of plastic scintillator hodoscope with front, side and top′
views. The hodoscope consisted of three layers of ∆E, E, E , the two of them wereused. The ∆E and E layers were 5-mm and 60-mm thick and were divided into 13
and 16 bars, respectively. Two PMTs were attached at the both ends of each bar. The
timing, charge, and position information were extracted from the signals of hodoscope.

2.4. DETECTORS FOR γ RAYS : GRAPE

2.4.1

21

Structure of GRAPE

Figure 2.6 shows a schematic view of the inner structure of the planar γ-ray detector usedin the GRAPE. Each detector contains two cylindrical crystals of 2-cm thickness and 7-cm
in diameter. The crystal surrounded by a cooler ring. The ring was made from aluminumin order to avoid the attenuation of γ rays from the surface of the cylindrical crystal. The
anode between the two crystals was energized with a positive voltage of typically 2500 V.
The cathode at the other end of each crystal was divided into 3 × 3 electrode segments anda 5-mm wide guard ring. The guard ring makes the electronic field uniform even in the edge
region of the cathode. The resulting effective volume of germanium crystal was 2-cm thickand 6 cm in diameter. The signal from each cathode was pre-amplified independently andevery nine signal from the crystal was summed up to reconstruct the net-charge signal. The
number of signals from one crystal was twenty and the total number from the GRAPE was
120 in the present experiment. These signals were processed to obtain the charge, timing
and position information. The detail of the signal processing is described in the section 2.6.
To obtain the charge information, we can choose two ways; the add back of each digitizedsignals from the cathode segments (digital sum) and the add back of two net-charge signals(analog sum). The advantage of the digital sum is the better resolution than that of theanalog sum while the advantage of the analog sum is the better efficiency than that of thedigital sum. The detail of deduction of the γ-ray energy is described in section 3.5.1. The
basic idea of the position extraction and the analysis using it are described in section 2.4.2
and 3.5.3.
The typical energy resolution is 2.5 keV (FWHM) and the full-energy-peak efficiency
taking the solid angle into account is 0.4%, respectively, for 1332 keV γ ray emitted from60
Co standard source located at the secondary target position.

2.4.2

Basic Concept for Extraction of γ-ray Interaction Position

The interaction position of γ ray was extracted based on a pulse-shape analysis of charge
pulses from segmented cathodes. Figure 2.7(a) shows a schematic of pulse shapes from thecathodes. There are two cases of hit patten as denoted by black dots in the first row of the
figure. In the left column the electron-hole pair is created near the cathodes and in the rightcolumn it is created near the anode. The letters A, B and C denote the different segments.
In the second and third rows, dotted curves show the contributions of the electron (e) andhole (h), while the solid curves show the pulse shape to be observed. It is assumed forsimplification that the drift velocities of the electron and hole are constant and same. The
shapes of the sum of the signals from cathodes A, B and C are identical (shown in the secondrow) for the two cases; however, the contributions of the electron and hole are different. On
the other hand, the shapes from B (the fourth row) differ between the two cases because
the polarity of the induced signals in its neighbor segments (the third row) also differ. It is

22

CHAPTER 2. EXPERIMENT

liq. N2

pre−amplifiers(a) A drawing of a GRAPE detector
cathode segment(2 cm x 2 cm)

1111111111111111
0000000000000000
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
guard ringcold contact
(5mm)

(5mm)

anode

11111111
00000000
00000000
11111111
00000000
11111111
00000000
11111111
00000000
11111111
00000000
11111111
00000000
11111111
00000000
11111111
00000000 6 cm
11111111
00000000
11111111
00000000
11111111
00000000
11111111
00000000
11111111
00000000
11111111
00000000
11111111
00000000
11111111
2 cm 2 cm(b) Schematic view of a crystal in a GRAPE detector
Figure 2.6: (a) A drawing of a GRAPE detector. (b) Schematic view of planar typegermanium detector used in GRAPE. Two crystals were in a chassis with an anodebetween them. The size of each crystal was 2 cm thick and 7 cm in diameter. The
anode was energized typically with a positive voltage of 2500 V. The cathode wasdivided into 3 × 3 segments and surrounded by a guard ring.

found through the simulations of the pulse shapes that the depth of the γ-ray interaction
point is characterized by two zero-cross timings picked up after (CR)2 (RC)4 shaping of the
net-charge (Tsum ) and the hit segment (Tseg ) signals.
The simulated correlations are shown in Fig. 2.7(b). Assuming the rotational invariance,
the correlations are divided into three groups, i.e., center, side and corner, as indicated in
the figure. Each dotted line corresponds to a certain point in the cathode plane and eachclosed circle corresponds the depth at that point. The depths are divided into four groups
and colored for easy viewing. As seen in the figure, the correlations are well discriminatedfrom each other, although those in the region around 6 mm are overlapping.

2.4. DETECTORS FOR γ RAYS : GRAPE

23

Figure 2.7: Schematic pulse shape from a segmented planar detector. Black dots inthe top row denote the positions where an electron-hole pair is created in the example.
The second and fourth rows show the pulse shapes, which are used for the pulse-shapeanalysis and the third row shows the induced pulse shapes which are essential for thedifference of the pulse shapes in the fourth row. (taken from ref. [56]).

For utilizing this method, three kind of circuits are developed: analog adders for thesum of the nine cathode signals, (CR)2 − (RC)4 shaping amplifiers, and discriminators forpicking off the zero-cross timings.

24

CHAPTER 2. EXPERIMENT

CenterSide

Tsum [ns]500
16-20 mm16-20 mm400
1-5 mm

11-15 mm

1-5 mm

11-15 mm6-10 mm6-10 mm300-50

Corner

16-20 mm

Tsum [ns]

500

0

50

100

Tseg - Tsum [ns]Corner400
1-5 mm

11-15 mmCenterSide

6-10 mm

300

-100

-50

0

50

100

Tseg - Tsum [ns]Figure 2.8: Zero-cross timing correlations between the summed cathode signal andhit cathode signal subtracted by summed signal. Each connected line corresponds tothe set of zero-cross timing correlations at a certain point in the cathode plane.

2.4.3

Consideration on the positioning of GRAPE

In order to reduce Doppler broadening, the position of GRAPE are considered. For Doppler
correction, γ-ray emission angle and the velocity of the outgoing particle are needed. The
emission angle is calculated in combination with the γ-ray detection point, the stopped position of the outgoing position in hodoscope and the γ-ray emission position in the secondary
target. The emission position, which is same as the position where a reaction occurs in mostcases, is should be on the line extrapolated using the extrapolated momentum vector of anincident particle. In the extrapolation, the reaction is assumed to occur at the center ofthe secondary target along the beam axis, since the position along the beam axis cannot be
determined due to the limitation of the setup. Because of this assumption, the ambiguitiesof the emission angle and the velocity persist in the Doppler correction. However, it is foundthat the effect of these ambiguities vanish when the γ-ray detector is located at a certain
position, described below.
The measured energy is affected by the Doppler shift and represented as,
Elab = Ecm /γu (1 − βu cos θu ),(2.1)

2.4. DETECTORS FOR γ RAYS : GRAPE

25

Figure 2.9: A result of the search for the position of the GRAPE. The curve shows
the positions of the detector where the Doppler broadening due to the ambiguity of thereaction position along the beam axis. The position of the GRAPE is determined as
close to the secondary target as possible.

where Ecm(lab) are the energy of the γ-ray in the center-of-mass frame (the laboratory frame),
β is velocity in units of light speed, and θ is the emission angle in the laboratory frame. The
subscript u(d) means that the reaction occurs at the upstream (downstream) end of thesecondary target. On the other hand, now, the reaction is assumed to occur at the centerof the secondary target. Then, the angle and the velocity are calculated to be θc and βc ,respectively, differing θu and βu . The conditions which realizes the smallest ambiguities areγu (1 − βu cos θu ) = γc (1 − βc cos θc ),γd (1 − βd cos θd ) = γc (1 − βc cos θc ),

(2.2)
(2.3)

and they are found to be satisfied when the detectors are located at large (backward) angles.
These conditions for any point in the secondary target are satisfied when the detector are
located at the position satisfied the conditions listed above. Figure 2.9 shows the result of
the search for the optimum detector positions. When the detector is located at the optimum
position on the curve, the conditions described above are satisfied and the effect due to theambiguity of the reaction position along the beam axis becomes small. The position of thedetectors should be as close to the secondary target as possible from the vantage point of thedetection efficiency. In a realistic way, the finite size of the detectors determine the positionon this curve, and GRAPE is placed at 140◦ to the beam axis, about 180 mm apart fromthe secondary target.

26

2.4.4

CHAPTER 2. EXPERIMENT

Signal Processing for GRAPE

Figure 2.10 shows a schematic flow of the signal processing for GRAPE. Each detector within
GRAPE has two crystals inside it and nine segmented cathodes for each crystals. This section
describes the signal processing used for GRAPE focusing on one crystal for simplicity. The
outputs from the nine cathodes are first processed by individual pre-amplifiers. The preamplified signals are summed up to reproduce a net-charge pulse by a analog adder, called
the sum amplifier. Since the gains of pre-amplifiers are not same, the gain for each signal
must be tuned so that the pulse height of each signal for a given energy deposition is sameas the others. We tuned the gains of totally 108 channels manually. The pulse height of eachsignal, however, is not exactly same. A more precise correction is needed in offline analysis(see Sec. 3.5.1). All ten signals, i.e. nine cathode signals and one net-charge signal, aretransferred to the flow A. Each signal is divided into two signals in order to obtain charge
and pulse-shape information. To obtain the charge information, the signal is integrated andshaped into a Gaussian pulse, and the pulse height is digitized by a peak-sensitive ADC(V785 VME module). The pulse-shape information is deduced as the zero-cross timing ofthe (CR)2 − (RC)4 -shaped pulse. The timing is digitized by a TDC, (V775 VME module).
The timing of the γ-ray emission is approximately replaced by the detection timing since thethe difference among the detection timings is at maximum 500 ps. The detection timing is
extracted from the net-charge pulse with fast shaping and amplitude-rise-time compensation(ARC) method (B). The ARC is realized by using the CFD with small delay time of 50 ns.
The detection timing is digitized by a TDC (C006 CAMAC module). The detection timingfrom each crystal is also used as a trigger indicated by Ge-i trigger, where i is the ordinal
number of the crystal. A trigger signal GRAPE is made when at least one γ ray was detectedin the GRAPE.
The details of the data acquisition from the VME and CAMAC modules is described in
section 2.6.

2.5

Secondary Target

The secondary target of helium was liquidized to be thick enough to perform the present
measurement with a low intensity RI beam. The helium target is cooled below a boilingpoint of 4.2 K at 1 atm. The typical density of the liquid helium at 1 atm and 4 K is
125 mg/cm3 , which is 700 times larger than that of the gas. Due to the higher target densityof the liquid helium compared to the gaseous, more target nuclei allows for better reaction
statistics. Figure 2.11(a) shows the schematic view of the liquid helium target system usedin the present experiment. The system consisted of a cryogenic refrigerator, a target celland two stages of thermal-conductive materials between the two. The first stage cools down
to 60 K and also cools down the heat radiation shield covering the second stage and target

2.5. SECONDARY TARGET

27

pre amp.

AB

sum amp.
fast amp.

z.c. discri.

A

NIM−ECL

TDC

3412

V767

V785
spectroscopic amp.
(N568B)

ADC
CFD

B

logic delay

OR935
fast amp.
(OR474)TDC
C006

Ge−i trigger

Ge−01 trigger
...

GRAPE

Ge−12 trigger
Figure 2.10: Schematic view of the electronic circuits for the signal processing of the
GRAPE.

28

CHAPTER 2. EXPERIMENT

cell. The shield had the window covered with 6-µm thick Havar foils. The second stage cools
down to 4 K and also cools down the target cell and the helium filled therein. The front viewand cross section of the target cell are shown in Fig. 2.11(b). The target cell had windowsof 24-mm diameter at the entrance and exit. The windows were covered with Havar foils
same as the shield. At the bottom of the target cell a thermometer was installed to monitor
the temperature of the helium. The temperature and pressure remained stable at around4 K and 1 atm. The target thickness was extracted to be 143±5 mg/cm2 from the velocity
difference of outgoing particles in the experimental data sets (w/ Liq.He and w/o Liq. He).

2.6

Signal Processing and Data Acquisition

The signals from the detectors generated in the reactions induced by the secondary beam
were processed by electronic circuits and taken by a general purpose computer on an eventby-event basis. Section 2.6.1 describes the process of the signals from each detector, section
2.6.2 describes the data acquisition system consisting of several general purpose computers,
and Sec. 2.6.3 summarizes the data set for the analysis.

2.6.1

Signal Processing and Trigger Condition

The signals from each detector were process by electronic circuits to deduce the charge,
timing and/or multiplicity information. The signal processing for GRAPE has been describedin Sec. 2.4.4. In this section, the signal processings for the plastic scintillation counter for the
beam monitor, PPACs and hodoscope, and the trigger for the data acquisition are described.
Figure 2.12 shows the circuit diagram for the signal processing of the plastic scintillation
counters. The diagram applies to the two plastic scintillation counters at F2 and F3 excluding the circuit for the trigger signals. The signal from the PMT attached to the plastics
scintillation counter was divided into two signals. One signal was delayed and digitized bya charge-sensitive analog-to-digital converter (QDC) to deduce the energy deposited in theplastic scintillators. The other signal was connected to a leading-edge discriminator, which
output the logic signal when the amplitude of the signal was higher than the threshold.
The logic signal was used for the stop signal of the time-to-digital converter (TDC) and thetrigger for the data acquisition.
The logical AND of two logic signals from the PMTs attached to the left (L) and right(R) sides of plastic scintillation counter was prepared as BEAM signal and the timing of thissignal was decided by that of the left PMT (PL-L). The BEAM signal was used for the logicalAND with the signals from the other detectors to identify that the reaction was induced by
the secondary reaction. The BEAM signal was thinned out by four orders of magnitude
making DSBEAM signal as a trigger to check the beam profiles during the experiment. The2.6. SIGNAL PROCESSING AND DATA ACQUISITION

Aluminized Mylar(a) Schematic view of the liquid helium target system

24mm(b) Schematic view of the target cell
Figure 2.11: Schematic view of the liquid target system and target cell.

29

30

CHAPTER 2. EXPERIMENT

PL−L

Lin. FI/FO

Att.

Delay

QDC

Discri.

Delay

TDC
Rate Driver

Delay

PL−R

Lin. FI/FODSBEAM
BEAM

Discri.

Delay

TDC

Att.

