Isoscalar and Isovector strengths of low-energy
dipole excitations in neutron-rich unstable 20O

Noritsugu Nakatsuka
Department of Physics, Kyoto University
A dissertation submitted in partial fulfillment of therequirements for the degree of Doctor of Science3

Abstract
The Isoscalar and Isovector strengths of low-energy dipole excitations on neutron-rich unstable nucleus 20 O has been investigated. Two excitation cross sections obtained from theinelastic scattering of the α particles, a dominant Isoscalar probe, and the Coulomb excitation using a gold target, a dominant Isovector probe were analyzed by the distorted-wave
Born approximation (DWBA) to extract the Isovector and Isoscalar dipole strengths independently. Two previously known 1− states with significant Isovector dipole strengths at−energies of 5.36(5) MeV (1−1 ) and 6.84(7) MeV (12 ) were also excited by the Isoscalar probe.
−
These two states were found to exhaust 2.70(32) % (1−1 ) and 0.67(12) % (12 ) of Isoscalardipole energy-weighted sum rule, respectively. Unexpectedly strong Isoscalar strength of 1−1
was observed. The diﬀerence in Isoscalar strengths indicated that they have diﬀerent underlying structures. This is the first study in unstable nuclei to reveal the Isospin characters of
low-energy dipole excitations.

5Contents
Chapter 1
1.1
1.2
1.3
1.4
1.5
1.6
1.7

Introduction
Dipole Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Low-energy Dipole Excitation . . . . . . . . . . . . . . . . . . . . . . .
Impact of the LED on Nuclear Physics . . . . . . . . . . . . . . . . . .
Present Experimental Data on the LEDs . . . . . . . . . . . . . . . . .
Isoscalar and Isovector strengths of the LEDs . . . . . . . . . . . . . . .
Present Theoretical Predictions on the LEDs . . . . . . . . . . . . . . .
The Isospin Structure of the LEDs on Unstable Nuclei . . . . . . . . .

13
13
14
18
19
19
21
25

Chapter 2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9

Experimental Apparatus
Radioactive Isotope Beam Factory (RIBF) at RIKEN Nishina Center .
In-beam γ-Ray Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . .
Production of Radioactive Isotope Beam . . . . . . . . . . . . . . . . .
Detector installed around and after the secondary reaction target . . .
Secondary Reaction Targets . . . . . . . . . . . . . . . . . . . . . . . .
Reaction Product Spectrometer: ZeroDegree . . . . . . . . . . . . . . .
Beam Line Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Detectors for γ Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Data Acquisition System . . . . . . . . . . . . . . . . . . . . . . . . . .

29
29
30
31
35
35
37
38
39
41

Chapter 3
3.1
3.2
3.3

Data Analysis
Calibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Analysis of the Beam Particle . . . . . . . . . . . . . . . . . . . . . . .
Analysis of the γ Rays . . . . . . . . . . . . . . . . . . . . . . . . . . .

43
43
50
51

Chapter 4
4.1

Distorted-Wave Born Approximation Analysis
Distorted-Wave Born Approximation Analysis . . . . . . . . . . . . . .

67
67

Chapter 5
5.1
5.2
5.3

Discussions
B(E2) ↑ Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B(E3) ↑ Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B(E1) ↑ Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79
79
80
80

Chapter 6

Conclusive Remarks

87

Chapter 7

Acknowledgements

89

A

Appendix: Extrapolating Optical Potentials

91

6

Contents
Bibliography97

7

List of Figures
1.1

1.2
1.3
1.4

1.5

1.6

1.7

1.8

1.9

1.10

2.1
2.2
2.3
2.4

A typical excitation spectrum of the photo-absorption reaction (forward) andthe particle (such as proton) scattering (backward). From the textbook by
M. N. Harakeh and A. van der Woude. . . . . . . . . . . . . . . . . . . . . .
A schematic view of the IVGDR. . . . . . . . . . . . . . . . . . . . . . . . .
The electric dipole excitation spectrum of 208 Pb. From A. Tamii et al. . . .
Experimentally measured low-energy dipole strengths in units of energyweighted sum rule fraction (EWSR (%)) plotted against mass number A.
Note that the data are plotted as Isovector dipole energy-weighted sum rule
fraction. From the review by D. Savran et al. . . . . . . . . . . . . . . . . .
The measured Isoscalar dipole strengths (top panel) in diﬀerential cross section, and Isovector dipole strengths (bottom panel) in B(E1) on 124 Sn. From
J. Endres et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The experimentally measured Isoscalar dipole strengths (left column) in
Isoscalar dipole energy-weighted sum rule (ISEWSR) fraction or crosssections, and Isovector dipole strengths (right column) in B(E1). From thereview by A. Bracco et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
An example of the predicted structure of the dipole excitation. The transition densities of the LED (top panel) and the typical GDR (bottom panel)predicted by the random-phase approximation (RPA). From N. Paar et al. .
An example of the comparison between the predictions and the measurement.
The Isovector dipole strength measured on 208 Pb (a), a predicted strength
using the microscopic quasiparticle-phonon (QPM) calculation (b), and another predicted strength using the relativistic time-blocking approximation(RTBA) calculation (c). From A. Tamii et al. . . . . . . . . . . . . . . . . .
A schematic velocity fields from diﬀerent collective modes. Neutron skinagainst the nuclear core (a), toroidal dipole (b), and compressional dipole(c). From A. Repko et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The previously measured decay γ-ray spectrum of 20 O from the Coulomb
excitation using a lead target. From E. Tryggestad et al. . . . . . . . . . . .
A map of the RIKEN Radio Isotope Beam Factory Facility. From Y. Yano.
A conceptual view of the in-beam γ-ray spectroscopy. . . . . . . . . . . . . .
A conceptual view of the RIKEN Nishina Center cyclotron complex. . . . .
The schematic view of the BigRIPS fragment separator and the ZeroDegree
Spectrometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13
14
14

20

21

22

23

24

25
26
29
31
32
33

8

List of Figures
2.5

2.6
2.7
2.8
2.9
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12

3.13
3.14

3.15

3.16
3.17
3.18

The measured beam profiles. Horizontal position (x) versus incident angle (x′ ) (top panel), vertical position (y) versus incident angle (y ′ ) (bottom
panel). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Detector setup around the secondary target. . . . . . . . . . . . . . . . . . .
The schematic view of the cryogenic system. Taken and modified to describe
the present configuration from H. Ryuto et al. . . . . . . . . . . . . . . . . .
Detectors installed at the ZeroDegree spectrometer. . . . . . . . . . . . . . .
Schematic view of signal processing. . . . . . . . . . . . . . . . . . . . . . . .
Definition of residual for the PPAC resolution estimation. . . . . . . . . . .
The measured residual distribution of a PPAC for the resolution estimation.
Definition of the scattering angle θ for the ZeroDegree spectrometer acceptance estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The measured acceptance of the ZeroDegree spectrometer. As a function of
θout (top panel), and θ (bottom panel). . . . . . . . . . . . . . . . . . . . . .
The measured liquid helium cell position. . . . . . . . . . . . . . . . . . . . .
The measured 20 O beam profile. Overlaid with the measured cell structures.
The particle-identification plot of the secondary beam. . . . . . . . . . . . .
The particle-identification plot of the reaction product after the secondary
target. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The LaBr3 :Ce energy spectrum from one detector module acquired with 88 Y
(top panel) and 60 Co (bottom panel) γ-ray checking source. . . . . . . . . .
The residuals of the LaBr3 :Ce detector as a function of input γ-ray energy.
From A. Giaz et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A view of the geometries used for the simulation generated by the Geant4 .
An example of walk eﬀect seen in the QTC electronics. Relative charge(corresponding to γ-ray energy) versus relative time to trigger (corresponding
to the time of flight). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The TOFγ distribution for the resolution estimation. . . . . . . . . . . . . .
The correlation plot of the γ-ray energy Eγ (horizontal) and its time of flight
TOFγ (vertical). From the 20 O+α (top panel), and 20 O+Au reaction (bottom
panel). The γ-ray energy Eγ is not Doppler-corrected. . . . . . . . . . . . .
The Doppler corrected γ-ray singles spectra, in black dots with statisticalerrors. The gray histograms overlaid are backgrounds from 24 O beam, lines
are the best-fit response functions. The fits are performed over 1900 keV toimprove the sensitivities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The γ-γ coincidence plots. The selected events where two γ rays are detected
simultaneously are plotted in order of the γ-ray energy Eγ1 > Eγ2 . . . . . .
The response functions generated by the Monte-Carlo simulation based ontheGeant4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Doppler corrected γ-ray coincidence spectra (Eγ1 ), where a γ ray (Eγ1 )is detected in coincidence with the γ ray from 2+
1 state (Eγ2 ), in black dotswith statistical errors. Lines are the best-fit response functions. . . . . . . .

34
35
37
39
42
44
45
46
48
49
49
50
51
53
54
55

57
58

59

61
62
65

66

9
4.1

4.2

4.3

4.4

5.1

5.2

5.3

A.1

The calculated angular distribution of scattered 20 O seen from the laboratory
system. Calculated by the DWBA code (solid line), angular-folded distribution by the detector and multiple Coulomb scattering (dashed line), andangular-folded distribution multiplied by the ZeroDegree acceptance (dasheddotted line). For the 2+ and 3− states. . . . . . . . . . . . . . . . . . . . . .
The calculated angular distribution of scattered 20 O seen from the laboratory
system. Calculated by the DWBA code (solid line), angular-folded distribution by the detector and multiple Coulomb scattering (dashed line), andangular-folded distribution multiplied by the ZeroDegree acceptance (dasheddotted line). For the 1− states. . . . . . . . . . . . . . . . . . . . . . . . . .
The calculated angular distribution of the decay γ ray. Seen from the restframe of the beam, and from the laboratory system (Lab. Frame). For the2+ , 3− states decay via the 2+
1 state. . . . . . . . . . . . . . . . . . . . . . .
The calculated angular distribution of the decay γ ray. Seen from the restframe of the beam, and from the laboratory system (Lab. Frame). For the
1− states decay via the 2+
1 state and the ground state. . . . . . . . . . . . .
The Isoscalar and Isovector dipole strengths in 20 O (right panel), in comparison with those in 16 O (left panel). The Isoscalar dipole strengths in unit of
ISD EWSR fraction (%) (top panel), and the Isovector dipole strengths in
B(E1)↑ (e2 fm2 ) (bottom panel). . . . . . . . . . . . . . . . . . . . . . . . . .
The transition density predicted below the neutron separation threshold energy. Calculated based on the RPA using SLy4 Skyrme interaction by T.
Inakura. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Isoscalar and Isovector dipole strengths measured in 40 Ca (left panel)and 48 Ca (right panel). The Isoscalar dipole strengths in unit of ISD EWSR
fraction (%) (top panel), and the Isovector dipole strengths in B(E1)↑ (e2 fm2 )(bottom panel). From D. Verya et al. . . . . . . . . . . . . . . . . . . . . . .

72

73

76

77

81

84

85

The optical potentials of 12 C, 16 O, 20 Ne, 24 Mg + 20 O systems and their
extrapolation to the α+20 O system (Extra. α). Real parts V (r)(top panel),and imaginary parts W (r) (bottom panel). Calculated from T. Furumoto et al. 93

11

List of Tables
The previous result of Isovector dipole strength in 20 O. From E. Tryggestad
et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

2.1
2.2
2.3
2.4

BigRIPS Magnet Settings. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
BigRIPS slit settings. The slits are most wide open at ± 120 mm. . . . . . .
Specifications of the liquid helium target cell. . . . . . . . . . . . . . . . . .
ZeroDegree spectrometer magnet settings. . . . . . . . . . . . . . . . . . . .

33
33
36
38

3.1

The acceptance of the ZeroDegree spectrometer parametrized by the sum of
two Fermi functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Walk correction parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Excited states and decay γ-ray energy, used to produce response functions. .
Measured Branching ratios (Br). . . . . . . . . . . . . . . . . . . . . . . . . .

47
56
64
65

1.1

3.2
3.3
3.4
4.1
4.2
4.3
4.4

5.1
5.2

A.1
A.2

A.3

The Woods-Saxon parameters and their extrapolation to the 20 O+α system.
Calculated from T. Furumoto et al. . . . . . . . . . . . . . . . . . . . . . . .
The accepted ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Measured cross sections in millibarns. . . . . . . . . . . . . . . . . . . . . . .
The Woods-Saxon Parameters of the background materials. Calculated fromT. Furumoto et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68
74
78
78

Estimated multipole strength. . . . . . . . . . . . . . . . . . . . . . . . . . .
Calculated Isoscalar dipole strength (ISD EWSR), and Isovector dipole
strength (B(Eλ)↑). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

The Woods-Saxon parameters of the global α potential. From N. Nolte et al.
10-range Gaussian parameters αi (At ), βi (At ) (left column), and parameters
extrapolated to α+20 O system (right column). Calculated from T. Furumoto
et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Extrapolation function coeﬃcients. . . . . . . . . . . . . . . . . . . . . . . .

92

83

94
95

13

Chapter 1

Introduction
1.1

Dipole Excitation

The electric dipole excitation is one of the most fundamental excitation mode of atomic nuclei.
The most well-known electric dipole excitation is called the Isovector Giant Dipole Resonance(IVGDR), strongly populated by the photo-absorption reaction. The IVGDR makes a broad
peak around the excitation energy of 10 to 30 MeV, and is interpreted as the collective
oscillation of protons against neutrons. A schematic view of the IVGDR mode is presented
in Fig. 1.2. The IVGDR is a general property of atomic nuclei existing in every nuclei fromhelium to uranium [1]. A typical excitation spectrum of the photo-absorption reaction ispresented in Fig. 1.1 (indicated as σγ ). In contrast to the particle scattering presented in
the backward graph in Fig. 1.1, the photo-absorption reaction selectively excites IVGDR
distributing around the excitation energy at ω = 20 MeV (indicated as E1 resonance in Fig.
1.1).

Fig. 1.1: A typical excitation spectrum of the photo-absorption reaction (forward) and theparticle (such as proton) scattering (backward). From the textbook by M. N. Harakeh andA. van der Woude [1].

14

Chapter 1 Introduction

Isovector GDR

Protons

Neutrons

Fig. 1.2: A schematic view of the IVGDR.

1.2

Low-energy Dipole Excitation

In neutron-rich medium to heavy nuclei, the electric dipole excitation is fragmented intoa lower energy region than the energy region of IVGDR, around the neutron separationthreshold energy. Those electric dipole excitations are called low-energy dipole excitation(LED), or Pygmy dipole resonance (PDR). The LED strength of 208 Pb, as an example, ispresented in Fig. 1.3 in unit of reduced transition probability B(E1)↑. The broad bumpcentered at around 13 MeV corresponds to the IVGDR (mentioned as GDR in Fig. 1.3).
There is another small bump structure around 7-9 MeV corresponds to the LED (mentioned
as PDR in Fig. 1.3). The neutron separation energy of 208 Pb is Sn = 7.367 MeV [2], and one
can see that the LED distributes just around Sn . Because the LED was observed in neutronrich nuclei, which have neutron skins, it was initially interpreted phenomenologically as theneutron skins oscillating against the nuclear core. The conceptual view of this oscillation
is presented in Fig. 1.9 -(a). However, recent studies pointed out that the LED cannot be
simply interpreted as such, as discussed in the following section.

Fig. 1.3: The electric dipole excitation spectrum of 208 Pb. From A. Tamii et al. [3].

1.2 Low-energy Dipole Excitation

15

Basic unit of the dipole excitation: Isovector dipole strength, Isoscalar dipole strength,and energy-weighted sum rule
The strength of the dipole excitation (or any excitation) is measured by the reduced transition
probability B(Eλ; Ji → Jf ), where Ji ,Jf is the initial and final spin of the state. The reduced
transition probability is the nuclear matrix element divided by the Clebsch-Gordan coeﬃcientaccording to the Wigner-Eckart theorem. The reduced transition probability B(Eλ; Ji → Jf )is given by [1]:∑bB(Eλ; Ji → Jf ) =| < Jf Mf |O(Eλ,µ)|Ji Mi > |2 ,µMf

by using the Wigner-Eckart theorem:bb< Jf Mf |O(Eλ,µ)|Ji Mi >=< Ji λMi µ|Jf Mf >< Jf ||O(Eλ)||Ji >,where < Ji λMi µ|Jf Mf > is the Clebsch-Gordan coeﬃcient, the reduced transition probability B(Eλ; Ji → Jf ) is rewritten to be:B(Eλ; Ji → Jf ) =1
2b| < Jf ||O(Eλ)||Ji >| .
2Ji + 1

According to this formula, the reduced transition probability of the initial spin state Ji to
the final state Jf and the initial spin state Jf to the final spin state Ji has the following
relationship:
2Ji + 1B(Eλ; Ji → Jf ) =B(Eλ; Jf → Ji ).
2Jf + 1
As a convention, the reduced transition probability of the excitation (Ji to Jf ) is denoted byB(Eλ; Ji → Jf ) ≡ B(Eλ) ↑,and the reduced transition probability of the de-excitation (Jf to Ji ) is denoted byB(Eλ; Jf → Ji ) ≡ B(Eλ) ↓ .
In this work, the reduced transition probability from the ground state to an excited stateB(Eλ)↑ is always used. The reduced transition probability B(Eλ)↑ carries information of
the wave function of the excited states, and determining it is the primary purpose of theexperiment. The reduced transition probability B(Eλ)↑ is determined by the excitation
cross section between the state σ(Ji → Jf ) through, for example, the distorted-wave Born
approximation as discussed in Sec. 4.1.
Isovector dipole strengthbIV (E1) is given by [1]:The Isovector dipole operator O
N
N
∑
Ne ∑
bIV (E1) = ZeO
rn Y1m (Ωn ) −rp Y1m (Ωp ),
A n=1
A n=1

where N , Z, A is the neutron, proton, mass number, and rp , rn is the individual proton andneutron coordinate. The reduced transition probability of the excitation B(E1)↑ is widely

16

Chapter 1 Introductionused as the standard unit of the Isovector dipole strength,
B(E1) ↑= B(E1; Ji → Jf ) =1bIV (E1)||Ji > |2 (e2 fm2 ).
| < Jf ||O
2Ji + 1In many references the Isovector dipole strength is simply denoted as B(E1)↑ without “Isovector” for convention. From the experiment, the B(E1)↑ is determined from the excitation crosssection of the dipole state via photon absorption (γ, γ ′ ) reaction:−+−
σ(γ; 0+g.s. → 1 ) −−−−→ B(E1; 0g.s. → 1 ) = B(E1) ↑,DWBA

where taking as an example the nuclei whose ground state is 0+ for simplicity, and γ stands
for the the probe used to excite the state. The isovector dipole is selectively excited via thephoto-absorption reaction. There are also other type of experiments such as inelastic proton
scattering (p, p′ ) [4, 5], depending on the experimental technique. In case of experiments
on unstable nuclei, where the real-photon absorption is hard to apply, the virtual-photon
absorption via the Coulomb excitation is widely used instead.
Isoscalar dipole strengthbIS (E1) is given by [1]:The Isoscalar dipole operator O
bIS (E1) =
O

A
(
)
∑
5
e ri3 − ⟨r2 ⟩ri Y1m (Ωi ),3n=1

where ri is the individual nucleon coordinate, and ⟨r2 ⟩ is the second moment of the nuclear
radius. This means the Isoscalar dipole strength depends on the nuclear radial function. The
reduced transition probability of the excitation BIS (E1)↑ is given by:
BIS (E1) ↑= B(E1; Ji → Jf ) =1bIS (E1)||Ji > |2 (e2 fm6 ).
| < Jf ||O
2Ji + 1The Isoscalar dipole operator is in higher order (r3 ) than the Isovector dipole operator (r), andthe dimension of the strength is diﬀerent. This leads to a rather complicated situation where
the Isoscalar and Isovector dipole strengths are not comparable. When an excited state hasboth Isoscalar and Isovector dipole strengths, this indeed is one of a specific character of theLED, the state is often expressed as “dual Isospin character”, or “Isospin split”. The standard
unit of Isoscalar dipole strength is complicated. One reason is that the reduced transition
probability BIS (E1)↑ depends on the nuclear radial function, and in that sense depends on themodel. Some references even use the excitation cross section σ(Ji → Jf ) as Isoscalar dipole
strength unit instead of BIS (E1)↑, however the excitation cross section σ(Ji → Jf ) is hard tocompare with other nuclei. Instead, the Isoscalar dipole energy-weighted sum-rule fraction isoften used. From the experiment, the BIS (E1)↑ is often determined from the excitation crosssection of the Isoscalar dipole state via the inelastic scattering of the α particle (α, α′ γ):−+−σ(α; 0+g.s. → 1 ) −−−−→ BIS (E1; 0g.s. → 1 ) = BIS (E1) ↑,DWBA

by taking advantage of the selectivity of the α particle on the Isoscalar excitation. Note that
unlike the Isovector dipole excitation, which is selectively excited by the photo-absorption
reaction, the Isoscalar dipole excitation cannot be selectively excited. Separating Isoscalardipole from other Isoscalar multipoles is a major experimental diﬃculty.

