Systematic analysis of inelastic alpha scatteringoﬀ self-conjugate A = 4n nuclei

Satoshi Adachi
Doctoral Dissertation
Department of Physics, Kyoto University
January, 2018

Abstract
Strength distributions in excitation-energy spectra of atomic nuclei provide insights
into the nuclear structure because they directly reflect nuclear wave functions. The
multipole decomposition analysis (MDA) for the inelastic alpha scattering is very useful
to determine the strength distribution in excitation-energy region where several states
overlap each other, and widely used to study the nuclear incompressibility and the cluster
structures in nuclei. However, the reliability of the MDA was shaken by the puzzle ofthe missing monopole strength in 12 C.
Therefore, it is very important to examine and solve this puzzle of the inelastic alphascattering, but no systematic measurement of the inelastic alpha scattering to examine
the reliability of the theoretical calculation used in the MDA has been carried out until
now. In the present work, we systematically measured the cross sections of the inelasticalpha scattering at Eα = 130 and 386 MeV exciting low-lying discrete states in 12 C,
16 O, 20 Ne, 24 Mg, 28 Si and 40 Ca for the first time. In addition, the comparison of themeasured cross sections with the “parameter-free” distorted-wave Born-approximation(DWBA) calculation was carried out. All of the adjustable parameters in the DWBA
calculation were determined by the electromagnetic transition strengths and the elasticalpha scattering, therefore there is no room for so-called “the puzzle of missing monopole
strength” if the consistency between the measured cross sections and the present DWBA
calculations is confirmed.
It was found that the DWBA calculation with the density-independent eﬀective αNinteraction at Eα = 386 MeV was better than that with the density-dependent eﬀective
αN interaction, and thus the inadequate density dependence in the eﬀective αN interaction caused so-called “puzzle of missing monopole strength” by overestimating the crosssections for the ∆L = 0 transition. This puzzle was not specific to the Hoyle state in 12 C
but universally observed in all of the ∆L = 0 transitions. We also studied ambiguitiesof the DWBA calculation from the distorting potentials, phenomenological interaction,transition densities, and coupled-channel eﬀects.
The present results should provide the unique and important information for theoretical studies to develop a new reliable αN interaction.

Contents
1 Introduction1

1.1

Inelastic alpha scattering as an useful probe to examine nuclear structures

1

1.2

Application of inelastic alpha scattering . . . . . . . . . . . . . . . . . . .

6

1.2.1

Giant resonances in nuclei . . . . . . . . . . . . . . . . . . . . . . .

6

1.2.2

Cluster structure in nuclei . . . . . . . . . . . . . . . . . . . . . . .

9

1.3

Missing monopole strength of the Hoyle state . . . . . . . . . . . . . . . . 16

1.4

Systematic measurements of inelastic alpha scattering . . . . . . . . . . . 22

2 Experiment

24

2.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2

Beam line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3

Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.1

Gas target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4

Magnetic spectrometer Grand Raiden . . . . . . . . . . . . . . . . . . . . 27

2.5

Focal plane detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5.1

2.6

Multi-wire drift chambers . . . . . . . . . . . . . . . . . . . . . . . 30

Faraday cups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.6.1

Beam-line polarimeter . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.7

Data acquisition system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.8

Summary of the measurements . . . . . . . . . . . . . . . . . . . . . . . . 36

3 Data Analysis

38

3.1

Particle identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2

Track reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.1

Eﬃciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3

Calibration of the excitation energy and the scattering angle

3.4

Background subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48i

. . . . . . . 45

3.5

Cross sections for the (α, α) and (α, α′ ) reactions . . . . . . . . . . . . . . 49

4 Results

52

5 DWBA analysis

61

5.1

5.2

5.3

Optical-model potential and the eﬀective αN interaction . . . . . . . . . . 61
5.1.1

Density distribution of nucleus . . . . . . . . . . . . . . . . . . . . 62

5.1.2

Determination of the eﬀective αN interaction . . . . . . . . . . . . 64

Macroscopic transition densities and potentials . . . . . . . . . . . . . . . 71
5.2.1

Transition densities . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2.2

Transition potentials . . . . . . . . . . . . . . . . . . . . . . . . . . 72

The DWBA calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6 Discussion
6.1

76

Comparison between experimental data and DWBA calculations . . . . . 76
6.1.1

Normalization factor for the calculated cross sections to fit theexperimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.2

Characteristics of the macroscopic transition densities and potentials . . . 79
6.2.1

6.3

6.4

Dependence on the nuclear size . . . . . . . . . . . . . . . . . . . . 81

Ambiguity in the DWBA calculations . . . . . . . . . . . . . . . . . . . . 86
6.3.1

Transition densities . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.3.2

Distorting potentials . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.3.3

Phenomenological αN interaction . . . . . . . . . . . . . . . . . . . 88

6.3.4

Coupled-channel eﬀect . . . . . . . . . . . . . . . . . . . . . . . . . 89

Determination of the transition strengths . . . . . . . . . . . . . . . . . . 91

7 Summary94

ACKNOWLEDGMENTS

96

A Data tables of cross sections

98

A.1 Elastic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
A.1.1 Eα = 130 MeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
A.1.2 Eα = 386 MeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
A.2 Inelastic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
A.2.1 Eα = 130 MeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106ii

A.2.2 Eα = 386 MeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
References

147

List of Tables

151

iii

Chapter 1

Introduction
1.1 Inelastic alpha scattering as an useful probe to
examine nuclear structures
Strength distributions in excitation-energy spectra of atomic nuclei provide insights into
the nuclear structure because they directly reflect nuclear wave functions. An excitation
strength is fundamentally the overlap between wave functions of ground and excitedstates, and an energy is an eigenvalue of a nuclear Hamiltonian associated with the wave
function. The excitation strength and energy are experimental observables, and can bedirectly compared with theoretical calculations of the nuclear structures. Therefore, thedetermination of the strength distribution is important to study the nuclear structure.
For example, the strength distributions of the isoscalar monopole transition in which
the transferred angular momentum (∆L), spin (∆S), and isospin (∆T ) are zero provide
important information on the incompressibility of the nuclear matter [1–10] and thenuclear cluster structures [11–13].
In order to deduce the strength distribution in atomic nuclei, several experimental
probes are proposed and utilized. Inelastic electron scattering is one of the useful probes
to determine the excitation strength distribution [14]. Since there is little theoreticalambiguity in the interpretation of the experimental data obtained by electromagnetic
probes due to the well-known property of the electromagnetic interaction, the inelastic
electron scattering is very useful for measuring the excitation energies and strengths tothe discrete states. However, the inelastic electron scattering induces transitions withvarious spin, parity, and isospin transfers, and thus this feature makes diﬃcult to explore

1

excited states and to determine their spins, parities, and isospins in excitation-energy
region where several states overlap each other. The typical excitation energies of lowlying discrete states are less than 10 MeV below particle-decay thresholds. The isoscalar
giant monopole resonances (see Sec. 1.2.1) are known to appear around 80A−1/3 MeV
with a width of several MeV [15], and the alpha condensed states (see Sec. 1.2.2) are
predicted to appear at a few-MeV above the nα decay threshold in the self-conjugate
A = 4n nuclei, for example around 33 MeV in 24 Mg.
In contrast to the inelastic electron scattering, inelastic alpha scattering is useful insuch excitation-energy region. The inelastic alpha scattering has selectivity to isoscalar
natural-parity transitions where transferred spin and isospin are ∆S = 0 and ∆T = 0
since both spin and isospin of the alpha particle are zero. Therefore, the inelasticalpha scattering at Eα > 100 MeV has been extensively measured to extract thestrength distribution of the isoscalar natural-parity transitions at KVI in the Netherlands (Eα = 120 MeV) [1, 2], at Texus A&M University in the United States (Eα = 96–
240 MeV) [3–5], and at the Research Center for Nuclear Physics (RCNP), Osaka University in Japan (Eα = 386 MeV) [6–13, 16, 17]. The reaction mechanism of the inelasticalpha scattering is relatively simple, and the cross sections are reasonably reproduced
by a simple folding-model calculation, and approximately proportional to the relevant
nuclear transition strengths. The angular distribution of the cross sections depends on
the transferred angular momentum, but is not sensitive to details of nuclear wave functions. Therefore, the multipole decomposition analysis (MDA) works well to obtain thestrength distribution of the isoscalar natural-parity excitations even from continuous
excitation-energy spectra where many states with large widths overlap each other if thetheoretical calculation correctly describes the measured cross sections.
Figure 1.1 shows the cross sections for the 208 Pb(α, α′ ) reaction at Eα = 386 MeV
calculated by the distorted-wave Born-approximation (DWBA) calculation with the single folding-model potential assuming that a single state at Ex = 14.5 MeV exhausts
the total energy-weighted sum-rule (EWSR) strength. Experimental double diﬀerential
cross sections [d2 σ/dΩdEx ]exp are fitted by the calculated cross sections of transitions
with various transferred angular momenta [d2 σ/dΩdEx ]DWBA
to search for the best pa∆Lrameters a∆L (Ex ) in the MDA as,[ 2]exp ∑[ 2]DWBA
d σ(θc.m. , Ex )d σ(θc.m. , Ex )=a∆L (Ex )
.
dΩdEx
dΩdEx
∆L
∆L

2

(1.1)

10 4

208

Pb(α,α´)dσ/dΩ (mb/sr)

10Eα = 386 MeV

L=0, T=0

Ex = 14.5 MeV

L=1, T=03

L=2, T=0
L=3, T=0

10 2

10

1

0

2.5

5

7.5

10

θcm. (degree)
Figure 1.1: The DWBA calculations of the diﬀerential cross sections for various transitions in 208 Pb(α, α′ ) reaction at Eα = 386 MeV. Taken from Fig. 1.3 inRef. [18]. The blue, red, green, and yellow lines show the angular distribution of the cross sections with diﬀerent transferred angular momentum ∆Land transferred isospin ∆T , respectively. Note that ∆L and ∆T are denotedby L and T .

3

The determined a∆L (Ex ) corresponds to the fraction of the EWSR strength for eachtransferred angular momentum ∆L, and thus the strength distribution is obtained asa function of the excitation energy. Figure 1.2 shows examples of the MDA for the
208 Pb(α, α′ ) reaction in Ref. [18]. The diﬀerential cross section at each excitation energy(a)Ex= 13.5 MeV

10 2

10

1

2

d σ/dΩdE (mb/sr MeV)

10 3

10

-1

0

10

θcm. (deg.)(b)
Ex= 22.5 MeV

10 2

10

1

2

d σ/dΩdE (mb/sr MeV)

10 3

5

10

-1

0

5

10

θcm. (deg.)
Figure 1.2: Examples of the MDA for the 208 Pb(α, α′ ) reaction at Eα = 386 MeV. Taken
from Fig. 4.5 in Ref. [18]. The solid circles show the experimental crosssections. The solid lines near the measured cross sections are the sum of
the calculated DWBA cross sections. The thick and thin-solid, dotted, anddashed curves are the DWBA cross sections for ∆L = 1, ∆L = 0, ∆L = 2,
and ∆L = 3, whereas the sum of the cross sections for ∆L > 3 is shown by
the dot-dashed line. (a) The result of the MDA at Ex = 13.5 MeV. (b) The
result of the MDA at Ex = 22.5 MeV.

4

was well reproduced by the sum of the calculated cross sections, and the strength distributions of the isoscalar natural-parity transitions were successfully obtained as shownin Fig. 1.3.

Figure 1.3: Strength distributions for the ∆L = 0, 1, 2 and 3 transitions in 208 Pb by theinelastic alpha scattering at Eα = 400 MeV. Taken from Fig. 3 in Ref. [19].
Note that ∆L is denoted by L. The lines in (b) show the fit to the strength
distribution.
In Secs. 1.2.1 and 1.2.2 we briefly describe several examples of the application of theinelastic alpha scattering to the giant resonances and the cluster structures in nuclei.

51.2 Application of inelastic alpha scattering
1.2.1 Giant resonances in nuclei
The isoscalar giant monopole resonances (ISGMR) and the isoscalar giant dipole resonances (ISGDR) are extensively examined by means of the MDA for the inelastic alphascattering [3–13, 17, 20]. The strength distributions of the ISGMR and the ISGDR weresuccessfully extracted from the continuous excitation-energy spectra by the MDA, andthese analyses precisely determined the nuclear incompressibility of the symmetric nuclear matter [5–7]. Incompressibility of the nuclear matter is of fundamental importancein nuclear physics to define the equation of state for the nuclear matter, which describes
a number of interesting phenomena from collective excitations of nuclei to supernova
explosions and radii of neutron stars.
The infinite nuclear matter incompressibility K∞ is defined by the second derivativeof the energy per particle E/A with respect to the density ρ at the saturation point ρnm ,d2
K∞ = 9ρ2 2
dρ

( )
h
,
ρ ρ=ρnm(1.2)

where h is the isoscalar part of the Hamiltonian density for the nuclear matter [21].
The infinite nuclear matter does not exist on earth, therefore K∞ can not be measured
directly. However, the centroid energy of the ISGMR EISGMR is related to the incompressibility of the finite nucleus KA as√
EISGMR =ℏ2 KA,m⟨r2 ⟩m(1.3)

where m is the nucleon mass and ⟨r2 ⟩m is the mean square mass radius of the groundstate, and the incompressibility of the finite nucleus can be parameterized using KA as
follows [22],KA = K∞ + Ksurf A−1/3 + Kτ [(N − Z)/A]2 + KCoul Z 2 A−4/3 .

(1.4)

The nuclear matter incompressibility K∞ was previously deduced from the systematic
measurements of the KA values in various medium-heavy nuclei. For example, KA in
208 Pb was determined from E
208 Pb was obISGMR according to Eq. (1.3), and EISGMR intained as the averaged excitation energy of the isoscalar monopole-strength distribution

6

shown in Fig. 1.3(a) which exhausted most of the EWSR strength.
Nowadays, the K∞ is theoretically estimated from various energy functionals of thenuclear matter with diﬀerent parameters, and the ISGMR strength distributions are also
calculated for diﬀerent energy functionals with a self-consistent RPA framework [23,24]. The energy functional which gives the best description of the ISGMR strength
distribution is considered to be the most appropriate, and it gives the reliable value of
K∞ . The presently accepted value of K∞ is 240 ± 10 MeV [25–27] as introduced inRef. [28].
The asymmetry term Kτ in Eq. (1.4) is also important in addition to K∞ because
Kτ is crucial for the asymmetric nuclear matter like neutron stars. The inelastic alphascattering oﬀ the Sn isotopes provides information for Kτ [7]. Figure 1.4 shows thestrength distributions of the isoscalar monopole transition obtained for the Sn isotopes
with the MDA. The diﬀerence KA − KCoul Z 2 A−4/3 for the Sn isotopes as a function of

Figure 1.4: Strength distributions obtained by inelastic alpha scattering of the Sn isotopes at Eα = 400 MeV. Taken from Fig. 2 in Ref. [7]. Error bars represent
the uncertainty in the MDA. The solid lines show Lorentzian fits to the data.
the asymmetry parameter [(N − Z)/A] are shown in Fig. 1.5. The values of KA was7

Figure 1.5: Systematics of the diﬀerence KA − KCoul Z 2 A−4/3 in the Sn isotopes asa function of “asymmetry parameter” [(N − Z)/A]. KCoul was fixed at−5.2 MeV [21]. Taken from Fig. 4 in Ref. [7].
derived from the strength distribution of the isoscalar monopole transition, and KCoul
was fixed at −5.2 MeV. As expressed in Eq. (1.4), the term with K∞ is independent
from the mass number A, and the surface term with Ksurf is also almost independent
from A because the mass range of the measured Sn isotopes is as narrow as A = 114–
124. Therefore, KA − KCoul Z 2 A−4/3 approximately has a quadratic relationship withthe asymmetry parameter. Finally, the value of Kτ = −550 ± 100 MeV was obtained inRef. [7].
In addition to the stable nuclei, inelastic alpha scattering oﬀ unstable nuclei comesto be measured recently [29, 30] because the ISGMR in unstable nuclei attracts interest
in nuclear physics to examine the nuclear incompressibility of the asymmetric nuclear
matter.
As described above, the reliable determination of the strength distribution of theisoscalar monopole transition in stable and unstable nuclei by the MDA of the inelasticalpha scattering is crucial to establish the equation of state for the nuclear matter.

8

1.2.2 Cluster structure in nuclei
Alpha clustering phenomena in atomic nuclei have been studied for long time as aninteresting research field in nuclear physics since the Gamow’s α-decay theory [31, 32].
Many theoretical and experimental eﬀorts have been devoted since 1930, although there
still remain many hot topics to be discussed from both the experimental and theoreticalsides.
The Bloch-Brink model developed in 1950s and 1960s is one of the famous microscopic
theories to describe the cluster structures [33]. This model was originated from theconsideration for the 2α cluster system in Ref. [34], and is generalized for the multi-αsystem. The multi-α system is described as an antisymmetrized wave function consistingof wave functions of alpha clusters at diﬀerent position, and each alpha -cluster wave
function consists of four Gaussian single-particle wave functions at the same position.
Moreover, Ikeda et al. proposed that the cluster structures would appear near alphadecay threshold energies. This is known as “threshold rule” [35] and schematically
represented by Ikeda diagram as shown in Fig. 1.6.
One of the most famous alpha cluster states is the 0+
2 state at Ex = 7.65 MeV in
12 C. This excitation energy of 7.65 MeV is only 0.38 MeV above the 3α decay thresholdenergy in 12 C. This state is called as the Hoyle state, and plays a very important rolein the nucleosynthesis in the universe [37].