Delay

QDC

Figure 2.12: Electronic circuit diagram for beam counter. The signal from a PMT was
divided into signals for charge and timing information. Both signals were delayed anddigitized by CAMAC modules. Logical AND of two timings from the PMTs attachedto the left (L) and right (R) side of the plastic scintillator was made for the BEAM
signal. The time was determined by the timing of PL-L. The BEAM signal was thinnedout to make a DSBEAM signal, that was a factor of 10−4 compared to the rate of the
BEAM signal. The circuits for the plastic scintillators at F2 and F3 were same, except
for the circuit for the trigger.

preampPPAC anode/cathodeTFA
discri.

delay

TDC

Figure 2.13: Electronic circuit diagram for the PPACs. The signals from the cathodes
and anode of a PPAC were pre-amplified and the timing was picked off by a TFA. The
timing signals were generated by leading-edge discriminators and digitized by TDCs.

pileup after a signal was killed by the precursory signal and the signal itself was register tothe event data in order to kill the data triggered by the precursory signal.
Figure 2.13 shows the electronic circuit diagram for PPAC. The signals from the cathodes
and anode, in total five signals per PPAC, were amplified by the pre-amplifiers located as
close to the PPAC as possible in order to improve the signal-to-noise ratio. The timing of
pre-amplified signals was picked up by the timing-filtering amplifiers (TFA), and the timing
signals were picked off by discriminators and digitized by TDCs.
The electronic circuit diagram for outgoing reaction products is shown in Fig. 2.14. Twosignals were extracted from the photomultiplier tubes attached to the both ends of eachplastic scintillation bar. One signal was attenuated and delayed to be in the integration
window of Fast Encoding and Reading Analog-to-digital converter (FERA) module, in order2.6. SIGNAL PROCESSING AND DATA ACQUISITION

Att.

delay

discri.

delay

FERA

FERA drv.

FERA

FERA drv.

hodoscope
TFC

31
memorymodulememorymoduleSUM
dE SUM

discri.
HODO

E SUM

discri.

Figure 2.14: Electronic circuit diagram for the hodoscope. Two signals are extracted
from each photomultiplier tube attached to both ends of each plastic scintillation bar.
One signal is integrated and digitized by FERA module to deduce the energy depositionin each bar, and the timing of the other is picked off by the discriminator and digitizedby a TFC-FERA combination. The signals are summed, and the multiplicity in eachbar is extracted from the discrimination of them. The multiplicity signal is used forthe requirement of the reaction product.

to deduce the energy deposited in each bar. The digitized data was read into a memory
module via FERA bus managed by a driver module (FERA driver). The other signal was
discriminated and delayed. The time difference from the trigger timing was converted tothe corresponding charge by time-to-charge converter (TFC) module. The converted charge
was integrated by FERA module and the digitized data was stored into the other memory
module.
The timing signals from the bars in ∆E- and E-layers were summed and the multiplicity
in each layer was obtained by discriminating the height of the summed signal. At least twohits, i.e. both signals from a single scintillation bar, in each layer was required. Although themultiplicity signal was logical OR rather than the signal consisting of logical AND processing,the multiplicity signal was used as the requirement of at least one scintillation bar firing sinceboth signals from two PMTs were coincident and the signal-to-noise ratio was good enoughin the energy region of interest corresponding to the energy deposited by the 12 Be or 13 B
nuclei.
The trigger for the data acquisition consisted of two signals. One is the DSBEAM
signal, described above for monitoring the beam profile during the experiment. The other is alogical AND between the BEAM, GRAPE, and HODO signals (COIN) to require the nuclear
reaction accompanied by de-excitation γ rays. The timing of the trigger was determined by
the BEAM signal. Typical rates of the COIN and DSBEAM trigger were 103 and 2 × 102
Hz, respectively.

32

2.6.2

CHAPTER 2. EXPERIMENT

Data Acquisition System

The digitized data were read out and stored into data storage by a data acquisition (DAQ)
system, “babarl DAQ”, consisting of two general purpose computers connected over Ethernet. Each computer concentrated on a specified function to reduce the dead time of thesystem. One computer (collector) read out the data from CAMAC and VME modules andsent the data to the other one (recorder). Then the recorder wrote the data to the magnetic
tape storage. Furthermore, the collector had two buffers to store the data in an effective
manner, i.e. when one buffer was full, the data stored in the other, to reduce the dead timeof the system due to the delay by the overhead of the network communication. The operating system of the collector was a real-time Linux system to guarantee that the responsetime was within a specified duration, e.g. 50 µs for the present DAQ system. The collector
checked the CAMAC LAM signal every 50 µs to start acquisition. The order of the readingof the modules was scheduled so as to read the data after the end of conversion.

2.6.3

Data Sets

Experimental data were taken under several different conditions as summarized in Table 2.2.
The principle data for single-proton transfer reaction were taken with the liquid 4 He target.
Data with empty target cell runs were taken to estimate the contribution from scatteringwith non-α materials.
Sets of data were obtained for each conditions by applying both the BEAM and COIN
trigger conditions. Also data for the detector calibration were taken using beam of 12,13,14 B
and 12 Be.

2.6. SIGNAL PROCESSING AND DATA ACQUISITION

Beam
Be 50 AMeV
12
Be 50 AMeV
12
Be 50 AMeV
12
B 50 AMeV
13
B 40 AMeV
14
B 50 AMeV
15
B 60 AMeV
16
B 50 AMeV
none
none
none
none
12

target/source
liquid helium
target cell w/o helium
none
none
none
none
none
none
60
Co
137
Cs
152
Eu
none

trigger
COIN ⊕ DSBEAMDSBEAMDSBEAMDSBEAMDSBEAMDSBEAMDSBEAMDSBEAM
GRAP E
GRAP E
GRAP E
GRAP E

Table 2.2: List of the data set for the analysis. The set is divided into four groups by
the type of measurement: the measurement of the transfer reaction, target thickness,
the timing calibration of beam-line counters, and the energy calibration of GRAPE.

33

34

CHAPTER 2. EXPERIMENTChapter 3
Data analysis
3.1

Overview

The goal of the analysis is to deduce the transferred angular momentum in the protontransfer reaction and to determine the spin and spectroscopic strength of each excited statein 13 B The analysis consists of three parts, as explained below.
1. Selection of 4 He(12 Be,13 B) reaction channel
In the first part of this chapter, the analysis procedure to select the reaction channels andto find ‘good’ (12 Be,13 B) events will be described. The timing measured with TDC modules
is calibrated and the walk of the timing originating from the pulse shape and height iscorrected. In order to select the reaction channel, the incident and outgoing particles are
identified on an event-by-event basis. The incident particles are identified using the timing
information measured with F2PL and F3PL, and the energy deposition measured with F3PL.
The outgoing particles are identified using the timing information measured with F2PL,
F3PL and hodoscope, and the energy deposited in the ∆E and E layer of the hodoscope.
The trajectory of the incident particle was deduced by using the PPACs located at F3. The
timing information of the PPACs is also used for the single hit event selection to eliminate
pileup events.
2. Deduction of the γ-ray yield and the angular distribution
In the second part, the analysis procedure to obtain the Doppler-shift corrected γ-ray energy
spectrum, the γ-ray yields and the angular differential cross section to each excited state is
described. The pulse height information measured with V785 ADC modules is calibratedusing the experimental data taken with standard sources. The Doppler shift of the γray energy is corrected by the combination of the γ-ray detection position deduced frompulse-shape analysis with the the scattering angle and velocity of the outgoing particle
deduced using the method detailed in the previous part. The resultant γ-ray energy spectrum
is decomposed to response functions and backgrounds to obtain the γ-ray yield for each

35

36

CHAPTER 3. DATA ANALYSIS

population of the excited state. The angular differential cross sections are deduced fromthe decomposition analysis of the γ-ray energy spectrum in combination with the angular
information of the outgoing particles.
3. Deduction of the transferred angular momentum and spectroscopic strengthfor each excited state
In the last part, the analysis of the angular differential cross sections using distorted wave
Born approximation will be described. The transferred angular momentum and the spectroscopic strength are deduced from the DWBA analysis using the phenomenological opticalpotentials and the optical potential obtained based on the single-folding model.

3.2

Time calibrations

In this section, a common procedure used for the calibration of the timing information
measured with TDC modules is described. All the channels of TDC modules are calibratedusing a time calibration module. The time calibration module generates start and stop
signal. The stop signal is randomly generated in a certain period, which is set to 20 ns in
the present study. Figure 3.1 shows a typical example of the time calibration, which is forthe timing signal of RF. In Fig. 3.1(a) and (b), the obtained time spectrum and the result offitting, respectively, are shown. A set of the mean values of peaks in Fig. 3.1(a) is fitted witha linear function. In Fig. 3.1(c), differences of the experimental data from the linear function
are shown. As shown in this figure, the differences become larger in the lower and upperchannel region, which is the so called ‘integral non-linearity’. To correcting the integral nonlinearity, the data set is fitted with a spline curve rather than the linear function. The curves
in Fig. 3.1(c) indicates the deviation of the spline from the linear function. The curve well
reproduces the deviation of the experimental data, so the spline is chosen as a calibration
function for the present calibrations.

3.2.1

Walk correction for the plastic scintillators and Germanium
detectors

The walk effect [59] in the timing information of F2PL and F3PL should be corrected inorder to achieve intrinsic timing resolutions. The leading edge timing of a pulse shape V (t)is approximated by a parabolic function of time t as follows,
V (t) = A(t − t0 )2 ,(3.1)

where t0 and A are the time of initial rise, i.e. the detection timing, and the pulse height ofthe signals, respectively. From this equation, the measured timing t is obtained as follows,t = t0 +sqrtV (t)
√
.
A

(3.2)

1000

(a)

800
600
400
200

deviation (ns)00Reference timing (ns)count (a.u.)

3.2. TIME CALIBRATIONS

37

250

(b)

200
150
100
50

1000

2000

3000
4000
TDC (ch)

0
0

1000

2000

3000
4000
TDC (ch)

0.04

(c)

0.02
0

-0.02
-0.04
-0.06
-0.08
-0.1
-0.12
-0.14
0

1000

2000

3000
4000
TDC (ch)Figure 3.1: Typical example of time calibration using a time calibration module. Thisis the case of the TDC for the time signal of RF. Figure (a) shows a time spectrum
obtained the time calibrator with the time period 20 ns. Figure (b) shows a result of thefitting a linear function to the spectrum. Figure (c) shows deviations from the linear
function. Curve shows deviations of the spline fitted to the experimental spectrum
from the linear function. See text for details.

The second term represents the difference from the timing when the particles reach at thedetector. Since the threshold has been set to Vthres the correction term depends on only thepulse height. Finally the walk-corrected timing T can be obtained by subtracting the walkeffect from the calibrated TDC value TT DC ,√Vthres.
(3.3)
t = tT DC − √AThe parameter of Vthres is obtained by fitting the eq. 3.3 to the experimental data.
The timing signal of the plastic scintillator hodoscope and Germanium detectors are also
affected by walk. For these detectors, the empirical equation similar to Eq. (3.3) was used:√
Vthres,(3.4)
T = TT DC −Ab38

CHAPTER 3. DATA ANALYSIS

where b is a multiplier factor for the pulse height A. The parameters of Vthres and b were
determined by fitting these equation to the experimental data set. For the hodoscope, theparameters are obtained by fitting the eq. 3.4 to the data set with the calibration beams of
12,13,14
B. The timing resolutions of ∆E- and E-layers of hodoscope and GRAPE after thetime walk correction were 121 ps, 106 ps, and ∼5 ns in one standard deviation

3.3

Analysis of incident particle

This section describes analysis procedure for the incident particles. The incident particle
is identified by using the information on the energy deposition (∆E) and the time-of-flight(TOF) measured with two plastic scintillators, F2PL and F3PL. The direction and positionof incident particle on the target was measured by two PPAC’s.

3.3.1

Identification of incident particles

The 12 Be particles were discriminated from beam contaminants on an event-by-event basis
by means of the TOF-∆E method. The TOF is measured along the flight path of 5.4 m
between F2PL and F3PL. The resolution of the TOF was deduced to be 117±1 ps (onestandard deviation) by using beam calibration run of 13 B beam of 50 AMeV with 0.2 %momentum uncertainty. Figure 3.2 shows a two dimensional plot of TOF vs. the energy
deposited in the log scale. In the figure, the rectangular region corresponding to the 12 Be
beam used for the present analysis, is also shown. The region includes 2 % of the contaminant9
Li; however, the contaminant is unimportant since the probability of the fusion reaction at
50 AMeV is negligible small. The events in the region above the 300 ch of ∆E was pile up
events. From the other contaminants, the 12 Be beam was discriminated clearly. The ratioof secondary beam 12 Be in the gate to the total was 80%.

3.3.2

Beam profile

As described in Sect. 2.3.2, each PPAC(A and B) had two cathodes, which provided verticaland horizontal position information. Each cathode had two electrodes at both ends of thedelay line. The position information, e.g. X for horizontal position, was deduced from thetiming information from the two electrodes, as
X = c × (T1 − T2 + Tof f set ) + Xof f set (mm),(3.5)

where T1,2 are the timing information. A conversion factor from time difference to thelength c is typically about 0.63 mm/ns, and was measured precisely when the PPACs were
fabricated. Tof f set and Xof f set are constants for the corrections. The former is the correction
for the difference of cable lengths inside detector, which is inherent to each PPAC, and was3.3. ANALYSIS OF INCIDENT PARTICLE

39

600

Energy deposited (a.u.)

500
400
300
200
100
040

45

50

55

60

65

70

75

80

TOF (ns)

Figure 3.2: A two dimensional plot of the TOF and energy deposited for the identification of incident particles. A box indicates the selected region for 12 Be.

measured in advance when the PPAC was produced. The latter includes two corrections: onefor the difference of cable lengths outside detector and the other for the detector geometry.
This correction was deduced by using experimental data from runs without the target.
Single-hit event selection
In order to avoid incorrect extraction of the position, the selection of single-hit events was
performed. In the single-hit event, a sum of two timing (T1 and T2 ) should be constantcorresponding to the length of the delay line. By using this feature, we selected a certain
period of (T1 + T2 ) for each cathode of all the PPAC’s to exclude multi-hit events. The sumsof (T1 + T2 ) are shown in Fig. 3.3, which were obtained for PPACa horizontal cathode (a)and vertical cathode (b) with the 12 Be beam. A peak observed in each spectrum corresponds
to single hit events, while the events having a smaller timing sum show up as multi-hit events.
The data processing windows adopted as corresponding to single-hit events are also indicatedin Fig. 3.3. The widths correspond to the ±4σ-region were obtained by fitting a Gaussianfunction to the peak areas.
Detection and Tracking Efficiency
Detection efficiency of PPAC’s were estimated by analyzing the experimental data of 12 Be
beam trigger, DSBEAM. Single-hit events for both X and Y cathodes of each PPAC wereselected. The efficiencies of PPAC’s are extracted from the number of counts of X and Y as

40

CHAPTER 3. DATA ANALYSIS
5

10

5

10

Counts [A.u.]

(a)

4

10

Counts [A.u.]