1.2 Low-energy Dipole Excitation

17

Isovector Energy-weighted sum rule
The most famous energy-weighted sum rule is the Isovector dipole energy-weighted sum rule
(EWSR), also known as Thomas-Reiche-Kuhn (TRK) sum rule. The Isovector dipole EWSR
is defined by [1]:Isovector dipole EWSR =all∑(En − E0 )B(E1; J0 → Jn ) = 14.8

n

NZ 2 2(e fm MeV),
A

where En is the excitation energy of the Jn state, E0 is the excitation energy of the initialstate, A, Z, N is the mass, proton, and neutron number. The Isovector dipole EWSR isextremely useful. What it says is that the sum of all Isovector dipole strength times theexcitation energy in nuclei B(E1; J0 → Jn ) × E is determined only by its mass, proton, andneutron number A, Z, N . The IVGDR is known to exhaust about 100 % of the Isovector
EWSR in wide range of nuclei, which means you already know the sum of IVGDR strength
without running the experiment. For conventional reason, the Isovector dipole EWSR isvery often denoted simply as EWSR without “Isovector dipole”. The Isovector dipole EWSR
fraction is often used as a unit of the LED strength:
Isovector dipole EWSR fraction =LED∑(En − E0 )B(E1; J0 → Jn )/EWSR (%).

n

The LED of medium to heavy neutron-rich stable nuclei is known exhaust generally about
1-2 precent of the EWSR, but it is significant given that it is the energy-weighted ratio tothe Isovector Giant dipole resonance.
Isoscalar dipole Energy-weighted sum rule
Just like the Isovector dipole EWSR, the Isoscalar dipole EWSR (ISD EWSR) is also known.
The Isoscalar dipole EWSR is give by [6, 1]:

Isoscalar dipole EWSR =all∑(En − E0 )BIS (E1; J0 → Jn )

n

(
)
3ℏ2 A
25 2 24
2=
11⟨r ⟩ − ⟨r ⟩ − 10ϵ⟨r ⟩ (e2 fm6 MeV),
32mN π3(1.1)

where mN is the atomic mass unit and A is the mass number of excited nuclei, ⟨r2 ⟩, ⟨r4 ⟩were the 2,4-th moment of the real part of the optical potential (or the nuclear potential,
namely the nuclear radial function. Further discussed in Sec. 4.1). In order to avoid thedependency on the nuclear radial function, the Isoscalar dipole strength is often measured inunit of the ISD EWSR fraction:
ISD EWSR fraction =LED∑(En − E0 )BIS (E1; J0 → Jn )/ISD EWSR (%)

n

Unlike the well-studied Isovector dipole EWSR, studies of the ISD EWSR are limited. According to the (α, α′ ) experiments performed in a set of nuclei, it is suggested that the18

Chapter 1 Introduction
Isoscalar giant dipole resonance (ISGDR) exists, just like the famous IVGDR. It is also suggested that ISGDR actually exhausts about 100 % of the ISD EWSR. Recently discussed is
the Isoscalar dipole strength in the LED region, which is known to exhaust a few percent of
the ISD EWSR, and discussed in the next section.

1.3

Impact of the LED on Nuclear Physics

The LED has plenty of attentions nowadays motivated mainly by two reasons. One is its
impact on the photo-absorption cross sections. There are several experimental data suggestingthat the LEDs can have sizable cross sections. In certain astrophysical environment, such as
the r-process nucleosynthesis [7], reactions between nuclei and low-energy γ rays just aroundthe energy of the LED can happen. Because the nuclei selectively interact with the γ rays
via the dipole excitation, the LED can have significant impact on astrophysical process. In
the r-process nucleosynthesis, the LEDs of the unstable nuclei, especially the neutron-rich
unstable nuclei can be important, and measuring the LEDs on such nuclei is a recent topic.
The other reason is its connection with the nuclear Equation of State (EoS). The EoS
attracts a lot of attention recently for its connection with the bulk properties of the neutronstar [8]. The connection between the EoS and the LED is still under discussions [9, 10, 11,
12, 13, 4, 14, 15, 16, 17], and the robustness of the connection depends on the structure of
the LED. One proposed interpretation of the structure of the LED is the vibration of excess
neutrons against nuclear core, phenomenologically depicted as the neutron-skin oscillating
against the nuclear core. Various theoretical calculations based on mean-field approximation
supports this picture, however the interpretation of the structure of the LED depends ontheoretical model. Instead of discussing the detailed structure of the LED, the connection
between the neutron-skin thickness and the LED is discussed on their empirical correlation
via the dipole polarizability αD defined by:∫ℏc
σabs (Ex )
αD =
dEx .
2
2π
Ex2
where the σabs (Ex ) is the photo-absorption cross section as a function of the excitation energy
Ex . According to this formula, the photo-absorption cross section σabs (Ex ) of the LED is significant, because the dipole polarizability αD depends on the inverse of the excitation energy
Ex and the contribution of the lower excitation-energy part is essential. The connection between the neutron-skin thickness and the EoS also relies on model-dependent theoretical calculations, and discussed based on their empirical correlations. Under those model-dependent
two-step correlation, the photo-absorption cross section σabs (Ex ) is predicted to be the key
quantity to put constraints on the EoS from the experiment [18, 19, 20, 21]. However the
goodness of the models has to be tested not only by their predictions on the photo-absorption
cross section σabs (Ex ), but also by the predicted structure of the LED. For these reasons,not only the LED strength but also unveiling the underlying structure of the LED is strongly
desired.

1.4 Present Experimental Data on the LEDs

1.4

Present Experimental Data on the LEDs

Measured LED strengths are reviewed by D. Savran et al. in 2013 [22], as presented in Fig.
1.4 in the unit of Isovector dipole EWSR fractions. Among those, there are several data of
the LED strengths measured on neutron-rich unstable nuclei. The strengths on neutron-rich
unstable nuclei are presented in a filled dot (68 Ni [23]) and filled squares (129−132 Sn, 133,134 Sb[24, 20]). Those data generally use the highly model-dependent multipole decompositions to
extract the dipole strengths and in that sense strongly model-dependent. Although thosedata possibly have such uncertainties, the LEDs in neutron-rich unstable nuclei are known
to exhaust 4-5 % of EWSR in typical. Meanwhile, the dipole strengths of tin isotopes in thestable region, 116−122 Sn [25, 26, 27], displayed in open squares in Fig. 1.4, exhaust less than
2 % of EWSR. These data seem to indicate the enhancement of low-energy dipole strengths
in the neutron-rich unstable region nuclei in the isotopic chain.
They had much attention because the enhancement is possibly derived from the thicker
neutron skin specific to neutron-rich unstable nuclei, based on the interpretation of the LED
mode as neutron-skin and core oscillation mode. If this interpretation is eﬀective, the LED
strength should increase as the neutron-skin thickness gets thicker. However, from the figure
one can see that no simple correlation of such dipole strength and mass number, or numberof excess neutrons, is observed. Therefore, one can hardly connect the dipole strengths
simply to the neutron-skin thickness. At this point, it is quite fair to say that the underlying
mechanism of the enhancement of the dipole strengths when more neutrons are added isstill under considerable debate. One reason is that these studies measured only the Isovector
dipole strength, and ignore the possibly existing Isoscalar strength in the lower-energy region,as observed on the neutron-rich stable nuclei. Following these discussions, an attempt to study
the Isospin character of low-energy dipole excitations in exotic nuclei is mandatory for theunderstanding of the structure of such excitations, and more interestingly to establish theconnection between the neutron-skin thickness and the low-energy dipole strength. For thesame purpose, investigating the systematic behavior in an entire isotopic chain towards moreneutron-rich unstable region is important.

1.5

Isoscalar and Isovector strengths of the LEDs

Despite of long-lasting eﬀorts both from the theoretical and experimental studies, the structure of the LED is still far from established. In recent years, new measurements demonstrated
that the LED has both Isoscalar and Isovector strength, the character called Isospin splitting.
These studies attract much interest, because they can be a new powerful tool to investigate
the structure of the LED. The first study of the LED focusing on their Isoscalar strength was
actually made in 1981 by M.N. Harakeh and A.E.L Dieplink on 16 O and 208 Pb [6], and thenfollowed in 1991 by T. D. Poelhekken et al. on 40 Ca, 58 Ni, 90 Zr, and 208 Pb [5]. These studies
found the Isoscalar dipole excitation populated by the inelastic scattering of the α particles.
However in these studies, the structure of the LED was studied by using only the Isoscalar
probe. More recently in 2000s, new experiments of comparing the Isoscalar probe, inelastic
scattering of the α particles (α, α′ γ), and Isovector probe, photon absorption (γ, γ ′ ), wasperformed on a series of neutron-rich stable nuclei, 40,48 Ca [28], 74 Ge [29], 124 Sn [30], 138 Ba

19

20

Chapter 1 Introduction

Fig. 1.4: Experimentally measured low-energy dipole strengths in units of energy-weighted
sum rule fraction (EWSR (%)) plotted against mass number A. Note that the data areplotted as Isovector dipole energy-weighted sum rule fraction. From the review by D. Savran
et al. [22].

[31], and 140 Ce [32]. The measured Isoscalar and Isovector strengths on 124 Sn is presented inFig. 1.5. This study discovered that a group of the LED, distributing below ∼7 MeV, waspopulated both by the Isoscalar and Isovector probe, while the other group distributing above∼7 MeV was populated almost exclusively by the Isovector probe. This specific characteris called Isospin splitting, and recently recognized as a unique character of the LED. Notethat the neutron separation energy of 124 Sn is Sn =8.489 MeV [2], which means the data is
restricted below the neutron separation energy Sn .
The measured Isoscalar and Isovector dipole strengths in a set of neutron-rich stable nuclei
are summarized in Fig. 1.6 [33]. The strengths are integrated within the displayed excitationenergy into two groups. It can be seen, independent of the nuclei, that the Isovector strengths(right column) generally distribute independent of the excitation energy, making contrast
with the Isoscalar dipole strengths (left column) whose distribution are concentrated onlower excitation energy. Of particular interest is its universality among the nuclei. Although
the available data are limited, the Isospin splitting is observed widely in medium to heavy
neutron-rich nuclei. This suggests that there is a coherent mechanism of generating lowenergy dipole excitation. However interestingly, these low-energy dipole excitations can beclassified into two components with diﬀerent underlying structures. Given the strong Isoscalardipole strength distributing in the lower-energy region, it is also suggested that the lowerenergy component has predominant Isoscalar nature, rather close to the pure Isoscalar dipole.
Meanwhile the higher energy component is suggested to have predominant Isovector nature,1.6 Present Theoretical Predictions on the LEDs

Fig. 1.5: The measured Isoscalar dipole strengths (top panel) in diﬀerential cross section, andIsovector dipole strengths (bottom panel) in B(E1) on 124 Sn. From J. Endres et al. [30].

closer to the IVGDR. However, at this point, the origin of this mechanism is yet to beunderstood, and further studies are strongly desired.

1.6

Present Theoretical Predictions on the LEDs

One reason that makes understanding of the LEDs diﬃcult can be found in the theories. Thereis no coherent theoretical model to predict the LED on every aspects from its excitation energy
to Isovector and Isoscalar strengths. The LEDs are located in rather higher excitation energy
from the shell-model point of view, and rather lower excitation energy from the mean-fieldcalculation point of view, such as the random-phase approximation (RPA). Many theories
attempted to reproduce the LEDs recently, and some showed that the LEDs have significantly
diﬀerent structure as the IVGDR. Theoretical calculations and their diﬀerence are intensively
reviewed in Ref. [34]. One example of such theoretical calculation is performed on 140 Ce by
N. Paar et al. in 2009 [35] using the self-consistent relativistic quasiparticle random phaseapproximation (RQRPA). Transition densities of the predicted states is presented in Fig. 1.7.
One can see the Isovector and Isoscalar nature of the dipole states by the line shape of thetransition density. When the proton and neutron transition density have similar shape with

21

22

Chapter 1 Introduction

Fig. 1.6: The experimentally measured Isoscalar dipole strengths (left column) in Isoscalardipole energy-weighted sum rule (ISEWSR) fraction or cross sections, and Isovector dipole
strengths (right column) in B(E1). From the review by A. Bracco et al. [33].

1.6 Present Theoretical Predictions on the LEDs
the same sign, it is Isoscalar. If these two have exactly the same shape with the same sign,
it is purely Isoscalar. When the proton and neutron transition density have similar shape
but with the diﬀerent sign, it is Isovector. If these two have exactly the same shape with the
opposite sign, it is purely Isovector.
The transition density of the typical GDR (bottom panel in Fig. 1.7) is characterized by
the protons and neutrons having similar line shape but with diﬀerent sign, and therefore it
is Isovector. Meanwhile, the transition density of the LED (top panel in Fig. 1.7: mentioned
as PDR in the original paper) has diﬀerent character: the protons and neutrons have similarline shape with the same sign, generating the Isoscalar nature of the LED. According to
the calculation, the lower excitation energy part of the LED is predominantly carried by theneutron excitations, that can be interpreted as the neutron-skin oscillation mode. While thehigher energy part of the LED belongs to the lowest energy tail of the GDR. However thepredicted Isoscalar and Isovector strengths agreed with the measured Isoscalar and Isovector
strengths only by the tendency, and further studies were clearly necessary.

Fig. 1.7: An example of the predicted structure of the dipole excitation. The transition
densities of the LED (top panel) and the typical GDR (bottom panel) predicted by the
random-phase approximation (RPA). From N. Paar et al. [35].
This partial agreement of theoretical predictions with the measured data are still common.
Another example of theoretical calculation on the dipole excitation spectrum of 208 Pb ispresented in Fig. 1.8. Note that this work focused only on the Isovector strength B(E1)↑.
They compared the measured Isovector dipole strength ((a) in Fig. 1.8) with two elaborated
theories, the microscopic quasiparticle-phonon (QPM) calculation ((b) in Fig. 1.8), andthe relativistic time-blocking approximation (RTBA) calculation ((c) in Fig. 1.8). One cansee that the theories could reproduce the experimental data qualitatively, but it is hard
to explain the whole strength distribution, especially the LEDs quantitatively. Another

23

24

Chapter 1 Introduction
important aspect is that there still are few works discussing both the Isoscalar and Isovector
strengths of the LEDs. Many works focus only on the Isovector strengths, and missing thepossibly significant Isoscalar strengths. Again the interpretation of those states, such as
the dominant excitations of the excess neutrons, based on their Isovector strengths are not
suﬃcient.

Fig. 1.8: An example of the comparison between the predictions and the measurement. The
Isovector dipole strength measured on 208 Pb (a), a predicted strength using the microscopic
quasiparticle-phonon (QPM) calculation (b), and another predicted strength using the relativistic time-blocking approximation (RTBA) calculation (c). From A. Tamii et al. [4].
It should also be pointed out that there possibly exists another type of the LED than theexcess neutron excitations. Recently a macroscopic theory predicts fully collective excitation
called the toroidal dipole mode [34, 36, 37, 38, 39, 40, 41, 5, 42, 28], and subject to a
lot of discussions. This toroidal dipole excitation is particularly interesting because several
theories predict their predominant contribution to the Isoscalar dipole strengths in the lowest
excitation energy region. The schematic velocity fields of the toroidal dipole excitation andother proposed models are presented in Fig. 1.9. The early interpretation of neutron skinoscillation against the nuclear core is presented in Fig. 1.9-(a), recently discussed toroidal
dipole in Fig. 1.9-(b), and compressional dipole in Fig. 1.9-(c). The compressional dipole
mode was discussed in the early studies of Isoscalar dipole excitations: M.N. Harakeh and
A.E.L Dieplink on 16 O and 208 Pb [6], and T. D. Poelhekken et al. on 40 Ca, 58 Ni, 90 Zr, and208
Pb [5]. However not followed by other studies. Although further studies both from thetheories and experiments are necessary, it is proposed that both collective toroidal mode andexcess neutron mode actually exists in the LEDs [28]. In several neutron-rich stable nuclei,

1.7 The Isospin Structure of the LEDs on Unstable Nuclei
it is very recently discussed that the high-resolution measurement can possibly distinguish
one mode from another [33].

Fig. 1.9: A schematic velocity fields from diﬀerent collective motions. Neutron skin oscillation
against the nuclear core (a), toroidal dipole (b), and compressional dipole (c). From A. Repko
et al. [39].

1.7

The Isospin Structure of the LEDs on Unstable Nuclei

Extending the study of low-energy dipole excitations into the neutron-rich unstable nuclei isnecessary to reveal the underlying structure, specifically its connection with the neutron-skinthickness and the EoS, the key characteristics of the neutron-rich nuclei. Driven by thosemotivations, the present study intended to study the Isospin character of the low-energy
dipole excitations on neutron-rich unstable 20 O.

1.7.1

Existing Data on 20 O

Indeed, Isovector strengths of the LEDs on 20 O were first investigated by E. Tryggestad et al.
in 2002 [43, 44]. They populated the low-energy dipole excitations via the Coulomb excitation
at the energy of 100 MeV/nucleon using a lead target. De-excitation γ rays were detectedusing the in-beam γ-ray spectroscopy. The measured decay γ-ray spectrum is presented inFig. 1.10. They found two low-energy dipole excitations slightly below the neutron separation−energy at the energies of 5.35 MeV (1−1 ) and 6.85 MeV (12 ), and the Isovector dipole strengths
B(E1)↑ were determined as presented in Tab. 1.1. Note that B(E1)↑ was determined by the
Coulomb excitation formalism, and not by the distorted-wave Born approximation (DWBA).
Possible contribution of the nuclear excitation was not discussed.
They compared the results with the shell-model predictions, and pointed out the existenceof unexpectedly strong Isovector strength. In the unit of EWSR fraction, the experimental
value was one order of magnitude larger than the shell-model predictions with WBP and
WB10 interactions, presented in the bottom row of Tab. 1.1. The origin of this exceptional
strengths was discussed in terms of the neutron excitations enhanced by the p-sd shell gap
reduction, corresponding to the weaker binding of valence neutrons. They performed the

25

26

Chapter 1 Introduction

Fig. 1.10: The previously measured decay γ-ray spectrum of 20 O from the Coulomb excitation
using a lead target. From E. Tryggestad et al. [44].

shell-model calculation under the truncation involving only the valence neutrons, in otherwords, the excitation involving the doubly-magic 16 O core is forbidden. The shell-model
prediction with WB10 interaction considering the p-sd shell gap reduction is presented in
the right column of Tab. 1.1. The agreement with the experimental values got better withthe p-sd shell gap reduction, but still the measured strengths were considerably stronger.
They have concluded that it was still hard to fully explain the strengths in the context of theshell model with p-sd shell gap reduction, and called for further studies of those states. For
completeness, the Isovector dipole strengths above the neutron threshold was investigatedby A. Leistenschneider et al. [45] in 2001 via the Coulomb excitation in combination with
the invariant-mass spectroscopy, and possible enhancement of the Isovector low-energy dipole
excitation was discussed.