12 C is the doorway nucleus in the heavy-

elements synthesis, and 12 C is synthesized by the triple α reaction via the Hoyle state.
The Hoyle state is formed by the alpha capture reaction following the 8 Be formation
due to the collision of two alpha particles. 12 C is synthesized by the radiative decay of
the Hoyle state with a probability of (Γγ + Γπ )/(Γα + Γγ + Γπ ) ∼ 4×10−4 . The level
diagram in 12 C is shown in Fig. 1.7
Recently, the MDA for the inelastic alpha scattering is extensively carried out to
12 C was found at
search for the α cluster states. For example, the missing 2+
2 state in

Ex = 9.84 ± 0.04 MeV in 12 C with a width of 1.01 ± 0.15 MeV by the MDA of the
12 C(α, α′ ) reaction as shown in Fig. 1.8. Although this 2+ state in 12 C was incontrovert2

ibly predicted by the 3α cluster model as an excited state of the Hoyle state [38–40], no
experimental attempts to search for this state had been successful for a long time [41–43].
The MDA of the inelastic alpha scattering worked well to extract the ∆L = 2 transition
strength in the broad bump around Ex = 10 MeV in 12 C as seen in Figs. 1.8(c) and (d)and to find the missing 2+
2 state for the first time.

9

Figure 1.6: Ikeda diagram showing the relation between threshold energy to each decay
mode and the cluster structure. Taken from Fig. 1 in Ref. [36]. The small
circles without alphabets show alpha particles. The threshold energy todecay to each configuration from the ground state is shown in the unit of
MeV below the drawing.

10

Figure 1.7: Level diagram in 12 C related to the triple α reaction. Taken from Fig. 2 inRef. [36].

12

(a)
20

C(α,α' )Eα = 386 MeV 60oθlab=0
40

10

20

2

dσ /dΩdE (mb/sr MeV)

30

0

(c)

0
40

L=0
L=1(T=0)
L=2
L=3

30
20

C(α,α' )Eα = 386 MeVoθlab=3.7(d)

L=0
L=1(T=0)
L=2
L=3

30
20

10
0

12

(b)

10
8

9

10
11
Ex (MeV)

12

0

8

9

10
11
Ex (MeV)12

Figure 1.8: Taken from Ref. [16]. (a) and (b) Excitation energy spectra of the 12 C(α, α′ )reaction at Eα = 386 MeV at θlab = 0.0◦ and 3.7◦ , respectively. (c) and(d) Decomposition to the cross sections with various transferred angularmomenta by the MDA. A broad bump around 10.0 MeV for L = 2 component
was observed.

11

It was pointed out that large isoscalar monopole transition strengths are a signatureof cluster excitation in atomic nuclei [44]. This was theoretically explained by using
the Bayman-Bohr theorem [45] in Ref. [46]. The clustering degrees of freedom are
inherently possessed even by simple shell-model wave functions and can be activatedby the monopole excitation. As an example, the Hoyle state, which has a spatiallydeveloped 3α cluster structure, is excited with a large isoscalar monopole strength ofB(E0; IS) = 121 ± 9 fm4 [47]. This strength is about three times larger than the
single-particle limit in 12 C [47]. For 24 Mg, several 0+ states are strongly excited byisoscalar monopole transitions in the inelastic alpha scattering as reported in Ref. [11].
The strength distribution of the isoscalar monopole transition was extracted as shownin Fig. 1.9. Several discrete peaks due to the 0+ state near the threshold energies for
20 Ne+α, 12 C+12 C, and 16 O+2α were observed in the ∆L = 0 strength distribution.

Actually, the theoretical calculation suggests these 0+ states have spatially developed
cluster structures [48]. Figure 1.10 shows the theoretical calculation for the strength
functions of the isoscalar monopole excitations in 24 Mg. The β and γ deformations, theisoscalar transitions, and the many particle-hole configurations in 24 Mg are taken into
account in the basis set (c) of the wave function. The gray thick line in the right panel
shows the strength function smeared by a Lorentzian with a width of 0.8 MeV. The
vertical dashed lines indicate cluster decay threshold energies. The smeared strength
function shows similar structure to the measured ∆L = 0 strength distribution in the
top panel of Fig. 1.9.
Recently it is suggested that the Hoyle state is a 3α condensed state, where the 3alpha particles are condensed into the lowest s orbit [49]. The authors claimed that thisstate is a gas-like state of alpha clusters and is akin to the Bose-Einstein condensate.
Figure 1.11 shows the theoretically calculated occupation of the alpha-particle orbitals
associated with the Hoyle state in 12 C compared with the ground state. The occupation
probability of the 0s (S1 ) orbital is more than 70% in the Hoyle state.
It is expected that such alpha condensed states exist not only in 12 C but also in otherself-conjugate A = 4n nuclei. It was predicted in Ref. [51] that the nα condensed stateswith J π = 0+ appear above nα decay threshold energies in the heavier self-conjugate
A = 4n nuclei up to n = 10. Figure 1.12 shows the predicted excited energies androot-mean-square radii of the nα condensed states with J π = 0+ in the self-conjugate
A = 4n nuclei.
At present, the candidates of the alpha condensed states are proposed and widely

12

6α

C + 3α∆L=012

C + 12C
O + 2α12416

Ne + α20

6

EWSR Fraction (%)/Channel

2
0
6

∆L=1

4
2
0
6

∆L=2

4
2
0
6

∆L=3

4
2
0

5

10

15

20

25

30

Excitation Energy (MeV)
Figure 1.9: Strength distributions for the ∆L = 0–3 transitions obtained from the MDA
for the 24 Mg(α, α′ ) reaction [11]. Taken from Fig. 3 in Ref. [11]. The blue
vertical line with each configuration show the threshold energy to decay into
its configuration. The red vertical lines show the excited energies of the
known states in 24 Mg.

13

C+3α

12

8

6α

O+2α

C+12C
12

(c)

16

Ne+α

(a)
(b)

10

20

S(E) (% of EWSR/MeV)

12

6
4
2
0

10

15

20

25

30 10
Excitation Energy (MeV)

15

20

25

30

Figure 1.10: Strength functions of the isoscalar monopole excitations in 24 Mg calculated
in Ref. [48]. Taken from Fig. 2 in Ref. [48]. Three Calculated strength
functions (a), (b), and (c) are obtained by diﬀerent AMD variational wave
function. See text and Ref. [48] for more details.

1.0

0.8

µλ

0.6

0.4

0.2

0

S1 D1 G1 S2 D2 G2 S3 D3 G3Ground state

S1 D 1 G 1 S2 D 2 G 2 S3 D 3 G3
Hoyle state

Figure 1.11: Occupation of the alpha-particle orbitals associated with the Hoyle state in
12 C compared with the ground state. Taken from Fig. 7 in Ref. [50].

14

9

20

8

15

7

<r2> (fm)E (MeV)

25

10
5

6
5

0

4
3

4

5

6

7

8

9

10

11

3

n

4

5

6

7

8

9

10

11

n

Figure 1.12: The theoretically predicted excited energies (left) and root-mean-square
radii (right) of the nα condensed states with J π = 0+ in the self-conjugate
A = 4n nuclei. Taken from Fig. 2 in Ref. [51]. The excited energies aremeasured from the nα decay threshold energies.
accepted in 8 Be and 12 C only. The ground state in 8 Be and the Hoyle state in 12 C
are successfully described by the alpha condensed wave functions [51, 52]. The alphacondensed states in other nuclei are still being experimentally searched for and mucheﬀort is devoted to find out these states [11, 12]. We recently proposed to search for
α condensed states in heavier self-conjugate A = 4n nuclei by measuring the isoscalarmonopole transition to the highly excited states above the nα decay threshold in the
inelastic alpha scattering. The reliable determination of the transition strength of theisoscalar monopole transition for the highly excited states is crucial to search for the αcondensed states.
In addition to the alpha condensed states, there are theoretical predictions of thecluster states called “molecular states” to exist, and these states are experimentally
searched for. A typical example is reported in 10 Be. Several excited states in 10 Be aresuggested to have a unique α-2n-α configuration. Two excess neutrons play a role ofcovalent particles in molecular orbits among the two alpha clusters and behave like avalence electron pair bonding two atoms [53–55]. The density distributions of the single+
10 B are shownparticle wave functions for valence neutrons in the 0+
1 and 02 states inin Fig. 1.13. From the experimental point of view, the measurement of the transition
strength of the isoscalar monopole transition in 10 Be is strongly desired to compare thetheoretical prediction with the experiment. However, the inelastic electron scattering oﬀ
10 Be cannot be measured because 10 Be is unstable. A new measurement of the inelasticalpha scattering oﬀ 10 Be and the MDA of the measured data should be done.

15

Figure 1.13: Density distributions of the single-particle wave functions for valence neu10 B(0+ ). Schematic figure of the molecular orbitals;
trons in 10 B(0+1 ) and2
π and σ orbitals around the 2α core are also shown at the bottom. Taken
from Fig. 3(a) in Ref. [55].

1.3 Missing monopole strength of the Hoyle state
The MDA is established on the assumption that the cross sections of the inelastic alphascattering are reasonably well described by the DWBA calculation and are approximately
proportional to the relevant transition strengths. However, a result contradictory to thelinear dependence was reported in the monopole excitation to the Hoyle state fromthe ground state in 12 C. The isoscalar monopole strength determined from the inelastic electron scattering exhausts about 15% of the energy-weighted sum-rule (EWSR)
strength [47, 56, 57], but that from the inelastic alpha scattering exhausts as small as
7.6% [20]. This discrepancy is called as “missing monopole strength of the Hoyle state”.
The cross sections of the inelastic electron scattering for the Hoyle state was measured
over a wide range of the momentum transfer q [56, 57]. From Fig. 1.14, the electromagnetic transition matrix element between the ground state and the Hoyle state in 12 C
was determined as Mπ = 5.37 ± 0.22 fm2 [56]. Recently, the cross sections for the Hoyle
state were measured at a very small q region, and the matrix element was updated tobe Mπ = 5.29 ± 0.14 fm2 as shown in Fig. 1.15 [57]. The black circles and the solidline show the previous experimental data and the global fit to the data. The red solid

16

Figure 1.14: The√square root of the reduced transition probability B(C0, q) multipliedby 4π/q 2 as a function of the square of the momentum transfer q [56].
The experimental data were obtained by the inelastic electron scattering at
E = 28–60 MeV and the scattering angles θlab = 105◦ –165◦ . The experimental data and the theoretical calculation fitted to the data are shownby the points and solid line, respectively. Note that the intercept at q = 0
corresponds to the electromagnetic matrix element Mπ divided by 6.

17squares and the dashed line show the new experimental data taken in Ref. [57] and the
new fit to the data. The theoretical predictions by diﬀerent cluster models are shownby the dotted and dash-dotted lines.

2

5

6 Z |Ftr(q)| / q [fm ]

6

2

7
12

12

C (e,e’) C

Ex = 7.654 MeV
+

01

+

02

4
3
2
1
0

1

0.5
2

-2

q [fm ]
Figure 1.15: The charge transition form factor Ftr (q) multiplied by 6Z/q 2 as a functionof the square of the momentum transfer q. Taken from Fig. 4 in Ref. [57].
The experimental data were obtained by the inelastic electron scattering at
E = 29–78 MeV and the scattering angles θlab = 69◦ –141◦ . See text andRef. [57] for more details. Note that the intercept at q = 0 corresponds tothe matrix element.
If one state at an excitation energy Ex exhausts 100% of the EWSR strength forthe isoscalar monopole transition, the matrix element Mπ for the transition follows therelation,
Ex |Mπ |2 =ℏ2
A⟨r2 ⟩,
2m(1.5)

where m, A, and ⟨r2 ⟩ denotes the mass of the nucleon, the mass number of the considered
nucleus, and the square of the charge root-mean-square radius of the nucleus. Therefore,
the matrix element Mπ = 5.29 fm2 corresponds to 14% of the EWSR strength.
The inelastic alpha scattering from 12 C at Eα = 240 MeV was measured in Ref. [20].
The authors of Ref. [20] performed the DWBA analysis using a macroscopic transition
density, and compared the experimental data with the DWBA calculation as shown in18

Fig. 1.16. They deduced the Hoyle state exhausts 7.6% of the EWSR strength.
1000

(a)

12

12 *
C(α,α
α,α’)
α,α C ,Eα=240 MeV(a) Ex= 10.3 MeV, Jπ=0+x10(b) Ex= 7.65 MeV, Jπ=0+100(c) Ex=10.84 MeV, Jπ=1-dσ/dΩ (mb/sr)

(b)
10

1

(c)
X0.1
0.1

0.01
0

5

10

15

20

25

30

θc.m. (deg)Figure 1.16: The diﬀerential cross sections for the 12 C(α, α′ ) reaction at Eα = 240
MeV compared with the DWBA calculations [20]. Taken from Fig. 8 in
Ref. [20]. (b) The solid circles and line represent the experimental data andthe DWBA calculation for the Hoyle state at Ex = 7.65 MeV, respectively.
(a) and (c) for the transition to the 0+ state at Ex = 10.3 MeV and 1−
state at Ex = 10.84 MeV, respectively.
In Ref. [58], the authors performed a double folding-model analysis of the inelastic
α + 12 C scattering at Eα = 104, 139, 172.5, and 240 MeV using a complex densitydependent interaction (CDJLM interaction, which was constructed on the M3Y-Parisinteraction [59]) and the reliable wave functions of the ground and Hoyle states in 12 Cfrom 3α resonating group method (RGM) calculation [40]. The monopole excitation
strength to the Hoyle state predicted by the 3α RGM calculations is 22.8% of the EWSR
strength and close to the strength reported from the inelastic electron scattering [56, 57].
Nevertheless, the calculated cross sections of the inelastic alpha scattering for the Hoyle
state using the 3α RGM wave functions were significantly larger than the measured crosssection at Eα = 104–240 MeV as shown in Figs. 1.17 and 1.18. The authors of Ref. [58]performed both the DWBA and the coupled-channel (CC) calculations at Eα = 104, 139,
172.5, and 240 MeV. The dashed and dotted lines in Figs. 1.17 and 1.18 represent the
DWBA and CC calculations with 22.8% of the EWSR strength, whereas the solid and19dash-dotted lines represent the DWBA calculation with 6.9% of the EWSR strength andthe CC calculation in which the imaginary part of the optical potential was increased toreproduce the experimental data.

Figure 1.17: The DWBA and CC calculations for the cross section of the 12 C(α, α′ )reaction exciting the Hoyle state at Eα = 104 and 139 MeV. The solid
circles are the experimental data, whereas the dash, dotted, solid, dashdotted lines are the theoretical calculations. Taken from Fig. 3 in Ref. [58].
This discrepancy is consistent with the previous results [20] that the excitation strength
for the Hoyle state determined by the inelastic alpha scattering is much smaller thanthat determined by the inelastic electron scattering. In contrast to the Hoyle state, the
12 C were well reproduced by the CC calculation at thecross sections to the 2+
1 state inintermediate energies as shown in Fig. 1.19.
It was claimed in Ref. [58] that this problem is due to the weakly bound structure of the
Hoyle state which significantly enhances the absorption in the exit α + 12 C∗ (0+2 ) channel,
and the reaction mechanism of inelastic alpha scattering might strongly couple to the
nuclear structure. Therefore, the approximate linear proportional relation between crosssections and relevant transition strengths might not be valid in the analysis of the inelastic alpha scattering. It is a serious problem suggesting that the strength distribution
determined by the MDA of the inelastic alpha scattering might not be reliable.

20

Figure 1.18: Same as Fig. 1.17, but at Eα = 172.5 and 240 MeV. Taken from Fig. 4 inRef. [58].

Figure 1.19: The DWBA and CC calculations for the cross section of the 12 C(α, α′ )reaction exciting the 2+
1 state at Eα = 104, 139, 172.5, and 240 MeV. The
solid circles are the experimental data, whereas the dotted and solid lines
are the theoretical calculations. Taken from Fig. 5 in Ref. [58].

211.4 Systematic measurements of inelastic alpha scattering
The MDA of the inelastic alpha scattering at Eα = 96–386 MeV has been extensively
performed to examine the nuclear structure as described in Sec. 1.2, however, the reliability of the MDA was shaken by the puzzle of the missing monopole strength discussed
in Sec. 1.3. It is very important to examine and to solve this puzzle of the inelasticalpha scattering, otherwise, it leads us to serious situation where we cannot reliably
examine the nuclear structure by means of the inelastic alpha scattering. Nevertheless,the systematic measurement of the inelastic alpha scattering to examine the reliabilityof the theoretical calculations has never been carried out until now.
In the present work, we measured the cross sections of the inelastic alpha scattering at
Eα = 130 and 386 MeV, exciting low-lying discrete states in 12 C, 16 O, 20 Ne, 24 Mg, 28 Si
and 40 Ca, and systematically compared the measured cross sections with the DWBA
calculation for the first time. These targets were selected because the ambiguity of thenuclear structure in the DWBA calculation becomes small in the self-conjugate even-evennuclei.
Transition-density distributions used in the present DWBA calculations were obtainedby the macroscopic models [60, 61], and the amplitudes of the transition-density distributions are determined from the electromagnetic transition strengths. The electromagnetictransition strengths for the discrete states were reliably determined by the electromagnetic probes like the inelastic electron scattering. Since the natural-parity transitions
induced by the electromagnetic interaction are the “electric” transitions, their electromagnetic transition strengths are sensitive to protons only. On the other hand, theisoscalar natural-parity transitions induced by the inelastic alpha scattering are sensitive to both of protons and neutrons. Because the proton distribution in self-conjugate
nuclei can be assumed to be same with the neutron distribution due to the isospin
symmetry, the isoscalar transition strengths in such nuclei can be estimated from theelectromagnetic transition strengths. Moreover, the spin and parity of the ground statesin even-even nuclei are 0+ , therefore the transferred angular momentum is uniquely
identified from the spin and parity of the final state.
A reliable calculation of cross sections is necessary to extract the transition strengthfrom the experimental data in the MDA. We performed the “parameter-free” DWBA
calculation in which all of the adjustable parameters are determined by the electromagnetic transition strengths and elastic scattering, and systematically compared with the22measured cross sections to test the applicability of the conventional DWBA calculation to the inelastic alpha scattering, for the first time. We also studied ambiguitiesof the DWBA calculations from the distorting potentials, phenomenological interaction,transition densities, and coupled-channel eﬀects.