4

10

3

10

102

3

10

2

10

10

10

1
10

(b)

20

30

40

50

60

1
10

20

30

40

50

60

Timing Ysum [ns]Timing Xsum [ns]Figure 3.3: Spectra of the average of timing ((T1 + T2 )/2) of both electrodes attachedto X and Y cathode of F3PPACa. Single hit event constitute a peak around 45–50 ns,
while multi hit events exhibit smaller sum. The selected single hit events are indicatedby the arrows in the figure.

below[55]:NX
N12 BeNYηY =N12 BeNX ⊗ NY,ǫ =N12 Be

ηX =

(3.6)
(3.7)
(3.8)

where ηX(Y ) and ǫ denote the detection efficiency of X(Y ) cathode and the detection efficiency
of one PPAC, respectively. NX , NY and NX ⊗NY denote the number of counts of X cathode,Y cathode and coincidences of them for 12 Be beam, whose number of counts is denoted as
N12 Be. The obtained efficiencies are listed in Table 3.1. Tracking efficiency of PPAC’s was
also extracted to be 75% under the condition that all cathodes of two PPAC’s registered asingle hit.
Position and Angular resolutions
A scattering angle of an ejectile and an emission angle for a γ-ray were provided by position
information of PPAC’s and plastic scintillator hodoscope. Thus, the position resolution

3.3. ANALYSIS OF INCIDENT PARTICLE

ηX (%)ηY (%)ǫ(%)

F3PPACa
92.6
92.9
86.3

41

F3PPACb
92.5
93.0
86.2

Table 3.1: The detection efficiencies of PPAC’s.

determines the angular resolution of incident and scattering angles and γ-ray energy. We
evaluate the intrinsic position resolution of PPAC, and the position resolution at the target
place.
Considering the propagation of errors, the position resolution was deduced as below:
TX1 − TX22
TX1 + TX2≡
2
(δTX1 )2 + (δTX2 )2=2= (δTX )2

Tdif f ≡

(3.9)

TX

(3.10)(δTdif f )2

δX = c × 2δTdif f= c × 2δTX ,(3.11)(3.12)(3.13)(3.14)

where the TX1 and TX2 are the timings from the both end of the delay line X. The resolution
of TX(Y ) was deduced from a set of equations below,p(δF 2T )2 + (δF 3T )2 + (δT OFF 2−F 3)2(3.15)
δTF 2−F 3 =p
δTF 2−F 3a =(δF 2T )2 + (δF 3aT )2 + (δT OFF 2−F 3a )2(3.16)p(δF 2T )2 + (δF 3bT )2 + (δT OFF 2−F 3b)2
(3.17)
δTF 2−F 3b =p
δTF 3−F 3a =(δF 3T )2 + (δF 3aT )2 + (δT OFF 3−F 3a )2(3.18)p(δF 3T )2 + (δF 3bT )2 + (δT OFF 3−F 3b)2
(3.19)
δTF 3−F 3b =p
δTF 3a−F 3b =(δF 3aT )2 + (δF 3bT )2 + (δT OFF 3a−F 3b)2
(3.20)q(δF 3a(b)TX(Y ) )2 + (δF 3a(b))2 ,(3.21)
δTF 3a(b)X(Y ) −F 3a(b) =where F 2, F 3, and F 3a(b) represent F2PL, F3PL, and F3PPACa(b), respectively. F 2T isthe timing of F2PL and so are F 3T and F 3a(b)T . The values of δT OFi are calculated withan assumption of the uniform distribution of ∆p/p = 0.2%, taking account for the energy loss
in materials plated in the beam line. The set of equations is redundant so that 3 different
solutions for δF 2T , δF 3T and δF 3a(b)T were obtained. An average of 3 solutions was
taken as the timing resolution. The obtained values are listed Table 3.2. The resolution of

42

CHAPTER 3. DATA ANALYSIS

σ(ps)

δF 2T
65.3

δF 3T
98.8

δF 3aT276

δF 3bT
281

Table 3.2: The list of time resolutions of plastic scintillators and anodes in PPAC’s.

δX (mm)δY (mm)F3PPACa
0.97
0.85

F3PPACb
0.53
0.54(FWHM)(FWHM)

Table 3.3: The list of timing and position resolutions of cathodes in PPAC’s.

TOF(F2-F3), calculated to be 118 ps (σ) by using the values listed in Table 3.2, is consistent
with the values obtained in analysis on the plastic scintillators (3.3.1). The time and position
resolution of each cathode were obtained by combining the obtained time resolution of theanode and Eq. (3.21).
The position resolution at the target was calculated as below:l+L
L
XF 3P P ACb − XF 3P P ACa
l
l
2
 2

L
l+L
2
(δXF 3P P ACb) +(δXF 3P P ACa)2 ,=ll

Xtgt =(δXtgt )2(3.22)(3.23)

where l is the distance between F3PPACa and F3PPACb and L is the distance between
F3PPACb and the secondary target. The deduced horizontal and vertical position resolution
was 7.3 mm (FWHM) and 4.3 mm (FWHM), respectively.
The angular resolution for incident beam particles was calculated as below:Xb − Xa′(3.24)
X = arctan
l

2

1
′ 2
(δX ) =
(δXb )2 + (δXa )2 ,(3.25)
2l(1 + {(Xb − Xa )/l} )where X ′ is the incident angle, Xa and Xb are the position on F3PPACa and F3PPACb,
respectively, and l is the distance between F3PPACa and F3PPACb. The deduced horizontal
and vertical angular resolution were 3.7 mrad (FWHM) and 3.4 mrad (FWHM) at forward
angle.
Emittance of incident beam particles
Positions measured by one pair of PPAC, F3PPACa and -b, were employed to deduce theangles and positions of incident beam particles on the secondary target on an event by event

50

50

40

40

30

30

20

20

Y’ (mrad)X’ (mrad)3.3. ANALYSIS OF INCIDENT PARTICLE

10
0

-10

43

10
0

-10

-20

-20

-30

-30

-40

-40

-50
-50 -40 -30 -20 -10

0

10

20

X (mm)

30

40

50

-50
-50 -40 -30 -20 -10

0

10

20

30

40

50

Y (mm)

Figure 3.4: Beam spot size and incident angles of the 12 Be beam at the secondary
target for the horizontal (left) and vertical (right) directions. The plot was made by
extrapolation of the position information obtained by PPAC’s.

basis. The angle of the incident particle is defined in Eq. (3.24). Figure 3.4 shows twodimensional plots between the positions and angles of the incident particles at the secondary
target position. The plots show a rather incomplete focus of the beam in the vertical direction
compared with the horizontal focus. In this experiment, the final focal plane was changed
downstream by about 1 m compared with the standard F3 position so that the vertical focus
was poor. Some of the beam particles which are out of the effective areas of PPACs havenot been measured due to the poor focus, and thus the resultant emittance seems to be cut,as shown in the right of Fig. 3.4. Nevertheless, the spot size 8.5 × 5.7 mm2 in r.m.s was
sufficiently small compared to the target size of 24 mm-diameter.
The angular spread of the beam was 8.3 mrad and 13.17 mrad in r.m.s. for horizontal
and vertical directions, respectively. These values are smaller than the angular acceptanceof the plastic scintillator hodoscope placed after the target.

3.3.3

Target position

Position of the secondary target in the plain perpendicular to the beam axis was confirmed
by using a defocused beam. Figure 3.5(a) shows the scatter plot for the position distribution
at the secondary target with the condition of the hodoscope firing, which is extrapolatedusing the position measured with PPACs. Since the length of the target cell was long enoughto stop the incident particles, the intensities at the position corresponding to the target cellare very few. The position distribution normalized by the unbiased position distribution of

CHAPTER 3. DATA ANALYSIS

50

50

40

40

30

30

20

20

10

10

Y(mm)

Y(mm)

44

0

-10

0

-10

-20

-20

-30

-30

-40

-40

-50
-50 -40 -30 -20 -10

0

10

20

30

40

50

-50
-50 -40 -30 -20 -10

X(mm)

0

10

20

30

40

50

X(mm)

Figure 3.5: Position distribution of the defocused beam at the secondary target plane
under the condition that the particles arrived at plastic scintillator hodoscope. The
left one shows a raw distribution and right one shows the normalized one. An ellipse
indicates the selected region for 12 Be which bombarded the liquid helium target.

the incident beam is shown in Fig. 3.5(b). The shape of the target cell is clearly seen in the
figure. The position of the secondary target was deduced by fitting Gaussian function to theprojection of the distribution. The position of the secondary target was checked twice, atthe beginning of the anterior half and the last half of the experiment.

3.3.4

Transmission

The transmission of the beam from the F3PL to the hodoscope was estimated from thecomparison of the number of counts of the particles which hit two PPACs and bombarded
the liquid helium target for the events taken under DSBEAM trigger. The bombarded
region was defined as below:(X − X̄)2(Y − Ȳ )2+≤ 1,(R + 3σX )2 (R + 3σY )2(3.26)

where X(Y ) and σX(Y ) denote the extrapolated position and its resolution at the secondary
target, R denotes the radius of the window of the secondary target, X̄(Ȳ ) denotes the
position of the secondary target, described in Sec. 3.3.3. The selected region for the particles
bombarding the secondary target is indicated by an ellipse in Fig. 3.5. The incident particles
which bombarded the helium target rather than the target cell, were 94% of the total incident
particles.

3.4. ANALYSIS OF OUTGOING PARTICLES

3.4

45

Analysis of Outgoing Particles

This section describes analysis procedure for the outgoing particles. In this experiment, notonly 13 B, which was physical interest, but also other various nuclei were made via reactions
on the secondary target. We need to identify the outgoing particles to investigate the natureof 13 B and to distinguish non-13 B events. The identification of the outgoing particles were
performed by combination of the ∆E-E-TOF information, which were provided by the plastic
scintillator hodoscope.
The scattering angle was deduced from the position of the outgoing particle at thehodoscope, in combination with the position information of PPACs. The position of theoutgoing particle was measured with the hodoscope.
For the Doppler correction, velocity (β) of an outgoing particle was extracted from thetiming information of the hodoscope in combination with the time-of-flight between F2 andF3, measured with plastic scintillators at F2 and F3 (see Sec. 2.3).
In the analysis, the number of the outgoing particles detected in the hodoscope is limited
to be one in order to select a two-body reaction.

3.4.1

Identification of Atomic Number

The atomic number Z of the outgoing particle was identified by using a combination of theenergy loss in the ∆E plastic scintillator bar and the TOF between the secondary target
and the E plastic scintillator bar. The ∆E depends on Z and the velocity β of a reaction
product and the relation can be written as
∆E ∝

Z2
.
β2

(3.27)The TOF is proportional to β, therefore Z can be determined by combination of the ∆E
and TOF information. Figure 3.6(a) shows a typical correlation between ∆E and TOF. Foran easy selection, Z was calculated by using the information of ∆E and TOF as below.
Z ∝ ∆E · β.

(3.28)The distribution of Z is shown in Fig. 3.6(b). The boron isotopes (Z = 5) are clearly
discriminated from the other isotopes. The window from Z = 4.5 to Z = 5.5 is chosen for
the analysis of the proton transfer reaction.

3.4.2

Identification of Mass Number

The mass number A of the outgoing particle was identified by using a combination of theenergy loss in the E plastic scintillator bar and the TOF between the secondary target andthe E plastic scintillator bar. The outgoing particle stops in the E bar, hence the E is close

46

CHAPTER 3. DATA ANALYSIS

107
2000

106
count (a.u.)dE (a.u.)

1500

Z=5
1000

Z=4
Z=3

105

104

500

Z=2
0
0.2

Z=1
0.25

0.3

0.35

0.4
v/c

1

2

3

4

5

Z

(a) velocity vs energy deposition(b) atomic number distribution

Figure 3.6: (a) Correlation between velocity and energy deposition in ∆E plastic
scintillators of outgoing particles. The velocity is extracted from the TOF between theF2 and F3 and the one between the F3 and plastic scintillator hodoscope. (b) Atomic
number distribution of outgoing particles is deduced from the correlation between thevelocity and energy deposition by using Eq. (3.28).

to kinetic energy. The relation between the velocity β, the A, and the kinetic energy, i.e. E
is approximately to be,
E ∝ Aβ 2 .

(3.29)The TOF is proportional to β, therefore A can be determined by combination of the E
and TOF information. Figure 3.6(a) shows a typical correlation between E and T OF forthe atomic number region from Z = 4.5 to Z = 5.51 , corresponding to boron isotopes. Foran easy selection, A was projected along the direction indicated in Fig. 3.6(a). The obtained
distribution of A is shown in Fig. 3.7(b). A peak corresponding to 13 B (A = 13) is clearly
discriminated from the neighbor peak. Assuming the Gaussian distribution for each A peak,a typical resolution was 0.25 in the mass number unit, hence the peaks are separated fromtheir neighbors by 4σ.
1

The atomic number Z is always physically an integer value. However, due to the experimental uncertainties particles appear to have non-integer atomic numbers

160

47

count (a.u.)light output (MeVee)

3.4. ANALYSIS OF OUTGOING PARTICLES

20000

Z=5
140

15000

120
10000
100
13

80

5000

B

12

B11
B

0.25 0.26 0.27 0.28 0.29

0.3

00.31 0.32

10

11

v/c
(a) velocity vs light output

12
A

13

14

(b) mass distribution for Z = 5

Figure 3.7: (a) Correlation between velocity and energy deposition in E plastic scintillators of outgoing particles. The velocity is extracted from the TOF between F2 andF3, and from F3 to the plastic scintillator hodoscope. (b) Mass number distributionof outgoing boron isotopes is deduced from the correlation between the velocity andenergy deposition by using Eq. (3.29).

3.4.3

Position of Outgoing Particles

The position information of the outgoing particle is extracted from the time difference between signals of two photo-multiplier tubes attached to both ends of each plastic scintillation
bar. The horizontal (X) and vertical (Y) position is deduced by using ∆E and E bars, respectively, from the formula below,
X = (T1 − T2 − c1 )/c2 + Xof f set ,(3.30)

where c1,2 are calibration coefficients for each scintillation bars and Xof f set is common correction constant for each layer. The calibration coefficient is deduced from the geometrical
relation between ∆E and E bars. A bar in ∆E layer can be divided into 16 sections corresponding to 16 E bars and a bar in E layer can similarly be divided into 13 sections.
Focusing on one pair of ∆E and E bars, the time difference of each bar is limited by thewidth of the paired bar. The position of E bars are well known geometrically. The centroidof the time difference distribution of the ∆E bar, therefore, corresponds to the locationof E bar, and the same for the E bar. The centroid is approximately obtained by fittingwith a Gaussian distribution. The correlation between the centroids and locations is fitted

48

CHAPTER 3. DATA ANALYSIS

with a linear function, and then the calibration coefficients c1,2 are obtained. The common
correction constant is deduced by comparing the position extracted by hodoscope with theposition extrapolated by the PPACs. In the run with no secondary target, the trajectory
after the last quadruple magnet is straight since the effect of multiple scattering in mylars
is ignorable.

3.4.4

Velocity of Outgoing Particles

The velocity of an outgoing particle is deduced from the timing information measured withhodoscope and plastic scintillators at F2 (F2PL) and F3 (F3PL) with a correction for theenergy deposited in the secondary target. The timing from the hodoscope is extracted as
the average of the two timings T1,2 of the fired E bar. The velocity before the correction is
calculated from the time-of-flight from the secondary target and the hodoscope T OFT gt−Hodo .
The T OFT gt−Hodo is obtained from the time difference between F3PL and the hodoscope(T OFF 3P L−Hodo) by subtracting the time-of-flight between F3PL and the secondary target(T OFF 3P L−T gt). The T OFF 3P L−T gt is estimated from the time-of-flight between F2PL and
F3PL, taking the effect of the energy deposits in the F3PL and PPACs into account. The
intrinsic resolution of the velocity is deduced from the experimental data and the resolution
to the velocity δβ/β is 0.23% (one standard deviation). The calculated velocity is corrected
for the energy loss in the secondary target. The correlation is approximated as the second
order polynomial. In the correction the reaction position along the beam axis is assumed tobe the center of the secondary target. Th effect of the ambiguity of this reaction position to
the Doppler correction is discussed in Sec. 2.4.3.
A window was set on difference between the velocities of incident and outgoing particles
in order to select the two-body reaction and reduce the background events since there were
events where the α particles broke up, rather than the α particles went to the 3 H particles.
Figure 3.9 shows a scatter plot of the scattering angle of the outgoing particles vs. thedifference of the velocities between the incident and outgonig particles. Since the velocitiesof the ougoing particles depend on the scattering angle of them due to the kimatics, thevelocity difference was corrected with a second-order polynomial for the easy selection. The
selected region as the two-body reaction events is indicated by two (red) solid lines.