1.7.2

Our goal

The previous studies on 20 O have two major issues. One is that they studied only the
Isovector dipole strength. Another is that they discussed the LED states using the shellmodel involving only the valence-neutron excitations. Recent theoretical calculations suggestthat the LED cannot be simply interpreted as valence-neutron excitations, and other collectivemode such as the toroidal mode can be involved. Theoretical studies based on mean-fieldcalculations [46, 47] predict that the LEDs on 20 O have large Isoscalar dipole strengths. Since
recent studies on neutron-rich stable nuclei, especially 48 Ca [28], indeed found strong Isoscalar1.7 The Isospin Structure of the LEDs on Unstable Nuclei

27

Table 1.1: The previous result of Isovector dipole strength in 20 O. From E. Tryggestad et al.
[43, 44].
Expt. results
Jπ
1−
1
1

−

Ex
5.05

6.85(5)B(E1) ↑
2.4[-03]3.5(9)[-02]

EWS[Expt.]:0.57(11)

WB10
Jπ

Ex

B(E1) ↑

1−
1

5.34

2.3[-03]

1−
2

6.55

1.7[-03]

1−
3
1−
4

7.20

7.6[-03]

7.89

1.8[-02]−
EWS[1−
1 → 13 ]:0.078

WB10 (p-sd gap reduction)
JπEx

B(E1) ↑

1−
1
1−
2
1−
3
1−
4
1−
5

4.10

3.7[-04]

5.47

2.5[-03]

6.06

8.9[-03]

6.75

1.4[-02]

7.45

2.4[-02]−
EWS[1−
2 → 14 ]:0.16

dipole strengths distributing around the lowest excitation energy, extending the research into
further neutron-rich unstable nuclei is strongly encouraged.
We aimed at measuring the Isoscalar and Isovector strengths of the LED in neutronrich unstable 20 O. The Isoscalar strength can give a new information to investigate theirstructures. In order to achieve this, we followed the experimental technique successfully
applied for the stable nuclei. The inelastic α scattering (α, α′ γ), an Isoscalar probe, andthe Coulomb excitation (γ, γ ′ ), an Isovector probe, was performed in inverse kinematics toextract the Isoscalar and Isovector strengths respectively. The excited states were identified
by detecting the de-excitation γ rays using the in-beam γ-ray spectroscopy technique. A
distorted-wave Born approximation (DWBA) analysis was performed to convert the crosssections of each excited states into the Isovector and Isoscalar dipole strengths.
The Isoscalar dipole strength has never been measured on unstable nuclei, and it is a newchallenge. Recently built Radioactive Isotope Beam Factory (RIBF) [48] of RIKEN Nishina
Center enabled us to access high-intensity beam of unstable nuclei. The RIBF enabled us toinvestigate neutron-rich unstable 20 O using the inelastic scattering of the α particles (Isoscalar
probe), where the typical cross section is a hundred times smaller than that of the previously
measured Coulomb excitation (Isovector probe), owning to unprecedented-intensity primarybeam of 48 Ca provided, reaching to 250 particle nA on average. 20 O is a good nucleus toinvestigate the Isospin structure of the LED. The limited level density enables us to accessthe bound excited states via the in-beam γ-ray spectroscopy, which is a powerful method toinvestigate the bound-excited state of the unstable nuclei in-beam. Actually, two LED states
located at 5.35 and 6.85 MeV in 20 O are bound excited states accessible via the the in-beam γray spectroscopy, providing a good reason to investigate. In order to detect such high-energy γrays, indeed the energy gets even higher because the γ rays are emitted in-flight (discussed inSec. 2.2), a large-volume LaBr3 :Ce scintillator array developed at INFN Milano was employed(discussed in Sec. 2.8), for the first time at the RIBF-level secondary beam energy of about
250 MeV/nucleon. The LaBr3 :Ce scintillator has three crucial advantages to our experiment:
large-volume (3.5ϕ × 8 inch) and high atomic number material for the detection eﬃciency,
good time resolution (< 1 nsec) for the particle and γ separation using the time of flight,
and stable response under high count rate (tested up to 250 kHz). The inelastic scattering

28

Chapter 1 Introductionof α particles on unstable 20 O combined with the in-beam γ-ray spectroscopy employing alarge-volume LaBr3 :Ce scintillator array is a completely new experimental challenge.

29

Chapter 2

Experimental Apparatus
2.1

Radioactive Isotope Beam Factory (RIBF) at RIKEN
Nishina Center

Recently built cyclotron complex of RIKEN Nishina Center enabled us to access high-intensity
beam of unstable nuclei via projectile fragmentation reaction, named Radioactive Isotope
Beam Factory (RIBF) [48]. This new generation cyclotron complex is capable of accelerating very heavy-mass nuclei, 238 U, to 345 MeV/nucleon. In combination with the projectile
fragmentation reaction, the facility provided us the opportunity to investigate a broad rangeof nuclei, represented by the extremely neutron-rich unstable nuclei, many of which were
never accessible before. The schematic view of the facility is presented in Fig. 2.1. The
facility was equipped with many devices already running. For the analysis of the projectile
fragments, a in-flight fragment separator (BigRIPS in the figure: used in the present work)
was installed. The facility was also equipped with several spectrometers for diﬀerent purposes, a zero-degree forward spectrometer (ZDS in the figure: used in the present work),a large acceptance spectrometer (SAMURAI in the figure), a high-resolution spectrometer(SHARAQ in the figure). Many experiments are proposed and performed in the RIBF since
its commissioning performed in 2008.

Fig. 2.1: A map of the RIKEN Radio Isotope Beam Factory Facility. From Y. Yano [48].

30

Chapter 2 Experimental Apparatus

2.2

In-beam γ-Ray Spectroscopy

Because unstable nuclei are produced as a energetic beam, in many cases studies on these
nuclei have to be carried out in inverse kinematics. One of the most powerful ways to study thebound excited states of those nuclei is in-beam γ-ray spectroscopy. In inverse kinematics, thenuclei of interest is excited in-beam by the target, and the excited nuclei decays immediately
again in-beam by emitting de-excitation γ rays. The in-beam γ-ray spectroscopy detects
those de-excitation γ rays.
A conceptual view of the in-beam γ-ray spectroscopy is presented in Fig. 2.2. The unstable
nuclei 20 O is produced as a beam by the fragment separator BigRIPS (Sec. 2.3), and excitedby two secondary targets (Sec. 2.5). A liquid helium target was used for inelastic scatteringof α particles:20

O(α, α′ γ)20 O,(2.1)

where γ stands for the de-excitation γ ray from the excited state. A gold target was used for
the Coulomb excitation:
20

O(Au, Au′ γ)20 O.

(2.2)

The Coulomb excitation is often expressed in alternative formula as:
20

O(γ, γ ′ )20 O,(2.3)

in the sense of the virtual-photon absorption. Here γ ′ stands for the de-excitation γ ray fromthe excited state. The excited 20 O* immediately decay by emitting de-excitation γ rays.
Note that this γ decay is specific to the bound-excited state, and therefore the measurable
energy of the excited state is experimentally limited to the neutron-separation energy of 7.6
MeV. The de-excitation γ rays are detected by the large volume LaBr3 :Ce scintillator array(Sec. 2.8) to determine the excitation cross sections:−+−
σ(α; 0+g.s. → 1 ) −−−−→ BIS (E1; 0g.s. → 1 ) → ISD EWSR (%).
DWBA

2
−
+
−
2
σ(Au; 0+g.s. → 1 ) −−−−→ B(E1; 0g.s. → 1 ) = B(E1) ↑ (e fm ).
DWBA

(2.4)

Note that the ground state of 20 O is 0+ . The excitation cross sections are converted into the
Isoscalar strength (ISD EWSR (%)), and The Isovector strength (B(E1)↑ (e2 fm2 )) using thedistorted-wave Born approximation (Sec. 4.1). Reacted 20 O is detected by the ZeroDegree
spectrometer (Sec. 2.6) for the particle identification, ensuring the reaction channel to be theinelastic scattering.
There are two technical diﬃculties in this method. One is Doppler shift, and the otheris Doppler broadening. The Doppler shift modifies the energy of those γ rays seen fromthe laboratory system depending on their emitted angle and the velocity of the emitter
nuclei. The other factor, the Doppler broadening, worsens the energy resolution of the γ rays
depending on angular granularity of detectors and thickness of a target. Those eﬀects were
estimated by a Monte-Carlo simulation discussed in the following section (Sec. 3.3.8).

2.3 Production of Radioactive Isotope Beam

31

20O*

Immediate
γ decay

20O

20O

Detected by
ZeroDegree
Spectrometer
Located at 0°30°
De-excitation
γ ray
Identified by
Fragment Separator
BigRIPSγ
Detected by
LaBr3 Array

Secondary target
Liq. He or Au

Fig. 2.2: A conceptual view of the in-beam γ-ray spectroscopy.

The early study using the in-beam γ-ray spectroscopy in combination with the Coulomb
excitation was performed by T. Motobayashi et al. in 1994 [49]. They measured the B(E2)↑32strength of the 2+
Mg using a lead target. This method was also suitable to study
1 state in
the low-energy dipole excitation, as precedented by E. Tryggestad et al. [43, 44].

2.3

Production of Radioactive Isotope Beam

The secondary 20 O beam was produced via projectile fragmentation reaction. A 48 Ca ion fromthe electron cyclotron resonance (ECR) ion source was initially accelerated by the RIKEN
Heavy Ion linear particle accelerator (RILAC) to be injected to the cyclotron complex. The
performance of the cyclotron is often represented by the K value defined by:K( Z )2A= Eu .

Where Z and A is the charge and the mass of the accelerated nuclei, and Eu is the energyof the accelerated nuclei per nucleon. For the acceleration of a 48 Ca beam, four cyclotrons
were used in cascade with the K = 2600 superconducting ring cyclotron (SRC) at the end.
The conceptual view of the cyclotron complex is presented in Fig. 2.3. With those powerful
cyclotrons, the 48 Ca beam was accelerated to an energy of 345 MeV/nucleon with the averageintensity of 250 particle nA. This high intensity and high beam energy allowed us to investigate 20 O using an inelastic α scattering, because the cross section is typically one hundred
times smaller than that of widely applied Coulomb excitation, such as Refs. [50, 49]. The
primary beam hit on a 15-mm-thick beryllium target, and the reaction product was identified
on event-by-event basis by the fragment separator named BigRIPS [51].

2.3.1

In-flight fragment separator: BigRIPS

The schematic view of the BigRIPS is presented in Fig. 2.4. The BigRIPS was composedof six normal-conducting dipole magnets labeled D1 to D6, fourteen superconducting triplet

32

Chapter 2 Experimental Apparatus

Fig. 2.3: A conceptual view of the RIKEN Nishina Center cyclotron complex.

quadrupole magnets labeled STQ1 to STQ14. A STQ magnet consisted of three superconducting quadrupoles packaged into one cryostat. Each STQ magnet located next to the dipole
magnet has a superconducting sextupole installed in the inner bore of the quadrupole. These
superconducting STQs enabled us to produce the highly energetic neutron-rich nuclei beam,
since they had higher magnetic rigidity and required large acceptance. The focal planes
were labeled as F0 to F8. The beryllium primary target was installed at F0, an achromatic
focal plane. The fragments were first selected according to the magnetic rigidity, Bρ, bythe slit installed at F1 dispersive focal plane. The magnetic rigidity satisfies the following
relationship:
0.3Bρ [Tm] =A
mu vγ [GeV/c],
Z

(2.5)

where A and q are mass and charge of the beam-like nuclei respectively, mu is atomic√ massunit, v is the velocity of the nuclei, and γ is the Lorentz factor given by γ = 1/ 1 − β 2 .
Here the fragments are assumed to be fully-stripped, so that q = Z where Z is the numberof protons. This assumption is applicable to light ions such as oxygen isotopes of the current
velocity.
In case of the projectile fragmentation reaction, every fragment approximately has thesame mu vγ. Therefore, by selecting the magnetic rigidity Bρ, the fragments with desired
A/Z were obtained at F1. In order to choose the fragments with desired Z and A, a wedgeshaped energy degrader made of 15-mm-thick aluminum was installed right after the slit at
F1. The fragments went through the degrader lost energy approximately proportional to Z 2 ,and the magnetic rigidity went smaller according to the energy loss. By choosing the desiredmagnetic rigidity using the D2 dipole magnet and the slit at F2 achromatic focal plane, thefragments went through second selection. This two-step selection enabled us to have higher
purity secondary 20 O beam. In order to reduce the light ion background such as triton, theslits at F5 dispersive focal plane and F8 achromatic focal plane were additionally used. The

2.3 Production of Radioactive Isotope Beam

33

BigRIPS settings are tabulated in Table 2.1 and 2.2.

Fig. 2.4: The schematic view of the BigRIPS fragment separator and the ZeroDegree Spectrometer.
The achieved 20 O secondary beam properties were 45% purity, 1.3×105 particle/s intensity,and 276(9) MeV/nucleon kinetic energy. The achieved beam profile at the secondary target
position is presented in Fig 2.5- top panel: horizontal position x (mm) and incident angle x′(mrad) and bottom panel: vertical position y (mm) and incident angle y ′ (mrad), measured
using the PPACs (See Sec. 3.1.2 for detail). The energy of the secondary beam was determined right before the secondary reaction target from the magnetic rigidity of the D6 dipole
magnet. The energy loss by the beam line detectors between D6 and the secondary target
was additionally taken into account. The fragments were identified event-by-event basis using
the Bρ-TOF-∆E method [51], which will be discussed in the following section (Sec. 3.2.1).
Table 2.1: BigRIPS Magnet Settings.
Dipole
Field

D1

D2

D3

D4

D5

D6

(Tm)

6.970

6.522

6.471

6.471

6.427

6.427

Table 2.2: BigRIPS slit settings. The slits are most wide open at ± 120 mm.
Focal Plane

F1 (Disp.)

F2 (Achr.)

F5 (Disp.)

F8 (Achr.)

(mm)

±64.2

±5.00

±50.0

±70.0

34

Chapter 2 Experimental Apparatus

60

40

x’ (mrad)

20

0

-20

-40

-60
-60

-40

-20

0
x (mm)

20

40

60

-40

-20

0
y (mm)

20

40

60

60

40

y’ (mrad)

20

0

-20

-40

-60
-60

Fig. 2.5: The measured beam profiles. Horizontal position (x) versus incident angle (x′ ) (toppanel), vertical position (y) versus incident angle (y ′ ) (bottom panel).

2.4 Detector installed around and after the secondary reaction target

2.4

35

Detector installed around and after the secondary
reaction target

The detector installed around the secondary target is presented in Fig. 2.6. The Large-volume
LaBr3 :Ce detector array consisted of eight detectors, and located at (r,θ)=(430 mm, 30◦ ) inpolar coordinate seen from the secondary target. Eight detectors were placed cylindricallysymmetric (ϕ symmetric in terms of polar coordinate) with each other. Two PPACs were
installed upstream of the secondary target, and one downstream. They were used to measure
the scattering angle of the reaction product in the secondary target. A plastic scintillator
was installed upstream of the secondary target, in order to measure the time-of-flight of thereaction product between the secondary target and the LaBr3 :Ce detectors.

Detectors installed around the secondary targets
Tracking

Time of flight

PPAC

Plastic Scintillator

Secondary target
Liq. He or Au

γ-ray

Tracking

LaBr3

PPAC

F8
/

γ
40 mm
500 mm/1058 mm

876 mmFig. 2.6: Detector setup around the secondary target.

2.5

Secondary Reaction Targets

In order to investigate the desired dipole excitations of 20 O, the secondary reaction targets
were installed at F8 achromatic focal plane, located at the end of the BigRIPS fragment
separator. Both liquid helium and gold target was built into an exchangeable system, and36

Chapter 2 Experimental Apparatus
exclusively occupied the target chamber.

2.5.1

A Liquid-helium target system

We employed a liquid helium target as an Isoscalar probe. The α particle had several advantages crucial to our measurement. It is an almost pure Isoscalar probe, because the α particle
is N = Z. Since the α particle is a spin zero particle, it selectively excites the natural parity
state, such as the E1 excitation. It can minimize the Coulomb excitation contribution for
its charge as low as two. Because the first excited state of the α particle is 20.21 MeV, it
can suppress the mutual excitation within the energy range of our measurement. In order togain high statistics even applicable to the unstable beam, we developed a very-thick liquid
helium target cell. The target system was based on Ref. [52], consisted of a 4K refrigerator
connected to a target cell via a two-stage copper thermal conductor. The schematic view of
the cryogenic system is presented in Fig.2.7. In order to increase eﬀective thickness, a 32 mm
thickness cell was developed specifically for this work. The cell had eﬀective diameter of 66
mm separated by 10 µm Havar foil windows [53], and surrounded by a radiation shield with
14 µm Aluminum foil windows. The eﬀective thickness was determined by comparing thebeam energy loss with a liquid helium filled data and an empty data. The measured mass
thickness was 317±28 mg/cm2 . The error was estimated from the temperature fluctuationof 0.03 K corresponds to the density fluctuation of 12 mg/cm3 [54] during the one-day datataking. The cell specifications are summarized in Tab. 2.3. The target area was constantly
kept in good vacuum, in order of 10−6 Torr or better, using a turbo-molecular pump. In order to avoid the contamination by oil gas adhered to the cell windows, an oil-free mechanicalpump was used as a backing pump.
Table 2.3: Specifications of the liquid helium target cell.
Cell size

ϕ66 × 32 mm

Cell Material

Aluminum

Cell window material

Havar alloy

A cell window thickness

10 µm

Radiation shield window material

Aluminum

A radiation shield window thickness

7 µm

Cell eﬀective area

ϕ40 mmCell eﬀective mass thickness

317±28 mg/cm2

2.5.2

A gold target

A gold target was employed for the Coulomb excitation, as a predominant Isovector probe.
The Coulomb excitation was widely applied to study the excited states of unstable nuclei [49].
In order to gain higher statistics, we employed a gold target as thick as 2.5 mm, equivalentof 4.8 g/cm2 . The application of the Coulomb excitation using a thick target was also
tested to be successful [50, 55] at our kinetic energy range of 250 MeV/nucleon, although the2.6 Reaction Product Spectrometer: ZeroDegree

Fig. 2.7: The schematic view of the cryogenic system. Taken and modified to describe thepresent configuration from H. Ryuto et al. [52].

appropriate estimation of nuclear excitation contribution was necessary. In comparison withwidely applied lead targets, such as Refs. [49, 50, 55], the gold target had an advantage. The
lead target mostly composed of doubly-magic isotope of lead, 208 Pb, while the gold target was
mostly composed of odd-even stable isotope of gold, 197 Au. Because of its odd-even nature,
the gold target tend to emit lower energy γ rays compared with the lead target. This γ raysfrom the target were major background component of the γ-ray spectrum, and because weneeded to detect γ rays from highly excited states, the background γ rays were desired tohave lower energy. The thickness was measured by the beam energy loss. The measured
thickness was 2.45±0.05 mm, whereas the catalog value was 2.5 mm. The error was definedby the discrepancy from the catalog value.

2.6

Reaction Product Spectrometer: ZeroDegree

The ZeroDegree spectrometer was used for the particle identification of the reaction productafter the secondary target. The schematic view of the ZeroDegree spectrometer is presented
in Fig. 2.4. The secondary reaction targets were installed at F8 achromatic focal plane. The
ZeroDegree spectrometer was composed of two normal-conducting dipole magnets labeled
D7 to D8, and eight superconducting triplet quadrupole magnets labeled STQ15 to STQ22.

37

38

Chapter 2 Experimental Apparatus
The magnets have the same design as those used in the BigRIPS fragment separators, except
for the sextuple magnets installed but not used. The ZeroDegree spectrometer settings aretabulated in Fig. 2.4. The magnetic rigidities depend on the secondary targets to reflect theenergy-loss of the beam particle.
Table 2.4: ZeroDegree spectrometer magnet settings.
Dipole Field

Secondary Target

D7

D8

(Tm)

α

6.268

6.250

(Tm)

Au

5.912

5.901

2.7

Beam Line Detectors

The beam line detectors (BLDs) were installed at focal plains and mainly used for the particle
identification of the fragments. The plastic scintillators were used to measure the time-offlight of the fragments. They are also used to measure the charge of the fragments fromthe energy loss. The position-sensitive detectors, parallel plate avalanche counters and amulti-wire drift chamber, were used to reconstruct the track of the fragments. The beam-line
detectors installed at the ZeroDegree spectrometer is schematically shown in Fig. 2.8.