23Chapter 2

Experiment
2.1 Overview
We systematically measured the cross sections of the inelastic alpha scattering at forward
angles including 0◦ at Eα = 130 and 386 MeV, exciting low-lying states in 12 C, 16 O,
20 Ne, 24 Mg, 28 Si, and 40 Ca.

A series of the measurements (E253, E308, E369, and E402) was performed at Research Center for Nuclear Physics (RCNP), Osaka University. The incident alpha beam
was accelerated by the coupled cyclotrons at RCNP, and transported to the target. Momenta of scattered alpha particles from the target were analyzed with the high-resolution
magnetic spectrometer Grand Raiden (GR) [62]. The scattered particles are detectedwith the focal plane detectors of GR.

2.2 Beam line
A 4 He2+ beam was accelerated to 130 MeV by the AVF cyclotron. The accelerated
beam was transported to the West experimental hall using the bypass beam line. For
the measurements with the 386-MeV 4 He2+ beam, 4 He2+ beam accelerated up to 87.1
MeV by the AVF cyclotron was next injected into the ring cyclotron and accelerated
again to 386 MeV. After the acceleration, the 386-MeV beam were transported to theWest experimental hall. The accelerated beam was transported to the target throughthe WS beam line in the West experimental hall. Figure 2.1 shows the AVF cyclotron,
the ring cyclotron, the bypass beam line, the West experimental hall, the WS beam line,
and the target position.

24

LAS
TargetGrand Raiden

WS
BLP-2

BLP-1

Ring
CyclotronWest Experimental Hall
N-BLP

Bypass Beam Line050 m
superconducting
solenoid magnets

N
AVF
Cyclotron

Figure 2.1: Overview of the AVF cyclotron and the ring cyclotron in RCNP. West experimental hall and the WS beam line are also shown. “WS” means the WS
beam line.

25

The 130-MeV 4 He2+ beam was achromatically transported to the target, whereas the386-MeV 4 He2+ beam was dispersively transported to the target in the measurements.
The excitation-energy resolution at Eα = 386 MeV was improved with the dispersive
beam transport [63–65]. However, the dispersive beam transport for the 20 Ne gas target+

could not be done because the energy spread of the 4 He2 beam from the ring cyclotron(∆E = 250 keV) was wider than the usual (∆E = 100 keV) when the 20 Ne gas target
was used. This wider energy spread caused by the condition of the accelerators. The
dispersive transport of such a beam with a wide energy spread results in unacceptably
large beam size on the target.

2.3 Targets
We used self-supporting foil targets for 12 C, 24 Mg, 28 Si, and 40 Ca. The SiO2 foil wasused as the 16 O target. The contribution from nat Si was subtracted using a nat Si target.
For the 20 Ne target, a cooling gas target system was used [66, 67].
Table 2.1 summarizes the target nuclei, thicknesses, and isotope enrichment.
Table 2.1: Thicknesses and isotope enrichment of the targets used in the present measurements with the 4 He2+ beams at Eα = 130 and 386 MeV. All the foil
targets are self-supporting.
Nucleus

State

Thickness

Enrichment

4 He2+ energy

foil
foil
gasd
foil
foil
foil
foil
foil

(mg/cm2 )
2.2
2.2
2.4
1.2
2.5
1.72
2.16
1.63

(%)
98.9a
99.8c
99.95
99.92
> 99.992.2a
92.2a> 99.9

130
386
(MeV)
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓

12 C

SiO2 b
20 Ne
24 Mg
24 Mg
28 Si
28 Si
40 Ca

a Natural abundance.
b The SiO

2 foil was used as thec Natural abundance of 16 O.
d See the text for more details

26

16 O target.

2.3.1 Gas target
The isotopically enriched 20 Ne gas with a purity of 99.95% was filled into the gas cell andcooled by liquid N2 in order to increase the target density. The entrance and exit windows
were sealed with aramid films with a thickness of 4 µm. The pressure and temperatureof the 20 Ne gas were monitored using the diaphragm pressure gauge (KM31, Nagano
Keiki Co., Ltd.) and the platinum resistance thermometer (R610-3, Chino Co., Ltd.)during the measurements. The gas pressure and temperature were at 78.0 ± 1.2 kPa
and 86.5 ± 0.4 K, respectively. The eﬀective thickness of the 20 Ne gas target along thebeam axis was calibrated by filling the gas cell with the CO2 gas at the pressure close
to that of the 20 Ne gas. We determined the thickness of the 20 Ne gas as 11.2 ± 1.7 mm
by comparing the measured cross section of the elastic alpha scattering from 12 C withthe known value. This thickness corresponded to 2.4 ± 0.4 mg/cm2 .

2.4 Magnetic spectrometer Grand Raiden
Scattered alpha particles were analyzed using the magnetic spectrometer Grand Raiden(GR) [62]. Figure 2.3(a) shows an overview of GR and its related instruments for theexperimental setup at θlab = 0◦ .
The GR has a magnet configuration of QSQD(M)D(+D). Here, the characters Q, S,
D, M represents the quadrupole, sextupole, dipole, and multipole magnets, respectively.
The QSQD(M)D(+D) are typically denoted as Q1, SX, Q2, D1, MP, D2, and DSR
in order. The design specification of GR are listed in Table 2.2. The MP are mainly
used to correct ion-optical aberrations, whereas the DSR is an auxiliary magnet formeasurements of the spin-rotation parameters. The MP and DSR, which are denoted by
the characters enclosed in parentheses, were not used to analyze the momenta of scatteredparticles in the present measurements. The DSR was used during the measurement at
θlab = 0◦ as a steering magnet to guide the incident 4 He2+ beam to Faraday cups.

2.5 Focal plane detectors
The two multi-wire drift chambers (MWDC1 and 2) and two plastic scintillation counters (PS1 and 2) were installed at the focal plane of GR. The detector system wasaligned along the focal plane at 45◦ with respect to the central ray of the spectrometer. Figs. 2.3(b) and 2.4 show the focal plane detectors, although the focal plane27

Target gas transferredby the dry pumpLiquid nitrogen

Vaporized nitrogen gas
Stepping motor

1.0 m

Moved by the
stepping motor

Target container
0.5 m

Gate valve 1

Removedhere.

Gate valve 2

Scatteringchamber

0.0 m

Pipes
LN2 reservoir
BeamTarget cell (Front view)
Foil targets

Figure 2.2: Schematic view of the gas target system, which is install on top of the scattering chamber [67]. 20 Ne gas was filled in the target gas cell and cooled by
the liquid N2 .

28

(a)

0

1

2

3m

D2
MP

(b)
Primary beam

DSR

MWDC1
MWDC2
PS1
PS2

FP-FC andBeam Viewer

D1Grand Raiden
scattered particles

Focal Plane
Detectors

to 0 deg. FC

Focal Plane
Polarimeter0 deg. FC

Beam Viewer

Q2
SX
Q1

Q1-FC

SC-FC

Scattering Chamer
4

He2+ Beam

Figure 2.3: (a) Overview of the Grand Raiden spectrometer and its related instruments.
The experimental setup at θlab = 0◦ is shown [68]. The solid lines with arrowheads in the spectrometer show the trajectories of primary beam particles,
and dashed and dotted lines show those of scattered particle. The primarybeam was guided to the Faraday cup, which was located at the downstreamof the spectrometer (0 deg. FC). (b) A schematic view around the focal
plane. In case of the spectrum measurements for low-lying excited states,
the primary beam was stopped at the focal plane Faraday cup (FP-FC),
which was placed at the upstream of the focal plane.

29

Table 2.2: Design specification of the magnetic spectrometer Grand Raiden
Configuration
Mean orbit radius
Total deflection angle
Focal plane length
Tilting angle of focal plane
Maximum magnetic field strength
Maximum magnetic rigidity
Horizontal magnification (x|x)
Vertical magnification (y|y)
Momentum dispersion
Momentum range
Momentum resolution
Acceptance of horizontal angle
Acceptance of vertical angle
Solid angle

Grand Raiden
QSQD(M)D(+D)
3m
162◦
150 cm45◦
1.8 T
5.4 T·m−0.417
5.98
15.451 m
5%37, 076±20 mr
±70 mr
∼ 5.6 msr

polarimeters were not used in the present measurements. The MWDCs have holes attheir high-momentum sides of the sensitive area, and the holes enable us to conduct the
primary beam particles to the downstream for the 0◦ measurements.
The MWDCs were used to determine the three-dimensional trajectories of scatteredparticles at the focal plane, and the PS1 and PS2 which were placed at just the downstream of the MWDCs were used to generate the data-acquisition trigger. The plastic
scintillators with a thickness of 10 mm were used as the PS1 and PS2, and the scintillation light was detected by the photo-multiplier tubes (PMTs) on the both sides of thePS1 and PS2.

2.5.1 Multi-wire drift chambers
Design specification of the MWDCs are shown in Table. 2.3. Each MWDC consists of
two anode planes, namely horizontal (X) plane and the 48.2◦ -tiled (U) plane, and theanode wires are placed with the 2 mm spacing in the anode plane. The anode wires
consists of the sense wires and the potential wires, and the spacing of sense wires are
6 mm in X-plane and 4 mm in U-plane as shown for X-plane in Fig. 2.5. High voltageof −4.8 kV was applied to the cathode planes of the MWDCs, and of −400 V wasapplied to the potential wires for the 20 Ne measurements. The wire configuration of the30

Primary BeamStandard Focal Plane Detectors
MWDC1
MWDC2
PS1 & PS2

Beam Viewer
Focal Plane Polarimeter
MWPC1
MWPC2

Carbon Slab

MWPC3
MWPC4
HS-X

Beam Viewer

Scattered particle

Electron
Sweeper
Faraday cup
Figure 2.4: Schematic view of the focal plane detectors and downstream instruments of
GR for the θlab = 0◦ measurement.
Table 2.3: Design specification of the MWDCs for GR placed at the focal plane. Supplied
voltages are for the 20 Ne measurements.
Wire configuration
Active area
Number of sense wires
Cathode-anode gap
Anode wire spacing
Sense wire spacing
Sense wire material
Potential wire material
Supplied voltage

Gas mixture
Gas seal
Pre-amplifier
Digitizer

MWDC
X(0◦ =vertical), U(−48.2◦ )1150W mm × 120H mm
192(X), 208(U)
10 mm
2 mm
6 mm (X), 4 mm (U)20µm ϕ Au-W
50µm ϕ Au-Cu/Be−4.8 kV (cathode)−400 V (potential)
0 V (sense)
Argon(71.4%)+Iso-butane(28.6%)+Iso-propyl-alcohol
12.5µm aramid film
LeCroy 2735DC
LeCroy 3377 drift chamber TDC

31

MWDCs for GR are shown in Fig. 2.6. The MWDCs were filled with a mixed gas which
consists of argon, iso-butane, and iso-propyl-alcohol. Iso-propyl-alcohol was mixed at
2 ◦ C vapor pressure into argon gas, and the argon and iso-butane gases were mixed with
71.4% and 28.6%. The MWDCs are placed at 45◦ with respect to the central ray of thespectrometer as shown in Fig. 2.4, therefore several sense wires typically generate thesignal per plane for one trajectory of a scattered charged particle as shown in Fig. 2.5.

SctteredParticle
Cathode Plane
10mm

2mm
d i -1

Anode WiresdiPotential Wired i +1

Sense Wire

Cathode Plane
Figure 2.5: Schematic view for an X-plane of the MWDC. Typical trajectory of a scattered particle is also shown by an arrow.

0

0

y

-48.2°

MWDC-X

4m

6mm

m

x

z

z = beam direction

MWDC-U

Figure 2.6: Wire configuration for the sense wires of the MWDCs. The wire numberingorder are shown by arrows.

32

2.6 Faraday cups
The incident 4 He2+ beam was stopped by the four Faraday cups, and beam currents
during the measurements were monitored. The zero-degree Faraday cup (0 deg. FC),
which was located at the downstream of the focal plane, was used for the measurements
at θlab = 0◦ , and the primary beam was transported to the 0 deg. FC through GR. Whenthe 0 deg. FC was used, the low-lying excited states at Ex < 2.5 MeV for Eα = 130 MeV
and at Ex < 7.5 MeV for Eα = 386 MeV could not be measured due to the geometricallimitation of the MWDCs. The lateral distances between trajectories of the primarybeam and scattered alpha particles exciting such low-lying states at the focal plane were
too short for the MWDCs to detect the scattered alpha particles separately from theprimary beam. To measure the (α, α′ ) spectra for such low-lying states at θlab = 0◦ , analternative Faraday cup called as the focal plane Faraday cup (FP-FC) was used. The
FP-FC was installed at the front of the MWDCs as shown in Fig. 2.3(b). Since the
FP-FC was placed very close to the sensitive area of the MWDCs, the FP-FC enabled
us to detect the scattered alpha particles exciting the low-lying states by the MWDCs.
However, huge background γ rays from the FP-FC hit both the MWDCs and the trigger
plastic scintillators, therefore the beam intensity was limited lower than 1.0 nA when
the FP-FC was used.
For the measurements at θlab = 2.5◦ –5.0◦ , the primary beam was stopped at the
Faraday cup behind the Q1 magnet of GR (Q1-FC) as shown in Fig. 2.3(a). The Faraday
cup installed in the scattering chamber (SC-FC) was used in the measurements at the
backward angles θlab ≥ 6.0◦ .
Charge collection eﬃciencies of the four Faraday cups were calibrated within 3% uncertainties by using a beam monitor (BLP-1) consisting of plastic scintillators locatedon the beam line.

2.6.1 Beam-line polarimeter
The beam-line polarimeters (BLP-1 and BLP-2) are located at the WS beam line asshown in Fig. 2.1. The BLP-1 was used to calibrate the charge collection eﬃciencies ofthe four Faraday cups in the present measurements. Each BLP consists of four sets ofplastic scintillation detectors as shown in Fig. 2.7.
The scintillation detectors were placed at the angles of 14.4◦ and 40.0◦ . An aramid
film with a thickness of 4 µm was used as the analyzer target to calibrate the charge

33

z

x

R

14.4°

40.0°

R'

y

Plastic
Scintillator
L'

L

PMT
Aramid target
Beam
Figure 2.7: Schematic top view of the BLP setup. The plastic scintillation detectors
labeled by “L” and “L′ ” are one set of the detectors. Similar to the horizontal(x-z) plane, there are also two pairs of the detector in the vertical (y-z)
plane, but the detectors in the vertical plane were not used in the present
measurements.
collection eﬃciency. An alpha particle flies to the detector at 14.4◦ when it is elasticallyscattered by a proton in the analyzer target, and the recoiling proton flies to the detector
at 40.0◦ . α + p elastic scattering was identified by detecting scattered alpha particle andrecoiling proton by a set of detectors in coincidence. The charge collection eﬃciencies ofthe four Faraday cups are calibrated by comparing the output of the four Faraday cupswith the counting rate of the coincidence signals from the BLP detectors.

2.7 Data acquisition system
The data-acquisition triggering system was constructed with LeCroy 2366 universal logic
modules (ULM), which were equipped with field programmable gate-array (FPGA)
chips [69], and the trigger signals were generated by the coincidence of the PS1 andPS2 signals. A schematic view of the trigger circuit is shown in Fig. 2.8. Since the focal
plane scintillators had the PMTs on both left and right edges, the PS1 and PS2 signals
were generated with the coincidence of both PMTs signals.
The overview of the data acquisition (DAQ) system was shown in Fig. 2.9. Each34

PS1-L

CFD

PS1-R

CFD

LeCroy 2366

Mean
Timer

Universal Logic Module

GR FP Trigger
To TDC (3377)
common stop

PS2-L

CFD

PS2-R

CFD

Mean
Timer

ADC gate

FERA FERATFC

TFC start

Analog signals

Figure 2.8: Schematic view of the trigger circuit and a part of circuit for analog signals.
event was constructed of the drift-time data from the MWDCs digitized by LeCroy
3377 TDC, the charge and timing signals from the trigger scintillators encoded with
FERA and FERET system, and the input register. The Flow Controlling Event Tagger(FCET) [70] attached the event header, event number, and input register words to thedata from LeCroy FERA and FERET systems, and the attached data was used to checkthe consistency of data in the subsequent event reconstruction.
Gigabit Ethernet

CPU
ScalerScaler

MWDC's

HSM
HSM
HSM
HSM
HSM
HSM
K2917
RM5576VME
FCET

linux on-line stationsaho-a,b

LeCroy 3377 System

Scintillators

Scaler

K3922

FCET
FERET
FERA

FERA DRVHD
Event build
Recording
Online analysis

CAMAC Crate

Figure 2.9: Overview of the data acquisition system.
The digitized data were transferred in parallel via ECL bus lines to the High Speed
Memory (HSM) modules in the VME crate. The stored data in the HSMs were transferred via gigabit Ethernet to the Linux-based DAQ server (saho-a). The data recording
to the storage, event building, online analysis were performed on the DAQ server.