3.4.5

Detection efficiency of Hodoscope

Acceptance of the experimental setup for the scattering particle will be described. The
acceptance was simulated including the angular and the position distribution of the incident beam, the size of the target cell and the position resolution of the hodoscope. In thesimulation, the acceptance ǫ for a certain angular range between θ1 and θ2 was defined as3.4. ANALYSIS OF OUTGOING PARTICLES

49

4500
4000

T1

count (a.u.)

3500

T2

3000
2500
2000
1500
1000
500
0-5

-4

-3

-2

-1

T1 - T2 (ns)(b) Typical plots of timing difference(a) A pair of ∆E and E bars

2

T1 - T2 (ns)

0

-2

-4

-6
4

6

8

10

12

14

E bar #(c) Typical correlation
Figure 3.8: (a) A pair of ∆E and E bars is selected. The emitted light in ∆E
bar reaches at the both ends and fires the photo-multiplier tubes. (b) Typical timing
difference of ∆E bar in coincidence with certain E bar. The centroid is approximately
obtained by fitting with Gauss distribution. (c) Typical correlation between the timedifference of ∆E bar and bars in E layer. The horizontal axis is just an ordering ofbars, i.e., the time difference is not on a straight line.

050

CHAPTER 3. DATA ANALYSIS

0dβcor (a.u.)

-0.01

-0.02

-0.03

-0.04

-0.050

1

2

3

4

5

6

7

8

9

10

scattering angle (deg.)

Figure 3.9: Scatter plot of the scattering angle of the outgoing particles vs. the difference between the velocities of incident and outgoing particles. The velocity difference
is corrected with a second-order polynomial for the easy cutting. The (red) solid lines
show the upper and lower limit for the selection of the two-body reaction.

below:
ǫAcc. =

Nmeasured,
Ngenerated(3.31)

where Nmeasured and Ngenerated are the numbers of measured and generated in the angularrange, respectively. The simulation procedure is listed below:• Generate the incident beam with angular and position distribution at the middle of thetarget. Each distribution was deduced to be approximately a Gaussian distribution
from the experimental results.
• Calculate the reaction point with an assumption that the reaction occurs uniformly
belong the beam direction. In this step, the horizontal and vertical reaction points arealso re-calculated.
• Generate the scattering angle and calculate the momentum vector of a outgoing particle. The scattering angle is generated to be isotropic in center-of-mass frame andconverted to one in the laboratory frame with the mean velocity of ∼30% light speed.
This assumption works enough to simulate the acceptance. The scattering angle islimited up to 8 degrees.

51

acceptance (a.u.)

3.5. ANALYSIS OF DE-EXCITATION γ RAYS

angle (deg)Figure 3.10: Acceptance of the hodoscope deduced from a Monte Carlo simulation.
The spacial and angular distribution of incident particle were taken from the experimental data. The count in each bin is divided by the number of generated events
corresponding to each bin. The distribution is flat in the angular region below six degrees, which are reduced from unity. The rejected component in flat region, however,has been excluded from the analysis. See the text for the detail.

• Check if the outgoing particle can go out of the target cell.
• Check if the outgoing particle stops in the 400×400 mm2 rectangle region at the hodoscope. For this check the position resolutions deduced from the experimental data
are used.
Figure 3.10 shows the simulated acceptance of hodoscope in the laboratory frame. The
count in each bin is divided by the number of generated events. The distribution is flat in
the angular region below six degrees, which are reduced from unity, and it decreases in thelarger angular region. The rejected component in flat region has been excluded from theanalysis, by setting the cut on the spacial distribution as shown in Fig. 3.5. The angulardifferential cross sections are deduced up to six degrees in the laboratory frame as describedin section 3.7.

3.5

Analysis of De-excitation γ Rays

This section describes the analysis of the de-excitation γ rays detected with GRAPE. Theenergy calibration is done using the experimental data taken with the standard sources (Sec.
3.5.1). To improve signal to noise ratio the timing of the germanium detector is measured

52

CHAPTER 3. DATA ANALYSIS

with an ARC method. The walk correction is done in Sec. 3.2.1. The selection of the timing
information for the rejection of accidental coincidence event is described in Sec. 3.5.2. The
pulse-shape analysis and the Doppler correction of γ-ray energies are described in Sec. 3.5.3
and Sec. 3.5.4, respectively.

3.5.1

Energy Calibration

The energy deposition of the emitted γ ray is obtained by summing the energies deposited in
two crystals packed in a detector. The energy deposition in each crystal can be obtained in
two ways. One is the add-back of the energies obtained from the cathodes’ outputs (digitalsum), the other is the energy obtained after correcting the net-charge signal (analog sum).
The advantage of the digital sum is better energy resolution, while the one of the analog sum
is better full-energy peak efficiency. In the present analysis, the analog sum is chosen since
the dominant contribution to the energy resolution for the γ-ray energy after the Dopplershift correction is not intrinsic energy resolution but the Doppler width. The comparison
between the digital and analog sum is described in Appendix B.
The procedure to deduce the analog sum consists of four processes: (1) conversion fromchannel to energy (usual energy calibration); (2) correction for the gain difference of the
pre-amplifiers among the segments; (3) correction for the induced-charge-like effect; (4)correction for the temporal gain drift.
The pulse-height data taken with V785 analog to digital conversion (ADC) modules
are calibrated to the reference energies of γ rays emitted from a 152 Eu standard source. In
the ideal case, since the AD conversion is performed with a linear function, the energy isa linear function of the output channel of the ADC module. In the realistic case, however,a linear function is not suitable for the calibration since the linearity of the V785 module
is guaranteed only in the range from 10% to 90% of dynamic range. A parabolic function
is usually used as a calibration function since the second-order term acts a correction for
the non-linearity. Although the deviation between the calibrated energy and its reference
becomes smallest for the interpolation using a parabolic function, the deviation is not trivial
for the extrapolation. In the present experiment the dynamic range of the ADC module is
set up to 5 MeV while the reference exists at most 1.5 MeV. From this point of view, thediscrepancy for the extrapolation with linear function is smaller than that with parabolic
function. Hence, the combination of the parabolic and linear functions seems better for thecalibration of the present data. The parabolic and linear functions are used for the low
and high energy region, respectively. For the analog sum, the hit segment is limited to thesegment nearest to the target in order to reject the effect of the gain difference in the analog
adder.
The gains of the pre-amplifiers are roughly tuned in the analog adder before summing
the segment signals. In this process the gains are precisely tuned not only for single segment3.5. ANALYSIS OF DE-EXCITATION γ RAYS

53

hit but also for many segments hit.
The charge collected by each cathode is added by the analog adder module to form thenet-charge signal as below,
9XQΣ =ǫi Qi ,
(3.32)
i=1

where QΣ and Qi are the net-charge signal and charge collected by i-th segment, respectively.
The coefficient ǫi is the total gain factor of the pre-amplifier and analog adder for i-th
segment signal. The relation between the real total charge induced in a crystal Qc and QΣ
is represented by using eq. (3.32) as below,
P
X
Qi
Qc =
Qi = QΣ P i.
(3.33)i ǫi Qii
As seen in eq. (3.33), the observed charge should be corrected by the charge from eachsegment in order to obtain the real charge. Since it is difficult to deduce the gain factor ǫitself, we define the relative gain factor ǫ′i = ǫǫ6i , taking the 6-th segment as reference. In
fact, the ǫ6 is already renormalized into the channel-to-energy (charge) conversion function
if the function is obtained under the condition that only 6-th segment is hit. By replacing
the charge Q by the energy E and using the charge-to-energy conversion function fi and fΣ
for i-th segment and net-charge signal, respectively, the total energy deposition in a crystal
is obtained from eq. (3.33) as below,
PEi(3.34)
Ec = EΣ P i ′ii ǫi EP
fi (Ni ),(3.35)= fΣ (NΣ ) P i ′i ǫi fi (Ni )where NΣ and Ni is the ADC channels for the net-charge and i-th segment signal, respectively.
The relative gain coefficient is obtained under the condition that the i-th segment is hit asshown below,fΣ (NΣ )(3.36)
ǫ′i =
Ec
hit=i

Even if the Ei is small below the threshold, the total energy Ec contains the information ofEi since the charge from i-th segment is added to the total energy before digitization. On
the other hand, the digital sum is smaller by Ei than the true energy deposition. That is
why the full-energy-peak efficiency of the analog sum is larger than that of the digital sum.
The induced-charge-like effect is found in the case of two crystals firing in a detector.
This effect is corrected by multiplying the obtained energy by a constant, which has been
determined by comparing the obtained energy with the γ-ray energy of 1408-keV from the152
Eu standard source. The correction for the temporal drift is performed by also multiplying

54

CHAPTER 3. DATA ANALYSIS

3

4000
3500

2

×10

counts/1keV (a.u.)difference (keV)

3000
1
0
−1

2500
2000
1500
1000

−2
−30

500
500

1000

1500

2000

Eγ (keV)(a) Energy deviation

2500

3000

00

500

1000

1500

2000

2500

3000

Eγ (keV)(b) natural background γ rays

Figure 3.11: (a) The energy deviation after the energy calibration of GRAPE. The
references are taken from the data with 152 Eu, 60 Co, 137 Cs standard sources and without
any source. (b) Background γ-ray energy spectrum. Above 1500 keV, the scale is ten
times larger than the one below.

the obtained energy by a constant on an event-by-event basis. All the events are separated
into several tens groups based on the event number. The constant has been obtained foreach group by referring the 511-keV γ ray.
The deviations between the calibrated energies and their reference are obtained by comparing the calibrated and referred energies taken with 152 Eu, 60 Co, 137 Cs, and without any
source, i.e. with background γ rays. Figure 3.11 shows (a) the energy deviations and (b) the
background γ-ray energy spectrum. The deviations are small enough and within ±1 keV up
to 3 MeV. The energy dependence of the resolution is shown in Fig. 3.12(a). The resolution
is not as good as those of other germanium detectors, since noises of segment signals are also
added. The contribution of in-phase noises to the analog sum is expected to be nine times
larger than to the digital sum, and the contribution of random noises three times larger.
However the resolution of around 2.5 keV is good enough to measure the in-flight decay γ
rays. The energy dependence is approximated by a empirical function as below,qσ(Eγ ) = σ02 + σE Eγ ,(3.37)
where σ0 is a constant contribution such as the electric noise of the circuit and the thermal
noise in the crystal, σE is a energy dependence, and Eγ is the γ-ray energy in MeV. The valuesof σ0 and σE are 2.35 and 1.32, respectively, were obtained by fitting to the experimental

3.5. ANALYSIS OF DE-EXCITATION γ RAYS

55

3
1.4

2.9

1.2

2.8
efficiency (%)

Resolution (keV)

2.7
2.6
2.5
2.4
2.3

1
0.8
0.6
0.4

2.2
0.2

2.1
20

200 400 600 800 1000 1200 1400
Energy (keV)(a) Energy resolution

00

200 400 600 800 1000 1200 1400
Eγ (keV)(b) Full-energy-peak efficiency

Figure 3.12: The energy resolution in one standard deviation and full-energy-peakefficiency of GRAPE with statistical errors. The curves show approximate functions.

data. The energy dependence of full-energy-peak efficiency is shown in Fig. 3.12(b). Belowaround 200 keV, the efficiency is rapidly decreased since the low energy γ rays are absorbed
in the beam pipe, chassis of the detector, cooler ring and the insensitive area of germanium
crystal itself. The energy dependence is well approximated by a empirical function of aE bwith a = 74.5 and b = −0.689 shown in Fig. 3.12(b). These tendency are well reproducedby the simulations using GEANT4 code described in section 3.6.

3.5.2

Timing Information

The γ-ray detectors are sensitive not only to the γ rays from reaction products but also tocharged particles, neutrons, γ rays from detectors, and the natural background. The timingof the germanium detector is used for checking and reducing the amount of such backgroundγ rays.
Figure 3.13(a) shows a correlation between the timing and the γ-ray energy measured
with GRAPE in coincidence with 13 B. The timing is obtained as the time difference between
the arrival timings of γ-ray to the germanium detector and of the incident particle to the secondary target. The latter timing is almost same as the timing of γ-ray emission. The arrival
timing of the secondary beam at the secondary target is estimated from the time informationof F2PL and F3PL. The horizontal axis is shifted with arbitrary offset for convenience. The
true coincident events are located at around 300 ns. Figure 3.13(b) is a projection to the

56

CHAPTER 3. DATA ANALYSIS

6000
2500
5000
2000
count (a.u.)Eγ (keV)

4000
3000
2000

1000

500

1000
00

1500

100

200

300

400

500

600

00

100

TARC - T0 (ns)(a) energy vs timing

200

300

400

500

600

TARC - T0 (ns)(b) timing distribution

Figure 3.13: (a) Typical scatter plot of γ-ray energy and timing difference after walkcorrection. The timing difference is between γ-ray detection timing and time when the
nuclear reaction occurred. The prompt peak is adjusted to be zero. (b) The timing
distribution of γ-ray detectors. The prompt timing window is set to the hatched region
and the pre- and post-prompt timing window is set to the region indicated by arrows.

horizontal axis of Fig. 3.13(a). As seen in Fig. 3.13, there are small amount of accidentally coincident events pre- and post-prompt region. To estimate the accidentally coincident
events, the timing spectrum for 12 Be is obtained. The pre- and post-prompt events are selectby setting the 100 ns timing window before 350 ns and after 450 ns region on the projected
timing spectrum. The obtained energy spectrum for the background events are modified in
the same manner for the Doppler correction of the energy spectrum for 13 B.

3.5.3

Extraction of γ-ray Interaction Position

The depth is extracted from the correlation between the zero-cross timing of the (CR)2 -(RC)4
shaped pulse of the net-charge (Tsum ) and segment (Tseg ) (see Sec 2.4.2). The experimentallyobtained correlation in the center region is shown in Fig. 3.14(b). The Tsum is measured
taking the timing when the incident particle arrives at secondary target as a starting point
so as to reduce the timing deviation due to the momentum deviation of the incident particle.
The offsets of Tsum and Tseg are tuned so as to minimize the distance of each experimental
datum to the nearest simulated point. The distance D is the norm of the vector between

500

500

480

480

460

460

440

440

420

420
Tsum

Tsum (ns)3.5. ANALYSIS OF DE-EXCITATION γ RAYS

400

400

380

380

360

360

340

340

320

320

300−80 −60 −40 −20

0

20

40

60

80

57

300
−100−80 −60 −40 −20 0

20 40 60 80 100

Tseg−TsumTseg−Tsum (ns)(a) Simulation(b) experiment

Figure 3.14: Correlation between Tsum and Tseg .

the experimental and simulated data calculated as below,qexpsim )2 + {(T exp − T exp ) − (T sim − T sim )}2 .
D = (Tsum− Tsum
seg
sum
seg
sum

(3.38)The experimental data is well reproduced by the simulation (Fig. 3.14(a)). The depth
is determined on an event-by-event basis by comparing the experimental datum with thesimulated one. The simulated depth which gives the minimum D is referred as the depth of
the interaction position. Figure 3.15 shows the distance distribution. When many segments
are fired, the segment where maximum energy is deposited is assumed to be hit first since
we cannot know which segment is hit first in the present setup.