2.7.1

Plastic scintillators

The plastic scintillator were installed at F3 (Achromatic), F5 (Dispersive), F7 (Achromatic),
and F8 (Achromatic) focal planes. The dimensions were 120×120×3 mm for the ones usedat the F5, F7, F8 and F11 achromatic focal planes, and 120×240 (horizontal)×3 mm for the
one used at the F5 dispersive focal plane. A scintillator was equipped with a photomultipliertube (PMT) at either end, in order to take coincidence between the two PMTs. In case of theF5 scintillator, the light propagation time diﬀerence between left and right hand side PMT
was used to detect the horizontal hit position of the fragments.

2.7.2

Parallel plate avalanche counters

Parallel Plate Avalanche Counters (PPACs) with delay-line readout [56] were used to measure
the hit position and to reconstruct the trajectory of the secondary beam. A PPAC had aneﬀective area of 240 mm (horizontal) × 150 mm (vertical). A module of one PPAC consistedof a cathode (horizontal) - an anode - a cathode (vertical) configuration, and two modules
were packaged into one PPAC for redundancy. A typical position is 0.5 mm (FWHM) [56].
For our experiment, isobutane gas (C4 H10 ) at several tens of torr was used, and the operation
voltage was typically 800 V. Because they were not applicable for the high count rate at 1×105
count/s, they were used for the calibration under lower count rate.

2.8 Detectors for γ Rays

39

Detectors installed at ZeroDegree Spectrometer
F11
D8

F10
STQ

Tracking
PPAC

F9

STQ

STQ

STQ

TOF
Plastic
Scintillator

STQ

F8
STQ

D7

Position
MWDC
TOF
Plastic
Scintillator
SecondaryTargetAu or LiqHe

Fig. 2.8: Detectors installed at the ZeroDegree spectrometer.

2.7.3

Multi-wire drift chamber

A Multi-wire Drift Chamber (MWDC) [57] was used to measure the hit position of the reaction product after the secondary reaction target. An advantage of the MWDC in comparison
with the PPAC was its operation capability at high count rate. For this reason, the MWDC
was installed at F9 dispersive focal plane to measure the hit position and identify the reaction product. The MWDC had U-V-V-U (U:+30◦ ,V: -45◦ ) wire configuration. The sensitive
area was 220 (horizontal) × 150 (vertical) mm, large enough to cover the particles transported within the ZeroDegree acceptance. A typical position resolution is 300 µm [57]. The
isobutane gas (C4 H10 ) at several tens of torr was used.

2.8

Detectors for γ Rays

2.8.1

Large-volume LaBr3 :Ce detector array

Since the decay γ rays were emitted from the fast beam of 20 O, the decay γ rays were Lorentz
boosted and focused on forward angle. This was usually called Doppler shift in terms of thein-beam γ-ray spectroscopy. The energy of the γ ray seen from the laboratory system Elab

40

Chapter 2 Experimental Apparatus
is shifted according to:√Elab
1 − β2=,E0
1 − β cos θlab

(2.6)

where the β is the velocity of the moving γ emitter divided by the speed of light, and θlabis the γ-ray emitted angle in the laboratory system. The calculated Doppler shifted γ-ray
angular distributions are presented in Fig. 4.3 and 4.4 in the group marked as Laboratory
frame. One can see that in the present beam energy, the γ ray has highest intensity around30◦ . This was the angle where the detectors should be located. The detail is discussed in thefollowing section (Sec. 4.1.4).
The γ-ray detectors were requested to have sensitivity up to the neutron separation energy,
7.6 MeV, of 20 O. This energy corresponds to 13 MeV after the Doppler shift at θlab =30◦ ,where the γ rays have high intensity. In order to detect the γ rays up to this high energy,
we employed a large-volume LaBr3 :Ce detector array developed at INFN Milano [58], which
was the key detector for this experiment. A LaBr3 :Ce crystal has 3.5ϕ×8 inch cylindricalvolume, among the largest volume ever produced. The crystal is coupled to a Hamamatsu
R10233-100SEL 3.5 inch photo-multiplier tube (PMT) to form one detector module.
The LaBr3 :Ce detector has three advantages crucial to our experiment. One is the good
intrinsic time resolution typically better than 1 nsec. This allowed us to separate the decay γ
rays and other particle backgrounds by selecting the time of flight from the secondary target
to the crystals. Another is the excellent linear response. Their linearity as a function of input
γ-ray energy was well tested up to 22 MeV (described in detail in [58]), which suﬃciently
covered our energy range. The other one is the stable detector response under high countrate. Because it takes only about 150 nsec for one pulse to end, including PMT response, thedetectors can be operated at high count rate. The detectors were tested up to 250 kHz countto maintain their response. The LaBr3 :Ce crystal is known to have better intrinsic energy
resolution compared with other often-used scintillators such as NaI or CsI, and this was also
an advantage. However in our experiment, the energy resolution was overwhelmed by the
Doppler broadening, and had little impact.
A disadvantage of the LaBr3 :Ce detector is the internal radioactivity derived from theradioactive Lanthanum isotope, mostly 138 La. 138 La decays via both β − decay and electron
capture decay. The β − decay produces γ ray at the energy of 0.789 MeV originated fromthe subsequent γ decay of 138 Ce, and β-ray continuum distributes approximately from 0.7
MeV to 1.0 MeV. Meanwhile the electron capture decay produces rather high-energy and also
intense 1.436-MeV γ ray derived from the subsequent γ decay of 138 Ba, and the characteristic
X ray at 32 keV in coincidence. The sum of simultaneous γ-ray and X-ray energy at ∼1.470
MeV is known to be observed in the internal activity spectrum [59]. The LaBr3 :Ce detector isalso known to have a contaminant, 227 Ac, due to its chemically analog nature to Lanthanum.
The α decay chain of 227 Ac makes several discrete peaks distributing from 1.8 MeV to 2.4
MeV. But because of the quenching eﬀect of α rays in the scintillator, they do not form
gaussian-shaped peaks correspond to their energies. These internal radioactivities produced
typically 1 kHz of event per one crystal, and can be a serious background. But since thedecay γ rays from the fast 20 O are Doppler shifted to higher energy in forward angle, theinternal backgrounds are relatively shifted to lower energy, resulted in better separation fromthe energy region crucial to our measurement. This gave us another point to apply the
LaBr3 :Ce detector to our experiment at forward angle. These internal radioactivity can also

2.9 Data Acquisition Systembe an advantage. In our case, the intense and discrete 1.436-MeV γ ray was used to checkthe stability of the crystal response.

2.8.2

X-ray shield

Because we used very thick gold target combined with intense beam, X rays from atomic
process was expected to be a strong background. In order to protect γ-ray detectors from X
rays, the target chamber, made of 3-mm-thick aluminum, was surrounded both by 1-mm-thicklead and 0.5-mm-thick tin foil. The eﬀects of X-ray shield are considered in a Monte-Carlosimulation as discussed in the following section (Sec. 3.3.2).

2.9

Data Acquisition System

The data acquisition system was based on RIBF-DAQ [60]. The system consisted of several
front-end computers reading data from VME modules, and a master event-builder computer
integrating each data from front-end computers to one unified data. The computers wereconnected via an Ethernet to communicate with each others. The system worked on thecommon trigger mode, and each front-end computers read data from modules in response to
the common trigger. The dead-time was common to all front-end computers.

2.9.1

Signal processing

The signals generated by the detectors were processed by NIM modules and digitized to store
charge and time information. The signal processing of the beam-line plastic scintillators andthe LaBr3 :Ce detectors are described in Fig. 2.9. The output signals of plastic scintillators
were send to the counting room via optical fibers and digitized by a charge-to-digital converter(QDC) CAEN V792 for the integrated charge information, and a time-to-digital converter(TDC) CAEN V1190. For the trigger system and the counting of the beam, the coincidenceof left-hand-side and right-hand-side signals was used.

2.9.2

Signal processing of the LaBr3 :Ce detectors

A charge-to-time converter (QTC) [61] module was used to integrate the pulse from the PMT.
The QTC module has several advantages compared with the conventional charge-to-digitalconverter (QTC). The QTC has a broad dynamic range and an excellent linearity, bothnatures are in perfect match with our requirement. Also in combination with broad dynamic
range time-to-digital converter (TDC) such as CAEN V1190, the signals can be processed
without any cable delay, which can severely attenuate the signals and potentially introduces
additional noises. For these advantages, we chose the QTC as the front-end electronics. The
broad dynamic range of the QTC module was realized by having three independent capacitorswith diﬀerent capacitances to integrate the pulse charge, resulting in three independent output
signals with diﬀerent dynamic range and resolution. The energy resolution was better as the
dynamic range was narrower. Those output signals are denoted as Hi GAIN, Medium GAIN,
and Low GAIN (in the order of narrow to broad dynamic range).
The signals from the LaBr3 :Ce detectors were first put into the active voltage divider(LABRVD [58]) to duplicate the signal, and then directly put into the QTC module. The

41

42

Chapter 2 Experimental Apparatus
QTC module was equipped with a preamplifier, leading-edge discriminator, low-pass filterand charge integration circuit already, and no other modules were necessary [61]. The output
pulse, whose width was proportional to the integrated charge, was directly put into the
TDC CAEN V1190 module to digitize the data. In this way, this signal processing consistedof minimum components to acquire the quality data. For the trigger system, the logicalconjunction among the array of the fast outputs from the built-in discriminator was used.

QDC CAEN V792

Delay
Pla-L

Opt Tr

Opt. Fiber

Optical Transmitter

Leading Edge Discriminator

Opt ReL.E.D.

TDC CAEN V1290

Optical Receiver

ScalerTrigger

Pla-R

Opt Tr

Opt. Fiber

Opt ReL.E.D.

TDC CAEN V1290
Delay

Hi GAIN

Active Voltage Divider

LaBr3

LABRVDQDC CAEN V792

QTC

TDC CAEN V1190

Medium GAINTDC CAEN V1190

Low GAIN

Fast Amp Output

TDC CAEN V1190
TriggerFig. 2.9: Schematic view of signal processing.

2.9.3

Trigger condition

Two types of triggers were mainly used. For the reaction-biased trigger, the coincidence of
LaBr3 :Ce, F3, F7, and F11 plastic scintillator signals were used. This condition guarantees the beam-like particles impinged to the secondary target, emitted something into the
LaBr3 :Ce array, and reached to the end of the spectrometer, which fulfilled the condition of
the inelastic scattering reaction. For the minimum-biased trigger, the F7 plastic scintillator
signal was used. Any beam-like particles passed the F7 plastic scintillator fulfilled this condition, so that the condition was minimum-biased. This trigger was used for the calibration
purpose, such as to estimate the acceptance of the ZeroDegree spectrometer. In the actual
data acquisition, the rate of minimum-biased trigger was reduced by the prescaler module
not to compromise the DAQ live rate.

43

Chapter 3

Data Analysis
3.1

Calibrations

3.1.1

Acceptance of the ZeroDegree spectrometer

Because we used the very thick 4.8 g/cm2 gold target, the angular distribution of the secondary beam was strongly broadened by the multiple Coulomb scattering. In order to calculate the absolute cross sections, the angular acceptance of the ZeroDegree spectrometer had
to be well understood. The angular acceptance of the spectrometer was measured using thesecondary beam transported with a special optics. The special optics had large horizontal
angle distribution to scan the acceptance of the spectrometer. The angular distribution wasmeasured event-by-event basis using three PPACs installed around the secondary target area.
PPAC position resolution
The position resolution of a PPAC was estimated using the secondary beam. The resolution
was evaluated as the residual distribution calculated from three PPACs installed around thesecondary target area as shown in Fig. 3.1. The residual was calculated from the following
formula where xn was the horizontal or vertical hit position measured by each PPAC:∆xres = x2 − (x3 − x1 ) ×

500
(mm).
500 + 1684

The residual had gaussian-like distribution as presented in Fig. 3.2. The distribution was wellfitted by a gaussian with σ=0.48 mm. The value was consistent with the typical resolution.
In the following analysis, each PPAC was assumed to have the same resolution as estimatedhere.
Scattering angular resolution
The scattering angle θ of the reaction product at the secondary target was measured by
the PPACs as shown in Fig. 3.3 where ⃗a = (x1 , y1 , z1 ) was the secondary beam vector and⃗b = (x2 , y2 , z2 ) was the scattered particle vector. They were directly determined from themeasured position of beam-like particles, and ⃗c = ⃗b − ⃗a. Here the scattering angle θ was
calculated as following:⃗a · ⃗c).
θ = arccos(|⃗a||⃗c|
The scattering angler resolution was estimated from the position resolution using the error

44

Chapter 3 Data Analysis

PPAC1

PPAC3

PPAC2
Measured Position
Calculated Position

Residual

500 mm1684 mmFig. 3.1: Definition of residual for the PPAC resolution estimation.

propagation relationship. The estimated resolution was 3.9 mrad. The value was reflectedon the distorted-wave Born approximation analysis discussed in the next Chapter 4.
The angular acceptance of the ZeroDegree spectrometer
The acceptance of the ZeroDegree spectrometer was measured as a function of scattering
angle θ. As an inevitable nature of the secondary beam, the beam itself was angled withrespect to the beam axis defined as θout in Fig. 3.3. Because of this θout contribution, theangular acceptance got worse than considering the contribution of the scattering angle θalone. In order to consider this eﬀect, we measured the angular acceptance as the function of
scattering angle θ in the following way. The angler acceptance of the ZeroDegree spectrometer
was defined in the first step as a function of θout , fAcc (θout ). This quantity was determined
independent of the nature of the secondary beam. In the next step we deduced the acceptancefunction as the scattering angle θ, fAcc (θ), by applying θout gate to have the average θout of
the secondary beam. The acceptance function was tested to reproduce the total transmissionof the secondary beam. Here the total transmission was defined as the total number of thesurvived beam divided by the total number of the injected beam. The survived beam meant
the beam detected at the end of the spectrometer (detected by the plastic scintillator at F11
achromatic focal plane).
The fAcc (θout ) was measured using the calibration beam with the α target. The calibration

3.1 Calibrations

45

×103

PPAC
Residual
Distribution

50

Counts / 0.1 (mm)

40

Fit

30

20

10

0
-10

-8

-6

-4

-2

0

2

4

6

8

10

Residual (mm)Fig. 3.2: The measured residual distribution of a PPAC for the resolution estimation.

beam was produced using the special optics to have broad θout distribution at the secondary
target position. The beam spacial distribution within the target plane was ignored. The
fAcc (θout ) was determined as following:fAcc (θout ) =NInjected∧Survived (θout ).
NInjected (θout )The function is plotted in the top panel of Fig. 3.4. The errors were determined fromthe binomial distribution. In the next step we considered the contribution of the angularspread of the secondary beam itself to reproduce the total transmission. The average angularspread of the secondary beam was ∼15 mrad. The acceptance function as the function of θ
corresponds to the fAcc (θout ) folded (or worsened) by the σ=15 mrad gaussian. In reality
f olding was performed by choosing the large θout gate using PPACs. The acceptance function
as a function of θ was defined in the same way:
fAcc (θ) =NInjected∧Survived (θ).
NInjected (θ)

The acceptance function as the function θ is presented in the bottom panel of Fig. 3.4.

46

Chapter 3 Data Analysis

secondary target
PPAC1

PPAC3

PPAC2

scattering
angle θθoutabeam axiscb
500 mm1058 mm

876 mmFig. 3.3: Definition of the scattering angle θ for the ZeroDegree spectrometer acceptanceestimation.

The function was well fitted by the sum of two Fermi functions,fAcc (θ) =

V1
V2
+
.
θ−r1
2
1 + exp( a1 ) 1 + exp( θ−r
a2 )

(3.1)

The parameters are tabulated in Tab. 3.1.
In order to confirm the validity of the acceptance function, we checked if the total transmission of the beam was reproduced by this function of not. The total transmission T was
defined by:
NInjected∧Survived
T =,NInjected
here Injected was defined by the number of the beam particle counted at F8 plastic scintillator(right before the targets), and Survived was defined by the number of the beam particle
counted at F11 plastic scintillator (at the end of the Zerodegree spectrometer). The totaltransmission of the beam with the α target was measured to be 91 %, and with the gold
target it was 85 %. Because of the multiple Coulomb scattering, calculated to be 7.7 mradin average, the total transmission was worse in the gold target. If the acceptance function
fAcc (θ) was good enough, the total transmission should be reproduced by:3.1 Calibrations

47

T =

NInjected∧Survived=
NInjected∫
fAcc (θ) × r(θ),θ

where r(θ) was the scattering angler distribution of the beam particle after the targets. Ther(θ) was measured by the minimum-biased trigger (F7 plastic scintillator trigger) data. The
total transmission of either target were reproduced within 2 %, and we had concluded thatthe acceptance function was reliable.
Table 3.1: The acceptance of the ZeroDegree spectrometer parametrized by the sum of two
Fermi functions.
V1

a1 (mrad)r1 (mrad)V2

a2 (mrad)r2 (mrad)

0.280

4.68

15.7

0.72

7.77

40.2

3.1.2

Beam position with respect to the liquid-helium cell

The position of liquid-helium cell with respect to the 20 O beam was measured using thecalibration beam. Since the eﬀective diameter of the cell was close to the beam diameter, it
was necessary to confirm if the beam actually hit the eﬀective area or not. Meanwhile thegold target had large area and all the beam was expected to hit the target. The measurement
followed the similar procedure as the acceptance measurement. Because the liquid-helium
cell was made of aluminum, when the beam particle hit the cell itself, the beam particle
lost large energy to be out of the acceptance of the spectrometer. In this way, the cell
could actually be seen by comparing the injected beam position distribution at the target
and the accepted beam position distribution. Here the accepted meant the beam particle
detected at the end of the ZeroDegree spectrometer (F11 Plastic scintillator). In order to
let the beam particle go through the whole cell structure, again the special optics to havelarge position distribution at the target position was used. The position of the beam particle
at the cell was reconstructed by the PPACs. The cell structure was diﬀerentiated by the
injected divided by the accepted beam position distribution. The measured target position
is presented in Fig. 3.5. The window area is indicated by brighter color, the window frame
area is indicated by darker color. The cell itself is indicated by the less event area. The
center of the target was (x, y) = (0, −2.0) in millimeter with respect to the beam axis. The
20
O beam position profile superimposed with the cell structure is presented in 3.6. It was
revealed that 11 % of the beam could possibly hit the window frame, made of 1-mm-thickaluminum. However due to the limited resolution of the tracking detectors, estimated to be
in order of a few millimeters, the eﬀect was presumably smaller than this. In order to accountfor this eﬀect, we introduced the eﬀective thickness of the window frame, as an extra target
the beam particle went through. The eﬀective thickness was evaluated by the the energy-lossof the beam particle went through this window frame region using the PPACs. The eﬀective
thickness was estimated to be 6.8 µm of aluminum, and considered to be an extra target.
This value was used in the distorted-Wave Born approximation analysis discussed in the next
chapter.

48

Chapter 3 Data Analysis

1
0.9
0.8

f acc(θout)

0.7
0.6
0.5
0.4
0.3

ZeroDegree
Acceptance

0.2
0.1
0
0

10

20

30
θ out (mrad)

40

50

1

Fit

0.9
0.8
0.7
f acc(θ)

0.6
0.5
0.4
0.3

ZeroDegree
Acceptance

0.2
0.1
0
0

5

10

15

20
25
θ (mrad)

30

35

40

45

Fig. 3.4: The measured acceptance of the ZeroDegree spectrometer. As a function of θout (toppanel), and θ (bottom panel).

3.1 Calibrations

49

60

Frame
Window

40

y (mm)

20

0

-20

-40

-60
-60

-40

-20

0
x (mm)

20

40

60

Fig. 3.5: The measured liquid helium cell position.

60

Frame
Window

40

y (mm)

20

0

-20

-40

-60
-60

-40

-20

0
x (mm)

20

40

60

Fig. 3.6: The measured 20 O beam profile. Overlaid with the measured cell structures.