35

2.8 Summary of the measurements
Table 2.4 summarizes the angular ranges for the cross-section measurements of the elasticalpha scattering at Eα = 130 and 386 MeV, respectively. The cross sections at Eα = 130
MeV were measured for all the targets except 20 Ne. The cross sections at backward
angles between θc.m. = 41.9◦ and 70.3◦ were measured using diﬀerent experimental setup
with the Si and CsI(Tl) detectors. The cross sections at Eα = 386 MeV were measured
for 12 C and 20 Ne only, although the cross sections for 24 Mg and 16 O were taken from
Refs. [11] and [12], respectively. Since the cross sections for 28 Si and 40 Ca at Eα = 386
MeV were not measured in the present experiment nor taken from previous works, aspecial treatment was done in the analysis of 28 Si and 40 Ca as described in Sec. 5.1.
Similarly, Table 2.5 summarizes the angular ranges for the cross-section measurementsof the inelastic alpha scattering at Eα = 130 and 386 MeV, respectively. The crosssections at Eα = 130 MeV were measured for all the targets except 20 Ne.

36

Table 2.4: Angular ranges for the cross section measurements of the elastic alpha scattering at Eα = 130 and 386 MeV.
Nucleus
12 C
16 O
20 Ne
24 Mg
28 Si
40 Ca

Elastic scattering
Eα = 130 MeV
Eα = 386 MeV
θc.m. (deg)θc.m. (deg)◦◦
5.0 –70.3
2.7◦ –34.2◦◦◦
4.7 –58.1
2.5◦ –28.5◦ a
Not measured
5.1◦ –27.4◦◦◦
5.4 –70.4
5.3◦ –28.8◦ b◦◦
8.0 –48.8
Not measured◦◦
4.1 –59.3
Not measured

a Ref. [12].
b Ref. [11].

Table 2.5: Angular ranges for the cross section measurements of the inelastic alpha scattering at Eα = 130 and 386 MeV.
Nucleus
12 C
16 O
20 Ne
24 Mg
28 Si
40 Ca

Inelastic scattering
Eα = 130 MeV
Eα = 386 MeV
θc.m. (deg)θc.m. (deg)◦◦
0.5 –18.2
0.4◦ –29.5◦◦◦
0.5 –17.0
2.4◦ – 7.0◦
Not measured
0.5◦ –13.9◦◦◦
0.5 –15.8
0.8◦ –12.0◦◦◦
0.5 –15.5
2.2◦ – 6.4◦
0.4◦ –14.8◦
2.1◦ – 6.2◦

37

Chapter 3

Data Analysis
3.1 Particle identification
Particle identification was performed applying two gates on the data. One was the gate
for the time diﬀerence between the data-acquisition trigger and the signal synchronized
to the acceleration by the AVF cyclotron (RF signal), which corresponds to the time offlight from the target to the focal plane. The other was the gate for the energy depositin the PS1.
Since the flight path of the scattered particle from the target to the focal plane depends
on the scattering angle and the momentum of the particle, the correction to the time
diﬀerence was performed to remove the dependence on the position at the focal plane(xfp ) related to the momentum and the scattering angle at the target (θtgt ) as shown in
Fig. 3.1. The raw and corrected time diﬀerence are shown in the left and right panels in
Fig. 3.2, respectively. Since the RF signal was downscaled by a rate divider circuit, two
prominent peaks due to the alpha particles were observed as shown in Fig. 3.2.
Particle identification by the energy deposit in the PS1 was performed using the signals
from the the PMTs attached to both edges of the PS1. Considering that the emitted
light in the scintillators is attenuated exponentially during the transmission to the PMT,
the geometric mean of the ADC values of the two PMTs on the left and right edges of thePS1 is approximately independent from the position where the scattered particle hits,
and is proportional to the energy deposit. Figures 3.3(a) and (b) show the correlation
between the ADC values from the left and right PMT and the horizontal positions,whereas Fig. 3.3(c) is the correlation between the geometrical mean of the ADC values

38

750

22

(a)

18

700

16
14

650

12
10

600

8
6

550

4

45

(b)

20

Time difference (ch)Time difference (ch)

750

40

700

35
30

650

25

600

20
15

550

10
5

0
500
−400−300−200−100 0 100 200 300 400
xfp (mm)500−2 −1.5 −1 −0.5 0 0.5
θtgt (deg)

750

750

140

(c)
700

120
100

650
600

80
60
40

550

20

0
500
−400−300−200−100 0 100 200 300 400
xfp (mm)Corrected time difference (ch)Corrected time difference (ch)

2

1

1.5

2

0

180

(d)

160

700

140
120

650

100
80

600

60
40

550

20

500
−2 −1.5 −1 −0.5 0 0.5
θtgt (deg)

1

1.5

2

0

Figure 3.1: Two demensional scatter plot of the time diﬀerence between the dataacquisition trigger and the RF signal versus the horizontal position at the
focal plane (xfp ) and the horizontal scattering angle (θtgt ). The data for themeasurement of highly excited region in 12 C from the inelastic alpha scattering at Eα = 386 MeV are shown. (a) and (b) show the time diﬀerence
before the correction, whereas (c) and (d) show after the correction.

39

104
Yield (count/ch)

Yield (count/ch)

103
102
10
1
0

500
1000
1500
Time difference (ch)

103
102
10
1
0

2000

500
1000
1500
Time difference (ch)2000

Figure 3.2: Example of the particle identification using the time diﬀerence. The regions
shown by the arrows correspond to the alpha particles. The left panel shows
the time diﬀerence before the correction, whereas the right shows that after
the correction.
from the left and right PMTs. The geometrical mean is almost independent from theparticle position although the signals from the PMTs attenuate with increasing the
distance between the particle position and the PMTs.
The loci with the higher ADC values in Fig. 3.3 were attributed to the alpha particles,
whereas those with the lower ADC values were due to the low energy-loss particles. These
low energy-loss particles were eliminated as seen in Fig. 3.4 by limiting the time diﬀerence
between the data-acquisition trigger and the RF signal (see Fig. 3.2). Figure 3.5 shows
the spectra of the geometric mean of the ADC values from left and right PMTs. The
small peak due to the low energy-loss particles in Fig. 3.5(a) disappears in Fig. 3.5(b)in similar to Fig. 3.4. The alpha particles were identified by selecting the region shownby the arrow in Fig. 3.5(b).

40

800

800

600

104

3

10

400

102

200

10

1
0
−400−300−200−100 0 100 200 300 400
xfp (mm)Geometrical mean (ch)

800

(b)
Right ADC (ch)Left ADC (ch)

(a)

104

600

3

10

400
200

102

10

1
0
−400−300−200−100 0 100 200 300 400
xfp (mm)

104

(c)
600

10

400

102

200

10

3

1
0
−400−300−200−100 0 100 200 300 400
xfp (mm)Figure 3.3: Example of the ADCs of the scattered particle at the PS1 for the measurement of the elastic and inelastic scattering from aramid films at Eα =386 MeV at θlab = 8.0◦ . (a) and (b) Two-dimensional scatter plots for the
ADC values from the left or right PMT versus the horizontal positions (xfp ).
(c) Two-dimensional scatter plot of the geometrical mean of the ADC values
from the left and right PMTs versus the horizontal position.

41

800

800

104

104

(b)

600

10

400

102

200

10

3

Right ADC (ch)Left ADC (ch)

(a)

1
0
−400−300−200−100 0 100 200 300 400
xfp (mm)Geometrical mean (ch)

800

600

10

400

102

200

10

3

1
0
−400−300−200−100 0 100 200 300 400
xfp (mm)

104

(c)
600

10

400

102

200

10

3

1
0
−400−300−200−100 0 100 200 300 400
xfp (mm)Figure 3.4: Same as Fig. 3.3, but gated by the time diﬀerence between the dataacquisition trigger and the RF signals.

104

(a)
Yield (count/2 ch)

Yield (count/2 ch)

104
103
102
10
1
0

200
400
600
Geometrical mean (ch)

103
102
10
1
0

800

(b)

200
400
600
Geometrical mean (ch)800

Figure 3.5: Example of the particle identification using the energy deposit in the PS1.
The region shown by the arrows correspond to the alpha particles. (a) The
geometric mean of ADCs without the gate of the time diﬀerence. (b) With
the gate of the time diﬀerence.

423.2 Track reconstruction
The three-dimensional trajectories of the particles were reconstructed by using the position information from the two sets of the MWDCs at the focal plane of GR. The position
parallel to the anode wires in each anode plane was determined from the vertical drift
lengths di−1 , di , · · · at hit wires as shown in Fig. 2.5. The positions on the X-plane of the
MWDC1 (X1) and the X-plane of the MWDC2 (X2) defined an X-tracking plane, andthose on the U-planes (U1 and U2) defined a U-tracking plane, and thus the trajectoryof the detected particles was determined as in Fig. 3.6.

rear-X

rear-Ulane plane
p
g
ing
ackin

U-tr

k

ac
X-tr

y
z

track
x
z = beam direction
front-X

front-U

Figure 3.6: Schematic view of the determination of the particle trajectory.
Since only the drift time to each sense wire can be directly measured, the calibrationof the drift time to the drift length was performed using the measurement of a continuum
spectrum in a highly excited region in 12 C. Schematic histograms which represents theconversion from the drift time to the drift length are show in Fig. 3.7.
As already discussed in Sec. 2.5.1, a typical scattered particle induces the signals at
two or more sense wires as shown in Fig. 2.5. We define a group of such adjacent sense
wires as a “cluster”. In the present analysis, all the anode planes were required to have
only one cluster which contains not less than two hit wires. The cluster which contains
only one wire was ignored because the background events such as X- and γ-rays usually43

counts/channel

10 2aroundwires

2000

close tocathode foil

1500
1000
500
0
200

300

400

500

TDC (X1)

600(TDC channel)

conversion

arbitrary unit

10 2
4000
3000
2000
1000
0

0

2

4

6

drift length

8

10
(mm)

Figure 3.7: Conversion from the drift times to the drift lengths of the MWDCs
cause the single hit wire.

3.2.1 Eﬃciency
The eﬃciency of each plane of the MWDCs is estimated as follows,
NX1∩U 1∩X2∩U 2,
NU 1∩X2∩U 2
NX1∩U 1∩X2∩U 2=,
NX1∩X2∩U 2
NX1∩U 1∩X2∩U 2=,
NX1∩U 1∩U 2
NX1∩U 1∩X2∩U 2,=
NX1∩U 1∩X2

ϵX1 =

(3.1)

ϵU 1

(3.2)

ϵX2
ϵU 2

(3.3)
(3.4)

where NX1∩U 1∩X2∩U 2 denotes the number of events in which the tracks of the particles were reconstructed by all of the four planes of the MWDCs, and NU 1∩X2∩U 2 ,(NX1∩X2∩U 2 , NX1∩U 1∩U 2 , and NX1∩U 1∩X2 ) in the denominator denotes the number of
events in which the tracks were reconstructed by the three planes except the X1 (U 1,
X2, and U 2) plane. The overall tracking eﬃciency ϵMWDC of the MWDCs was calculated44

by ϵMWDC = ϵX1 × ϵU 1 × ϵX2 × ϵU 2 .
The position dependence of the tracking eﬃciency was examined. Since the position at
the focal plane could be related to the excitation energy in the inelastic alpha scattering,
the eﬃciency of each plane of the MWDCs for α particles inelastically scattered fromthe 12 C target at Eα = 386 MeV and θlab = 8.0◦ was obtained as a function of excitationenergy. As shown in Fig. 3.8, the tracking eﬃciency of the MWDCs was almost constantto the excitation energy.

1Efficiency

0.8

0.6

0.4

0.2

0

10

12.5

15

17.5

20

22.5

25

27.5

30

Excitation energy (MeV)

Figure 3.8: Example of the detection eﬃciency of each plane of the MWDCs for alphaparticles inelastically scattered from the target at Eα = 386 MeV and θlab =8.0◦ as a function of excitation energy. The black, red, green, and bluelines show the eﬃciencies for X1, U1, X2, and U2 planes, respectively. The
histogram with error bars shows the total tracking eﬃciency.

3.3 Calibration of the excitation energy and the scatteringangle
The excitation energy was calculated from the momentum and scattering angle of thedetected particle assuming the two-body scattering at the incident energy of Eα = 386

45

or 130 MeV. Since only the momentum deviation from the central ray can be measured
with the magnetic spectrometer, the energy scale of the calculated excitation energy wascalibrated using the peak position for the known discrete excited states in the measured
spectra.
Scattering angles within the angular acceptance of GR were calibrated using the sieve
slit as reported in Ref. [68]. Figure 3.9 shows the sieve slit used for the 20 Ne measurements. The large hole at the center of the slit is designed to be placed on the incident
beam axis. The upstream side of the slit was at the 577.5 mm downstream from the target. The hole spacings in the horizontal and vertical directions corresponded to 0.496◦
and 1.190◦ from the target, respectively.

5 mm

12 mm

Beam
Figure 3.9: Drawing of the sieve slit used for the calibration in the 20 Ne measurements.
The large hole at the center of slit is designed to be placed on the incident
beam axis. The dash-dotted lines are drawn to guide the eyes.
Figures 3.10 and 3.11 show the data from five measurements with diﬀerent magnetic
field strengths using the sieve slit to calibrate the scattering angle for the 20 Ne measurements. The field strength of GR was set at five diﬀerent values with the steo of 0.8%.
The particles which were scattered at the target and passed through a hole of the sieve
slit were detected at the focal plane. The two dimensional scatter plot of the measured
angle at the focal plane (θfp) versus the horizontal position (xfp ) is shown in Fig. 3.10.
The particles passed through the holes at the same horizontal angle form one locus asseen in Fig. 3.10. The measured angle (θfp ) and the position (xfp ) were converted to the46

scattering angle at the target (θtgt ) and the momentum diﬀerence from the central ray
using the optical properties of GR, and calibrated using the geometrical information of
the sieve slit. The two dimensional scatter plot of the calibrated scattering angle at thetarget (θlab ) versus the momentum diﬀerence from the central ray is shown in Fig. 3.11.

Measured angle θfp (deg)The angular acceptance of GR was limited to |θtgt | ≤ 0.8◦ and was divided into four
6

102

4
2

10

0
−2

1

−4
−6
−400

−200

0
Position xfp (mm)

200

400

Scattering angle θtgt (deg)Figure 3.10: Two dimensional scatter plot of the measured horizontal angles at the focal
plane of GR (θfp ) versus the position at the focal plane (xfp ). The data fromfive measurements with diﬀerent magnetic field strengths are superimposed.
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
−2

10

1

10−1

−1.5

−1

−0.5
0
0.5
Momentum (%)

1

1.5

2

Figure 3.11: Two dimensional scatter plot of the scattering angles at the target (θtgt )versus the momentum diﬀerence from the central ray. The data from fivemeasurements with diﬀerent magnetic field strength are superimposed. The
dashed lines are drawn at |θtgt | = 0.8◦ , and the data in the region between
the two lines were analyzed to obtain the cross sections.
angular bins with a width of 0.4◦ to obtain the cross sections.