3.5.4

Doppler Correction

The γ rays are emitted from reaction products moving at a velocity v/c ∼ 0.29. Hence themeasured γ-ray energy is shifted because of the Doppler effect. The Doppler shift is corrected
for by using the angular information of the γ-ray emission and the velocity of the reaction
product. The γ-ray energy in the laboratory frame Eγlab and the one in the rest frame Eγcm isconnected by the velocity of the reaction product β and the gamma-ray emission angle withrespect to the momentum vector of reaction product in the laboratory frame θγlab as follows,Eγcm = γ 1 − β cos θγlab Eγlab ,(3.39)

58

CHAPTER 3. DATA ANALYSIS

×103
600

count (a.u.)

500
400
300
200
100
00

10 20 30

40 50 60 70 80

90 100D (ns)

Figure 3.15: Distance between the simulation and experimental data in the correlation
between Tsum and Tseg .

pwhere γ is the Lorentz factor 1/ (1 − β 2 ). The angle θγlab is determined by the geometricalinformation for each germanium detector and the pulse shape analysis described in Sec. 3.5.3.
A typical example of the Doppler-shift correction is shown in Fig. 3.16. Figure 3.16(a)shows the energy spectrum before the Doppler-shift correction. The broad peak at around1600 keV corresponds to the γ decay of the 2.1-MeV excited state in 12 Be. The energy
spectrum has been obtained by selecting the prompt region. The energy spectrum after
Doppler-shift correction for the energy spectrum shown in Fig. 3.16(a) with the pulse-shapeanalysis is shown in Fig. 3.16(b). The correction has been performed on an event-by-event
basis. Two peaks corresponding to the decay 2+ and 1− of 12 Be are seen in the Doppler-shiftcorrected spectrum. The energy resolution of 1.3% (FWHM) has been achieved after the
Doppler-shift correction with the pulse-shape analysis.
Due to the finite resolutions of the emission angle and of the reaction product velocity, theγ ray energy peaks are broader compared to the intrinsic energy resolution of the detectors,
which is called Doppler broadening. The Doppler broadening of Eγcm is approximated basedon Eq. (3.39).
!2
!2 
!2


2

∆Eγcm
∆Eγlabβγ 2 (β − cos θγlab )β sin θγlab
∆β
lab 2
∆θ
=
+
+
γ
Eγcm
Eγlab
1 − β cos θγlabβ
1 − β cos θγlab(3.40)
Three components contribute the energy resolution: the intrinsic resolution of the germanium detector ∆Eγlab /Eγlab ; the ambiguity of the velocity of the reaction product ∆β/β; the3.5. ANALYSIS OF DE-EXCITATION γ RAYS

59

counts / 10keV (a.u.)

5000

4000

3000

2000

1000
800 1000 1200 1400 1600 1800 2000 2200 2400 2600
Elab (keV)(a) Energy spectrum for 12 Be before Doppler correction
5000+

Be(2 →0+)124500
counts / 10keV (a.u.)

4000
3500

1.3% (FWHM)

3000
2500
-

Be(1 →0

12

2000

+

1500
1000
500
1000

1500

2000

2500

3000

3500

ECM (keV)(b) Energy spectrum for 12 Be after Doppler correction
Figure 3.16: Typical example of the Doppler-shift correction with the GRAPE. (a)Energy spectrum for 12 Be before Doppler-correction. Around 1600 keV the broad peakcorresponding to the γ decay of 2.1-MeV excited state in 12 Be is seen. (b) The highresolution energy spectrum for 12 Be nuclei after Doppler correction is obtained withthe pulse-shape analysis. The energy resolution of 1.3% (FWHM) is achieved for the γrays emitted from the fast moving (β ∼ 0.3) nuclei. Not only the transition at 2.1 MeV
from the first excited state but also the one at 2.7 MeV from the well known intruder
state is clearly seen.

60

CHAPTER 3. DATA ANALYSIS

accuracy of the γ-ray emission angle ∆θγlab . For an estimation of the energy resolution, therealistic condition deduced from the experimental data is taken, i.e., ∆Eγlab /Eγlab = 0.46%,∆β/β = 0.52%. The angular resolution is assumed to be ∆θγlab = 91mrad which corresponds to the 10 mm ambiguity of the γ-ray interaction position while the resolution of the
γ-ray position is assumed to be 3 mm in the simulation taking the position ambiguity of
the incident particle into account. The estimated overall energy resolution is 1.5%, which isconsistent with the experimental value 1.3%.

3.6

γ-ray Yield

This section describes a procedure for deducing the populations of excited states. The populations are calculated from intensities of transition γ rays deduced from a decompositionof the γ-ray energy spectrum with response functions of GRAPE. The decomposition is
performed by fitting the response functions to the experimental energy spectrum with parameters of normalization factors. The response functions are obtained by means of MonteCalro simulation with a toolkit for the simulation, GEANT4 taking into account the actual
configurations of germanium crystals and of the experimental equipments such as the γ-ray
detector cases, the support materials of GRAPE, vacuum chambers, the secondary target
cell, and the heat shield.
There is incomplete collection of the induced charge in germanium detector used in
GRAPE so that a tail arises at the low energy side of the total energy peak. The shapeof the tailing is approximated to be an exponential function of the γ-ray energy, and its
probability is represented as follows, sim
Eγ − Eγ0sim,(3.41)P (Eγ ) ∝ exp
αEγ0
where Eγ0 and Eγsim indicate the original energy deposition, and the energy include the effectof the incomplete collection, respectively, and α is a scaling factor. The fraction of themodified events and scaling factor α are deduced by fitting the simulated response function
to the experimental data taken with a standard source 137 Cs.
The simulated response function is folded with a Gaussian distribution with intrinsic
resolution dependent on the γ-ray energy. The intrinsic resolution ∆Eγlab is approximated to
be a function of the γ-ray energy as follows,q∆Eγsim = σ02 + σE Eγsim ,(3.42)
where σ0 is energy independent resolution due to the circuit noise and σE is energy dependent
factor.
Figure 3.17(a) shows the comparison between the experimentally obtained and simulated
response functions for 661-keV γ rays from a 137 Cs standard source. The experimentally3.6. γ-RAY YIELD

61

105

104

counts/1keV (a.u.)counts/1keV (a.u.)

104

103

103
102

10

102
1
0

100 200 300 400 500 600 700 800

05001000 1500 2000 2500

Eγ (keV)Eγ (keV)(a) 137 Cs(b) 60 Co

3000

Figure 3.17: The experimental and simulated response functions of the GRAPE forthe γ rays from 137 Cs standard source. The experimental response function is well
reproduced by the simulated one. The (black) thick solid line shows the experimentallyobtained response function including the background γ rays. The (blue) dotted line
shows the simulated response function for 661-keV γ rays and the (green) dotted line
shows the response function for the background γ rays. The (red) thin solid line shows
the response function fitted to the experimental one. (b) The comparison for the
standard source 60 Co. The (blue) dotted curves show the simulated response functionsfor 1173- and 1333-keV γ rays. The legends for the other curves are same as those of(a).

obtained response function indicated by the (black) thick solid line is well reproduced by thesimulated one indicated by the (red) thin solid line for the whole energy region including
the region between the full-energy peak and the Compton edge. Each response function
includes the experimental response function for the natural background γ rays indicated bythe (green) dotted line (the same spectrum shown in Fig. 3.11(b)). The intensity of thenatural background γ rays is fixed to reproduce the response function above 680 keV. The(blue) dotted line shows the response function simulated with GEANT4 taking the energydependent energy resolution, threshold of the electric circuit, and effect of the insufficientcharge collection discussed above into account. Figure 3.17(b) shows the comparison for thecase of the standard source 60 Co. There are two simulated response functions for 1173- and1333-keV γ rays indicated by (blue) dotted curves. The experimental energy spectrum isalso well reproduced by the combination of the simulated response functions and that for

62

CHAPTER 3. DATA ANALYSIS

the natural background γ rays.
In the case of the simulation of the in-flight γ-ray measurement, the momentum andspacial distributions of the incident beam, the energy loss of the incident and the outgoingparticles in the secondary target, the position resolutions of PPACs, GRAPE, and hodoscope,and the resolution of the velocity are taken into account.
In order to extract yields of the transition γ-rays, the γ-ray energy spectrum is decomposed into the simulated response functions and reasonable background with intensity
parameters. The intensity parameters are deduced by means of maximum likelihood estimation. The Poisson distribution is adopted to the likelihood functions in order to be applicableto γ-ray energy spectra with low statistics, i.e.,L(y; ν) =NYy νi exp(−yi )
i

i=1

νi !

,

(3.43)

where L represents likelihood function, y = (y1 , y2 , . . . , yN ) experimental yields, andν = (ν1 , ν2 , . . . , νN ) simulated yields. The νi should be a sum of simulated response
functions and reasonable backgrounds. When we involve the an intensity parameter αj forthe j-th simulated response function fEj (x), the νi is written as
νi =

n
X

αfEj (xi ) + βi ,(3.44)

j=1

where xi is the mean energy of the i-th bin and βi is the background contribution at xi . As
the backgrounds taken into account for the decomposition, contributions from contaminant
γ-rays in accidental coincidences, such as natural background γ rays, and γ rays originatedfrom the decay of isomeric state of 12 Be, and an exponential background represented as
β1 exp(−β2 xi ). In the estimation the likelihood ratio λ = L(y; ν)/L(y; y) is maximized.
This is equivalent to minimizing the quantity
χ2 = −2 ln λ = −2 ln L(y; ν) + 2 ln L(y; y),(3.45)

which is asymptotically obeys a χ-square distribution.
As the simplest case, the decompositions of the γ-ray energy spectra for standard sources,137
Cs and 60 Co, are demonstrated in Fig. 3.17. Figure 3.17(a) and (b) shows the spectra
taken with 137 Cs and 60 Co, respectively. In each figure, the thick solid line shows a sum of the
response functions and backgrounds, the thin solid line response functions, and the dashedline a background γ-ray spectrum shown in Fig. 3.17(c). The background γ-ray spectrum
is obtained after the experiment without any standard sources, where many significant γlines are identified; from the decay of 40 K and the Pb group nuclei as natural background;from the decay of the Fe group nuclei produced during the experiment via the reaction on
the nuclei in the vacuum chamber and in the secondary target cell; from the decay of 24 Na3.7. ANGULAR DISTRIBUTION AND DWBA ANALYSIS

63

1.4

efficiency (%)

1.2
1
0.8
0.6
0.4
0.2
00

200 400 600 800 1000 1200 1400
Eγ (keV)Figure 3.18: The comparison between the experimental and simulated efficienciesof the GRAPE. The open circles show the efficiencies obtained from the experimental
data with the statistical errors. The details for the experimental data is described in
section 3.5.1. The (red) solid curve shows the simulated efficiency curves. They are
consistent within 10% errors.

in NaI(Tl) detector (not used in the present study) irradiated during the experiment. The
exponential background, which comes usually from unspecified origin, is not needed for thedecompositions. Comparing the γ-ray emission numbers of the sources in the literature andin the simulation, there is a 10% deviation between them as shown in Fig. 3.18. The extrapolation of the efficiency curve to the higher energy is valid based on the fact that thetrend of the efficiency is well reproduced by the simulation for the whole energy region up to
1.4 MeV and the shape of the simulated response functions reproduce the experimental one
well. Hence, the response function obtained from the simulations reasonably reproduce theexperimental data of GRAPE and it enable estimation of the γ-ray yields by the decomposition of the experimental spectra into simulated response functions within 10% systematic
uncertainty.

3.7

Angular Distribution and DWBA Analysis

The angular distribution of outgoing particles is essential for deducing angular momentum.
The distribution is obtained by applying the decomposition analysis again. In this procedure,
the spectra, which will be decomposed, are obtained under a restriction on the scattering

64

CHAPTER 3. DATA ANALYSIS

angle of the outgoing particles. After the decomposition of the spectra, deduced strengths of
the response functions should indicate products of the differential cross section and the solid
angle for the corresponding excited state. This method has the merit that the statisticalerror can be small even if the statistics under the total energy peak are small since the thetotal number of events in the spectrum is still large. The decomposition for each angle isperformed through the same procedure as described in Sec. 3.6. The intensity of the γ rayfrom the j-th excited state, which is proportional to the population of the j-th excited state,at the scattering angle θ is obtained as below,
νi (θ) =nXαj (θ)fEj (xi ) + βi (θ),(3.46)

j=1

where the αj (θ) is the the notation is same as Eq. (3.44). The fEj is the same response
function for the de-excitation γ ray of the j-th excited state used in the decomposition to
extract the γ ray yields.
In order to determine the transferred angular momentum in the reaction, a distorted
wave Born approximation is applied to the analysis of the angular distribution. For thiscalculation, we used the phenomenological optical potentials deduced from the elastic scattering on the stable nuclei, since the elastic scattering data on 12 Be and 13 B has not been
available. The optical potentials of entrance and exit channels are referred to Ref. [60] andRef. [61], respectively, and their shape is assumed to be a Woods-Saxon shape:f (x) =V
,
1 + exp x−Ra(3.47)

where V , R, a is the depth, radius and diffuseness parameter. The R is calculated as
R = r · A1/3 by using the reduced radius r. These potentials were obtained via α elastic
scattering and 3 He elastic scattering on 12 C in the intermediate-energy range. Table 3.4 lists
the optical potentials used in the DWBA calculations. The calculations are performed byusing the computational code DWUCK5. The form factor was calculated from the singleparticle wave function in a Woods-Saxon potential in which the potential depth was tuned
to reproduce the binding energy of the transferred nucleon to the excited state. In Fig. 3.19,
the calculated angular distributions are shown. In each figure, the dashed curve indicatesthe original angular distribution and the solid curve indicates the one calculated from theoriginal one with consideration of the angular resolution for the scattering angle. Figure (a),(b) and (c) is DWBA predictions for the single-particle states, of which spectroscopic factors
are set to unity, at 4.83-MeV excitation energy with the transferred angular momenta of∆L = 0, 1, 2, that is, by transferring a proton onto the s1/2 , p1/2 and d5/2 orbitals on 12 Be
nucleus, respectively.

3.8. SINGLE FOLDING POTENTIAL

entrance
exit

V
(MeV)
77.25
69.36

rV
av
W
(fm)
(fm) (MeV)
1.2765 0.801
13.97
1.2526 0.7598 17.17

65

rW
aW
(fm)
(fm)
1.8846 0.426
1.7139 0.6738

12
12

C+α
C + 3 He

Table 3.4: Global optical model parameters used in DWBA calculation. The parameters are calculated from the formula including the energy dependence deduced in Ref.
[60, 61].