50

Chapter 3 Data Analysis

3.2

Analysis of the Beam Particle

3.2.1

Particle identification of the secondary beam

The TOF-∆E-Bρ method was used for the particle identification. The time of flight wasmeasured by the plastic scintillators installed at F7 and F3 achromatic focal planes. The
flight length was 46.6 m. The ∆E was measured by the plastic scintillator at F7. At this mass
region, the ∆E was enough to separate the charge of the particle. The Bρ was measured by
the horizontal position at F5 dispersive focal plane, using the plastic scintillator. The time
diﬀerence of the left-hand-side PMT and right-hand-side PMT was used to determine theposition. Although the resolution was not as good as the PPACs, the plastic scintillator could
operate at higher count rate. It indeed had suﬃcient resolution to completely separate theoxygen mass region. The A/Z was calculated following Eq. (2.5). The particle-identification
plot is shown in Fig. 3.7. The vertical axis is Z and the horizontal axis is A/Z. The 20 O wasisolated from other nuclei, although high count rate caused pile-up of the measured charge
appeared as vertical line.

12

20

O

10

Z

8

6

4

2

2.4

2.5

2.6
A/Z

2.7

2.8

Fig. 3.7: The particle-identification plot of the secondary beam.

3.2.2

Particle identification of the reaction product

The same TOF-∆E-Bρ method as discussed in the BigRIPS fragment separator was used
for the particle identification. The time of flight was measured by the plastic scintillators
installed in F8 and F11 achromatic focal planes. The flight length was 36.5 m. The ∆E wasmeasured by the plastic scintillator at F11. The Bρ was measured by the horizontal position

3.3 Analysis of the γ Rays

51

at F9 dispersive focal plane using the MWDC. The particle-identification plot is presented
in Fig. 3.8. The 20 O was clearly separated from other nuclei. In order to choose the inelastic
scattering channel, both the secondary beam and the reaction product was chosen to be 20 O
using these particle identification plots.

12

20

O

10

Z

8

6

4

2

2.4

2.5

2.6
A/Z

2.7

2.8

Fig. 3.8: The particle-identification plot of the reaction product after the secondary target.

3.3

Analysis of the γ Rays

3.3.1

Calibration of the LaBr3 :Ce γ-ray detector array

The LaBr3 :Ce detector array was calibrated using γ-ray checking sources, 60 Co (1.173 and1.333 MeV), 88 Y (0.898 and 1.836 MeV), and 244 Cm-13 C [62] (6.130 MeV). The sources thatemits relatively higher energy γ rays were used in order to obtain good calibration at higher
energy of interest like 8 MeV. Because the LaBr3 :Ce crystal have rather strong internalradiation background especially below 1 MeV, the 0.898-MeV γ ray from the 88 Y source
was excluded from the calibration. The spectrum with checking sources from one detector
module is presented in Fig. 3.9 top (with 88 Y) and bottom (with 60 Co). One can see strong
contributions from internal activity along with the γ rays from the sources, as mentioned
in Sec. 2.8. The calibration was performed including any eﬀects from the PMT, QTC and
TDC. The QTC ASIC chip had three gain range to realize wide dynamic range. For our
experiment, the Medium gain range was tuned to cover the dynamic range of interest, 1 to
10 MeV, by adjusting the PMT high voltage bias. The γ-ray peaks from Medium gain werefitted with gaussian plus linear background. Four centroids from each source in the unitof TDC channels were fitted by a linear function to obtain the channel to MeV conversion
parameters. The linear response of the LaBr3 :Ce detector including PMT was tested in Ref.

52

Chapter 3 Data Analysis[58] to be excellent below 10 MeV. The residuals versus input γ-ray energies of the LaBr3 :Cedetector with LABRVD voltage divider, namely our configuration, is presented in Fig. 3.10
as red dots. One can see that the residuals are lower than ∼ 10 keV below 10-MeV γ-ray
input. The response of the QTC Medium gain output to the input pulse charge was tested inRef. [61]. It is proved be suﬃciently linear ( < 0.2 % in deviation from linear fit) in the whole
dynamic range of Medium gain. Therefore, using the linear function as a calibration function
was well justified below 10 MeV. The residual of the linear fit was estimated and included in
the systematic error. However this eﬀect was overwhelmed by the limited resolution due tothe Doppler broadening as seen in Fig. 3.17.

3.3.2

A Monte-Carlo simulation of the γ ray

A Monte-Carlo simulation was built with the Geant4 [63] simulation tool kit to obtain response functions of the LaBr3 :Ce γ-ray detector. As a base of the simulation, the geometriesof the detectors, targets, target structure, target chamber, target chamber support structure,and X-ray shields were considered. The schematic view of the simulated geometries is presented in Fig. 3.11. In order to validate the simulated geometries, the photo-peak eﬃciencyof the standard γ source was compared with the simulation. The calibration data was takenwith the 88 Y standard source attached to either the liquid helium cell or the gold target. The
1.836-MeV photo peak was fitted with the gaussian plus background function to exclude theinternal radiation background. The photo-peak eﬃciency was determined by the area of thegaussian divided by the number of emitted 1.836-MeV γ ray. The number of emitted γ rays
were calculated from the source strength of the day the data was taken. The photo-peakeﬃciency of the LaBr3 :Ce array was measured to be 0.86±0.02 (%) for the gold target, and
0.92±0.02 (%) for the liquid-helium cell. The error derived from the standard source itself,
which had uncertainty of 2 %.
The simulated eﬃciency was 0.86 % for the gold target, and 0.91 % for the liquid-helium
cell. The simulated eﬃciencies were within the errors of the measured values, therefore we
had concluded that the simulation was suﬃciently reliable. The minor diﬀerence between
the simulated value and measured value was most consistent with the slight diﬀerence ofthe source position. The diﬀerence of the source position had strong eﬀect on the eﬀective
thickness of the X-ray shields seen from the γ ray.
Response functions generated by the Monte-Carlo simulation
Since the γ rays were emitted from the moving source, the simulation including the Doppler
eﬀect was critically important to understand the γ-ray spectrum. The simulation from themoving γ-ray emitter was constructed as following: The beam of 20 O at 276(9) MeV/nucleon
was injected into the targets and continuously lose its energy. The energy loss was calculatedby the LISE++ code [64]. The 20 O was excited randomly at some point within the target andimmediately emitted the γ ray (0.1 ps half-life assumed for all levels). The energies of the γrays seen from the detector system were Doppler-shifted according to the energy of the beam
particle when the decay took place. The γ rays were anisotropically emitted according to the
angular distribution of the decay pattern, although this eﬀect was minor (see Sec. 4.1 fordetail). The emitted γ rays reacted randomly with the detectors and surrounding materials
according to the electromagnetic reaction cross sections. The deposited energy within the3.3 Analysis of the γ Rays

53

104

LaBr3:Ce Spectrum
Y γ source

88

103

Counts/5 (keV)

138

-

La β
Internal

88Yγ
898 keV

102

138

La EC
Internal

10

88

227

Yγ
1836 keVAc αInternal

1
500

1000

1500

2000

2500

3000

3500

4000

E γ (keV)

LaBr3:Ce Spectrum
60

10Counts/5 (keV)

138

-

La β
Internal

10

Co γ source

3

2

60

Co γ
1173 keV60

Co γ1333 keV138

La EC
Internal

10

227

Ac α
Internal

1
500

1000

1500

2000

2500

3000

3500

4000

E γ (keV)

Fig. 3.9: The LaBr3 :Ce energy spectrum from one detector module acquired with 88 Y (toppanel) and 60 Co (bottom panel) γ-ray checking source.

54

Chapter 3 Data Analysis

Fig. 3.10: The residuals of the LaBr3 :Ce detector as a function of input γ-ray energy. From
A. Giaz et al. [58].

detector crystal was directly interpreted as measured energy. In this simulation, the Dopplerbroadening eﬀect was automatically included via the energy loss of the γ emitter in the targets
and the solid angle (or the geometries) of the detectors. The simulated energy spectra were
Doppler-corrected exactly in the same manner as the measured spectra as discussed in thefollowing section (Sec. 3.3.5). The simulated Doppler-corrected spectra of several 20 O states
are presented in Fig. 3.17. The injected number of γ ray was 1.0×107 . A photo peak wasformed at 1673 keV γ ray along with single-escape plus Compton-edge peak at lower energy.
At 2400 keV, the photo peak and other structures are fused to make one peak, with shoulderlike structure at lower-energy. At even higher energy γ rays above 5000 keV, the photo peak
was fused together with single-escape peak, double-escape peak, and other peak-like structure
like Compton edge to make one broad peak. Therefore the fit using these response function
was critically important to measure the yields of emitted γ rays, as discussed in the following
section (Sec. 3.3.8).

3.3.3

Time walk correction

Because we used a leading edge discriminator built in QTC for wide dynamic range, the γ-ray
energy resolution was strongly worsened by the walk eﬀect (or slew eﬀect). An example oftime (with respect to the F8 plastic scintillator) and charge correlation plot of one LaBr3 :Ce3.3 Analysis of the γ Rays55
φ130
t=3 Alt=1 Pbt=0.5 Sn

Chamber
SupportTargets

φ150
t=3 Al

Beam

γ ray

t=1 Pbt=0.5 SnAll units in (mm)Fig. 3.11: A view of the geometries used for the simulation generated by the Geant4 .

signal is presented in Fig. 3.12, from the gold target data. The LaBr3 :Ce signal was selectedby requiring both the secondary beam and the reaction product was identified to be 20 O.
The walk eﬀect was seen as a curve with later time (upward in vertical axis) at lower energy(smaller in horizontal axis), and otherwise at higher energy. The γ-ray appeared as a strong
curve at earlier time (downward in vertical axis) , and other particle background followed.
The curve was fitted detector-by-detector with the following function:f (ch) = c1 + c2 log |ch − c5 | + c3 (log |ch − c5 |)2 + c4 (log |ch − c5 |)3 ,where c1 to c5 are free parameters of the fits. The spectra were corrected with these functions.
The parameters are tabulated in Tab. 3.2.
The corrected time and charge correlation plots are presented in Fig. 3.14.

3.3.4

Time-of-flight gate for background suppression

Because the LaBr3 :Ce detectors are located right downstream of the secondary target, astrong background arising from particles was expected. The particle background was sup-

56

Chapter 3 Data Analysis
Table 3.2: Walk correction parameters.
c1

c2

c3

c4

c5

LaBr3 :Ce 1-308.518-48.4155

7.10991-0.382417

1506.79

LaBr3 :Ce 2-125.818-135.244

20.5509-1.07287

1425.76

LaBr3 :Ce 3-282.322-58.9498

8.72505-0.466398

1437.45

LaBr3 :Ce 4-304.511-44.41716.12928-0.315045

1524.11

LaBr3 :Ce 5-275.558-64.4813

9.20238-0.466202

1421.49

LaBr3 :Ce 6-63.7103-152.694

22.3570-1.12759

1357.95

LaBr3 :Ce 7-238.655-78.5006

11.9033-0.638964

1481.01

LaBr3 :Ce 8-351.215-20.5423

2.02194-0.0816067

1392.68

pressed using the time of flight diﬀerence between the γ ray and the particle. The excellent
time resolution of the LaBr3 :Ce crystals (<50 ps [58]) made this possible. The time of flight
was calculated between the plastic scintillator at F8 achromatic focal plane and the LaBr3 :Cedetectors. Because the target itself was not a detector, the time of flight from the target to
the LaBr3 :Ce detectors, TOFγ , was calculated as following:
TOFγ = TOFtot − TOFbeam = TOFtot −Lbeam− ∆βbeam (nsec),c · βbeam

here the TOFtot was the measured time of flight from the plastic scintillator to the LaBr3 :Cedetectors. The TOFbeam was the time of flight of the beam from the plastic scintillator to thetarget, and this quantity was calculated from the βbeam and Lbeam = 1.12 m, the distance
between the plastic scintillator and the target. The βbeam was measured by the time of flightbetween the plastic scintillators at F3 and F7 achromatic focal planes. The ∆βbeam was theenergy loss of the beam in the target. Assuming the reaction took place at the center of thetarget, the energy loss of the beam from the surface to the center of the target. The energy
loss was calculated by LISE++ code [64]. This term was introduced to improve the time offlight resolution, however not significant.
The TOFγ and γ-ray energy correlation plot is presented in Fig. 3.14-(a):20 O+α and (b):20 O+Au. Because we needed the TOFγ only for the separation of particle background,the TOFγ was plotted in a relative value in vertical axis with faster TOFγ downwards, andnot absolute time of flight. The de-excitation γ rays from 20 O appeared as horizontal line
around TOFγ =0. In the energy above 8000 keV, it is clear that the γ-ray line was the fastest
in time of flight, followed by the particle background. There appeared some backgrounds notcorrelated with the time of flight below 8000 keV, but γ-ray line was clearly separated. At
around TOFγ =-2, there was another line in correlation with the time of flight. The line wasmost consistent with the γ-ray background originated from the beam line detectors installedat F8 achromatic focal plane area. The γ-ray spectrum was obtained by applying |TOFγ | < 1
nsec gate. The TOFγ resolution was evaluated by fitting the TOFγ with gaussian at 6000
keV to 8000 keV energy region. The TOFγ distribution is presented in Fig. 3.13 along withthe fit line. The resolution was 610 ps in FWHM. This value included any eﬀects from thedetectors and analysis, and not the intrinsic resolution of the LaBr3 :Ce crystal alone.

3.3 Analysis of the γ Rays

57

-400

Relative time to trigger (ns)

-410

-420

-430

-440

20

O+Au

-450

-460

1600

1800

2000 2200 2400
Relative charge (ch)

2600

2800

Fig. 3.12: An example of walk eﬀect seen in the QTC electronics. Relative charge (corresponding to γ-ray energy) versus relative time to trigger (corresponding to the time offlight).

3.3.5

Doppler correction

The γ-ray energy was Doppler corrected on an event-by-event basis. The Doppler-correction
formula is given in Eq. 2.6:
1 − β cos θlab
E0 = Elab √,(3.2)
1 − β2where Elab is the detected energy, and E0 is the energy in the rest frame. For the Doppler
correction, one needed to know the γ-ray emitted angle θlab and the velocity divided by
the speed of light, β, of the emitter. Because the LaBr3 :Ce detectors employed were notsensitive to the hit position of the γ ray, the exact angle θlab was unknown. For this reason,

58

Chapter 3 Data Analysis

500

TOFγ
Resolution

Counts / 0.05 (ns)

400

Fit

300

200

100

0
-5

-4

-3

-2

-1

0

1

2

3

4

5

TOF (ns)Fig. 3.13: The TOFγ distribution for the resolution estimation.

θlab was approximated by the center angle of the crystal, θlab = 30◦ . This approximation
worsened the energy resolution, and one of the cause of so-called the Doppler broadening.
The eﬀect of the Doppler broadening was evaluated by the Monte-Carlo simulation discussed
in the following section (Sec. 3.3.8). The other factor, β was determined event by event as
βbeam . But because the 20 O beam continuously lost its energy in the secondary target, andthe exact point of the target at which the reaction took place was unknown. This meantthat the exact β of the beam when the reaction took place was unknown. Here the β wasapproximated assuming all the reaction took place at the half in thickness of the targets.
This approximation was another contributor of the Doppler broadening.

3.3 Analysis of the γ Rays

59

20
15
10

TOFγ (ns)

5
0
-5
-10

(a) 20O+α-15

Not Dopp. Corr.
-20
0

2000

4000

6000
8000
E γ (keV)

10000

12000

14000

20
15
10

TOFγ (ns)

5
0
-5
-10

(b) 20O+Au-15

Not Dopp. Corr.
-20
0

2000

4000

6000
8000
E γ (keV)

10000

12000

14000

Fig. 3.14: The correlation plot of the γ-ray energy Eγ (horizontal) and its time of flight TOFγ
(vertical). From the 20 O+α (top panel), and 20 O+Au reaction (bottom panel). The γ-ray
energy Eγ is not Doppler-corrected.

60

Chapter 3 Data Analysis

3.3.6

The Doppler-corrected γ-ray spectra

The Doppler corrected γ-ray spectra are presented in Fig. 3.15-(a):20 O+α and -(b):20 O+Au
as black dots with statistical error bars. Because the resolution was restricted by the Doppler
broadening, we needed inputs from the existing data to further resolve the spectrum. The
excited states of 20 O was investigated by several diﬀerent reactions: a Coulomb excitation
with the lead target [43, 44], a proton inelastic scattering [65], a 19 O(d,p) one-neutron transfer
reaction [66], a 18 O(t,p) two-neutron transfer reaction [67], a fragmentation reaction of 36 S
[68], a β decay of 20 N [69], a 10 Be(14 C,α) transfer reaction [70], and a 14 C(7 Li,p) transfer
reaction [71]. With respect to our reaction, the inelastic scattering, the spectrum of thegold target was expected to be consistent with the Coulomb excitation with the lead target[43, 44]. When comparing the 20 O+α and 20 O+Au spectra, there are three peaks remarkably
stronger in the gold target, but weaker in α target at 3.68 MeV, 5.35 MeV, 6.85 MeV. Thismeans these states are excited stronger by the Coulomb potential (the gold target), but
weaker by the nuclear potential (the α target). This is consistent with the expected dipole
nature of these excited states, as discussed for example in [72]. Indeed, the present spectra
were consistent with these results.
The γ-γ coincidence analysis
The decay scheme of the excited states was examined by the γ-γ coincidence analysis. Because our setup detected de-excitation γ rays, the decay scheme and the branching ratio
was necessary to extract the cross sections. Unlike the existing study in Ref. [43, 44] using
the theoretical prediction as a decay scheme, the present highly intense beam enabled us to
examine the decay scheme via the γ-γ coincidence analysis. The decay event emitting two γrays was selected by multiplicity = 2 in the LaBr3 :Ce array. Two γ rays were sorted in orderof energy and plotted as presented in Fig. 3.16-(a) 20 O+α and -(b) 20 O+Au. From theseplots, it was seen that some states decay via the 2+
1 state at 1.67 MeV, appeared as a broad
horizontal line in Fig. 3.16. For example, the peak at 6.85 MeV appeared in the γ singles
plots did not appear in the γ-γ coincidence plots. This means the peak at 6.85 MeV derived
from the direct decay to the ground state. Another peak at 5.18 MeV appeared both in theγ singles and the γ-γ coincidence plots (most clearly seen in Fig. 3.16-(b) 20 O+Au becauseof its high statistics). This means the peak at 5.18 MeV derived from the state at 6.85 MeV
decay via the 2+
1 state at 1.67 MeV. For a further decomposition, fits were necessary. The
fits are discussed in the following section (Sec. 3.3.8).
Another important information obtained from these plots was that there was no other decaybranch than the 2+
1 state at 1.67 MeV observed within statistical limits. We considered two−
1− candidate states decay via 2+
2 state at 4.07 MeV. If the 1 candidate state at 6.85 MeV
decay via the 2+
2 state at 4.07 MeV, the emitted gamma-ray energies should be 2.78 MeV+
and 4.07 MeV (1− → 2+
2 → 0g.s. decay branch), or 2.78 MeV, 2.04 MeV and 1.67 MeV++−+(1 → 22 → 21 → 0g.s. decay branch). If these decay branches were significant, there should
be a characteristic peak at 2.78 MeV in the γ singles plot and a locus at (Eγ1 , Eγ2 )=(4.07
MeV, 2.78 MeV) or (2.78 MeV, 2.04 MeV) in the γ-γ coincidence plots. However such
events were not observed. For the other 1− candidate state at 5.35 MeV, if the 1− at 5.35
MeV decay via the 2+
2 state at 4.07 MeV, the emitted gamma-ray energies should be 1.28

3.3 Analysis of the γ Rays

61(a) Singles 20O+α103

Fit Total-Counts / 40 (keV)

12 6.84 MeV-

11 5.36 MeV
3- 5.62 MeV

102

2+2 4.07 MeV

101

2+1 1.67 MeV

χ2/ndf=1.08

1000

3000

5000
E γ (keV)

7000(b) Singles 20O+Au
Fit Total-Counts / 40 (keV)

12 6.84 MeV-

11 5.36 MeV

103

3- 5.62 MeV
2+2 4.07 MeV

102

1000

2+1 1.67 MeV

χ2/ndf=1.42

3000

5000
E γ (keV)

7000

Fig. 3.15: The Doppler corrected γ-ray singles spectra, in black dots with statistical errors.
The gray histograms overlaid are backgrounds from 24 O beam, lines are the best-fit response
functions. The fits are performed over 1900 keV to improve the sensitivities.