47

3.4 Background subtraction
Beam particles passing through the target spread out due to multiple Coulomb scatteringand hit beam ducts. These particles caused continuous background in excitation-energy
spectra at forward angles. These background events were removed using the vertical
position information of scattered particles at the focal plane because alpha particles
scattered from the target were vertically focused near the focal plane whereas backgroundparticles scattered at diﬀerent positions were not focused [71].
An (α, α′ ) spectrum measured using the 20 Ne gas target is shown in Fig. 3.12(a).
Several discrete peaks for the ground and excited states in 12 C, 14 N, 16 O, and 35 Cl wereC : elastic
12

Ne : 2+1 state20

14

O : elastic
16

35

200

20

300

Cl : elastic

400

N : elastic(a)
Ne : elastic

500

100
−1.5

−1

−0.5

−1.5

−1

−0.5 00.5
1
1.5
Excitation energy (MeV)

400
300

0

0.5

1.5

2

2.5

3

2

2.5

3

20

1

Ne : 2+1 state(b)

Ne : elastic

0
−2
500

20

Yield (Counts/10 keV)

Yield (Counts/10 keV)

600

200
100
0
−2

Figure 3.12: Excitation-energy spectra for the 20 Ne(α, α′ ) reaction at Eα = 386 MeV
measured at θlab = 10.8◦ . (a) The open histogram represents the excitationenergy spectrum obtained using the gas cell filled with 20 Ne. The hatchedhistogram represents the normalized spectrum obtained using the empty gas
cell. (b) A background-free 20 Ne(α, α′ ) spectrum obtained by subtracting
the background events originated from aramid films.
observed. These nuclei are contained in the aramid films at the entrance and exit windows of the gas cell. In order to subtract the background events due to the aramid48

windows, the background measurements with the empty gas cell was carried out. The
pressure of the empty gas cell was kept at the order of 10−2 Pa. The measured background spectrum is shown by the hatched histogram in Fig. 3.12(a). The backgroundfree spectra were successfully obtained by subtracting background spectra as seen in
Fig. 3.12(b). The normalization factor for the background subtraction was calculated
by taking into account the beam intensities, target thicknesses, tracking eﬃciencies, anddata-acquisition eﬃciencies. Similarly, the background-free excitation energy spectra for
16 O were obtained by subtracting the excitation energy spectra for nat Si from those fornat SiO .
2

3.5 Cross sections for the (α, α) and (α, α′ ) reactions
The obtained excitation-energy spectra for the (α, α′ ) reactions at Eα = 130 MeV at

2

4

6

8

10

Excitation Energy (MeV)

2 BG
from FC
1

2

4

6

8

9.31 (0+)

4.12 (4+)
4.24 (2+)

6.43 (0+)
7.35 (2+)
7.62 (3−) −8.36 (3 )11.52 (2+)

10

8.28 (0+)

Ca(α,α’)Eα = 130 MeV
θlab = 0.0º

3

0

4

Excitation Energy (MeV)

6.29 (3−)

4

40

2

6.01 (4+)

12.05 (0+)

Yield (counts/20 keV)0125.90 (1−)

6.69 (0+)

4.98 (0+)

1
0

×103

Si(α,α’)Eα = 130 MeV
θlab = 0.0º

210

4.49 (5 )28

1.78 (2+)

Yield (counts/20 keV)

3

8

5

Excitation Energy (MeV)

×104
4

6

10

8.37 (4+)
8.58 (2+)

012Mg(α,α’)Eα = 130 MeV
θlab = 0.0º

15

−

10

Excitation Energy (MeV)24

8.09 (2+)

2

×103
20

3.74 (3−)
3.90 (2+)

8

46.88 (3−)
7.93 (2−)
8.90 (1−)
9.93 (1−)

6

6

3.35 (0+)

9.64 (3−)

28Eα = 130 MeV
θlab = 0.0º

9.84 (2+)
10.36 (4+)

10

4

0
4

12

O(α,α’)

Yield (counts/20 keV)Eα = 130 MeV
θlab = 0.0º

14

16

6.05 (0+)
6.13 (3−)+
7.12 (1−) 6.92 (2 )×103C(α,α’)

Yield (counts/20 keV)

127.65 (0+)

6

4.44 (2+)

Yield (counts/20 keV)

×104

1.37 (2+)θlab = 0.0◦ are shown in Fig. 3.13. The typical energy resolution of the (α, α′ ) spectra

6

8

10

Excitation Energy (MeV)

Figure 3.13: Excitation-energy spectra for the (α, α′ ) reactions at Eα = 130 MeV measured at θlab = 0.0◦ .

49

at Eα = 130 MeV and 386 MeV was about 85 keV and 95 keV at the full width at half
maximum (FWHM) for the solid targets except 20 Ne, respectively. On the other hand,the energy resolution of the 20 Ne(α, α′ ) reaction at Eα = 386 MeV was about 250 keV
at FWHM. This wider energy spread was caused by the condition of the accelerators.
From these excitation energy spectra, the cross sections of the (α, α) reaction at Eα =130 and 386 MeV were obtained as follows.
The number of the scattered alpha particles exciting an excited state was counted in
the excitation energy spectrum. The cross section at the scattering angle θlab in thelaboratory frame [dσ/dΩ(θ)]lab was determined from the number of the detected alphaparticles Y , the number of the target nucleus Ntgt , the number of the incident alphaparticles Nbeam , the solid angles for the spectrum ∆Ω, the tracking eﬃciency ϵtrack , thecharge-collection eﬃciency ϵFC , and the data-acquisition eﬃciency ϵDAQ .
[

]
Y
dσ
1
=
(θlab )·.
dΩNtgt · Nbeam · ∆Ω ϵtrack · ϵFC · ϵDAQ
lab

(3.5)

The cross section in the laboratory frame [dσ/dΩ(θlab )]lab was converted to that in thecenter-of-mass frame [dσ/dΩ(θc.m. )]c.m. using the relation between the laboratory andcenter-of-mass frames as follows,
[

]
[
]
dσ
dσ
(θc.m. )= g(θlab )(θlab )
,
dΩ
dΩ
c.m.
lab

(3.6)

where g(θlab ) is the conversion factor. g(θlab ) and θc.m. are calculated from the relativistic
kinematics using the masses of the target nucleus and the alpha particle, the excitationenergy of the state, and the scattering angle in the laboratory frame θlab . The errors
were estimated from uncertainties of all the quantities in Eq. (3.6).
The measured cross sections of the elastic (α, α) reactions at Eα = 386 and 130 MeV
relative to the Rutherford cross sections (Rutherford ratio) are presented in Figs. 3.14
and 3.15, respectively. Similarly, the cross sections of the inelastic (α, α′ ) reactions at
Eα = 386 and 130 MeV were obtained as presented in the next chapter.

50

Figure 3.14: Measured cross sections of the elastic alpha scattering oﬀ 12 C and 20 Ne at
Eα = 386 MeV relative to the Rutherford cross sections (Rutherford ratio)
compared with the theoretical calculation. The solid circles with error bars
show the measured cross sections. The solid and dashed lines represent
the results of calculation with the density-independent (DI) and densitydependent (DD) αN interactions, respectively. For the details about thetheoretical calculations, see Chap. 5.

Figure 3.15: Same as Fig. 3.15, but for 12 C, 16 O, 24 Mg, 28 Si, and 40 Ca at Eα = 130 MeV.

51

Chapter 4

Results
The measured (α, α′ ) cross sections for the ∆L = 0, ∆L = 2, and ∆L = 3 transitions at
Eα = 386 and 130 MeV are shown in Figs. 4.1 and 4.2, Figs. 4.3 and 4.4, and Figs. 4.5
and 4.6, respectively. The excited states of which the cross sections were measured in
the present work are listed in Table 4.1. Figures 4.7 and 4.8 show those for ∆L = 4
and ∆L = 1, respectively. In Figs. 4.2–4.8, the cross sections of the excited states
whose electromagnetic excitation strengths from the ground states were reported in theprevious studies [47, 72–76] are presented, whereas the measured cross sections of theexcited states whose electromagnetic transition strengths are unknown are shown in
Fig. 4.9.

52

Table 4.1: List of the excited states identified in the present work.
Nucleus

Jπ

Ex

12 C

0+
1
2+
1
0+
2
3−
1
1−
1
0+
1
0+
2
3−
1
2+
1
1−
1
2+
2
4+
1
2+
3
0+
3
0+
1
2+
1
4+
1
3−
1
0+
2
0+
1
2+
1
4+
1
2+
2
4+
2
0+
2
2+
3
3−
1
3−
2

(MeV)
0.00
4.44
7.65
9.64
10.84
0.00
6.05
6.13
6.92
7.12
9.84
10.36
11.52
12.05
0.00
1.63
4.25
5.62
6.73
0.00
1.37
4.12
4.24
6.01
6.43
7.35
7.62
8.36

12 C
12 C
12 C
12 C
16 O
16 O
16 O
16 O
16 O
16 O
16 O
16 O
16 O
20 Ne
20 Ne
20 Ne
20 Ne
20 Ne
24 Mg
24 Mg
24 Mg
24 Mg
24 Mg
24 Mg
24 Mg
24 Mg
24 Mg

Eα
130 386
(MeV)
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓a
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓b
✓
✓b
✓
✓
✓
✓
✓b
✓
✓b
✓
✓b
✓
✓b

Nucleus

Jπ

Ex

24 Mg

2+
5
1−
1
0+
3
2+
7
0+
1
2+
1
4+
1
0+
2
0+
3
3−
1
2+
2
2+
3
1−
1
2+
4
1−
2
3−
2
2+
5
0+
1
0+
2
3−
1
2+
1
5−
1
1−
1
3−
2
2+
2
0+
3
4+
1
2+
3

(MeV)
9.00
9.15
9.31
10.36
0.00
1.78
4.62
4.98
6.69
6.88
7.93
8.26
8.90
9.48
9.93
10.18
10.51
0.00
3.35
3.74
3.90
4.49
5.90
6.29
8.09
8.28
8.37
8.58

24 Mg
24 Mg
24 Mg
28 Si
28 Si
28 Si
28 Si
28 Si
28 Si
28 Si
28 Si
28 Si
28 Si
28 Si
28 Si
28 Si
40 Ca
40 Ca
40 Ca
40 Ca
40 Ca
40 Ca
40 Ca
40 Ca
40 Ca
40 Ca
40 Ca

a Ref. [12]b Ref. [11]53Eα130 386
(MeV)
✓
✓b
✓
✓b
✓
✓b
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓

16

dσ/dΩ (mb/sr)12C(α,α'), Eα = 386 MeV
0+2 (7.65 MeV)

102

102

20

O(α,α'), Eα = 386 MeV
0+3 (12.05 MeV)

102

10

10

10

1

1

1

10−1

10−1

10−1

10−2

0

5

10

15

20

10−2
25
0

dσ/dΩ (mb/sr)

Mg(α,α'), Eα = 386 MeV
0+2 (6.43 MeV)

102

10

10

1

1

10−1

10−1

10−2

0

5

10

15

10

15

10−2
20
0

5

10

15

20

28

24

102

5

Ne(α,α'), Eα = 386 MeV
0+2 (6.73 MeV)

10−2
20
0

Si(α,α'), Eα = 386 MeV
0+2 (4.98 MeV)

5

10

15

20

θc.m. (deg)Figure 4.1: Measured cross sections of the (α, α′ ) reaction at Eα = 386 MeV for the∆L = 0 transitions whose electromagnetic transition strengths are known
from the previous studies [47, 72–76]. The solid circles with error bars show
the measured cross sections. The solid lines with error bands and dashedlines are the theoretical calculations with the density-independent (DI) anddensity-dependent (DD) αN interactions, respectively. The error bands areshown for the theoretical calculations with DI interactions only. For details
about the theoretical calculations, see Chap. 5.

Figure 4.2: Same as Fig. 4.1, but the cross sections for the ∆L = 0 transitions at Eα =130 MeV.

54

dσ/dΩ (mb/sr)12C(α,α'), Eα = 386 MeV

103
102

16

O(α,α'), Eα = 386 MeV

103

2+1 (4.44 MeV)

2+1 (6.92 MeV)

102

10

10

1

10

10

−1

10

10−2

10−2
35
0

10−2

dσ/dΩ (mb/sr)

1
−1

10

15

25

30

2

4

6

8

10 12 14

103

103

103

102

102

102

10

10

10

1

1
Ne(α,α'), Eα = 386 MeV
2+1 (1.63 MeV)

−2

10

0

2

4

6

8

10 12 1424

Mg(α,α'), Eα = 386 MeV

103

2+5 (9.00 MeV)

102
10

10

Mg(α,α'), Eα = 386 MeV

102+1 (1.37 MeV)

−2

10

0

2

4

6

8

10 12 1428

Si(α,α'), Eα = 386 MeV

103

2+1 (1.78 MeV)

102

10−2

1

10
10 12 14

10

0

2

4

6

8

10 12 1440

Ca(α,α'), Eα = 386 MeV
2+1 (3.90 MeV)

10

−2

8

2+3 (7.35 MeV)

10

−1

6

10 12 14Mg(α,α'), Eα = 386 MeV

102

1

4

8

24

103

−1

2

6

10

1

0

4

−1

−1

−2

2

24
−1

10

10

0

1

20
−1

10

dσ/dΩ (mb/sr)

20

2+3 (11.52 MeV)

10

1

5

O(α,α'), Eα = 386 MeV

102

−1

0

16

103

0

2

4

6

8

10 12 14

10−2

0

2

4

6

8

10 12 14

θc.m. (deg)Figure 4.3: Same as Fig. 4.1, but the cross sections for the ∆L = 2 transitions at Eα =386 MeV.

55

Figure 4.4: Same as Fig. 4.1, but the cross sections for the ∆L = 2 transitions at Eα =130 MeV.

56

dσ/dΩ (mb/sr)12C(α,α'), Eα = 386 MeV
3−1 (9.64 MeV)

102

102

102

10

10

10

1

1

10

10−1

10−2
0

dσ/dΩ (mb/sr)

1
16

−1

5

10

15

20

25

O(α,α'), Eα = 386 MeV
3−1 (6.13 MeV)

10−2
30
0

5

10

15

20

10−1
10−2
20
0

102

102

102

10

10

10

1

1
24

Mg(α,α'), Eα = 386 MeV
3−1 (7.62 MeV)

10−1

10−2

0

5

10

15

10−2
20
0

5

10

15

20

1
40

24

10−1

Ne(α,α'), Eα = 386 MeV
3−1 (5.62 MeV)

Mg(α,α'), Eα = 386 MeV
3−2 (8.36 MeV)
5

10

15

10−1
10−2
20
0

Ca(α,α'), Eα = 386 MeV
3−1 (3.74 MeV)
5

10

15

20

θc.m. (deg)Figure 4.5: Same as Fig. 4.1, but the cross sections for the ∆L = 3 transitions at Eα =386 MeV.

Figure 4.6: Same as Fig. 4.1, but the cross sections for the ∆L = 3 transitions at Eα =130 MeV.

57

dσ/dΩ (mb/sr)

102

102

16

O(α,α'), Eα = 130 MeV
4+1 (10.36 MeV)

10

102

24

Mg(α,α'), Eα = 130 MeV
4+1 (4.12 MeV)

10

10

1

1

1

10−1

10−1

10−1

24

10−2

0

5

10

15

20

2

10

15

20

10−2
25
0

5

10

15

20

25

10

28

Si(α,α'), Eα = 130 MeV
4+1 (4.62 MeV)

10

10

1

1

10−1

10−1

10−2

5

2

10

dσ/dΩ (mb/sr)

10−2
25
0

Mg(α,α'), Eα = 130 MeV
4+2 (6.01 MeV)

0

5

10

15

20

10−2
25
0

20

Ne(α,α'), Eα = 386 MeV
4+1 (4.25 MeV)
5

10

15

20

θc.m. (deg)Figure 4.7: Same as Fig. 4.1, but the cross sections for the ∆L = 4 transitions at Eα =130 and 386 MeV.

58

dσ/dΩ (mb/sr)12C(α,α'), Eα = 130 MeV
1−1 (10.84 MeV)

102

102

O(α,α'), Eα = 130 MeV
1−1 (7.12 MeV)

102

10

10

10

1

1

1

10−1

10−1
EWSR 1%

10−2

0

5

10

15

10−2
20
0

Mg(α,α'), Eα = 386 MeV
1−1 (9.15 MeV)

10

102

10

15

28

Si(α,α'), Eα = 130 MeV
1−1 (8.90 MeV)

10−2
20
0
102

10

10

10−1

1

1

10−1
EWSR 0.3%

10−3

0

5

10

15

5

10

15

20

28

Si(α,α'), Eα = 130 MeV
1−2 (9.93 MeV)

10−1
EWSR 1%10−2200Mg(α,α'), Eα = 130 MeV
1−1 (9.15 MeV)

EWSR 0.3%

5

1

10−2

24

10−1
EWSR 6%24

dσ/dΩ (mb/sr)

16

EWSR 2%

5

10

15

10−2
20
0

5

10

15

20

dσ/dΩ (mb/sr)

θc.m. (deg)40Ca(α,α'), Eα = 130 MeV
1−1 (5.90 MeV)

102
10
1

10−1
EWSR 7%

10−2

0

5

10

15

20

Figure 4.8: Same as Fig. 4.1, but the cross sections for the ∆L = 1 transitions at
Eα = 130 and 386 MeV. Note that the calculated cross sections are normalized to exhaust 1%, 6%, 0.3%, 1%, 2%, and 7% of the EWSR strengths
16 O(1− ), 24 Mg(1− ), 28 Si(1− ),of the isoscalar dipole transitions in 12 C(1−
1 ),
1
1
1
−
−
28 Si(1 ), and 40 Ca(1 ), respectively. For details about the theoretical cal2
1
culations, see Chap. 5 and Sec. 6.4.

59

dσ/dΩ (mb/sr)

102

102

24

Mg(α,α'), Eα = 130 MeV
0+3 (9.31 MeV)

10

102

24

Mg(α,α'), Eα = 386 MeV
0+3 (9.31 MeV)

10
1

1

10−1

10−1

10−1

0

5

10

15

20

10−2
25
0

2

dσ/dΩ (mb/sr)

4

6

10−2
8 10 12 14 16
0

2

10

10

10
1

5

10

15

20

25

2

10

28

Si(α,α'), Eα = 130 MeV
2+5 (10.51 MeV)

1040

Ca(α,α'), Eα = 130 MeV
0+3 (8.28 MeV)

10

1

1

10−1

10−1

28

Si(α,α'), Eα = 130 MeV
3−2 (10.18 MeV)

10−1

10−2

0

5

10

15

20

10−2
25
0

5

10

15

20

10−2
25
0

5

10

15

20

25

θc.m. (deg)210dσ/dΩ (mb/sr)2Si(α,α'), Eα = 130 MeV
0+3 (6.69 MeV)

10

1

10−2

28

40

Ca(α,α'), Eα = 130 MeV
4+1 (8.37 MeV)

10
1

10−1

10−2

0

5

10

15

20

25

Figure 4.9: Same as Fig. 4.1, but the cross sections for the inelastic alpha scatterings at
Eα = 130 and 386 MeV where the corresponding electromagnetic transition
strengths are not reported in the previous studies [47, 72–76]. Note that thecalculated cross sections with the DI interactions (solid lines) are fitted tothe experimental data.