10-1

10-2
0

5

10

15

θCM (deg)(a) ∆L = 0

20

25

1

dσ
(mb/sr)dΩCM1dσ(mb/sr)dΩCMdσ(mb/sr)
dΩCM

1

10-1

10-2
0

5

10

15

θCM (deg)(b) ∆L = 1

20

25

10-1

10-2
0

5

10

15

20

25

θCM (deg)(c) ∆L = 2

Figure 3.19: Three DWBA predictions with ∆L = 0, 1, 2 in the center-of-mass frame.
The solid lines show the angular distributions used for the analysis, which are obtainedby folding the predicted distributions indicated dashed lines with angular resolution.

3.8

Single folding potential

In order to deduce the spectroscopic strength and to estimate its potential dependence,
another optical potential was obtained in the framework of the folding model [62]. Since theinelastic scattering data of α particle on 12 Be is also obtained in the present experiment, theoptical potential is obtained by optimize the parameters so as to reproduce the differential
cross section of the excitation to the first 2+ state in 12 Be. The real part of the folding
optical potential is obtained by folding the nucleon-nucleon interaction by the ground state
density distribution in the projectile and target nucleus.
A simple approach for this model, known as single folding model, is to perform the inte-

66

CHAPTER 3. DATA ANALYSIS

gration for one nucleus. In the present case, the integration for the α particle is assumed to
be already done. Furthermore, the density dependence of the effective α-nucleon interactionin the target nuclei is taken into account according to the recipe proposed in Ref. [63]. Thenthe effective α-nucleon interaction is obtained as follows,v(s, ρ(r ′ )) = −(v + iw) exp(−s2 /t2 )(1 − α {ρ(r ′ )}

2/3

),

(3.48)

where s = |r − r ′| is the distance between the center of mass of the α particle and a target
nucleon and ρ(r ′ ) is the ground state density of the target nucleus, such as 12 Be. The
parameters of v, t, α have been determined [63] by analyzing the experimental deferential
cross section of alpha elastic scattering on 58 Ni at 240 MeV incident energy, and they are
36.15 MeV, 1.88 fm, and 1.9 fm2 , respectively. The parameter w, which is the imaginary
part of the effective interaction, is determined by analysing the inelastic scattering on 12 Be
measured in the present experiment, as described below.
The single folding optical potential is obtained by folding the effective α-nucleon interaction by the ground state density of the 12 Be nucleus as follows,Z
U(r) = v(s, ρ(r ′ )) · ρ(r ′ )dr ′,(3.49)
where ρ(r ′ ) is the ground state density same as that in Eq. (3.48).
The ground state density of the 12 Be nucleus is obtained from a relativistic mean-fieldcalculation by using the computational code TIMORA. The calculated density is shown in
Fig. 3.20(a). Figure 3.20(b) shows the real part of the obtained optical potential together
with the approximate Woods-Saxon shape optical potential (Eq. (3.47)). The (black) solidline shows the single folding potential calculated using Eq. (3.49) and the (red) dashedline shows the approximate Woods-Saxon potential fitted to the single folding one. The
depth (V ), reduced radius (rv ), and diffuseness (av ) of Woods-Saxon shape are -72.57 MeV,
1.088 fm, and 0.9351 fm, respectively, obtained by fitting to the calculated single folding
potential. Although there are small discrepancies at the inner and surface region between two
potentials, since they are almost the same as the general trend the Woods-Saxon potential isa good approximation to the single folding potential. The resultant Woods-Saxon potential
is utilized for the DWBA calculation.
The imaginary part of the single folding potential is deduced so as to reproduce thedifferential cross section of the 2.1-MeV state in the 12 Be nucleus excited by inelastic scattering. The shape of the imaginary part is assumed to be the same as that of the real partbased on Eq. (3.48), i.e., the reduced radius r and diffuseness a is common for the real andimaginary part. A computational code, DWUCK4, is employed for the DWBA calculation.
The calculated differential cross section is fitted to the experimental data with two parameters, the depth W of the imaginary part and the normalization of the intensity, so as tominimize the χ2 value.

3.8. SINGLE FOLDING POTENTIAL

0.2

67

70
60

0.14

50

depth (MeV)

0.16
-3

density (fm )

0.18

0.12
0.1
0.08
0.06

40
30
20

0.04
10

0.02
00

2

4

6

8

10

12

R (fm)(a) Calculated density distribution of 12 Be

00

2

4

6

8

10

12

R (fm)(b) Single-folding and Woods-Saxon potential

Figure 3.20: (a) Calculated density distribution of 12 Be by using TIMORA computational code without further optimizing. (b) The real parts of single folding opticalpotential and approximate Woods-Saxon potential. The (black) solid line shows the
single folding potential and the (red) dotted line shows the approximate Woods-Saxon
potential. See the text for the detail.

Table 3.5 lists the parameters of the approximate Woods-Saxon optical potential deduced
from the single-folding model. The first raw lists the parameters for the real and imaginary
part of the optical potential for the entrance channel of 12 Be+4 He. The parameters of r anda for the exit channel are taken from the entrance channel by assuming the ground state
density is same as that of 12 Be with R(13 B)/R(12 Be) = (13/12)1/3 . The depths of the realand imaginary part for the exit channel are varied to estimate the statistical error of thespectroscopic factor. In the table 3.5, the best-fit parameters are listed. See Sec. 4.3.2 forthe detail.

68

CHAPTER 3. DATA ANALYSIS

dσ (mb/sr)
dΩcm

102

10

1

0

5

10

15

20

25

θcm (deg)
Figure 3.21: The differential cross section of the inelastic scattering on 12 Be exciting
2+
1 state. The open circle shows the experimental data obtained by decompositionof the γ-ray energy spectrum measured in coincidence with 12 Be. The (red) dottedline shows the DWBA prediction for ∆L = 0 with the single folding potential. The
imaginary part of the potential is optimized so as to represent the experimental data.
The details are in the text.

3.8. SINGLE FOLDING POTENTIAL

124

Be+ HeB+3 H

13V (MeV)-72.57-113.6rV (fm) aV (fm)
1.09
0.94
1.09
0.94

W (MeV)-42.69-32.6

rW (fm)
1.09
1.09

69

aW (fm)0.94
0.94

Table 3.5: Optical potential parameters obtained in the frame work of the single
folding model. First and second rows show the potential parameters for the entrance
and exit channels, respectively. The reduced radius and diffuseness are common for thetwo channels. The depth of the imaginary part for the entrance channel is adjusted so
as to reproduce the inelastic scattering data. The depths for the exit channel is thebest-fit value. See the text for the detail.

70

CHAPTER 3. DATA ANALYSIS

Chapter 4
Experimental Results
This chapter descries the experimental results concerning the excited states of 13 B. In section 4.1, the energy spectrum of 13 B is obtained by applying the window of the atomic number
and the mass number on the reaction products. We have found several peaks corresponding
to the de-excitation gamma rays from 13 B. In order to obtain the transition strength of each
excited state to the ground, the energy spectrum is decomposed with response functions
simulated with GEANT4 and experimentally obtained backgrounds (Sec. 4.2). The angular
distributions of the differential cross sections are extracted as the yield of γ rays obtained inevery 0.5 degree angular regions to which 13 B is scattered (Sec. 4.3). The distributions areanalyzed with DWBA, and the spin information and the spectroscopic strength are deducedin Sec. 4.3.1 and Sec. 4.3.2, respectively.

4.1

Energy Spectrum after Doppler Correction

The γ-ray energy spectrum of 13 B via the proton stripping reaction is obtained. The windowsof the atomic number Z and that of the mass number A are set as 4.5 ≤ Z ≤ 5.5 and12.5 ≤ A ≤ 13.5, respectively. The Doppler shift is corrected on an event-by-event basis byusing the γ-ray emission angle to the scattered 13 B deduced from the positions measured
with PPACs, hodoscope and GRAPE, and the velocity of the scattered 13 B deduced fromthe timings measured with F2PL, F3PL and hodoscope. In order to select the two-bodyreaction, the difference between the volocities of the incident and outgoing particles arelimitted by setting the window on the velocity difference spectrum. We have found threepeaks at around 3.7, 4.1 and 4.8 MeV, corresponding to the excitation energy (3.68, 3.73),
4.13, and 4.83 MeV. A doublet of 3.68- and 3.73-MeV γ rays has not been resolved by thepresent energy resolution of GRAPE. There is very few events above 5 MeV, which represents
that no γ line having more than 5 MeV energy exists, since the neutron separation energy of
13
B is 4.88 MeV. At the low energy region around 1–1.5 MeV, there seems to be relatively
wide peak, which is found also in energy spectra for the other boron isotopes such as 10,11,12 B

71

72

CHAPTER 4. EXPERIMENTAL RESULTS

exp.

100

background4829

counts/10keV

80

4131
3681+3713
3483+3535

60

40

20

0

1000

2000

3000
Eγ (keV)

4000

5000

6000

Figure 4.1: Doppler corrected γ-ray spectrum in coincidence with 13 B particles. The
spectrum is decomposed with the sum of response functions of the GRAPE for thede-excitation and background γ rays. The resulting response functions are shown withhatched areas. The response functions for two doublets around 3.5 and 3.7 MeV aresummed. The response function for the background includes the natural backgroundand the decay of the 12 Be isomer (see the text).

shown in Fig. 4.2. The wide peak, therefore, seems to be background γ rays in coincidencewith the beam particles although the origin has not been unspecified.

4.2

Cross Sections for Excited States in 13B

The cross sections for the excited states in 13 B are deduced by the decomposition analysis
described in section 3.6. The γ-ray spectrum shown in Fig. 4.1 is decomposed into the
response functions and backgrounds. The histogram shows the experimental spectrum. The
simulated response functions for the de-excitation γ rays and the response function for the
background γ rays are shown by the color and shade in the legend. The decomposition is
performed in the energy region above 700 keV since there is no significant peak in the region
below 700 keV. From the decomposition of the experimental spectrum, the number of theemitted γ rays are obtained. The number of the γ rays corresponds to the the number of4.2. CROSS SECTIONS FOR EXCITED STATES IN 13 B

350

10

B

counts / 10keV (a.u.)

300
250
200
150
100
50
0
0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Eγ (keV)

11

B

counts / 10keV (a.u.)

1200
1000
800
600
400
200
0
0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Eγ (keV)

1600

12

B

counts / 10keV (a.u.)

1400
1200
1000
800
600
400
200
0
0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Eγ (keV)Figure 4.2: Doppler corrected γ-ray spectrum in coincidence with 10,11,12 B particles.
The known γ lines are observed.

73

74

CHAPTER 4. EXPERIMENTAL RESULTS
Ex (keV)
3483
3535
3681
3713
4131
4829

σ(< 6◦ ) (mb)0.16 ± 0.04
0.17 ± 0.04
0.65 ± 0.06
0.60 ± 0.06
0.43 ± 0.04
0.88 ± 0.03

Table 4.1: List of the cross sections for the excited states in 13 B via the proton transfer
reaction on 12 Be. The cross sections are obtained as the integration up to ??? degrees
of scattered 13 B in laboratory frame. The errors are statistical only. The energies ofthe excited states are values from (see, e.g., Ref. [64] [NNDC]).

the 13 B excited to the corresponding energy since the γ-ray transitions between the excitedstates are small within the statistical errors based on the fact that the no γ-line between the
excited states is observed. The cross sections are calculated in combination with the numberof the γ rays, beam particles and target thickness.
The obtained cross sections are listed in Table 4.1, with the statistical errors evaluatedin the decomposition. The cross sections are obtained by the angular integration of 13 B up
to the scattering angle of 6 degrees in the laboratory system. The systematic error for theabsolute magnitude due to the geometrical condition of the GRAPE is estimated to be 10%
as described in Sec. 3.6. The two doublets at 3.5 and 3.7 MeV could not be resolved within
the present energy resolution. The strengths in the doublet have strong correlation and more
systematic errors. In the present study of the proton transfer reaction, the population of
4.83 MeV excited state is strongly populated, which may indicate the 4.83 MeV excited stateis of a proton single particle nature.

4.3

Angular Distributions for Excited States in 13B

The angular differential cross sections for each excited state is extracted via the decomposition analysis of the energy spectra for every 0.5-degree angular region up to 6 degrees inlaboratory frame, as described in Sec. 3.7. The angular acceptance of the hodoscope is almost 100% for every angular region so that the correction is not necessary. Figure 4.3 shows
the deduced differential cross sections of each excited state converted to the ones in thecenter-of-mass frame. The errors are statistical only. The angular differential cross sectionsof the 4.83-MeV excited state shown in Fig. 4.3(f) has the forward peak, which indicates thetransfer angular momentum is zero. In Sec. 4.3.1, the spin and parity of each excited stateis assigned based on the DWBA analysis of the differential cross section. Section 4.3.2 describes the spectroscopic strength and its statistical error deduced from the DWBA analysis

4.3. ANGULAR DISTRIBUTIONS FOR EXCITED STATES IN 13 B
Eex
(keV)
3483
3535
3681
3713
4131
4829

reduced χ2 for ∆L =
0
1
2
1.88 2.31
2.30
1.31 1.80
1.79
2.01 5.38
4.91
2.54 2.74
2.44
3.99 2.55
2.26
0.68 46.88 43.7575Jπ(1/2, 3/2, 5/2)+(1/2, 3/2)−(1/2, 3/2)−
1/2+

Table 4.2: List of the reduced χ2 obtained by fitting the each DWBA prediction using
the single folding model to the experimental data and the deduced spins and parities(J π s). The first column lists the excitation energy and the second, third and forthcolumn list the reduced χ2 for the transferred angular momentum ∆L = 0, 1, and 2,
respectively. The last column lists the spin and parity assigned only to the 4.83-MeV
excited state and the candidate given to the other states with parenthesis.

using the single folding potential described in section 3.8.

4.3.1

Spin and Parity Assignment

The DWBA calculations are performed with the transferred angular momenta (∆L) of 0, 1,
and 2. The 13 B has five protons and eight neutrons while the core of 12 Be has for protons.
In the present reaction of proton transfer, the low-lying states have a configuration of 12 Be⊗ (πpsd)1 so that the last proton is expected to be transferred to one of the orbits in psd
shell with ∆L = 0, 1, 2, i.e. to the p1/2 (∆L = 1), s1/2 (∆L = 0), and d5/2 (∆L = 2). The
strength of each predictions is normalized be minimize the χ2 value.
Figure 4.4 shows the differential cross section and the DWBA prediction with the global
optical potential for the 4.83-MeV excited state. The experimental data are reproduced bythe ∆L = 0 DWBA prediction very well. Since the spin and parity of the core nucleus 12 Be
is 0+ , the spin and parity of 1/2+ is assigned to the 4.83-MeV excited state for the firsttime.
The DWBA predictions for the 4.83-MeV excited state using the single folding potential
is shown in Figure 4.3(f). The potential parameter set is deduced from the inelastic scattering
data of 12 Be as described in section 3.8. The experimental data is well reproduced by the
DWBA calculation again. The spectroscopic strength is 0.20 ± 0.02(stat.) ± 0.1(syst.) as
described in the next section. The other excited states are analyzed using the single folding
potential with the best-fit parameters. The DWBA predictions for all the excited states areshown in Fig. 4.3. The minimized χ2 values are listed in Table 4.2. The second, third, andforth columns list the χ2 values for each excited state with the ∆L =0, 1, and 2, respectively.