62

Chapter 3 Data Analysis+
MeV and 4.07 MeV (1− → 2+
2 → 0g.s. decay branch), or 1.28 MeV, 2.04 MeV and 1.67++
MeV (1− → 2+
2 → 21 → 0g.s. decay branch). If this branch was significant, there should
appear a characteristic peak at 1.28 MeV or 2.04 MeV in the γ singles plot and a locus at(Eγ1 , Eγ2 )=(4.07 MeV, 1.28 MeV) or (2.04 MeV, 1.67 MeV) in the γ-γ coincidence plots.
However such events were not observed. Through the γ-γ coincidence analysis and existing
data, the present spectra were turned out to be most simply explained by a set of states andtheir decay to the ground state and the decay via the 2+
1 state at 1.67 MeV. The spectra
were resolved by fits discussed in the following section (Sec. 3.3.8).

(a) 20O+αE γ 2 (keV)

4000

E γ 1>E γ 2

3000
2000

2+1
→

1000

E γ 2 (keV)4000(b) 20O+Au

2000

4000

6000

2000

4000

6000

E γ 1>E γ 2

3000
2000

2+1
→

1000

E γ 1 (keV)
Fig. 3.16: The γ-γ coincidence plots. The selected events where two γ rays are detected
simultaneously are plotted in order of the γ-ray energy Eγ1 > Eγ2 .

3.3 Analysis of the γ Rays

3.3.7

Estimation of γ-ray backgrounds originated from the targets

Because we used very thick targets, understanding the γ-ray backgrounds from the excitedtargets was important. In order to estimate the γ-ray contribution from the targets, we used24
O beam. Because no bound excited states were known in 24 O [73, 74], the γ-ray spectrum
from 24 O+α or Au reactions were expected to represent the backgrounds originated from thetargets. The background spectra are imposed in Fig. 3.15-(a):24 O+α and -(b):24 O+Au asgray histograms. There were no peak-like structure above the energy of 2000 keV in either
target, and well fitted by an exponential curve.

3.3.8

Decomposition of the γ-ray spectra

The γ-ray spectra were decomposed by the fits with response functions generated by theMonte-Carlo simulation. As inputs for the simulation, known excited states from the Coulomb
excitation with the lead target [43, 44] were used. For the branching scheme, the decay to
the ground state and the 2+
1 state at 1.67 MeV was considered as the result of the γ-γ
coincidence analysis. There was one peak at 3.94 MeV not reported in Refs. [43, 44] but
observed in the present spectrum of either target, and also in the γ-γ coincidence plot. The
state was most consistent with a (3− ) state at 5.62 MeV decay via the 2+
1 at 1.67 MeV state,
10
14
14
7
reported by the Be( C,α) transfer reaction [70], the C( Li,p) transfer reaction [71], andthe fragmentation reaction of 36 S [68]. The state was included in the fits as well. Since allthese references consistently observed that the (3− ) state decay fully via 2+
1 , only this branch
was included. The excited states incorporated in the fits are tabulated in Tab. 3.3. The
response functions for the fits were generated by the Monte-Carlo simulation using the Eγ
listed in Tab. 3.3 as input γ-ray energies. Simulated response functions are presented in Fig.
3.17.
The fits
The χ2 -minimum fits were performed under the condition that the energy of each response
function was fixed, while the hight and branching ratios were free parameters. Following theresults of 24 O data, the exponential function was used as a background function. Because
the 2+
1 at 1.67 MeV had orders of magnitude higher statistics and inhibit the convergence ofthe fits, the energy above 2000 keV were fitted. The fits are presented in Fig. 3.15 as lines.
The fits revealed that the spectra could be explained by five excited states, 2+
1 at 1.67 MeV,
−
−
−
2+
state
at
4.07
MeV,
(3
)
state
at
5.62
MeV,
1
state
at
5.36
MeV,
and
1
2
1
2 state at 6.84
MeV.
−
2
For the 1−
1 and 12 states, we extracted our own energy value by searching for the χ minimum energy of each state while all other parameters were fixed. The errors were determinedfrom the calibration residuals. The values were consistent with the reported values [43, 44]
within systematic errors. The branching ratio of the 1− states were determined from theγ-γ coincidence analysis of high-statistics 20 O+Au data. In the analysis, the γ-ray spectra
detected in coincidence with the 2+
1 (Fig. 3.18 in black dots with statistical errors) were fitted
in the same way. The results are presented as lines in Fig. 3.18. The branching ratios were
determined by taking the ratio of the yield of 1− → 0+ γ ray and 1− → 2+ γ ray. The yieldof 1− → 0+ γ ray was determined from the fit of γ singles spectrum. The yield of 1− → 2+
1

63

64

Chapter 3 Data Analysis
Table 3.3: Excited states and decay γ-ray energy, used to produce response functions.
Ei (MeV)πJn,iEf (MeV)πJn,fEγ (MeV)

1.67a

2+
1

0.00

0+
g.s.

1.67

4.07a

2+
2

1.67

2+
1

2.40

0.00

0+
g.s.

4.07

4.46

0+
2

1.67

2.79

5.00b

(3− )

1.67

2+
1
2+
1

0.00

0+
g.s.

5.00

1.67

2+
1

3.68

0.00

0+
g.s.

5.35

1.67

2+
1

3.94

1.67

2+
1

5.18

0.00

0+
g.s.

6.85

b

b

5.35

5.62c
b

6.85

1−
1
(3− )
1−
2

a

Values from Ref. [75].

bValues from Refs. [43, 44].

c

Values from Refs. [70].

3.33

γ ray was determined from the fit of γ-γ coincidence spectrum by multiplying the yield of++
1− → 2+
1 γ ray by the detection eﬃciency of 21 γ ray. The detection eﬃciency of one 21
γ ray was determined from the Monte-Carlo simulation to be 0.0343. In the simulation, theangular correlation between the two γ rays were ignored. The determined branching ratios
are summarized in Tab. 3.4. The errors considered were the statistics and the fits. The
branching ratios could also be determined from the γ singles spectrum fit of the 20 O+Au
spectrum, actually they were consistent within errors with each other.
The branching ratio of the 1−
2 state at 6.84 MeV was consistent with reported value inRef. [43, 44]. However the 1−
state
at 5.36 MeV was not consistent with the reported value.
1
This was most consistently due to the inclusion of the (3− ) state at 5.62 MeV. The (3− )−state decay via the 2+
1 state emits 3.94-MeV γ ray, in neighborhood with the 11 state decay+
via the 2+
1 state emits 3.68-MeV γ ray. Similar situation happened for the 22 state at 4.07
MeV. The 2+
2 state at 4.07 MeV decay to the ground state was too close to the 3.94-MeV
γ ray, produced by the (3− ) state decay via the 2+
1 state. They could not be separated by
the fits. For the cross section determination discussed in the following section (Sec. 4.1.5),+++
we used the reported branching ratio (2+2 → 0 : 28 %, 22 → 21 : 72 %) in Ref. [43, 44] andreferences therein.
Determination of the number of γ rays from the fits
The response-function fit enabled us to easily determine the number of γ rays Nγ in thehistograms. Now a simulated response function, generated by injecting 107 γ rays of energy
Ei , is taken as a function of the detected γ energy Eγ , FEi (Eγ ) (Shown in Fig. 3.17). The

3.3 Analysis of the γ Rays

65

35000

Doppler-Corrected
Response Functions

Counts / 25 (keV) / 107 events

30000

6850 keV
5350 keV
5180 keV

25000

3941 keV
3680 keV

20000

2400 keV
15000

1673 keV

10000

5000

1000

2000

3000

4000

5000

6000

7000

8000

E γ (keV)

Fig. 3.17: The response functions generated by the Monte-Carlo simulation based ontheGeant4.

measured histogram as a function of γ energy H(Eγ ) was fitted directly using the simulated
response functions:
H(Eγ ) = cE1 FE1 (Eγ ) + cE2 FE2 (Eγ ) + · · · + cEn FEn (Eγ ),where cE1 to n are coeﬃcients determined by the fits.
This means the number of detected γ rays of energy Ei , NEi , is given by:
NEi = cEi × 107 .

Table 3.4: Measured Branching ratios (Br).
Ei (MeV)Jnπ

5.36(5)1−
1

6.84(7)

1−
2

Ef (MeV)π
Jn,fBr (%)

1.67

2+
1

26(6)

0.00

0+
g.s.

74(2)

1.67

2+
1

48(6)

0.00

0+
g.s.

52(3)

Chapter 3 Data Analysis

Counts / 200 (keV)6660(a) Gate on 2+1

50

20

O+α

40

Fit Total-

12 6.84 MeV

30-

11 5.36 MeV
3- 5.62 MeV
2 +2 4.07 MeV

20

χ2 /ndf=1.05

10Counts / 200 (keV)060(b) Gate on 2+1

50

20

40

O+Au
Fit Total

30-

12 6.84 MeV-

11 5.36 MeV

20

3- 5.62 MeV
2 +2 4.07 MeV
χ2 /ndf=1.27

10
0
1000

2000

3000

4000

5000

6000

7000

8000

E γ 1 (keV)
Fig. 3.18: The Doppler corrected γ-ray coincidence spectra (Eγ1 ), where a γ ray (Eγ1 ) is
detected in coincidence with the γ ray from 2+
1 state (Eγ2 ), in black dots with statisticalerrors. Lines are the best-fit response functions.

67

Chapter 4

Distorted-Wave Born
Approximation Analysis
4.1

Distorted-Wave Born Approximation Analysis

A widely-used distorted-wave Born approximation analysis (DWBA) was performed to extract the dipole strengths. The program ECIS97 [76] was used. Inputs for the calculation,
optical potentials and transition potentials, were based on theoretical calculations. We had
extracted not only 1− strengths, but also 2+ and 3− strengths to check the consistency of
our analysis.

4.1.1

Optical potentials

Because the elastic scattering data on the unstable nuclei were strictly limited, we need to relyon the theory to obtain the optical potential. We employed the potential model proposed by
T. Furumoto et al. described in Refs. [77, 78, 79]. This potential was successfully applied tothe analysis of unstable nuclei in a similar beam energy region [50]. The 4 He was not coveredby the potential model because of unsolved theoretical issues. In this work, we extrapolated
the 4 He+20 O optical potential from heavier N = Z systems, 12 C+20 O, 16 O+20 O, 20 Ne+20 O,
and 24 Mg+20 O. Following the Refs. [77, 78, 79], the real part of optical potential V and theimaginary part of W is described respectively as 10-range Gaussian form:( 2 ))10 (∑r
αi exp − 2
V ≃
,
γ
i
i=1
W ≃

10 (
∑
i=1

(

r2
βi exp − 2
γi

))
,

where αi , βi are parameters that depend on target nuclei (t), mass and charge of projectile
nuclei (p) (20 O), and incident energy E,
αi = αi (Ap , Zp , At , E),
βi = βi (Ap , Zp , At , E),

68

Chapter 4 Distorted-Wave Born Approximation Analysis
and γi is a function of index i defined as:
γi = 0.45

i + 8 1/3
1/3
(Ap + At + 1).
18

In the present analysis, αi and βi of 4 He+20 O system was extrapolated from 12 C+20 O,
16
O+20 O, 20 Ne+20 O, and 24 Mg+20 O parameter sets using quadratic functions. For thedetailed discussion, see App. A. The sign of the real and imaginary part of the opticalpotential follows the notation in Refs.[77, 78, 79] defined by:
U (r) = V (r) + iW (r) + VCoul (r),where VCoul (r) is the Coulomb potential. In order to use the macroscopic Isoscalar dipole
transition potential proposed by M. N. Harakeh and A. E. L. Dieplink [6, 1], the opticalpotentials obtained were fitted and parametrized by the Woods-Saxon formula. The WoodsSaxon parameters are tabulated in Tab. 4.1.
Table 4.1: The Woods-Saxon parameters and their extrapolation to the 20 O+α system.
Calculated from T. Furumoto et al[77, 78, 79].
V

RR

aR

W

RI

aI

System

(MeV)

(fm)

(fm)

(fm)

(fm)

(fm)

20

O+α

-60.182.143

0.4680-89.482.940

0.872220

O+Au-173.35.951

0.6294-505.3

6.190

1.277

4.1.2

Transition potentials

Macroscopic vibrational transition potentials from deformed optical potential model [80] were
employed in the present analysis. Because of the relatively high incident energy, for thegold target, the nuclear excitation had to be properly included in addition to the Coulomb
excitation. And for the α target, the Coulomb excitation had to be included in addition to the
nuclear excitation. For this purpose, both nuclear and Coulomb potentials were consideredin either target, the gold or α target.
E2 and E3 transitions
The macroscopic transition potential of l-th multipole (l = 2, 3) using the deformed opticalpotential model is given by [1, 80]:
−βl,R RR d√
V (r)Pl4π dr−βl,I RI d
W (r)Pl ,
+i √
4π dr

∆U (r) =(4.1)

where the βl,R , βl,I is the real and imaginary nuclear deformation respectively, RR , RI is
the real and imaginary radius of the optical potential respectively, V (r), W (r) is the realand imaginary part of the optical potential respectively, and Pl is the l-th order Legendre

4.1 Distorted-Wave Born Approximation Analysis

69

polynomial. As a relationship between real and imaginary nuclear deformation, we used a
simple assumption [6]:βl,R RR = βl,I RI .

(4.2)

Inverse kinematics
For the purpose of avoiding the misunderstanding between the inverse kinematics, i.e. thepresent experiment, and the normal kinematics, the DWBA analysis was carried out in thenormal kinematics at initial stage. The observables were converted into the inverse kinematics
where necessary. In the following DWBA analysis, the target, represented by t, was the 20 O,
and the probe, represented by p, was the gold or the α particle.
Transition strength B(El) ↑The Coulomb deformation βl,C is directly connected to the B(El) ↑ strength via the widelyused macroscopic model [81, 80]:
B(El) ↑=

( 3Zeβl )2l,C RC(e2 fm2l ),(4.3)
4π
here Z is the charge of the excited nuclei, e is the elementary charge, and RC is the Coulomb
radius given by:
RC = r0C A1/3 (r0C = 1.3 (fm)).

(4.4)

Reduced radius
For the reaction between the heavy nucleus, the radius is replaced by the reduced radius
following the prescription described in Ref. [80] to account for the size of the nuclei itself.
The reduced radius is defined as:
R′ = r0 (At1/3+ Ap1/3 ),

(4.5)

1/3

is the probe nuclei, 197 Au. Using

1/3

where the At is the target nuclei, here 20 O, and Ap
the invariance of deformation length δl ,′
δl = βl,R R = βl,R
R′ .

(4.6)

The deformations were also replaced by reduced deformations,
′
βl,R
=

1/3

At
1/3

At

1/3

+ Ap

βl,R .

(4.7)

The similar reduced radius is also introduced to the Coulomb deformation βl,C given by:
′
βl,C
=

(

)l

1/3

At
1/3

At

1/3

+ Ap

βl,C .

(4.8)

The reduced radius was introduced for 20 O+Au system. However, in 20 O+α system, thereduced radius was not introduced, following the standard prescription that the α particle
should be treated as a point-like particle.

70

Chapter 4 Distorted-Wave Born Approximation Analysis

E1 transition
Because the low-energy dipole states were generally known to be the mixture of Isoscalar andIsovector excitation, both eﬀect had to be taken into account. In the analysis, we assumed
that the Coulomb potential coupled only to the Isovector E1 excitation, and the nuclear
potential coupled only to the Isoscalar E1 excitation. The nuclear potential can possibly
excite Isovector dipole excitation through the Isospin mixing, however this eﬀect was smallenough to be ignored in the present analysis. In order to extract Isoscalar dipole strength,
Isoscalar dipole transition potential proposed by M. N. Harakeh and A. E. L. Dieplink [6, 1]
was used. This transition potential was first used in the analysis of low-energy dipoles in 16 O
and 208 Pb [6], and recently used in 40,48 Ca [28]. The Isoscalar dipole transition potential isgiven by:
( d2
−β1,R ( 2 d
5
d
d ))
3r
+ 10r − ⟨r2 ⟩R+ϵ r 2 +4
V (r)
RR
dr
3
dr
dr
dr
(4.9)
( d2
−β1,I ( 2 d
5 2 d
d ))
+i
3r
+ 10r − ⟨r ⟩I+ϵ r 2 +4
W (r).
RI
dr
3
dr
dr
dr
√
√
Note that the right hand side of this formula is multiplied by a factor of 2l + 1 = 3 fromthe original one given in Refs. [6, 1]. This is necessary to take into account the summation
over magnetic substates, also pointed out in other work such as Ref. [82] by Youngblood
et. al.
The ϵ is given by:∆U (r) =

(
ϵ=

4
5
+
E0
E2

)

ℏ2
,
3mN A

(4.10)

where
E2 = 65A− 3 MeV,

(4.11)

E0 = 80A− 3 MeV,

(4.12)

1and
1were the excitation energies of the Isoscalar giant quadrupole and Isoscalar giant monopole
resonances. These terms were due to the center of mass corrections [6, 1]. ⟨r2 ⟩, ⟨r4 ⟩ were the
2nd and 4th moment of the real part of the optical potential defined as:∫
V (r)r4 dr2⟨r ⟩R = ∫,(4.13)
V (r)r2 dr∫
V (r)r6 dr⟨r ⟩R = ∫.
V (r)r2 dr
4(4.14)For the relationship between the real and imaginary deformation, the same relationship
Eq. (4.2) was introduced, following the prescription in Ref. [6]. The Isoscalar and Isovector
E1 strengths were independently determined so as to reproduce 20 O+Au and 20 O+α cross4.1 Distorted-Wave Born Approximation Analysis

71

sections. In other words, the Coulomb deformation β1,C and nuclear deformation β1,R was
independently tuned to reproduce both 20 O+Au and 20 O+α cross sections at the same time.
As an unit of Isovector E1 strength, B(E1) ↑ (e2 fm2 ) was employed using the same macroscopic model in Eq.(4.3). For the unit of Isoscalar strength, we used the Isoscalar dipole
energy-weighted sum rule fraction (%) (ISDEWSR fraction). The ISDEWSR [6, 1] is defined
by:
(
)
3ℏ2 A
25 2 2
4
2
ISDEWSR =11⟨r ⟩R − ⟨r ⟩R − 10ϵ⟨r ⟩R (e2 fm6 MeV),
32mN π3(4.15)

where mN is the atomic mass unit and A is the mass number of excited nuclei. The
maximum of the nuclear deformation β1,R is limited by the ISDEWSR. Note that this formula
is again multiplied by a factor of three compared with the original formula given in Refs. [6, 1],
to account for the summation over magnetic substates [82]. The maximum β1,R exhausting
100% of ISDEWSR is given by:
2
β1,R
=

6πℏ2
R2
,
2 2
2
mN AEx 11⟨r4 ⟩R − 25
3 ⟨r ⟩R − 10ϵ⟨r ⟩R(4.16)

where Ex is the excitation energy and R is the nuclear radius given by:
R = r0 A1/3 (r0 = 1.2 (fm)).

(4.17)The nuclear deformation β1,R is converted into ISDEWSR fraction using this relationship.

4.1.3

Angular distributions of scattered particles

The angular distribution of the scattered 20 O was calculated by the DWBA and used toestimate the eﬀect of acceptance. The calculated angular distributions are presented inFig. 4.1 (2+ , 3− states) and Fig. 4.2 (1− states) as red-solid lines. The calculated angular
distribution was folded by the resolution and multiplied by the acceptance function Eq.(3.1)
to calculate the accepted ratio. The integrated angular resolution of the detectors (4.0 mrad)
and the multiple Coulomb scattering in the targets (1.0 mrad in the liquid helium system,
8.0 mrad in the gold target) was used as a folding angle. The calculated angular distribution
was folded by the Gaussian with σ (rad) as the folding angle. The folding was given by:
( dσ
dΩ

∫ 2π

)
(θ)

=
fold

∫ π
dϕ

0

sin θdθ
0

( dσ
dΩ

)
(θ)
calc

√

( α2 )
exp − 2 ,
2σ
2πσ 2
1

(4.18)

where α was:()α = arccos cos θfold cos θ + sin θfold sin(θ) sin ϕ ,(4.19)
)
(
)
(
dσ
dσ
(θ)
was the folded diﬀerential cross section, and dΩ(θ)
was the calculated
and dΩ
foldcalc
diﬀerential cross section.
The folded angular distributions are presented in blue-dashed lines in Fig. 4.1 and Fig.
4.2. Even in case of the relatively thin α target, the folding wiped out all the structures in
the calculated distribution. The folded distributions multiplied by the acceptance function
are presented in green dash-dotted lines. The accepted ratios are tabulated in Tab. 4.2.