60

Chapter 5

DWBA analysis
The theoretical model calculation is necessary to extract the transition strength from theobtained cross sections, and we performed the DWBA analyses using the computer code
ECIS-95 [77]. The measured cross sections were compared with the DWBA calculations
using the known values of the electromagnetic transition strengths in order to checkthe reliability of the theoretical calculation, which are widely used in the previous [3–13, 16, 17]. In this chapter, the formalism of the present calculations are explained.
In the present DWBA calculation, the optical-model potentials for the elastic alphascattering were used as the distorting potentials. The same distorting potential was usedin the entrance and exit channels for each nucleus. The optical-model potentials and thetransition potentials between the ground and excited states were calculated by folding
the macroscopic densities with the eﬀective αN interaction.

5.1 Optical-model potential and the eﬀective αNinteraction
An optical-model potential U (r) for elastic alpha scattering is obtained by folding the
ground-state density distribution of the target nucleus with eﬀective αN interaction
u [|r − r ′ |, ρ0 (r′ )] as follows,∫
U (r) =[]
dr ′ ρ0 (r′ ) u |r − r ′ |, ρ0 (r′ ) ,

61

(5.1)

where r ′ and r denote the position of a point-like nucleon in the target and an incident
alpha particle, respectively. ρ0 (r′ ) is the ground-state density distribution of a target
nucleus. Figure 5.1 shows the definition of the spacial coordinates r and r ′ .

Targetαr'rFigure 5.1: Definition of the spacial coordinates in Eq. (5.1).
u [|r − r ′ |, ρ0 (r′ )] is empirically parameterized by five adjustable parameters [78] as[]u |r − r ′ |, ρ0 (r′ ) =

[
]
′ 2
2
2/3
−v 1 + βρ0 (r′ ) e−|r−r | /αv
[
]
′ 2
2
2/3
−iw 1 + βρ0 (r′ ) e−|r−r | /αw ,(5.2)

where v and w are the depth parameters, β is the density-dependence parameter, andαv and αw are the range parameters for real and imaginary parts of the αN interaction,
respectively.

5.1.1 Density distribution of nucleus
The density distributions of the ground states of the target nuclei were calculated fromthe charge distributions obtained by the electron scattering measurements. The charge
distributions of 12 C, 16 O, 24 Mg, 28 Si, and 40 Ca were parameterized in the form of thesum of Gaussian functions [79], while that of 20 Ne was in the form of the two-parameter
Fermi model [80]. In the form of the sum of Gaussian function, the charge distributionρ̃c0 (r) is expressed as follows [79],ρ̃c0 (r) =

∑
i

with

{
[
]
[
]}
r − Ri 2
r + Ri 2
Ai exp −+ exp −,γγ[

Ai = ZeQi 2π

(
)]−1
2Ri2
γ 1+ 2
,
γ

3/2 3

62

(5.3)

(5.4)

where Ri and Qi are the position and the amplitude of the Gaussians and the values∑of Qi are normalized such that i Qi = 1. Ri , Qi , and γ in Eq. (5.3) were determined
to reproduce the cross sections of the elastic electron scattering. In the form of thetwo-parameter Fermi model, the charge distribution ρ̃c0 (r) is expressed as follows [80],ρ̃c0 (r) =ρ0,
1 + exp [4.4(r − c)/t](5.5)

where ρ0 is the saturation density, and c and t are determined to reproduce the crosssections of the elastic electron scattering. The calculated charge distributions are nor∫malized to satisfy
4πρ̃c0 (r′ )r′2 dr′ = Ze.

(5.6)

We calculated the point-proton distributions of the target nuclei by unfolding theircharge distributions with the proton charge form factor [81–83]. The charge distribution
is expressed using the point-proton distribution ρp and the charge distribution of protonρ̃c,proton
as follows,0∫ρ̃c0 (r) =ρp (r ′ )ρ̃c,proton(r − r ′ )dr ′ .
0

(5.7)

The Fourier transform of Eq. (5.7) is given by,
Fc (q) = Gpc (q) · Fp (q),(5.8)

where q, Fc (q), Fp (q), and Gpc (q) are the momentum transfer, the charge form factorof nucleus (Fourier transform of charge distribution), the form factor of point-proton(Fourier transform of point-proton distribution), and the charge form factor of proton(Fourier transform of proton charge distribution), respectively. Gpc (q) is also called as
Sachs electric form factor of proton [84], and parameterized in Ref. [82] as follows,
Gpc (q) =1 − 0.24τ
,1 + 10.98τ + 12.82τ 2 + 21.97τ 3(5.9)

where τ = q 2 /4m2nucleon and mnucleon is the mass of the nucleon. The point-proton distributions are obtained by performing the inverse Fourier transformation of Fc (q)·[Gpc (q)]−1.

Since Eq. (5.9) has a node at τ = 1/0.24, it is inappropriate to use Eq. (5.9) directly
for the calculation. In the present analysis, another parameterization was utilized in the63calculations as follows [85],Gpc (q) =

0.312
1.312
0.709
0.085
+
−
+
.
2
2
2
1 + q /6.0 1 + q /15.02 1 + q /44.08 1 + q 2 /154.2(5.10)

Figure 5.2 shows the experimental data compared with Eq. (5.9), whereas Fig. 5.3 shows
the comparison of Eqs. (5.9) and (5.10). As seen in Fig. 5.3, Eqs. 5.9 and 5.10 were satisfactory consistent each other. The point-neutron distributions were assumed to be samewith the point-proton distributions for the self-conjugate A = 4n nuclei. The calculated
density distribution of the nuclei are shown in Fig. 5.4. The density distributions of
58 Ni, 90 Zr, 116 Sn, 144 Sm, and 208 Pb were calculated by assuming that the point-protonand point-neutron distributions have the same shape and normalizing ρ0 (r′ ) to satisfy∫
4π ρ0 (r′ )r′2 dr′ = A. A is the mass number of the nucleus. The density distribution
heavier than 40 Ca were used to determine the interactions for 28 Si and 40 Ca in Sec. 5.1.2.

5.1.2 Determination of the eﬀective αN interaction
The parameters in Eq. (5.2) were determined so as to reproduce the cross sectionsof the elastic alpha scattering. We assumed the same range parameters for real andimaginary parts of the interaction (αv = αw ). The density-dependence parameter β was
fixed at −1.9 or 0.0. The parameter value of β = −1.9 was proposed in Ref. [78], andwidely used for the analysis of inelastic alpha scattering [4, 6–13, 16–18, 44, 71]. On theother hand, the parameter value of β = 0.0 means that the interaction is independentof the density of the target nucleus. For comparison between density-dependent andindependent interactions, we determined two diﬀerent sets of the interaction parameterswith β = −1.9 and 0.0 for each nucleus. In this work, the αN interactions with β = 0.0
are denoted by “DI” (density independent), whereas those with β = −1.9 are denotedby “DD” (density dependent).
The calculated cross sections with the best-fit parameter values to the elastic alphascattering data at Eα = 386 and 130 MeV are shown in Figs. 3.14 and 3.15, respectively.
The cross sections calculated with the DI and DD interactions are almost the same.
Tables 5.1 and 5.2 list the best-fit parameters for the DI and DD interactions at Eα = 130
and 386 MeV, respectively. The mass dependence of these parameters are shown in
Figs. 5.5–5.8.
As described in Sec. 2.8, there are no available data of the elastic alpha scattering oﬀ64

p

Gc (q)

Figure 5.2: Charge form factor of proton (Gpch ) [86–88] compared with Eq. (5.9). Taken
from Fig. 4.7 in Ref. [82].

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0

10

20

30

40

50

60
2

70

80

-2

Momentum transfer q (fm )Figure 5.3: Charge form factor of proton (Gpch ) parameterized by Eqs. (5.9) and (5.10).
The red solid line and the blue dashed line show Eqs. (5.10) and (5.9), respectively.

65

ρ(r) (fm−3)
ρ(r) (fm−3)
ρ(r) (fm−3)
ρ(r) (fm−3)

0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0

12

16

20

24

28

40

58

90

116

144

208

C

O

Ne

Ca

Si

Mg

Zr

Ni

Sn

1

Sm

1

2

3

4

5

6

7

8

9

2

3

4

5

6

7

8

9

10

Pb

10

1

2

3

4

5

6

7

8

9

10

r (fm)

Figure 5.4: Calculated density distributions of the nuclei which are used to analyze thepresent data.

66

Table 5.1: Parameterization for the density-independent (DI) and density-dependent(DD) αN interaction at Eα = 130 MeV.
NucleusInt. type

12 C

DI
DI
DI
DI
DI
DD
DD
DD
DD
DD

16 O
24 Mg
28 Si
40 Ca
12 C
16 O
24 Mg
28 Si
40 Ca

28 Si and 40 Ca at E

v
(MeV)
22.35
18.48
26.61
39.73
36.72
47.65
48.67
57.85
74.36
71.14

w
(MeV)
9.90
7.34
14.21
17.82
19.25
21.56
19.56
31.64
34.24
37.69

α v = αw
(fm)
2.16
2.26
2.06
1.93
1.97
1.99
2.04
1.86
1.79
1.81

β
(fm2 )
0.0
0.0
0.0
0.0
0.0
−1.9
−1.9
−1.9
−1.9
−1.9

α = 386 MeV. Therefore, we could not determine the parameters of

the interactions for these nuclei by fitting the cross sections of elastic alpha scattering.
We fitted the elastic alpha scattering data of 58 Ni, 90 Zr, 116 Sn, 144 Sm, and 208 Pb at
Eα = 386 MeV taken from Refs. [6, 18, 19] and determined the parameters for these
nuclei as shown in Figs. 5.5 and 5.6. Although the parameters scattered for the lighter
nuclei than 24 Mg, we could see the approximate linear relation between the parametersand A1/3 for the heavier nuclei. We interpolated the parameters for the heavier nuclei
as drawn by the solid lines in Figs. 5.5 and 5.6, and estimated the parameters for 28 Si
and 40 Ca at Eα = 386 MeV.

67

Table 5.2: Parameterization for the density-independent (DI) and density-dependent(DD) αN interaction at Eα = 386 MeV.
NucleusInt. type

12 C

DI
DI
DI
DI
DI
DI
DI
DI
DI
DI
DI
DD
DD
DD
DD
DD
DD
DD
DD
DD
DD
DD

16 O
20 Ne
24 Mg
28 Sia40 Caa58 Ni
90 Zr
116 Sn144 Sm
208 Pb
12 C
16 O
20 Ne
24 Mg
28 Sia40 Caa58 Ni
90 Zr
116 Sn144 Sm
208 Pb

v
(MeV)
18.89
20.54
20.96
14.37
12.63
11.99
13.99
13.22
12.24
11.78
11.62
41.29
45.84
46.19
32.41
33.99
33.24
34.53
32.93
30.09
28.32
28.00

w
(MeV)
13.39
11.07
19.69
9.05
8.94
8.63
8.51
7.74
7.33
6.84
6.97
28.24
24.21
43.00
20.53
21.15
20.54
21.05
19.51
18.20
16.49
16.75

α v = αw
(fm)
2.01
1.99
1.89
2.15
2.11
2.13
2.11
2.17
2.21
2.23
2.29
1.81
1.78
1.65
1.91
1.87
1.89
1.85
1.91
1.96
1.99
2.08

β
(fm2 )
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
−1.9
−1.9
−1.9
−1.9
−1.9
−1.9
−1.9
−1.9
−1.9
−1.9
−1.9

a The interaction parameters were determined from theglobal trend. See text for details.

68

25

2.8

Re. part
Im. part

20

Ne

20

208

2.4

24

Mg

2.2

58

Ni

116

144
Zr Sn Sm

90

Pb

16

O

2
1.8

12

C

Depth (MeV)

Range (fm)

2.6

20

Ne

15
10

0

2 2.5 3 3.5 4 4.5 5 5.5 6
A1/3

C
16

O

5

1.6

12

24

Mg

58

Ni 90Zr116Sn

208

144

Sm

Pb

2 2.5 3 3.5 4 4.5 5 5.5 6
A1/3

2.3
2.2
208
2.1
Pb
116
Sn
90
24
2
Mg
Zr
12
144
C
Sm
1.9 16
O
58
1.8
Ni
1.7
20
1.6
Ne
1.5
2 2.5 3 3.5 4 4.5 5 5.5 6
A1/3

Depth (MeV)

Range (fm)Figure 5.5: Depth and range parameters of the phenomenological density-independent(DI) αN interaction at Eα = 386 MeV as a function of A1/3 . The solid circles
show the range parameters αv = αw , whereas the upward and downward
triangles show the real and imaginary depth parameters v and w, respectively.
The solid lines are the results of the linear fitting to the parameters at A =24–208. The dotted lines connect each parameter points to guide the eyes.

50
20
Ne
Re. part
45 12C40
Im. part
35
30
25
20 16O 24
Mg 58Ni 90 116
15
Zr Sn144Sm208Pb
105
0
2 2.5 3 3.5 4 4.5 5 5.5 6
A1/3

Figure 5.6: Same as Fig. 5.5, but for the DD interaction at Eα = 386 MeV.

69

2.8

Depth (MeV)

Range (fm)

2.6
16

2.4

12

C

2.2

O
24

Mg
28

Si

2

40

Ca

1.8
1.6
2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6
A1/3

45
28
Re. part
Si
40
40
Ca
Im. part
35
24
Mg
30
25
20
12
C
15
16
O
10
5
0
2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6
A1/3

2.3
2.2
16
O
2.1
12
C
2
24
Mg
40
1.9
28
Ca
Si
1.8
1.7
1.6
1.5
2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6
A1/3

Depth (MeV)

Range (fm)Figure 5.7: Same as Fig. 5.5, but for the DI interaction at Eα = 130 MeV.

80
70
60
50
40
30
20
10
0

Re. part
Im. part

24

Mg

12

C

28

Si

40

Ca

16

O

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6
A1/3

Figure 5.8: Same as Fig. 5.5, but for the DD interaction at Eα = 130 MeV.

70

5.2 Macroscopic transition densities and potentials
5.2.1 Transition densities
We used the macroscopic charge transition densities for isoscalar transitions calculated
from the standard macroscopic models [60, 61]. The charge transition density ρ̃Jf ,Ji (r′ )(λ)

for a transferred angular momentum λ with the initial-state spin Ji and the final-statespin Jf is given as follows,
(
)
d
(0)
ρ̃Jf ,Ji (r′ ) = −α0 3 + r′ ′ ρ̃0 (r′ ),dr(λ = 0)

[

( 2
)]
β1
5 ′2 d
d
d
(1)
′
′2 d
ρ̃Jf ,Ji (r ) = − √3r+ 10r − ⟨r ⟩ ′ + ϵ r ′2 + 4 ′ρ̃0 (r′ ),
dr′
3
dr
dr
dr
3R

(5.11)(λ = 1)(5.12)ρ̃Jf ,Ji (r′ ) = −δλ(λ)

dρ̃0 (r′ ),dr′(λ ≥ 2)(5.13)

where α0 is the dimensionless deformation parameter and δλ is the deformation length.
ρ̃0 (r′ ) is the ground-state charge density distribution. β1 , R, and ⟨r′2 ⟩ are the collective
coupling parameter for the isoscalar dipole resonance, the half-density radius of the
Fermi charge distribution, and the root-mean-square radius of the ground-state charge
distribution, respectively.
If one excited state at the excitation energy Ex with the transferred angular momentum λ exhausts 100% of the EWSR strength, α0 , β1 , and δλ are calculated from the sum
rule limit in Ref. [60], Ref. [61], and Refs. [89–91] as follows,
2πℏ2,
AmEx ⟨r′2 ⟩
6πℏ2
R2
β12 =
,
mAEx 11 ⟨r′4 ⟩ − 25/3 ⟨r′2 ⟩2 − 10ϵ ⟨r′2 ⟩⟨⟩λ(2λ + 1)2 2πℏ2 r′2λ−22
δλ =,(λ + 2) AmEx ⟨r′λ−1 ⟩2

α02 =(5.14)(5.15)(5.16)

where A and m are the mass number of the target and the nucleon mass, respectively,and ϵ = (4/E2 + 5/E0 ) ℏ2 /3mA. E2 and E0 are the excitation energies of the giant
quadrupole and monopole resonances, respectively.

71

The values of α0 and δλ were determined so as to reproduce the known electromagnetic transition strengths [47, 72–76]. We need to pay attention to the fact that theelectromagnetic probes are sensitive to protons only in electric transitions, therefore theelectric transition strength carried by protons [B(Eλ)] is generally diﬀerent from theisoscalar strength [B(Eλ; IS)] carried by both protons and neutrons [44]. B(Eλ) andB(Eλ; IS) with a multipolarity of λ are described as,
2Jf + 1
|Mp (Eλ)|2 ,2Ji + 1
2Jf + 1B(Eλ; IS) =|Mp (Eλ) + Mn (Eλ)|2 ,2Ji + 1
B(Eλ) = e2

(5.17)(5.18)

where Mp and Mn are the proton and neutron transition matrix elements, respectively.
We assumed Mn (Eλ) = Mp (Eλ) = M (Eλ) considering the approximately conserved
charge symmetry in the light self-conjugate nuclei in the present work, and it leads tobe,
B(Eλ; IS) =4B(Eλ).
e2(5.19)M (Eλ) is calculated from the charge transition density ρ̃Jf ,Ji (r′ ) as follows,(λ)√ ∫ (0)M (E0) = 4π ρ̃Jf ,Ji (r′ )r′4 dr′ ,∫(λ)
M (E1) = ρ̃Jf ,Ji (r′ )r′5 dr′ ,∫(λ)
M (Eλ) = ρ̃Jf ,Ji (r′ )r′λ+2 dr′ (λ ≥ 2).