10

10

10

1

1

1

dσ (mb/sr)dΩdσ (mb/sr)dΩ

CHAPTER 4. EXPERIMENTAL RESULTS

dσ (mb/sr)
dΩ

76

10-1

10-1

10-1

10-2
0

5

10

15

20

25

10-2
0

5

θcm (deg)

10

15

20

25

10-2
0

θcm (deg)(a) 3.483 MeV(b) 3.535 MeV

1
dσ (mb/sr)dΩ

1dσ (mb/sr)dΩ

1dσ (mb/sr)
dΩ

10

10-1

10-1

10-1

10-2
0

10-2
0

10-2
0

15

20

θcm (deg)(d) 3.713 MeV

25

5

10

15

15

20

25

(c) 3.681 MeV

10

10

10

θcm (deg)

10

5

5

20

θcm (deg)(e) 4.131 MeV

25

5

10

15

20

25

θcm (deg)(f) 4.829 MeV

Figure 4.3: Differential cross section of each excited state in 13 B. The errors are
statistical only. The (red) solid, (green) dashed, and (blue) dotted line shows the
DWBA prediction using the single folding optical potential with the transferred angularmomentum ∆L = 0, 1, and 2, respectively. See the text for the detail.

4.3. ANGULAR DISTRIBUTIONS FOR EXCITED STATES IN 13 B

d σ (mb/sr)d Ω CM

10

1

10-1

0

5

10
15
θcm (deg)20

25

Figure 4.4: The angular distribution of 13 B for the 4.83-MeV excitation via the protontransfer reaction 4 He(12 Be,13 B∗ ). The open circle shows the experimentally obtained
angular distribution and the (red) solid, (green) dashed, and (blue) dotted curves show
the DWBA predictions with ∆L = 0, 1, 2, respectively, with global optical potentials.
The DWBA predictions are folded by the angular resolution measured in the present
experiment.

77

78

CHAPTER 4. EXPERIMENTAL RESULTS

Figure 4.3(c) shows the differential cross section and the DWBA prediction for the
3.68-MeV state. The experimental data seem to be reproduced by the ∆ = 0 DWBA
calculation. However, the finite energy resolution of the GRAPE, the 3.68-MeV exited state
is not discriminated well from the 3.71-MeV excited state. The spin and parity of 1/2+ isa candidate, but the other spins are not rejected in this experiment. The candidates are
obtained as described below.
The DWBA analysis for the other states cannot determine due to the statistical errors.
However, the possible assignments of the spins of the other several excited states are obtained,
where the parities known from the previous experiment in Ref. [43]. In the reference, theparities were determined from the transferred angular momentum deduced from the angulardifferential cross sections in the two neutron transfer reaction. The 3.54-, 3.71-, and 4.13MeV excited states have negative parities and the 3.48- and 3.68-MeV excited states havepositive parities. Since the 4.13-MeV excited state has negative parity, the transferredangular momentum ∆L = 1 satisfies parity conservation. Then the candidates for the 4.13MeV excited state are limited to 1/2− and 3/2− . The 3.73-MeV excited state has also thenegative parity, then the candidates are 1/2− and 3/2− . The 3.68-MeV excited state has
positive parity. Then the candidates are 1/2+ , 3/2+ and 5/2+ assuming ∆L = 0 or ∆L = 2.
The 3.48- and 3.53-MeV excited states may not be of a proton-single-particle nature since
the populations of the 3.48- and 3.53-MeV excited states are very small. The spins forthese states cannot be, therefore, not suggested by the present study. The possible spin
assignments given by the present experiment are listed in the last column of Table 4.2.

4.3.2

Spectroscopic Strength

In the previous section, the experimental angular distribution is compared with the DWBA
predictions obtained from the phenomenological optical potentials and the transferred angular momenta are deduced. In this section, the result of the spectroscopic strength for each
excited state is described.
The spectroscopic strength has a strong dependence in a choice of optical potentials. Fora qualitative evaluation of the spectroscopic strength, one needs to estimate how much thespectroscopic strength depends on the optical potentials. For this purpose, in addition to thephenomenological optical potential, a set of single-folding optical potentials for the entrance
channel 12 Be+4 He and the exit channel 13 B+3 H are used. The single-folding potential ismade based on the density-dependent effective interaction, which is described in Sec. 3.8. The
radii of the real and imaginary part are the same values for the entrance channel. The density
distribution of 13 B is assumed to be same as one of 12 Be with R(13 B)/R(12 Be) = (13/12)1/3 .
All the diffusenesses are same. The depth of the real part for the entrance channel isobtained from the single-folding model. The depth of the imaginary part is determined so
as to represent the angular differential cross section data of the inelastic scattering on 12 Be,

4.3. ANGULAR DISTRIBUTIONS FOR EXCITED STATES IN 13 B

79

-30

1

-34

∆ χ2 =W (MeV)

-32

-36

-38
-118

-116

-114
-112
V (MeV)

-110

-108

Figure 4.5: The contour plot of χ2 distribution. Horizontal and vertical axes indicate
the depth of the real and imaginary part for the exit channel. The cross indicates theminimum point of the χ2 value. The closest contour corresponds to the curve where∆χ2 = 1.
12
populating the 2+
Be. The depths of the real and imaginary part for the exit channel is
1 insearched in order to estimate the statistical error for the spectroscopic strength. The search is
performed in a energy range from -108.0 to -117.9 MeV and from -29.0 to -38.9 with 0.1 MeV
step for the real and imaginary part, respectively. Figure 4.5 shows the χ2 map as a functionof the potential depths for the exit channel. The horizontal and vertical axes indicate thereal (V) and imaginary (W) potential depths of the exit channel, respectively. The χ2minimum is found at (V, W ) = (−113.6, −32.6) in units of MeV. Using these potentials
the spectroscopic strength is determined to be 0.20. Its error is estimated to be 0.02 bycomparing with the spectroscopic strength obtained using the potential parameter set at∆χ2 = 1 (shown in Fig. 4.5 with a red thick contour). The systematic error is determined to
be 50% by comparing with the spectroscopic strength of 0.10 obtained by using the global
optical potential. The deduced spectroscopic strength for the 4.83-MeV excited state is
0.20 ± 0.02(stat.) ± 0.1(stat.).

80

CHAPTER 4. EXPERIMENTAL RESULTS

Chapter 5
Discussion
The γ transitions from the 4.83 MeV excited state and other excited states are observed in
the 4 He(12 Be,13 B) reaction, and the 4.83 MeV excited state is found to have the spin andparity of 1/2+ state. In this chapter, the properties of the excited states, in particular the
proton intruder state, and the structure of excited 13 B are discussed. In section 5.1, thepopulations are compared with the previous experimental studies. In section 5.2 and 5.3,
the theoretical interpretations of the proton intruder state at 4.83 MeV are discussed. Ashell model calculation is demonstrated with the interaction which represent the intruder
state in 11,12 Be. The calculation in psd model space with maximum 3~ω excitation cannot
reproduce the experimentally obtained proton intruder state (section 5.2). In Sec. 5.3, theeffect of a deformed 12 Be core is studied. In this case the proton intruder state is simply
expressed to have a configuration of (πsd)1 ⊗12 Be. The Nilsson orbital of [220 21 ] provides a
possible configuration and explains the low excitation energy of the proton intruder state.

5.1

Comparison with the Previous Works

In this section, the previous works are summarized and the properties of the excited stateof 13 B are characterized based on the reaction selectivity and the present result.
The previous works which are compared with the present study are listed below.
(a). Two neutron transfer reaction on 11 B [43] – expected to populate two neutron particle
state.
(b). Neutron knockout reaction on 14 B [44] – expected to populate one neutron hole state.
(c). β decay of 14 Be followed by neutron emission [46] – expected to populate one proton
particle one neutron hole state.
(d). Multi-nucleon transfer reaction of 16 O [50] – expected to populate one or more proton
excited state

81

82

CHAPTER 5. DISCUSSION
Ex
(MeV)
3.48
3.53
3.68
3.71
4.13
4.83

present
populations
Jπ
0.19±0.05
0.20±0.05
0.74±0.07 (1/2,3/2,5/2)+
0.68±0.07(1/2,3/2)−
0.49±0.04(1/2,3/2)−
1
1/2+

(a)

(b)
(c)
populations
0.06 0.60±0.14
0.19
1
0.38
1
0.25
1
0.04±0.04
0.03

(d)
Jπ
(3/2+ )
(5/2+ )

1

(1/2− )

Table 5.1: List of the relative populations of the excited states. In the relative
populations, intensities are normalized by the most intense population. The errors are
statistical only. In the last column, the spins and parities which have been assigned orsuggested (with the curry bracket) in the present and previous studies are listed. See
the text for the meaning of the label (a), (b), (c) and (d). For the reaction (b) the
relative intensity is at θlab = 10◦ and for the reaction (d) at θc.m. = 5.4◦ .

The relative populations of the present and these reactions are listed in Table 4.1.
The 4.83 MeV excited state is strongly populated in the present reaction and reaction(d) while is is hardly populated in the other reactions. The present reaction and reaction(d) are expected to populate proton excited state, and the present reaction, in particular, is
expected to populate proton single particle states. The reaction (a), (b) and (c) are expected
to populate the excited states related to the neutron excitation. The proton single particle
nature of the 4.83 MeV excited state is indicated by these comparison. The spin and parityof this state has been assigned to be 1/2+ by the DWBA analysis of the angular differential
cross sections. The spin and parity of 1/2− was suggested to this state by the reaction (d)in Ref. [50], which is different from the assignment of 1/2+ by the present study. In Ref.
[50] discussed the spin and parity based on the reaction mechanism and the ground state
configuration of 16 O. However, the spin and parity was not uniquely determined.
The 4.13 MeV excited state is populated most strongly in reaction (a) and somewhat
strongly in the present reaction. Concerning the selectivity of reaction (a), the 4.13 MeV
excited state is a two neutron excited state. The transferred angular momentum of two
neutrons is reported to be ∆L = 2 in the reaction (a), and the state has negative parity, suggesting the possible spins of 1/2, 3/2, 5/2, 7/2. To form negative parity, two neutrons shouldbe transferred to the same shell such as (νp)2 or (νsd)2 . Since the two neutrons transferred
to p-shell is expected to form the ground state, the 4.13-MeV excited state may have a
configuration of (νsd)2 ⊗11 B. In the present reaction the proton should be transferred to thep-shell in order to produce negative parity, forming the configuration of (πp)1 ⊗12 Be(0+).
Concerning the existence of the neutron 2~ω configuration in the ground state of the 12 Be

5.2. COMPARISONS WITH SHELL MODEL CALCULATION

83

nucleus, the proton transfer to the p3/2 orbit allows the configuration of (νsd)2 ⊗11 B forming
the spin and parity of 3/2− . Based on this discussion, the 3/2− assignment may be more
likely although the 1/2− assignment can not be excluded.
The 3.71 MeV excited state is reported to have negative parity via the reaction (a). In
the present reaction the transferred angular momentum is non-zero according to the angular
distribution. Assuming the possible transferred momentum ∆L = 1, the possible spins are
1/2 and 3/2. The relatively large population shows this excited state is of the single particle
nature. The second lowest proton single particle state in 13 B is expected to have the spin
and parity of 1/2− , which is populated by the proton transferred to the πp1/2 orbital.

5.2

Comparisons with Shell Model Calculation

There are few theoretical predictions for the excited states in 13 B. In Ref. [44], the resultof a shell model calculation for 13 B has been shown, however, the calculation is performed
from the viewpoint of the neutron structure. In order to compare with the shell-model
predictions, a shell model calculation is performed focusing on the proton shell structurewith the calculation code, OXBASH [65]. The model space is psd-shell considering the
maximum excitation of 3(2)~ω for 13 B(12 Be). The interaction includes the effect of thetensor force and reproduces intruder states in the neutron-rich nuclei such as 11,12 Be [15].
The spectroscopic factor is calculated as the overlap between each excited state of 13 B andthe ground state of 12 Be. The predicted energy, spin, parity and spectroscopic factor of thestates in 13 B are shown in Fig. 5.1 with the experimental ones.
Focusing on the first six predicted excited states, the excitation energies seem to be predicted in 2 MeV energy difference. However, the shell model cannot predict the spectroscopic
strength.
The shell model calculation predicts four 1/2+ states below 10 MeV excitation energy.
Three lower states are predicted at 3.7, 6.5 and 9.1 MeV with spectroscopic strengths of
0.01, 0.03 and 0.003, respectively. These strengths are too small to explain the experimental
result. The fourth 1/2+ state with a spectroscopic strength of 0.30 is close to the experimental
spectroscopic strength; however its excitation energy is 9.5 MeV. These results of the shellmodel calculation may indicate that the theoretical approach based on the effect of the tensor
force does not explain the experimental result, or the model space consisting of the psd shelland the excitation of 3~ω are not large enough.

5.3

Property of 1/2+ Intruder State

A simple expression for the observed 1/2+ state with proton single particle nature is (πsd)1 ⊗12 Be.
Figure 5.2 shows the proton orbitals in the deformed potential of 13 B. The deformed poten-

84

CHAPTER 5. DISCUSSION

1/2+ 0.30

91/2+ 0.003

Ex (MeV)1087

1/2+ 0.03

6
5

1/2

4

++

3

4829
4131
3713
3681
3535
3483

+

1/2- 0.87
3/2- 0.15
3/2+ 0.002
5/2+ 0.01
1/2+ 0.01

2
1
0

3/2-

exp.

3/2- 0.44

S.M.

Figure 5.1: The experimental result is shown at the left with each adopted excitationenergy, spin, and parity. The shell model predictions are shown at the right with eachpredicted spin, parity and spectroscopic strength. For the predictions, only four lower
1/2+ excited states and the lowest 1/2− , 3/2± , 5/2+ excited states are shown. See text
for the detail.