72

Chapter 4 Distorted-Wave Born Approximation Analysis

20

O+Au 2+ 4.07 MeV
DWBA
Angle Folded
ZDS Acceptedd σ /d Ω (mb/sr)

103

102

10
0

0.02

0.03 20

0.04

0.05

O+α 2+ 4.07 MeV

0.06

DWBA
Angle Folded
ZDS Acceptedd σ /d Ω (mb/sr)

10

0.01

3

102

10
0

0.01

0.02

0.03
θ lab (rad)
20

0.04

0.05

0.06

-

O+Au 3 5.62 MeV
DWBA
Angle Folded
ZDS Acceptedd σ /d Ω (mb/sr)

103

102

10
0

0.01

0.02

103

0.03 20

0.04-

0.05

O+α 3 5.62 MeV

0.06

d σ /d Ω (mb/sr)

DWBA
Angle Folded
ZDS Accepted

102

10
0

0.01

0.02

0.03
θ lab (rad)

0.04

0.05

0.06

Fig. 4.1: The calculated angular distribution of scattered 20 O seen from the laboratory system. Calculated by the DWBA code (solid line), angular-folded distribution by the detector
and multiple Coulomb scattering (dashed line), and angular-folded distribution multiplied by
the ZeroDegree acceptance (dashed-dotted line). For the 2+ and 3− states.

4.1 Distorted-Wave Born Approximation Analysis

104

20

73

-

O+Au 1 5.35 MeV

d σ /d Ω (mb/sr)

DWBA
Angle Folded
ZDS Accepted

103

102

0

0.01

0.02

0.03 20

0.04-

0.05

O+α 1 5.35 MeV

0.06

DWBA
Angle Folded
ZDS Acceptedd σ /d Ω (mb/sr)

103

102

10
0

0.01

0.02

104

0.03
θ lab (rad)
20

0.04

0.05

0.06

-

O+Au 1 6.85 MeV

d σ /d Ω (mb/sr)

DWBA
Angle Folded
ZDS Accepted

103

102

d σ /d Ω (mb/sr)

103 0

0.01

0.02

0.03 20

0.04-

0.05

O+α 1 6.85 MeV

0.06

DWBA
Angle Folded
ZDS Accepted

102

10

0

0.01

0.02

0.03
θ lab (rad)

0.04

0.05

0.06

Fig. 4.2: The calculated angular distribution of scattered 20 O seen from the laboratory system. Calculated by the DWBA code (solid line), angular-folded distribution by the detector
and multiple Coulomb scattering (dashed line), and angular-folded distribution multiplied by
the ZeroDegree acceptance (dashed-dotted line). For the 1− states.

74

Chapter 4 Distorted-Wave Born Approximation Analysis
Table 4.2: The accepted ratio.
Accepted (%)

2+ (4.07 MeV)

3− (5.62 MeV)

1−
1 (5.36 MeV)

1−
2 (6.84 MeV)

20

O+α

90

86

87

87

20

O+Au

73

71

78

77

4.1.4

Angular distributions of γ rays

The angular distribution of the de-excitation γ rays was also estimated by the DWBA analysis
via the statistical tensor [83, 84]. The angular distribution of γ ray wL (θγ ) with respect tothe beam axis emitted from multiple radiation of 2L -th order with initial spin J and the final
spin J0 was give by:
even
)
∑
1 (
wL (θγ ) =1+Fλ (L J0 J)⟨AJλ0 ⟩Pλ (cos θγ ) ,4π(4.20)

λ>0

where ⟨AJλ0 ⟩ was the statistical tensor of λ-th order averaged by the scattering angle of theγ emitter particle defined by:∫1
dσJ⟨Aλ0 ⟩ =(4.21)dΩ (θ)AJλ0 ,σ0dΩhere σ0 was the total cross section (or a normalization factor) given by:∫dσ
σ0 = dΩ (θ).
dΩ

(4.22)The statistical tensor AJλ0 was directly calculated by the ECIS code. Pλ (cos θγ ) was theλ-th order Legendre polynomial, and Fλ (L J0 J) was so-called F-coeﬃcient, defined in the
textbook by M. E. Rose [85]. The F-coeﬃcient can be disassembled using Racah coeﬃcientW [86] and Clebsch-Gordan coeﬃcient C⟨L 1 L − 1|λ 0⟩ into:
Fλ (L J0 J) = (−1)J0 −J−1 (2J + 1) 2 C⟨L 1 L − 1|λ 0⟩ W (J J L L ; λ J0 ).
1(4.23)In the analysis, the γ-ray angular distribution was produced using these formulas, considering the initial spin J, final spin J0 , and the lowest-order multipole radiation allowedby the initial and final spin parity. The statistical tensor AJλ0 up to the fourth order wasconsidered. The calculated distribution functions wL (θγ ) are presented in Fig. 4.3 for the+−2+ , 3− states decay via the 2+
1 state, and in Fig. 4.4 for the 1 states decay via the 21 state
and the ground state. Only the lowest electric multipole transition was considered: i.e. the
3− → 2+ decay E1 transition, for the 2+ → 2+ decay E2 transition, and for the 1− → 0+or 2+ decay E1 transition was considered. The functions presented were multiplied by thesin (θγ ) to account for the Jacobian. Both the distribution in the center-of-mass system andthe laboratory system, the group marked as Lab. Frame, was presented. The LaBr3 :Cedetectors were located at 30◦ ≃ 0.5 (rad) in the laboratory frame. It could be seen that at

4.1 Distorted-Wave Born Approximation Analysis

75

30◦ , the eﬀect of the decay pattern was small. These angular distributions were built into
the Monte-Carlo simulation of the decay γ ray, and reflected to the response functions to be
fitted. The diﬀerence in the yield of the fits between the angular distribution included andnot included (isotropic) simulation, was generally small like a few percent, but included in
the present analysis.

4.1.5

Cross sections

The cross sections σ of each excited state was determined by the following formula:σ=Nγ,
Ntgt × Nbeam × Racc

(4.24)

where the Nγ was the integrated number of γ rays emitted. The value was determinedfrom the fits, as a normalization factor of the simulated response function, and summed overbranches. Angular distribution calculated from the DWBA analysis was included. The Ntgt
was the number of target nuclei, determined from the thicknesses of the targets. The Racc
was the accepted ratio of reacted 20 O nuclei by the ZeroDegree spectrometer calculated fromthe DWBA analysis. The Nbeam was the number of eﬀective beam defined by:
Nbeam = NF8Pla × RPurity × RDAQ livetime × ϵBLD ,(4.25)

where the NF8Pla was the integrated number of beam counted by the plastic scintillator
installed at F8 achromatic focal plane, closest detector from the secondary target, using thescaler. This integrated number included any beam-like particle passed the detector. The
integrated number was reduced by the achieved purity of the secondary beam, RPurity , andthe DAQ live time RDAQlivetime , and the eﬃciency of the beam-line detectors ϵBLD . The
eﬃciency of the beam-line detectors derived from the MWDC installed at F9 dispersive focal
plane, due to the unstable gas pressure. The cross sections determined in this way was used as
an input to the DWBA analysis to determine the reduced transition probability, or strength,
B(El) ↑.
The errors of cross sections were estimated using the propagation of errors. Considerederrors were: the statistical error (up to 1%) and uncertainty in the fits (up to 15 %) in Nγ ,the uncertainty in the thicknesses of the targets ( up to 10 %) in Ntgt , and the uncertainty
in the accepted ratio (up to 5 %) Racc . These errors were merged into the errors of crosssections according to the propagation of errors give by:
X1
X1 ± σ1=±(X2 ± σ2 ) × (X3 ± σ3 )X2 × X3√( σ )2 ( X σ )2 ( X σ )2
1
1 2
1 3
,
+
+
2
X2 X3
X2 X3
X2 X32(4.26)

where σi is an error of a measured value Xi (i = 1, 2, 3). The estimated cross sections aretabulated in Tab. 4.3.
Estimation of the contributions from the liquid helium cell system
In the deduction of the 20 O+α cross sections, the contributions from the cell window material,
the heat shield, and the window frame were additionally considered. The DWBA analysis
was performed exactly in the same manner as the targets. The thicknesses of the materials

76

Chapter 4 Distorted-Wave Born Approximation Analysis↓ Lab. Frame

Isotropic-w L (θ γ ) sinθ γ (a.u.)

3 →2+ 20O+α-

3 →2+ 20O+Au

0

0.5

1

1.5

θ γ (rad)

2

2.5

3

↓ Lab. Frame

Isotropicw L (θ γ ) sinθ γ (a.u.)

2+→2+ 20O+α
2+→2+ 20O+Au

0

0.5

1

1.5

θ γ (rad)

2

2.5

3

Fig. 4.3: The calculated angular distribution of the decay γ ray. Seen from the rest-frameof the beam, and from the laboratory system (Lab. Frame). For the 2+ , 3− states decay via
the 2+
1 state.

4.1 Distorted-Wave Born Approximation Analysis

77↓ Lab. Frame

Isotropic-w L (θ γ ) sinθ γ (a.u.)

1 →0+ 20O+α-

1 →0+ 20O+Au

0

0.5

1

1.5

θ γ (rad)

2

2.5

3

↓ Lab. Frame

Isotropic-w L (θ γ ) sinθ γ (a.u.)

1 →2+ 20O+α-

1 →2+ 20O+Au

0

0.5

1

1.5

θ γ (rad)

2

2.5

3

Fig. 4.4: The calculated angular distribution of the decay γ ray. Seen from the rest-frame of
the beam, and from the laboratory system (Lab. Frame). For the 1− states decay via the 2+
1state and the ground state.

78

Chapter 4 Distorted-Wave Born Approximation Analysis
Table 4.3: Measured cross sections in millibarns.
Ex (MeV)Jnπσ(20 O+α)σ(20 O+Au)

4.07(4)2+
2

1.11(16)5.10(62)5.36(5)1−
10.376(43)7.92(35)

5.62(5)(3− )3.10(28)

7.10(55)

6.84(7)1−
2

0.079(13)5.42(35)

are summarized in Tab. 2.3. The radiation shields made of aluminum foil with thickness of
7 × 2µm was represented by 27 Al, and the cell windows made of Havar foil with thickness of
10 × 2µm was represented by 59 Co. Note that the Havar foil, actually was an alloy contained
42.5 % of cobalt, was represented by cobalt. The contribution from the window frame, where
the eﬀective thickness was estimated to be 6.8 µm was also represented by 27 Al. The opticalpotentials of 27 Al and 59 Co was deduced using the model proposed by T. Furumoto et al. in
Refs. [77, 78, 79], and fitted by the Woods-Saxon formulas. The optical potential parameters
are tabulated in Tab. 4.4.
Table 4.4: The Woods-Saxon Parameters of the background materials. Calculated from T.
Furumoto et al. [77, 78, 79].
V

RR

aR

W

RI

aI

System

(MeV)

(fm)

(fm)

(fm)

(fm)

(fm)

20

O+27 Al-278.42.408

0.7687-377.1

3.003

1.153

20

O+59 Co-294.23.258

0.8285-497.3

3.731

1.253

The acceptance of the scattered 20 O, and the angular distribution of the γ rays wereanalyzed in the same manner as discussed in the previous chapter. The contributions ofthese materials in Nγ was estimated to be up to 5 %. These contributions were subtracted
from Nγ when deducing the cross sections and the transition strengths.

79

Chapter 5

Discussions
Reduced transition probability B(El) ↑5.0.1

The measured cross sections were converted into the reduced Transition Probability B(El) ↑by the DWBA analysis. For the E2 and E3 transitions, both the Coulomb and nuclear
excitation contribution were considered by tuning the Coulomb deformation parameter βl,Cand the nuclear deformation parameter βl,R so as to reproduce both the 20 O+Au and 20 O+αcross section using the same deformation parameters. This was an advantage of using two
probes with diﬀerent Coulomb/nuclear potential strength. The Coulomb deformation parameter βl,C determined was converted into the B(El) ↑ value by the macroscopic modelEq.(4.3). For the E1 transition, we approached in the same manner under the assumptionthat the Coulomb potential contributed only to the Isovector E1 strength, and the nuclear
potential contributed only to the Isoscalar E1 strength. The Coulomb deformation parameter
β1,C and the nuclear deformation parameter β1,R was tuned to reproduce both the 20 O+Au
and 20 O+α cross sections at the same time. The Coulomb deformation parameter β1,C wasconverted into the B(E1) ↑ value using the same model Eq.(4.3), and the nuclear deformation
was converted into the ISDEWSR fraction using the sum-rule limit deformation Eq.(4.16).
The errors of B(El) ↑ were estimated from the errors of cross sections. The upper (lower)
limit B(El) ↑ were determined to reproduce the upper (lower) limit cross sections, using thesame DWBA analysis. The estimated multipole strengths are tabulated in Tab. 5.1.
Table 5.1: Estimated multipole strength.
Ex (MeV)Jnπ

4.07(4)2+
2

5.36(5)1−
1

5.62(5)(3− )6.84(7)1−
2

5.1

ISD EWSR (%)B(Eλ) ↑ (e2 fm2λ )28.8 ± 6.6

2.70(32)

3.57(20) × 10−2
1.19(10) × 103

0.67(12)3.79(26) × 10−2

B(E2) ↑ Strength

The B(E2) ↑ of the 2+
2 state at 4.07 MeV was first deduced and compared with the reported
value to check the consistency of our analysis. The extracted values were: β2,C = 0.23 and80

Chapter 5 Discussions
β2,R = 0.32. In terms of cross sections, the Coulomb contribution in the 20 O+α system
was 2.0 %, and that in the 20 O+Au system was 48 %. The rest of the cross sections was
exhausted by the nuclear excitations. The interference between the Coulomb and nuclear
potential was investigated by switching on and oﬀ the either potential. The nuclear andCoulomb interference contribution was negligible (< 1%). The extracted B(E2) ↑ value was
28.8 ± 6.6 e2 fm4 .
2
4
The B(E2)↑ of the 2+
2 state at 4.07 MeV was reported to be 30 e fm in Refs. [43, 44],
measured by the Coulomb excitation with the lead target. The value was reported without
error. Our value was consistent within errors, and hence we had concluded that our approach
including the optical potential model was usable for our purpose. It should be mentionedout that considering the nuclear excitation was necessary at this incident energy region of
250 MeV/nucleon, because almost half of the cross sections were exhausted by the nuclear
excitation even in the high-Z gold target.

5.2

B(E3) ↑ Strength

The B(E3) ↑ of the possible (3− ) state at 5.62 MeV was deduced to test the spin-parity
assignment. We followed the same way as extracting B(E2) ↑. However, more than 99 % of
the cross section was exhausted by the nuclear excitation in either target, the gold or the αtarget. This meant for B(E3) ↑, neither probe, even the high-Z gold target, was sensitive
to the Coulomb deformation β3,C . Thus we couldn’t use the same way as B(E2) ↑, namely
to independently determine the β3,C and β3,R taking the advantage of the two probes with
diﬀerent Coulomb/nuclear potential strength. We deduced the β3,R respectively from eachcross section, and introduced a simple relationship between the β3,C and β3,R to determine
the β3,C respectively given by:
β3,R R = β3,C RC ,(5.1)

where R was the nuclear radius Eq. (4.17) and RC was the Coulomb radius Eq. (4.4). The
deduced nuclear deformations were: β3,R = 0.56 for 20 O+α, and β3,R = 0.59 for 20 O+Au
system. The extracted B(E3) ↑ values were: 1.30(12) × 103 e2 fm6 for 20 O+α, and 1.19(10) ×
103 e2 fm6 for 20 O+Au system. These values were consistent with each other within errors.
This result suggested the spin-parity assignment of 3− was actually correct, because otherspin-parity could hardly give similarly consistent strengths. This consistency gave stronger
support on the nuclear potential model, because the cross sections depend almost purely on
the nuclear excitation.
The B(E3) ↑ strength in 20 O was compared with strengths of other even-even oxygenisotopes at similar energy region. The B(E3) ↑ strength of 16 O at 6.13 MeV was reported
to be 1.49(7)×103 e2 fm6 [87], and the B(E3) ↑ strength of 18 O at 5.10 MeV was reported tobe 1.30(4)×103 e2 fm6 [88]. These B(E3) ↑ strengths were similar to the strength observedin 20 O, and this gave another support for the validity of the 3− spin-parity assignment.

5.3

B(E1) ↑ Strengths−
The B(E1) ↑ of the 1−
1 state at 5.36 MeV and the 12 state at 6.84 MeV was determinedin the same manner as extracting the B(E2) ↑. The Isovector contribution (Coulomb ex-5.3 B(E1) ↑ Strengths

81

citation) and the Isoscalar contribution (nuclear excitation) was determined respectively, toreproduce each cross section simultaneously. For the 1−
1 state at 5.36 MeV, the Coulomb
excitation contribution to the cross sections was: 81.4 % for the 20 O+Au, and 2.2 % for the
20
O+α system. The extracted strengths were: Isoscalar strength of 2.70(32) %, and Isovector
strength of 3.57(20) × 10−2 e2 fm2 . For the 1−
2 state at 6.84 MeV, the Coulomb excitation
contribution to the cross sections was: 94.9 % for the 20 O+Au, and 9.7 % for the 20 O+α system. The extracted strengths were: Isoscalar strength of 0.67(12) %, and Isovector strengthof 3.79(26) × 10−2 e2 fm2 .
The extracted B(E1) ↑ strengths, namely the Isovector strengths, were compared with thereported value in Refs. [43, 44], using the similar experimental method. The reported values
were obtained using the lead target for the Coulomb excitation and the beam energy was 100
MeV/nucleon, and the B(E1) ↑ strengths were extracted using the virtual photon theory,
diﬀerent from the present DWBA. The reported B(E1) ↑ strengths were: 6.2(16)×10−2 e2 fm2−2 2
e fm2 for the 1−
for the 1−
2 state at 6.84 MeV. Despite of
1 state at 5.36 MeV, and 3.5(9)×10
the diﬀerences in the method applied to both measurement and analysis, the B(E1) ↑ for the
1−
2 state was consistent with the present value within errors. However the reported B(E1) ↑for the 1−
1 state at 6.84 MeV was 60 % smaller than the present value. The diﬀerence was+most likely due to the diﬀerence in branching ratio of the 1−
1 → 21 decay mode, originatingfrom the inclusion of the (3− ) → 2+
1 decay in the present fit.

5.3.1

Comparison of Isoscalar and Isovector strengths with N =Z nuclei
in the isotopic chain, 16 O
5O(α, α’ γ )ISD EWSR (%)

4

16

3
2
1
0
0
1

B(E 1)↑ (10 e2 fm2)

2

-2

-2

B(E 1)↑ (10 e2 fm2)

ISD EWSR (%)

5

3
4
5
4

O(γ , γ ’)

16

5

6

E x (MeV)

7

8

4

-

20

11

O

3

-

2

12

1
0
0
1

2
3
4
5
4

20

O
5

6

E x (MeV)

7

8

Fig. 5.1: The Isoscalar and Isovector dipole strengths in 20 O (right panel), in comparison
with those in 16 O (left panel). The Isoscalar dipole strengths in unit of ISD EWSR fraction(%) (top panel), and the Isovector dipole strengths in B(E1)↑ (e2 fm2 ) (bottom panel).
The extracted Isoscalar and Isovector strengths of 20 O are plotted along with those of 16 O

82

Chapter 5 Discussions
in Fig. 5.1. The Isoscalar strength on 16 O was studied using (α, α′ ) reaction by M.N. Harakeh
and A.E.L. Dieplink [6] in 1981, and the Isovector strength was studied using (γ, γ ′ ) reaction
by C.P. Swann in 1970 [89]. On 16 O, there was only one dipole excitation observed withinsimilar excitation energy, at 7.12 MeV. This state has strong Isoscalar strength exhausting
4.2 % of ISD EWSR with almost negligible Isovector strength, and therefore regarded as pureIsoscalar dipole excitation described as the macroscopic squeezing mode [6, 1]. Comparing
this state in 16 O with the states in 20 O, the integrated Isovector strength (ΣB(E1) ↑) drastically increased from 0.492×10−2 in 16 O to 7.36(33) × 10−2 in 20 O, meanwhile the integrated
Isoscalar strength (Σ ISD EWSR) did not show large diﬀerence from 4.2 % in 16 O to 3.37(34)% in 20 O. This clearly indicates the dual character of those states in 20 O, and suggests thatthose states in 20 O have diﬀerent structure from the state in 16 O. Comparing two statesin 20 O one another, the large diﬀerence in the Isoscalar strength suggests that these states
have diﬀerent structure. It is worth mentioning here that this diﬀerence is observed in the
Isoscalar strength, while the Isovector strengths do not diﬀer much. This clearly emphasizethat studying the Isoscalar strength is especially important to understand the structure of
low-energy dipole excitations, even in neutron-rich unstable nuclei. It is because in neutronrich unstable nuclei, the enhancement of the LED strength tends to be discussed in termsof the neutrons against core oscillation mode, however significantly strong Isoscalar dipole
strength is hardly explained as such.