(5.20)(5.21)(5.22)

Therefore, we can determine the amplitudes α0 , β1 , and δλ in Eqs. (5.14)–(5.16) fromthe known electromagnetic transition strengths B(Eλ) and monopole transition matrix
elements. Tables 5.3–5.6 show the excited states measured in the present work whose
electromagnetic transition strengths are reported in Refs.[47, 72–76] and the amplitudes
in Eqs. (5.14) and (5.16) which were determined to reproduce the B(Eλ) values.

5.2.2 Transition potentials
Considering the density dependence of the interaction u [|r − r ′ |, ρ0 (r′ )] to the lowest
order, the transition potential δUλ (r) was obtained by folding the transition density

72

Table 5.3: Electromagnetic transition strengths B(E0) [47, 72–76] and the deformation
parameters α0 for the λ = 0 transitions. α0 is determined to reproduce B(E0)for each transition.
Nucleus
12 C
16 O
16 O
20 Ne
24 Mg
28 Si
40 Ca

Ex
(MeV)
7.65
6.05
12.05
6.73
6.43
4.98
3.35

Jπ
0+
2
0+
2
0+
3
0+
2
0+
2
0+
2
0+
2

B(E0)
(e2 fm4 )30.3 ± 2.2
11.4 ± 0.8
16.2 ± 0.7
55 ± 3044.9 ± 5.4
46.2 ± 5.4
7.3 ± 0.2

α00.27 ± 0.01
0.102 ± 0.004
0.121 ± 0.003
0.15 ± 0.04
0.108 ± 0.006
0.089 ± 0.005
0.0199 ± 0.003

Table 5.4: Electromagnetic transition strengths B(E2) [47, 72–76] and the deformation
lengths δ2 for the λ = 2 transitions. δ2 is determined to reproduce B(E2) for
each transition.
Nucleus
12 C
16 O
16 O
16 O
20 Ne
24 Mg
24 Mg
24 Mg
24 Mg
24 Mg
28 Si
28 Si
28 Si
28 Si
40 Ca
40 Ca
40 Ca

Ex
(MeV)
4.44
6.92
9.84
11.52
1.63
1.37
4.24
7.35
9.00
10.36
1.78
7.93
8.26
9.48
3.90
8.09
8.58

Jπ
2+
1
2+
1
2+
2
2+
3
2+
1
2+
1
2+
2
2+
3
2+
5
2+
7
2+
1
2+
2
2+
3
2+
4
2+
1
2+
2
2+
3

B(E2)
(e2 fm4 )38 ± 237 ± 10.67 ± 0.06
18.2 ± 0.8
330 ± 16426 ± 933 ± 2
11.6 ± 4.6
3.2 ± 0.7
6.5 ± 1.6
327 ± 67.6 ± 1.1
6.9 ± 2.7
3.4 ± 1.6
90 ± 626.2 ± 1.8
12.3 ± 1.6

73

δ2
(fm)
1.42 ± 0.04
1.02 ± 0.01
0.127 ± 0.006
0.67 ± 0.01
2.04 ± 0.05
1.93 ± 0.02
0.53 ± 0.02
0.31 ± 0.06
0.17 ± 0.02
0.24 ± 0.03
1.38 ± 0.01
0.19 ± 0.01
0.20 ± 0.04
0.178 ± 0.04
0.46 ± 0.02
0.250 ± 0.009
0.17 ± 0.01

Table 5.5: Electromagnetic transition strengths B(E3) [47, 72–76] and the deformation
lengths δ3 for the λ = 3 transitions. δ3 is determined to reproduce B(E3) for
each transition.
Nucleus
12 C
16 O
20 Ne
24 Mg
24 Mg
28 Si
40 Ca
40 Ca

Ex
(MeV)
9.64
6.13
5.62
7.62
8.36
6.88
3.74
6.29

Jπ
3−
1
3−
1
3−
1
3−
1
3−
2
3−
1
3−
1
3−
2

B(E3)
(e2 fm6 )(7.2 ± 0.4) × 102(1.38 ± 0.04) × 103(1.8 ± 0.3) × 103(1.5 ± 0.3) × 103(2.6 ± 0.7) × 103(4.4 ± 0.7) × 103(1.72 ± 0.08) × 103(2.0 ± 0.4) × 103

δ3(fm)
1.84 ± 0.05
1.59 ± 0.02
1.2 ± 0.1
0.90 ± 0.097
1.19 ± 0.16
1.25 ± 0.10
1.35 ± 0.03
0.46 ± 0.05

Table 5.6: Electromagnetic transition strengths B(E4) [47, 72–76] and the deformation
lengths δ4 for the λ = 4 transitions. δ4 is determined to reproduce B(E4) for
each transition.
Nucleus
16 O
20 Ne
24 Mg
24 Mg
28 Si

Ex
(MeV)
10.36
4.25
4.12
6.01
4.62

Jπ
4+
1
4+
1
4+
1
4+
2
4+
1

B(E4)
(e2 fm8 )(3.4 ± 1.2) × 103(3.8 ± 0.8) × 104(2.6 ± 0.4) × 103(3.5 ± 0.8) × 104(1.8 ± 0.2) × 104

74

δ4
(fm)
0.64 ± 0.11
1.28 ± 0.13
0.28 ± 0.02
1.01 ± 0.12
0.56 ± 0.03

ρJf ,Ji (r′ ) with u [|r − r ′ |, ρ0 (r′ )] as follows,(λ)∫δUλ (r) =[×

dr′ ρJf ,ji (r′ )
(λ)

]
′
′
[
]
′
′
′ ∂u [|r − r |, ρ0 (r )]u |r − r |, ρ0 (r ) + ρ0 (r ).
∂ρ0 (r′ )(5.23)The transition density ρJf ,Ji (r′ ) is given by unfolding the charge transition density(λ)ρ̃Jf ,Ji (r′ ) with the proton-charge form factor.
(λ)

5.3 The DWBA calculation
From the formalism described in Secs. 5.1 and 5.2, we calculated the optical-model
potential for the entrance channel U (r) and the transition potential δUλ (r). The opticalmodel potential for the exit channel were assumed to be same as that of the entrance
channel in the present calculation. Thus, we performed the DWBA calculations forthe transitions whose strengths are known using the computer code ECIS-95 [77], andthe calculated cross sections are shown by the solid and dashed lines in Figs. 4.1–4.8.
The solid and dashed lines in the figures show the calculated cross sections using the
DI and DD interactions, respectively. The errors of the cross sections arose from theuncertainties of the electromagnetic transition strengths are drawn for the DI interaction
only for simplicity.
It should be noted that there is no adjustable parameter in the present DWBA calculation because the αN interaction and transition densities are already adjusted by theelastic alpha scattering data and the known B(Eλ) values.

75Chapter 6

Discussion
6.1 Comparison between experimental data and DWBA
calculations
The measured cross sections of the inelastic alpha scattering are compared with thecalculation using the DI and DD interactions in Fig. 4.1–4.7. Since the experimental
cross sections were obtained for the angular bins with a width of 0.4◦ , the calculated
cross sections at each angle are averaged over the angular range of ±0.2◦ .
As shown for the ∆L = 0 transitions in Figs. 4.1 and 4.2, the calculated cross sections
with the DD interactions are systematically larger than the measured cross sections.
This discrepancy is consistent with the previous results in Refs. [20, 58]. However, it
should be noted that the DWBA calculations using the DD interaction overestimate thecross sections for not only the Hoyle state but also most of the 0+ states in 16 O, 20 Ne,
24 Mg, 28 Si, and 40 Ca. This result suggests that the “missing monopole strength” is not

specific to the Hoyle state but universal for the ∆L = 0 transitions. On the other hand,the calculated cross sections with the DI interactions are systematically smaller thanthose with the DD interactions, and are close to the measured cross sections. Especially,
the calculation at Eα = 386 MeV reasonably well reproduces the experimental data,
and does not exhibit the puzzling situation about the missing monopole strength for the
Hoyle state.
16 O and 40 Ca are much smaller thanThe measured cross sections of the 0+
2 states in

the calculations as seen in Fig. 4.2. The angular distributions of the measured crosssections for the 0+
2 states are quite diﬀerent from the calculations. The reason of the76discrepancy is still unclear. Note that both of these states are the first excited states inthe double closed nuclei.
For the ∆L = 2 transitions shown in Figs. 4.3 and 4.4, the DWBA calculations using
the DI and DD interactions give almost the same results, and reasonably reproduce theangular distributions of the cross sections except for the several states. For the 2+
2 state28 Si, the calculated angular distributions of the cross sectionsin 16 O and the 2+
3 state inare considerably diﬀerent from the experimental data. The measured cross sections
for these 2+ states are smaller than 1 mb/sr. Since the direct couplings between the
ground states and these states are weak, the multi-step processes would be importantto reproduce the cross sections for these states.
+
24 Mg, the angular distributions of the measured cross
For the 2+
3 and 25 states in

sections slightly shift to forward angles compared with the DWBA calculations. This
shift would be understandable if we assume that the radii of these states are larger thanthose of the usual states. In Refs. [92, 93], it was theoretically shown that the angular
distribution of inelastic cross sections for the ∆L = 0 transition is not sensitive to theradial expansion of the excited states. However, the recent theoretical study suggeststhat the situation in the ∆L = 2 transition is diﬀerent from the ∆L = 0 transition [94].
The shift of the angular distribution due to the radial expansion of the excited states ismore visible in the ∆L = 2 transition than in the ∆L = 0 transition, and is more visible
at lower reaction energies than at higher reaction energies. This is consistent with thepresent results that the shift in the angular distribution is more visible at Eα = 130
MeV than at Eα = 386 MeV.
For the ∆L = 3 and 4 transitions shown in Figs. 4.5–4.7, the diﬀerences between thecalculations with the DI and DD interactions are small, although the DI interaction givesslightly smaller cross sections.

6.1.1 Normalization factor for the calculated cross sections to fit theexperimental data
The calculated cross sections for the (α, α′ ) reactions at Eα = 386 MeV are fitted to theexperimental data by applying the normalization factor R. If the present DWBA calculation is correct, R should be unity. The normalization factor R is shown in Fig. 6.1 forthe ∆L = 0, 2, and 3 transitions. The R values for the DI interaction are systematicallycloser to the unity than those for the DD interaction, although the R values for the 0+
3

77

2.5

∆L = 0

DI

2

DD

R

1.5
1

Si : 0+2 (4.98)

24

2.5

∆L = 2

DI

2

28

2

Mg : 0+ (6.43)

Ne : 0+2 (6.73)

16

20

C : 0+2 (7.65)
120

O : 0+3 (12.05)

0.5

DD

R

1.5
1

Ca : 2+1 (3.90)

40

Si : 2+1 (1.78)

28

5

3

Mg : 2+ (9.00)
24

Mg : 2+ (7.35)
24

1

Mg : 2+ (1.37)

Ne : 2+1 (1.63)

24

16

20

O : 2+3 (11.52)

O : 2+1 (6.92)

16

12

0

C : 2+1 (4.44)

0.5

2.5

∆L = 3

DI

2

DD

R

1.5
1

-

Ca : 31 (3.74)

40

2

Mg : 3- (8.36)
24

1

Mg : 3- (7.62)
24

Ne : 3-1 (5.62)
20

O : 3-1 (6.13)
16

12

0

C : 3-1 (9.64)

0.5

Figure 6.1: Normalization factors for the calculated cross sections for the (α, α′ ) reactions
at Eα = 386 MeV to fit the experimental data. The solid squares andcircles are the normalization factors for the calculations with the DI and
DD interactions. The errors arise from the known electromagnetic transitiontransitions [47, 57, 72–76]

78

20 Ne and in 28 Si significantly deviate from the unity.
state in 16 O and, the 0+
2 states in
20 Ne calculated with the DI interaction is conThe cross section for the 0+
2 states insistent with the experimental data at 0◦ within the error due to the uncertainty from theelectromagnetic transition strength. However, the disagreement between the calculationand the experiment around the diﬀraction minimum causes the deviation of R from theunity.
28 Si was not measured at 0◦ . Since the cross
The cross section of the 0+
2 state insections near 0◦ are crucial for the reliable determination of the isoscalar monopoletransition strength, further measurements for these states should be done.
As discussed above, the DWBA calculation with the DI interaction gives a better description for inelastic alpha scattering than those with the DD interaction. Furthermore,
the DI interaction at Eα = 386 MeV is more applicable to fit the experimental data thanthat at Eα = 130 MeV. This situations is naturally understood from the well-known factthat direct processes are dominant and the reaction mechanism becomes simple aboveE ∼ 100 MeV/u.

6.2 Characteristics of the macroscopic transition densities
and potentials
The transition densities and potentials in the present folding-model analyses should be
examined in order to clarify the reason why the DI and DD interactions give the diﬀerent
results for the ∆L = 0 transitions (see Figs. 4.1 and 4.2). Figure 6.2 shows the transition
densities and potentials calculated based on the macroscopic model for the transitions
between the ground and excited states in 12 C at Eα = 386 MeV.
The DI and DD interactions give similar potentials for the 2+
1 state, whereas these
two interactions give significantly diﬀerent transition potentials for the 0+
2 state. The
transition density for the ∆L = 0 transition has sizable values around the center ofa nucleus (r = 0). Since the DD interaction is much weaker than the DI interaction
around the origin due to the density eﬀect, the DD transition potential for the ∆L = 0
transition is much shallower than the DI transition potential in the inner region. On
the other hand, the transition density for the ∆L = 2 transition is almost zero at theorigin. Therefore, the DI and DD transition potentials are almost same for the ∆L = 2
transition.

79

0.15

0

ρλ(r) (fm− 3)

0.05

0.1

−0.05

0.05
0

δU(r) (MeV)

0+1 − 0+2 coupling

−0.2

0
−2
−4
−6

−0.1

−0.15

0+1 − 2+1 coupling−0.05

δU(r) (MeV)ρλ(r) (fm− 3)

0.2

Eα = 386 MeV

−8

12
10
8
6
4
2
0
−2

Eα = 386 MeV

δU(r) (MeV)

δU(r) (MeV)

DI (Re. part)

0
−2
−4
−6

Eα = 130 MeV

−8
−10

0

2

4

6

8

10

12
10
8
6
4
2
0
−2

DI (Im. part)DD (Re. part)DD (Im. part)Eα = 130 MeV
0

2

4

6

8

10

r (fm)r (fm)

Figure 6.2: Transition densities given by Eqs. (5.11) and (5.13), and the transition potentials with the phenomenological αN interaction. The top panels show thetransition densities, and the middle and bottom show the transition poten+++
12 C at E = 386 MeV andtials for the 0+α
1 − 21 and 01 − 02 transitions in
Eα = 130 MeV, respectively. The solid and dashed lines show the real andimaginary parts of the transition potentials with the DI interaction. The dotted and dot-dashed lines show the real and imaginary parts of the potentials
with the DD interaction.

80

Historically, the density dependence of the phenomenological αN interaction was introduced to obtain better description of elastic alpha scattering at backward angles [78].
Although the depth and range parameters used for the DI and DD interactions are
determined to fit elastic alpha scattering, cross sections of elastic alpha scattering are
sensitive to the nuclear wave functions at the surface only because of the strong absorption [95]. Since the interaction parameters are basically tuned at the surface where thenuclear density is relatively low, in the present work, the DD interaction gives a good
description for the ∆L = 2 transition whose transition density enhances near the surface,but does not for the ∆L = 0 transition in which the interior transition density gives asizable contribution to the transition potential. The weak absorption by the DD transition potential for the ∆L = 0 transitions causes the systematic overestimation of thecross sections and “missing monopole strengths”. However, the R values for the ∆L = 0
transitions in Fig. 6.1 shows that the DWBA calculation with the DI interaction is still
unsatisfactory, and this fact suggests that the treatment of the density dependence inthe present eﬀective αN interaction given in Eq. (5.2) is not suitable to describe theinelastic alpha scattering.
+
It is noteworthy that the transition potential for the 0+
1 –02 transition from the recent

full microscopic calculation using the realistic N N interaction [96] is similar to the
DI transition potential for the ∆L = 0 transition as shown in Fig. 6.3. Note thatthe sign of the potential in Fig. 6.3 is diﬀerent from that in the present calculation in
Fig. 6.2 due to the diﬀerent definition of the sign. This fact suggests that a sophisticatedeﬀective αN interaction could be constructed from the realistic N N interaction, andsuch an interaction might give better description than the present empirical eﬀective
αN interaction.

6.2.1 Dependence on the nuclear size
We found that the DI interaction at Eα = 386 MeV is better than the DD interaction
to reproduce the experimental cross sections, but the DD interaction was widely used toanalyze the alpha scattering data, especially for the study of the ISGMRs [3–9, 17]. The
diﬀerence between the DI and DD interactions should be checked by calculating the cross+
40 Ca, 90 Zr, 116 Sn, and 208 Pb. All the procedures tosections to the 0+
2 and 21 states incalculate the cross sections are same as in Sect. 5.3, but the transition strengths are set
at 100% of the EWSR strength. The calculated cross sections and transition potentials

812

coupling potential (MeV)

0
-2
-4
-6
-8
-10
-12
Real part
Imaginary part

-14
-16
0

1

2

3

4

5

6

7

8

radius (fm)+
12 C(α, α′ ) reaction
Figure 6.3: The transition potential for the 0+
1 –02 transition in theat Eα = 172.5 MeV in Ref. [96]. Taken from Fig. 4 in Ref. [96].