5.3. PROPERTY OF 1/2+ INTRUDER STATE

85

Figure 5.2: Proton orbitals in deformed potential of 13 B provided by deformed 12 Be
core, essentially same as Fig. 1.1. The [220]1/2 orbital indicated by red curve comes
down with large deformation parameter and [110]1/2 orbital (the lowest one) comes
down too.

tial is provided by the deformed core of 12 Be. At the large deformation, the orbitals having
the spin and parity of 1/2+ in the sd-shell are represented by an asymptotic quantum number [220]1/2, [211]1/2, and [202]1/2. In case of a spherical nucleus, the lowest 1/2+ state
has a configuration of s1/2 ⊗ 12 Be(g.s.), while the lowest one arises with a configuration of(π[220 12 ])1 to the 12 Be(g.s.) core in case of a deformed nucleus.
As described above, at large deformation, the π[220 12 ] orbital gains energy. The energy
gain of the deformed system to the spherical system is estimated. The energy gain of theorbital itself is estimated to be 11.8 MeV from the Nilsson diagram where ~ωdef ormed ∼~ωspherical ∼ 41Ȧ−1/3 is ∼17.4 MeV at δ ∼ 0.5 (recently calculated to be 0.7 [66]). Theenergy difference estimated from the semi-empirical mass formula consists of the differenceof the surface energy and the Coulomb energy, since the volume should be conserved under
the change of the shape, and the symmetry and paring energies do not depend on the nuclear
shape. The Coulomb energy itself is much smaller than the orbital energy and surface energy
and is ignored in this estimation. The surface area of a spheroid is described as below,pp
S = 2π[b2 + ab sin−1 1 − (b/a)2 / 1 − (b/a)2 ],
(5.1)

86

CHAPTER 5. DISCUSSION

where a represents the length of the symmetry axis, b the one of the axis perpendicular tothe symmetry axis, and volume conservation requires the equation of ab2 = R03 , where R0
is the radius when the nucleus is spherical. The spherical surface energy is represented by−17.2A2/3 with the surface area S0 = 4πR02 in empirical mass formula. Then the energy
difference of the surface energy is estimated to be −17.2A2/3 (S/S0 − 1) ∼ −3.4 MeV for
A = 13 at δ ∼ 0.5. The total energy gain of 1/2+ orbital is calculated to be ∼8 MeV, which
explains the low excitation energy of 1/2+ state at 4.83 MeV. The experimentally obtained
spectroscopic factor can be compared with the spherical 2s1/2 component in the deformed[220 21 ] orbital, since the proton is transferred by an s-wave (∆L = 0), which is assumed
in the DWBA analysis. The spectroscopic factor of the spherical 2s1/2 component in the
deformed [220 12 ] orbital is S ∼ 0.3 at δ = 0.4 [18], which also explains the experimental
result of C 2 S = 0.20
Based on the discussion above, the 1/2+ intruder state at 4.83 MeV is a signature of adeformed field and indicates the importance of a deformation in this nuclei.
Recently the excited states of 13 B were studied with a method of antisymmetrized molecular dynamics (AMD) [66]. A proton intruder state having the spin and parity of 1/2+ werepredicted as the first 1/2+ state and found to be largely deformed (β ∼ 0.7) state. Although
the excitation energy of ∼8 MeV is larger than the experimental result the proton intruder
configuration and the deformation of the state explain the present result.

Chapter 6
Conclusion
The in-beam γ-ray spectroscopy of neutron-rich unstable nucleus 13 B was performed withthe proton transfer reaction on the 12 Be in order to investigate proton single-particle statesin 13 B nucleus. The new γ-ray detector array GRAPE was developed to perform the highresolution in-beam γ-ray spectroscopy. The typical energy resolution was 1.3% (FWHM) for
the Doppler-shift corrected γ rays emitted from the fast-moving (β ∼ 0.3) 12 Be. The corresponding position resolution was less than 10 mm (FWHM). The experiment was performed
at RIPS course in RIKEN. The secondary beam of 12 Be was produced and selected from thefragments of the 18 O beam particles bombarding the 9 Be target. The 12 Be beam bombarded
the thick liquid helium target of 143 mg/cm2 . The incident and outgoing particles were measured with two plastic scintillation counters and plastic scintillator hodoscope, respectively.
They were identified on an event-by-event basis by using T OF − ∆E and T OF − ∆E − E
method, respectively.
The γ decay of the 4.83-MeV excited state to the ground state was observed. The
spin and parity of 1/2+ has been uniquely assigned for this state by DWBA analysis of
the angular differential cross section for the first time. The spectroscopic strength has been
0.20 ± 0.02(stat.) ± 0.10(syst.). The other γ decays of the 3.68-, 3.73-, and 4.13-MeV excitedstates were also observed. The transferred angular momenta for these excited state cannot
be uniquely determined. The comparison among the present and previous works, however,gives the qualitative information and possible spin assignments. The 3/2− -assignment for
the 4.13-MeV excited state has been suggested based on the comparison with the present
reaction and two neutron transfer reaction, considering the configuration of this state. The
spin of 1/2, however, has been also a candidate for this state. The 1/2− -assignment forthe 3.73-MeV excited state has been suggested based on its negative parity, considering
the strong population and configuration of this state. The spin of 3/2, however, is also a
candidate for this state. The possible assignment of (1/2, 3/2, 5/2)+ has been given for the
3.68-MeV excited state.
The comparison with a shell model calculation using OXBASH has been performed.

87

88

CHAPTER 6. CONCLUSION

There are four 1/2+ states predicted, however, none of them can explain the observed 1/2+
state from the point of view of the excitation energies and/or spectroscopic strengths. The
effect of the deformed mean field was qualitatively examined considering a deformed potential
provided by the deformed core of 12 Be. The spin and parity of 1/2+ is characterized by
the asymptotic orbitals of [220]1/2, [211]1/2, and [202]1/2. The [220]1/2 orbital gains theenergy at the large deformation. Its energy is around 7 MeV calculated from a the changeof asymptotic single particle orbital energy and effect of the deformation in empirical mass
formula. The observed 1/2+ state at 4.83-MeV excitation energy is well represented by theconfiguration of (π[220]1/2)1⊗12 Be.
For the disappearance of the magic number, there are three theoretical interpretations,such as the tensor force, weakly bound system, and nuclear deformation. From the present
study, the importance of nuclear deformation is pointed out especially for the proton intruder
state in the neutron-rich nucleus 13 B.
We can also investigate the effect of the deformation in ‘island of inversion region. Thereis a well known deformed nucleus 32 Mg. Then 33 Al is expected to have a proton intruder
state like 13 B. In the 33 Al case, these will exist negative parity intruder state from pf -shell.
In this region, the effects of tensor force and weakly bound nature succeeds to represent theexperimental results. The effect of nuclear deformation in these region needs to be studied
in order to clarify the relationship among these effect and to expand our knowledge of thenuclear structure in unstable nuclear.

Acknowledgments
I would like to take this opportunity to place on record my sincere thanks to many people
without whom this thesis would not have been at all possible.
I would like to express my sincere gratitude to my thesis adviser Prof. S. Shimoura, for
his continuous encouragement and supports through my research career. He has given me
good opportunities to study the structure of unstable nuclei using proton transfer reactions
but also many experimental nuclear physics. I have learned from him not only knowledgeof nuclear physics but also attitudes to solve scientific problems. I am deeply grateful tomy supervisor Prof. T. Murakami for his encouragement and helpful discussion. I would
like to thank to Prof. K. Imai for his guidance to the field of nuclear physics and for his
encouragement and proofreading of this thesis.
I would address my thanks to all the member of R337n-1 collaboration: Prof. S. Shimoura, Dr. H. Iwasaki, Dr. M. Kurokawa, Dr. S. Michimasa, Dr. N. Aoi., Dr. H. Baba,
Mr. K. Demichi, Prof. Z. Elekes, Dr. T. Fukuchi, Dr. T. Gomi, Dr. S. Kanno, Prof. S. Kubono,Prof. K. Kurita, Mr. H. Hasegawa, Prof. E. Ideguchi, Prof. N. Iwasa, Mr. Y.U. Matsuyama,
Dr. K.L. Yurkewicz, Dr. T. Minemura, Prof. T. Motobayashi, Prof. T. Murakami, Dr. M. Notani, Prof. A. Odahara, Dr. A. Saito, Prof. H. Sakurai, Dr. E. Takeshita, Dr. S. Takeuchi,
Mr. M. Tamaki, Prof. T. Teranishi, Dr. Y. Yanagisawa, Dr. K. Yamada, and Prof. M. Ishihara for their great contributions and collaborations to complete this work, successfully. I
am also grateful to the staff member of RIKEN Ring Cyclotron for their operation of theaccelerator during the experiment. I would like to appreciate variable discussion for theoretical calculations with Prof. T. Suzuki and Prof. Y. Kanada-En’yo. I thank to Dr. S. Bishop
for the proofreading of the published paper and to Mr. D. Kahl for the proofreading of thisthesis.
I would like to thank Dr. Y. Fukao, Dr. K. Miwa, Mr. T. Nakura, Dr. H. Okada,Dr. S. Terashima, and Dr. M. Togawa. I spent my fruitful and enjoyable time with them in
my graduate school days in Kyoto. I also express my thanks to Prof. N. Saito, Prof. N. Nakamura, Prof. H. Funahashi, and Prof. M. Yosoi, for their guidance.
I spent much time in Center for Nuclear Study, University of Tokyo, in my graduate
school days. I would like to appreciate fruitful discussions and enjoyable talks, singing,playing softballs, and drinking with all the researchers and students in CNS. I would like to

89

90

CHAPTER 6. CONCLUSION

thank Prof. T. Kawabata, Dr. H. Baba, Dr. S. Michimasa, Dr. A. Saito, Dr. Y. Wakabayashi,
Dr. F. Kajihara, Dr. K. Nakanishi, Dr. M. Niikura, and Dr. T. Gunji for many fuitful discussion and advices. I am also grateful to secretaries of CNS, Ms. M. Hirano, Ms. I. Yamamoto,Ms. Y. Endo, Ms. Y. Kishi, Ms. K. Takeuchi, Ms. Y. Otsu, Ms. Y. Sohma, Ms. Y. Fujiwara,
Ms. M. Yamaguchi and Ms. N. Teranishi for their kind help for my research life.
I would like to express my grate thanks to my parents, brother, sister, grand mothers(one passed away in December, 2008) for their continuous encouragements, patience andsupports.
Finaly, I would like to address my heartfelt acknowledgment to Miwa for her cheeringand respect on my studies and life.

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Appendix A
Data Table of Angular Differential
Cross Section
θCM
dσ/dΩ
(deg.) (mb/sr)
1.36
6.98
3.16
7.07
5.15
4.84
7.17
2.83
9.20
2.28
11.2
1.77
13.3
1.38
15.3
0.793
17.4
0.529
19.4
0.282
21.6
0.131
23.6
0.108

error
(mb/sr)
1.13
0.63
0.42
0.29
0.22
0.18
0.14
0.105
0.078
0.059
0.040
0.032

Table A.1: Data table of angular differential cross section for 4.83-MeV excited state.

97

98APPENDIX A. DATA TABLE OF ANGULAR DIFFERENTIAL CROSS SECTION

Appendix B
Comparison between analog anddigital sum of GRAPE
We used the analog sum for deducing the γ-ray energy in this thesis. In this chapter, we
compare the properties of analog and digital sum. Before going into details, we remember thedefinition of the analog and digital sum. Those of a crystal are obtained using the equations
below,P
Ei
Ea = EΣ P i ′ ,(B.1)
i ǫi Ei
X
Ed =
Ei ,
(B.2)
i

where, Ea(d) is the analog (digital) sum, Ei is a measured energy in segment i, and ǫ′i is a gain
correction factor (see Sec. 3.5.1). In the analysis, the threshold at 80 keV has been set for
all the segments so as to align the thresholds among them. Although the energy deposition
below 80 keV is not included in the digital sum, it is included in the analog sum since the
add-back is performed before the digitization. The correction factor for EΣ in Eq. (B.1) is
not exactly correct, however, the effect is small, especially when the total energy depositionis large. Then the full-energy peak efficiency of the analog sum is larger than that of thedigital sum. Figure B.1(a) shows the comparison of the full-energy peak efficiencies between
the analog and digital sum. The data of the analog and digital sum are shown by (red) open
circle and (blue) crosses, respectively. The efficiencies of the analog sum is 20% larger thanthose of the digital sum.
On the other hand the energy resolution of the analog sum is worse than that of thedigital sum, since the noise of each segment is coherently added when the noise is in-phase.
Figure B.1(b) shows the comparison of the energy resolutions between the analog and digitalsum, which are shown by (red) open circles and (blue) crosses, respectively. The resolutionof the digital sum strongly depends on the γ-ray energy due to the deviation of the energy
calibration.

99

APPENDIX B. COMPARISON BETWEEN ANALOG AND DIGITAL SUM OF GRAPE
100

3
1.4

2.8
2.6
Resolution (keV)

efficiency (%)

1.2
1
0.8
0.6
0.4

2.4
2.2
2
1.8
1.6
1.4

0.2
0

1.2
200 400 600 800 1000 1200 1400
Eγ (keV)(a) efficiencies

10

200 400 600 800 1000 1200 1400
Energy (keV)(a) energy resolutions (1σ)

Figure B.1: Comparison between the analog and digital sum. The data of the analog
and digital sum are shown by (red) open circles and (blue) crosses, respectively. (a)Comparison of the full-energy peak efficiencies. (b) Comparison of the energy resolution(one standard deviation).

Appendix C
Tailing function of Full Energy Peakin Germanium Detector
A treatment of the γ-ray energy obtained by the simulation is described. Figure C.1(a) shows
the measured response function (black histogram) and fitted response functions taken with60
Co standard source. In the measured response function, the tail at the low energy side ofeach full-energy peak is observed. Such tails are not found in the original simulated response
function (blue lines), which causes an overestimation of the full-energy peak efficiencies since
the response functions are not fitted correctly. Therefore, we need to modify the simulated
response functions.
The modification has been performed by using a probability distribution described below.
(Ec − E0 )(E ≤ E0 ),(C.1)
P (E) ∝ exp αE0where E0 is a energy deposition of γ ray obtained by the simulation, Ec is the modified
energy deposition, and α is a constant determined by fitting the modified response function
to the experimental one. The Ec is randomly generated from the probability distribution
P (E) in the range from 0. to E0 . This modification is applied to a certain fraction ǫ of theevents. Figure C.1(b) shows a energy spectrum obtained by the modification of the energy
spectrum in Fig. C.1(a), where α = 2.0941 and ǫ = 5.900 × 102 are used. These parameters
are obtained by fitting a modified response function to the experimental one for the γ-ray
energy spectrum of 137 Cs standard source. The modified responses function reproduce theexperimental one for 60 Co standard source well.
The probability function is obtained assuming the insufficient collection of the charge ina germanium crystal. A probability that charge Q is collected at a given timing t, is assumed
to be an exponential function as:t
P (t) ∝ exp −
,
(C.2)
τ

101

APPENDIX C. TAILING FUNCTION OF FULL ENERGY PEAK IN GERMANIUM
102
DETECTOR

104
counts/1keV (a.u.)counts/1keV (a.u.)

104

3

10

102

1000

3

10

102

1100

1200

1300

1400

Eγ (keV)(a) before modification

1500

1000

1100

1200

1300

1400

1500

Eγ (keV)(b) after modification

Figure C.1: Demonstration of the treatment of the tailing. The energy spectrum forthe 60 Co is fitted with two types of simulated response functions (blue lines); before (a)and after (b) modification for the tailing. The green lines show the response function
for the background γ rays.

where τ is a time constant. An average time of the charge collection t̄ for the charge Q is
obtained as below,
Q
t̄ =
,
(C.3)
vQ
where vQ is an average velocity of the charge collection. The velocity is approximated to
be Q0 /TQ for the induced charge Q0 with an average collection time TQ for Q0 . Now, sincevQ is assumed to be constant for a given charge Q0 , the probability of the charge collection
obeys Eq. (3.28) by replacing t by t̄ = Q/vQ . Then we obtained the probability distributionof the collected charge Qc as below,
TQ Q0 − Qc
.
(C.4)
·
P (Q) ∝ exp −τQ0Since the charges Q0 and Qc is proportional to the energy E0 and Ec , respectively, we obtain
Eq. (3.33) by defining α ≡ TQ /τ . Eq.