5.3.2

Comparison with random phase approximation (RPA) calculation

The theoretical calculation based on the random-phase approximation was performed by T.
Inakura [90]. The calculations used two types of Skyrme interactions, SLy4 and SkI2 [91] forcomparison. The SLy4 interaction is most often applied in the calculation of neutron-rich nuclei in recent calculation, and the SkI2 interaction is known to reproduce the strength of thelow-energy dipole excitations well. Independent of the interactions, the calculation predicts
one dipole excitation below neutron separation threshold energy. The calculated values aretabulated in Tab. 5.2. One can see that the protons and neutrons are in-phase inside of thenuclei (r≤ 3 fm) and outside of the nuclei the protons and neutrons are again in-phase with
opposite sign. This transition density exhibits the typical character of the low-energy dipolemode, discussed based on several theories. However, both neutrons and protons distributeoutside of the nuclei, and the state is hardly interpreted as neutron skin oscillation. Interestingly, according to the prediction, this state is carried by proton excitations. The largest
contribution was one-particle-one-hole state of the proton excitation of 1p1/2 → 1d2/5 , carrying 60 % of the total strength, and the next largest contribution was one-particle-one-holestate of the proton 1p1/2 → 2s1/2 , carrying 20 % of the total strength. The transition density
calculated using the SLy4 interaction is presented in Fig. 5.2. The previous studies based onshell model calculation predicted the neutron excitation [43, 44], making clear contrast withthe present study predicting the proton excitations. According to the prediction, the proton
excitations without any neutron excitations involved lead to the dominant Isovector character of this state with weak Isoscalar strength. The opposite is true: the neutron excitations
without protons involved lead to predominant Isovector character. The proton excitations
producing a dominant Isovector character has yet to be discussed in neutron-rich unstable
nuclei. This study can be particularly important in search for the valence-neutron excitation

5.3 B(E1) ↑ Strengths

83

or the neutron skin mode.
The calculated Isoscalar and Isovector strengths were smaller than the measured strengths,
independent of the interactions. The calculation using the SkI2 predicts stronger strengths
closer to the measured values than that using the SLy4. This tendency is consistently known
in this type of calculation. The relationship between the predicted Isoscalar and Isovector
strength was similar to that observed for the 1−
2 state at 6.84 MeV, independent of the−interactions. The other 11 state 5.36 MeV had much stronger Isoscalar strength, and it ishard to compare with the predictions. Following these considerations, we deduced that the1−
2 state at 6.84 MeV is generated by the proton excitations.
The other 1−
1 state at 5.36 MeV may originate from more complex eﬀects. The present RPA
considers only the superposition of one-particle-one-hole state. One possibility is that thisstate cannot be expressed as such, and may derive from two-particle-two-hole state or evenhigher states. Another theoretical possibility is the pairing correlation [92, 93, 34], anothereﬀect which the present RPA does not consider. It is known that the pairing correlation aﬀects
the orbital occupancy of the valence neutrons (small occupancy of the 2s1/2 orbit appears).
In this case the valence neutrons can be involved in the state, and further theoretical studies
are strongly desired. In any scenario, the strong Isoscalar strength must be explained. It issuggestively said that the close Isoscalar strength with that observed on 16 O may be relevant
20
to its structure. The 1−
O exhausts 2.70(32) % of the ISD EWSR,
1 state at 5.36 MeV in−
16
and the 1 state at 7.12 MeV in O exhausts 4.2 % of the ISD EWSR. Meanwhile the20
Isovector strength diﬀers much, 1−
O has 3.57(20) × 10−2 e2 fm2 ,1 state at 5.36 MeV inand 0.492×10−2 e2 fm2 in 16 O. Because 16 O is a N = Z nuclei, the Isovector excitation isprohibited. Thus the state is considered to be the collective compressional dipole mode as
discussed in Refs. [6, 1]. Similar structure as the one in 16 O has a possibility to exist also in
20
O. Since 20 O is a N > Z nuclei, the state can have strong Isoscalar strength with decent
amount of Isovector strength.
Table 5.2: Calculated Isoscalar dipole strength (ISD EWSR), and Isovector dipole strength(B(Eλ)↑).
Interaction

Ex (MeV)

ISD EWSR(%)

B(Eλ)↑(e2 fm2 )

SkI2

6.51

0.43

3.49×10−2

SLy4

5.52

0.22

1.02×10−2

5.3.3

Comparison with heavier neutron-rich nuclei

Comparing the Isoscalar and Isovector strengths in 20 O with other neutron-rich nuclei cangive another insights into their structure. The observed Isoscalar and Isovector strengths
in a set of nuclei are reviewed in Fig. 1.6. It can be seen that the dipole strengths have acommon tendency among observed nuclei. The Isoscalar strength quickly decreases as theexcitation energy gets higher, and the Isovector strength distributes rather independent of
the excitation energy. Note that the Isoscalar strengths in some nuclei are not given in ISD
EWSR fraction but in the cross section, however the tendency itself is comparable. Although
there were only two, the states observed in 20 O share this tendency. This is suggested to beChapter 5 Discussions

r2δ ρ (L=1) (fm-1)

84

0.04

20

O SLy40.02-

1 below S n

0
5

10

r (fm)

-0.02
-0.04

protons
neutrons

-0.06
-0.08
Fig. 5.2: The transition density predicted below the neutron separation threshold energy.
Calculated based on the RPA using SLy4 Skyrme interaction by T. Inakura [90].

common among neutron-rich nuclei, and the low-energy dipole excitations may be categorized
into two structurally diﬀerent states. In order to unveil the Isovector dominant part of thistendency, distributing around higher excitation energy, not only the neutron excitations, butalso the proton excitations are suggested to be important, since the RPA calculation in 20 O
predicts the Isovector dominant state to be generated by the proton excitations. In orderto understand the other part with strong Isoscalar strength and certain Isovector strength,
more theoretical studies are necessary. Those theories have to explain the especially strong
Isoscalar strength, and its connection with the pure Isoscalar strength observed in N = Z
nuclei.
It is also interesting to compare the results in oxygen isotopic chain with the calcium
isotopic chain, so far the only case where both the Isoscalar and Isoscalar strengths areobserved within the isotopic chain. The measured Isoscalar and Isovector strengths in 40 Ca(N = Z) and 48 Ca (N > Z) are presented in Fig.5.3. It is observed that the integrated
Isoscalar strength between 40 Ca and 48 Ca doesn’t diﬀer much: 4.3(3) % in 40 Ca and 4.93(13)% in 48 Ca in the unit of ISD energy-weighted sum rule. On the other hand, the integrated
Isovector strength diﬀers much: 0.020(3) % in 40 Ca and 0.39(4) % in 48 Ca in the unit ofenergy-weighted sum rule (or TRK sum rule). These characters are similar with that observedin the oxygen isotopes. The integrated Isoscalar strengths are: 4.2 % in 16 O and 3.37(34) %in 20 O in the unit of ISD energy-weighted sum rule. The integrated Isovector strengths are:
0.060 % in 16 O and 0.634(30) % in 20 O in the unit of energy-weighted sum rule. It has to be
expressed that either strength has the same order of value between the calcium and the oxygen
chain. This suggests that there is a similar mechanism of generating the low-energy dipole
excitations between the oxygen and calcium isotopes. One theoretical proposition discussed
in the calcium isotopes is a collective excitation called toroidal dipole mode [28, 42]. This

5.3 B(E1) ↑ Strengths
collective excitation is predicted to exist in the lowest excitation energy range among the
low-energy dipole excitation, and recently attracts a lot of attention. Further theoreticalstudies in the context of collectivity in the oxygen isotopes may be the key to understandtheir strong Isoscalar strength.

Fig. 5.3: The Isoscalar and Isovector dipole strengths measured in 40 Ca (left panel) and 48 Ca(right panel). The Isoscalar dipole strengths in unit of ISD EWSR fraction (%) (top panel),and the Isovector dipole strengths in B(E1)↑ (e2 fm2 ) (bottom panel). From D. Verya et al.
[28].

85

87

Chapter 6

Conclusive Remarks
The Isoscalar and Isovector strengths of the low-energy dipole excitations were investigated on
20
O, for the first time in unstable nuclei. The inelastic scattering of α particles, a predominant
Isoscalar probe, and the Coulomb excitation, a predominant Isovector probe, at the incident
energy of 276(9) MeV/nucleon was used to populate the low-energy dipole excitations. Inbeam γ-ray spectroscopy technique was used to identify the dipole excitations. Two excitation
cross sections from two probes were converted into the Isoscalar and Isovector dipole strengths
via distorted-wave Born approximation (DWBA) analysis. It was revealed that the two LED−
states at the energies of 5.36(5) MeV (1−1 ) and 6.84(7) MeV (12 ) states have similar Isovector2−2dipole strengths, B(E1)↑ = 3.57(20) × 10−2 (11 ) and 3.79(26) × 10−2 (1−2 ) e fm respectively, however significantly diﬀerent Isoscalar dipole strengths, exhausting 2.70(32) (1−1 ) and−0.67(12) (12 ) % respectively of Isoscalar dipole energy-weighted sum rule (ISD EWSR). The
diﬀerence in the Isoscalar dipole strengths suggested that these two states have diﬀerent underlying structures. Comparison with a theoretical prediction based on the random-phaseapproximation (RPA) suggested that 1−
1 state with smaller Isoscalar dipole strength may
derive from the proton excitations, unlike the previous studies predicting the valence-neutron
excitations based on the shell model. The other 1−
2 state, however, has strong Isoscalar dipole
strength similar to that observed in N = Z nuclei in the oxygen isotope, 16 O, and is hardly
reproduced by the theory. Similar LED states with strong Isoscalar dipole strengths are also
observed in neutron-rich calcium isotope, 48 Ca. Further theoretical studies that explainsstrong Isoscalar strengths are desired.

89

Chapter 7

Acknowledgements
I’d like to thank Dr. T. Murakami, my supervisor at Kyoto University, and Dr. H. Baba,
my supervisor at RIKEN, for providing me the opportunity to work on this experiment. I’dlike to thank the members of the Nuclear Hadron group at Kyoto University, Dr. T. Nagae,
Dr. T. Kawabata and other members for long-lasting supports in my career as a Ph.D student.
I’d like to thank my fellow students at Kyoto for the discussions.
I’d like to thank Mr. S. Koyama for his devotion to the cryogenic system and its stable
operation. I’d like to thank Dr. A. Giaz and other members of INFN Italy for providing us thelarge volume LaBr3 :Ce detectors. I’d like to thank Dr. R. Avigo and Mr. J. Tscheuschner for
their dedicated eﬀorts on the experiment campaign. I’d like to thank T. Isobe, S. Nishimura,
D. Suzuki and many other members of RIKEN for the support in the operation of the experiment. I’d like to thank Dr. T. Aumann, Dr. A. Bracco, Dr. T. Inakura and Dr. O. Wieland
for the discussions on the publication.
I’d like to thank my collaborators on this work, Dr. S. R. Banerjee, Dr. C. Ceaser,
Dr. F. Camera, Dr. S. Ceruti, Mr. S. Chen, Dr. V. Derya, Dr. P. Doornenbal, Ms. A. Horvat,
Dr. K. Ieki, Dr. N. Imai, Dr. T. Kawabata, Dr. N. Kobayashi, Dr. Y. Kondo, Dr. M. KurataNishimura, Mr. S. Masuoka, Dr. M. Matsushita, Dr. S. Michimasa, Dr. B. Million,
Dr. T. Motobayashi, Dr. T. Nakamura, Dr. T. Ohnishi, Dr. H. J. Ong, Dr. S. Ota,
Dr. H. Otsu, Mr. T. Ozaki, Ms. A. Saito, Dr. H. Sakurai, Dr. H. Scheit, Ms. F. Schindler,
Dr. P. Schrock, Dr. Y. Shiga, Ms. M. Shikata, Dr. S. Shimoura, Dr. D. Steppenbeck,
Dr. T. Sumikama, Ms. I. Syndikus, Dr. H. Takeda, Dr. S. Takeuchi, Dr. A. Tamii,
Mr. R. Taniuchi, Dr. Y. Togano, Mr. J. Tsubota, Dr. H. Wang, Dr. K. Wimmer,
Mr. Y. Yamaguchi, Dr. K. Yoneda, and Dr. J. Zenihiro.
Above all, the greatest thanks to my family for letting me do anything I wanted during mystudenthood.

91

A

Appendix: Extrapolating Optical
Potentials
The real part of optical potential V and the imaginary part of W is described respectively
as 10-range Gaussian form [77, 78, 79]:
( 2 ))10 (∑r
V (r) ≃
αi exp − 2,γii=1
W (r) ≃

10 (
∑
i=1

( 2 ))rβi exp − 2,γiwhere αi , βi are parameters depend on target nuclei (t), mass and charge of projectile nuclei(p) (20 O), and incident energy E,
αi = αi (Ap , Zp , At , E),
βi = βi (Ap , Zp , At , E).
For this work, Ap = 20, Zp = 8, E ≃ 276 MeV/nucleon, and γi is a function of index i definedas:i + 8 1/3
1/3
γi = 0.45(Ap + At + 1).
18
Refs. [77, 78, 79] do not cover α particle due to unresolved theoretical issue. In order to guess
α optical potential, we took an extrapolation approach. The 10-range gaussian parameters
αi , βi were extrapolated from heavier mass N = Z systems, 12 C+20 O, 16 O+20 O, 20 Ne+20 O
and 24 Mg+20 O. The parameters αi , βi of the 12 C+20 O, 16 O+20 O, 20 Ne+20 O, 24 Mg+20 O
systems are tabulated in Tab. A.2. These parameters are directly given by the code given in
Refs. [77, 78, 79].
The dependency of αi , βi on the target mass number At was extrapolated using quadratic
functions:
αi (At ) ≃ a1 A2t + a2 At + a3 ,βi (At ) ≃ a1 A2t + a2 At + a3 .
The coeﬃcients a1 , a2 , a3 were determined by fitting four αi (At ), βi (At ) points, 12 C+20 O,
16
O+20 O, 20 Ne+20 O, 24 Mg+20 O in the left column of Tab. A.2. The coeﬃcients a1 , a2 , a3

92

A Appendix: Extrapolating Optical Potentials
determined by the fits are tabulated in Tab. A.3. The αi (At ), βi (At ) for the α+20 O system
was obtained by substituting At = 4 in the extrapolation functions. The results are tabulated
in the right column of Tab. A.2. The extrapolated α potential was fitted with Woods-Saxon
formula to be applied to the Harakeh-Dieplink form factor [6].
Comparison with the global α potential.
The optical potentials are plotted in Fig. A.1. We compared the extrapolated α potential
with the global alpha optical potential described in Ref.[94]. The global optical potential forthe α particle was developed to reproduce the elastic scattering of α particle above 80 MeV.
The global α potentials are given by the following set of formulas.
V,
1 + exp ((r − RR A1/3 )/aR )W
W (r) =,
1 + exp ((r − RI A1/3 )/aI )Z
V = 101.1 + 6.501 1/3 − 0.248E (MeV),A
W = 26.82 − 1.706A1/3 + 0.006E (MeV),
V (r) =

RR = 1.245 (fm),
RI = 1.570 (fm),aR = 0.817 − 0.0085A1/3 (fm),aI = 0.692 − 0.020A1/3 (fm),where E is the total energy of the α-particle beam, A, Z is the mass and charge number of thetarget nuclei. In our case the set of formulas corresponds to the 276-MeV/nucleon α-particle
beam hit on the 20 O target.
The calculated global α potential is plotted in Fig. A.1 (Global α). The Woods-Saxon
parameters for our experiments are tabulated in Tab. A.1. This global α potential has been
widely used for the analysis of stable nuclei data in this energy region, such as Ref. [28].
However one can see that from Fig. A.1, the potential was as large as the 12 C system.
It turned out that the potential was not applicable to the current energy region of 276
MeV/nucleon.
Table A.1: The Woods-Saxon parameters obtained from the global α potential. From N.
Nolte et al. [94].
V

RR

aR

W

RI

aI

System

(MeV)

(fm)

(fm)

(fm)

(fm)

(fm)

20

-154.91.245

0.7939-28.811.570

0.6377

O+α

93

0

-50

V(r) (MeV)

Extra. α
-100

-150

12

C

16

O

20

Ne

24

Mg

-200

Global α-250

A -20O (276 MeV/u)
Optical Potential V(r)

-300
0

2

4

6

8

10

r (fm)

0-50

Extra. αW(r) (MeV)

-100
-150
-200

C

16

O

20

Ne

24

Mg

Global α-250

A -20O (276 MeV/u)
Optical Potential W(r)

-300
-350
0

12

2

4

6

8

10

r (fm)
12

16

Fig. A.1: The optical potentials of C, O, 20 Ne, 24 Mg + 20 O systems and their extrapolation to the α+20 O system (Extra. α). Real parts V (r)(top panel), and imaginary parts
W (r) (bottom panel). Calculated from T. Furumoto et al. [77, 78, 79].

94

A Appendix: Extrapolating Optical Potentials

Table A.2: 10-range Gaussian parameters αi (At ), βi (At ) (left column), and parameters extrapolated to α+20 O system (right column). Calculated from T. Furumoto et al.[77, 78, 79].
From Refs. [77, 78, 79].
12

C+20 O

16

O+20 O20

Ne+20 O

Extrapolated
24

Mg+20 O

α+20 O

α12141.68

2005.32

1883.33

1785.152475.91α2-25944.1-24458.1-23195.1-22245.7-29761.8

α3145457138302132460128315164454α4-490968-470572-454354-443285-546186α51090120

1051350

1020460

9991491194590α6-1639860-1587660-1544450-1512450-1775560

α716655701616050

1572400

1536830

1786550α8-1102330-1071960-1043210-1017720-1171100α9433009423000412689402870453669α10-77033.4-75851.0-74442.2-72984.1-79064.3

β11398.361294.141205.35

1128.33

1645.94

β2-16875.3-15625.8-14563.9-13651.1-19861.8

β394024.887173.3

81332.5

76321.3110407β4-315086-292566-273295-256757-368867β5695884647248605432569482811754

β6-1046150-974830-913060-859663-1215130β710680209971529350218807041234160β8-714248-668588-627811-591531-819468β9285370268490253034238977323352β10-52547.5-50004.1-47589.8-45335.7-58082.2

95

Table A.3: Extrapolation function coeﬃcients.
αi (At ) ≃ a1 A2t + a2 At + a3
αi

a1

a2

a3

α1

0.596513-51.2637

2671.42α2-8.38445

610.795-32070.8

α347.028-3124.69

176200α4-145.725

9227.81-580766

α5272.712-17412.4

1259880α6-315.621

21998.6-1858510α7217.994-18595.4

1857440α8-76.1481

9805.14-1209100α92.96989-2625.08

464122α104.30777

183.839-79868.6

βi (At ) ≃ a1 A2t + a2 At + a3
βi

a1

a2

a3

β1

0.424968-37.7705

1790.23β2-5.25997

457.72-21608.6

β328.7547-2508.95

119982β4-93.4782

8221.62-400258β5

198.209-17661.1

879227β6-279.967

25609.3-1313080

β7258.589-24911.1

1329670β8-146.579

15500.1-879123β944.1019-5453.58

344461β10-4.51931

763.944-61065.6

97

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