+are shown in Fig. 6.4 for the 0+
2 states and in Fig. 6.5 for the 21 states. We can find thatthe diﬀerence between the cross sections calculated with the DI and DD interactions issmaller for heavier nuclei as shown by the values denoted as “ratio” in Fig. 6.4 which
are the ratios of the calculated cross sections at 0◦ with the DD interaction to thosewith the DI interaction. We can understand this result from the fact that the inelasticalpha scattering comes to less sensitive to the inner region with increasing nuclear radii
because the alpha particles are absorbed before they reach the inner region.
Eventually, the diﬀerence between the DI and DD interactions decreases with increasing nuclear size, however, it is not negligible for the determination of the nuclear
incompressibility. Although the shapes of the angular distribution of the cross section
calculated with the DD interaction is quite similar to those with the DI interaction, theabsolute values of the cross sections with the DD interaction are larger than those with
the DI interaction. Thus, the shape of the strength distribution of the isoscalar monopoletransition estimated by the MDA with the DD interaction are similar to those with the
DI interaction, but the isoscalar monopole transition strengths estimated with the DD
interaction are much smaller than those with the DI interaction. For the determinationof the nuclear incompressibility, we need to reliably determine the averaged energy over82

the isoscalar monopole transitions exhausting most of the EWSR strengths as discussed
in Sec. 1.2.1. If the observed EWSR fraction is much smaller than the EWSR limit due
to the puzzle of the missing monopole strengths, the estimated KA values must be reconsidered. Therefore, the density dependence of the phenomenological αN interaction
should be treated carefully even for the heavy nuclei such as 208 Pb.

83

10

102
10
1

30
25
20
15
10
5
0
−5
− 10

+

40

Ca : 0 2
5

10

θc.m. (deg)

15

(c)

0

2

4

6

8

r (fm)

10 12

104

14

(e)

ratio = 1.53

3

10

102
10
1

+

116

Sn : 0210− 1 05

10

θc.m. (deg)15

δ U(r) (MeV)δ U(r) (MeV)

(g)

15
10
5
0
−5
− 10

0

2

4

6

8

r (fm)

10 12

10

102
10
1

30
25
20
15
10
5
0
−5
− 10

14

+

90

Zr : 025

10

θc.m. (deg)

15

20

(d)

0

2

4

6

8

r (fm)

10

12 14

104

(f)

3

ratio = 1.45

10

102
10
1

+

208

Pb : 0210− 1 0

20

25
20

(b)

ratio = 1.58

3

10− 1 020

δ U(r) (MeV)δ U(r) (MeV)d σ /d Ω (mb/sr)

ratio = 2.01

3

10− 1 0d σ /d Ω (mb/sr)

104

(a)

d σ /d Ω (mb/sr)d σ /d Ω (mb/sr)

104

5

10

θc.m. (deg)

10
8

15

20

DI (Re. part)6
4

DI (Im. part)2
0

DD (Im. part)DD (Re. part)

−2
−4

(h)
0

2

4

6

8

r (fm)

10

12 14

Figure 6.4: (a),(b),(e), and (f); Calculated cross sections of the inelastic alpha reaction to40
90
116 Sn, and 208 Pb at E = 386 MeV. The solid andthe 0+
α
2 states in Ca, Zr,
dashed lines are the results with the DI and DD interactions, respectively.
Each transition strength is set at 100% of the EWSR strength.
(c),(d),(g), and (h); The transition potentials for the reactions of (a),(b),(e),and (f), respectively.

84

104

(a)

3

10

102
10
1
−1

10

+

40

Ca : 2 1

0

5

10

θc.m. (deg)

15

2
0
−2
−4
−6
−8
− 10
− 12
− 14
2

4

6

8

r (fm)

10 12

1

+

90

Zr : 210

5

10

θc.m. (deg)

15

2
0
−2
−4
−6
−8
− 10
− 12
− 14

20

(d)
0

2

4

6

8

r (fm)

10

12 14

104

d σ /d Ω (mb/sr)

(e)

3

10

102
10
1

+

116

Sn : 2110− 1 05

10

θc.m. (deg)

2
0
−2
−4
−6
−8
− 10
− 12
− 14

15

(g)
DI (Re. part)DI (Im. part)DD (Re. part)DD (Im. part)

0

2

4

6

8

r (fm)

10 12

(f)

3

10

102
10
1

14

1
0
−1
−2
−3
−4
−5
−6
−7
−8

+

208

Pb : 2110− 1 020

δ U(r) (MeV)d σ /d Ω (mb/sr)

10

14

104

δ U(r) (MeV)

102

10

20

(c)
0

(b)

3

10

−1

δ U(r) (MeV)δ U(r) (MeV)d σ /d Ω (mb/sr)d σ /d Ω (mb/sr)

104

5

10

θc.m. (deg)

20

(h)
0

2

4

6

8

r (fm)

10

Figure 6.5: Same as Fig. 6.4, but for the 2+
1 states.

85

15

12 14

6.3 Ambiguity in the DWBA calculations
Several approximations are introduced to the present DWBA analyses. (i) The macroscopic transition densities are used instead of the realistic microscopic transition densities. (ii) The distorting potentials for the exit channels are assumed to be the same withthe entrance channels. (iii) The interactions are approximated by the Gaussian-type
phenomenological αN interaction. (iv) The coupled-channel eﬀects are ignored.
These approximations might cause errors in the present analyses. Ambiguities fromthese approximations are discussed in the following by comparing the present DWBA
calculations with the reliable microscopic calculations. However, such reliable macroscopic calculations are available only for several nuclei such as 12 C and 16 O, therefore,
the ambiguities in 12 C are discussed in this section.
Only the calculations with the DI interaction at Eα = 386 MeV are shown because
the situation does not essentially change even for other cases; for example, the DD
interaction or at Eα = 130 MeV.

6.3.1 Transition densities+−
The macroscopic transition densities for the 2+
1 , 02 , and 31 states given by Eqs. (5.11)

and (5.13) are shown by the solid lines in the bottom panels of Fig. 6.6. In addi+tion, the dashed lines represent the microscopic transition densities for the 0+
1 –21 and+0+
1 –02 transitions from the α-particle condensate wave functions (so-called THSR wave−
functions) [97, 98] and the transition density for the 0+
1 –31 transition from 3α RGM

calculations [40]. The amplitudes of the microscopic transition densities are normalized
to reproduce the known electromagnetic transition strengths [47, 57, 72–76]. The microscopic transition densities are considerable diﬀerent from the macroscopic transitions
densities.
+−
The cross sections for the 2+
1 , 02 , and 31 states at Eα = 386 MeV obtained fromthe DWBA calculations using the microscopic and macroscopic transition densities arecompared in the top panels of Fig. 6.6. The solid lines are the same calculations as
described in Sect. 5.3 using the DI interaction and the macroscopic transition densities.
The dashed lines are the calculations using the DI interaction and the microscopic transition densities. The diﬀerences between the solid and dashed lines are very small for+−all of the 2+
1 , 02 and 31 states in spite of the diﬀerences in the transition densities.

although the macroscopic transition densities shown by the solid lines in the bottom

86

12

dσ/dΩ (mb/sr)

102

102

C(α,α'), Eα=386 MeV

102

10

10

10

1

1

1

10−1

10−2

10−1

2+1 (4.44 MeV)

10−1

0+2 (7.65 MeV)

10−2
0 5 10 15 20 25 30 35
0

3−1 (9.64 MeV)

10−2
10 15 20 25
05

5 10 15 20 25 30

θc.m. (deg)0.05

ρλ(r) (fm− 3)

0.3
0.2
0.1

−0.05

0

−0.1

−0.1

−0.15

−0.2 0

1

2

3

4

5

0.3

0

0+1 − 2+1

6

7

−0.2 0

0+1 − 3−1

0.2
0.1

0+1 − 0+2

0
−0.1

1

2

3

4

5

6

7

−0.2 0

1

2

3

4

5

6

7

r (fm)+−
Figure 6.6: Calculated cross sections and transition densities for the 2+
1 , 02 , and 31
states in 12 C at Eα = 386 MeV. The solid circles with error bars are theexperimental data. The solid lines show the DWBA calculations using the
DI interaction and macroscopic transition densities which are same with thesolid lines in Figs. 4.1, 4.3, and 4.5. The dashed lines show the DWBA
calculations using the DI interaction and the microscopic transition densities
from Ref. [40, 97, 98]. The dotted lines are the same DWBA calculations
shown by the dashed lines except that the distorting potentials for the exit
channels are calculated by folding the density distributions of the excitedstates with the DI interaction. The dashed and dotted lines are almost the−same for the 2+
1 and 31 states. In the bottom panels, the solid lines show themacroscopic transition densities given by unfolding the macroscopic chargetransition densities [Eqs. (5.11) and (5.13)] with the proton charge form
factor, whereas the dashed lines are the microscopic transition densities (Seetext).

87

panels are considerably diﬀerent from the microscopic transition densities. These results
show that the ambiguities due to diﬀerences in transition densities are negligibly small
in the present DWBA calculation.

6.3.2 Distorting potentials
In order to examine ambiguities from the distorting potentials, we replace the distorting
potentials for the exit channels with the diagonal potentials for the excited states. The
diagonal optical-model potentials are calculated by folding the density distributions of
the excited states from Refs. [40, 97, 98] with the DI interaction. The calculated crosssections are represented by the dotted lines in the top panels of Fig. 6.6. The cross−sections for the 2+
1 and 31 states do not change even if the distorting potentials are

replaced. For the 0+
2 state, on the other hand, the cross sections slightly change when
the distorting potentials are replaced. This result is explained from the fact that theradius of the 0+
2 state is much larger than that of the ground state [40, 97, 98] and thus,
the distorting potential for the exit channel is very diﬀerent from that for the entrance
channel. However, the variation of the cross sections due to the distorting potential isacceptably small.

6.3.3 Phenomenological αN interaction
The present DWBA calculation using the single-folding potentials assumes the α particle
is point particle, and the phenomenological αN interaction is empirically determined toreproduce elastic alpha scattering. This interaction is really phenomenological, andhence it does not established on the realistic N N interaction. The present DWBA
calculation should be compared with the calculation using the realistic N N interaction.
In the following calculations, we used the diagonal potentials for the excited states as
the distorting potential for the exit channels as performed in Sec. 6.3.2.
For comparison, we carried out the DWBA calculation using the N N G-matrix interaction by the Melbourne group [99]. The transition potentials were calculated by doubly
folding the projectile density distribution and the target transition-density distribution
with the N N G-matrix interaction. The density distributions used with the N N Gmatrix interaction are taken from the 3α RGM calculation [40] as the same prescription+−
12 C with
with Ref. [96]. The calculated cross sections for the 2+
1 , 02 , and 31 states in

the Melbourne and DI interactions are shown by the solid and dashed lines in Fig. 6.7,

88

respectively. The dashed lines calculated with the DI interaction are same with thedotted lines in the top panels of Fig. 6.6. Both the calculations reasonably reproduce
the diﬀraction pattern of the measured cross sections. The cross sections with the DI
interaction are systematically smaller than those with the Melbourne interaction, anddσ/dΩ (mb/sr)

are close to the experiment.
2+1 (4.44 MeV)

102
10

12

C(α,α'), Eα = 386 MeV
0+2 (7.65 MeV)

102
10

10

1

1

1

−1

−1

Melbourne (DWBA)−1−2

DI (DWBA)

10

10

−2

10

0

5

10

15

20

25

30

10
35

0

5

10

15

3−1 (9.64 MeV)

102

10
20

10−2
25
0

5

10

15

20

25

30

θc.m. (deg)Figure 6.7: Calculated cross sections of the inelastic alpha scattering at Eα = 386 MeV+−
12 C. The solid and dashed lines show
exciting the 2+
1 , 02 , and 31 states inthe DWBA calculations with the Melbourne [99] and DI interactions, respectively. The solid circles with error bars are the experimental data.

6.3.4 Coupled-channel eﬀect
The coupled-channel (CC) eﬀect is ignored in the present DWBA calculation, but it
might not be negligible. Especially, it is pointed out that the coupling between the+
12 C is very strong [40, 100]. The CC calculation for the inelastic
0+
2 and 22 states in+−
12 C using the DI interaction at
alpha scattering exciting the 2+
1 , 02 , and 31 states in

Eα = 386 MeV was compared with the DWBA calculation in the top panels of Fig. 6.8.
The distorting and transition potentials are calculated by using the wave functions from+the 3α RGM calculation for the 3−
1 state [40] and the THSR wave functions for the 01 ,+++2+
1 , 02 , 22 , and 41 states [97, 98].

The dotted lines show the CC calculation taking into account the coupled-channel
+
+
−
+
+
eﬀects between the 6 states (0+1 , 21 , 02 , 31 , 22 , and 41 ), while the dashed lines show

the CC calculation taking into account the coupled-channel eﬀects between the 5 states
except the 2+
2 state. The solid lines show the DWBA calculation, which is the same−with the dashed lines in Fig. 6.7. The CC eﬀects for the 2+
1 and 31 states are negligibly

small, and the DWBA and CC calculations give similar results except for the 0+
2 state.
The DWBA and 5-state CC calculations give similar results even for the 0+
2 state, but
the 6-state CC calculation gives smaller cross section at forward angles than the other

89

12

dσ/dΩ (mb/sr)

102

2+1 (4.44 MeV)

10

10

1

1

10−1

10−1

10−2

C(α,α'), Eα=386 MeV

102

0+2 (7.65 MeV)102

3−1 (9.64 MeV)

10
1
10−1

DI

10−2
0 5 10 15 20 25 30 35
0

5

10−2
10 15 20 25
0

5 10 15 20 25 30

θc.m. (deg)122dσ/dΩ (mb/sr)

10

2+1 (4.44 MeV)

2

10

10

10

1

1

10−1

10−1

10−2

10−2
0 5 10 15 20 25 30 35
0C(α,α'), Eα=386 MeV

0+2 (7.65 MeV)102

3−1 (9.64 MeV)

10
1

Melbourne
5

10−1

10−2
10 15 20 25
0

5 10 15 20 25 30

θc.m. (deg)Figure 6.8: Calculated cross sections of inelastic alpha scattering at Eα = 386 MeV+−
12 C. The top panels are the DWBA
exciting the 2+
1 , 02 , and 31 states inand CC calculations with the DI interaction, and the bottom panels are thecalculations with the Melbourne interaction [96]. The solid lines show the
DWBA calculation. The dotted and dashed lines respectively show the 6state and 5-state CC calculations (see text). The dotted and dashed lines−
are almost the same for the 2+
1 and 31 states. The solid circles with error
bars are the experimental data.

90

calculations. In addition, the diﬀraction pattern of the angular distribution in the 6state CC calculation slightly shifts to forward angles. This result reflects the fact that+the coupling between the 0+
2 and 22 states is strong.

The reduction of the cross section for the 0+
2 state is essentially the same phenomenon
with the enhanced absorption in the α+ 12 C(0+2 ) channel proposed in Ref. [58]. It should
be noted that the DWBA calculations are better to reproduce the experimental data than
the 6-state CC calculation. One possible explanation is that the coupling between the+0+
2 and 22 states in the present CC calculation is too strong. The transition strength+
between the 0+
2 and 22 states has never been measured, therefore the experimental study

is strongly desired.
Recently, a full microscopic calculation of the inelastic alpha scattering from 12 C
using the Melbourne N N G-matrix interaction was performed [96]. The CC calculations
reported in Ref. [96] are shown in the bottom panels of Fig. 6.8. The dotted and dashedlines show the 6-state and 5-state CC calculations, respectively. On the other hand,the solid lines show the DWBA calculations same with the solid lines in Fig. 6.7. The
situation is quite similar to the calculation with the DI interactions. The coupling to+−+
the 2+
2 state is negligibly small for the 21 and 31 states but not for the 02 state. The

6-state CC calculation gives smaller cross sections of the 0+
2 state than the 5-state CC
calculations at forward angles, and the diﬀraction pattern of the angular distribution
in the 6-state CC calculation slightly shifts to forward angles. The full microscopic CC
calculation slightly underestimates the experimental cross sections for the 0+
2 state at
forward angles and overestimates at backward angles. The three-nucleon force possibly
decreases the cross section at backward angles and improves the calculation as discussed
in Ref. [96].
As discussed above, the CC eﬀects are negligibly small in most cases of the inelasticalpha scattering although the CC eﬀects sometimes give sizable modification to crosssections. The DWBA calculation using the DI interaction at Eα = 386 MeV reasonably
12 C.
reproduces the experimental results even for the 0+
2 state in

6.4 Determination of the transition strengths
Among the excited states observed in the present work listed in Table 4.1, there are
the excited states whose electromagnetic transition strengths are not reported in theprevious studies. As discussed in Sect. 6.1 that the DWBA calculation with the DI91

interaction gives a better description for the inelastic alpha scattering than that with
the DD interaction for the excited states whose transition strengths are known. Thus,
we should determine the transition strengths for those states from the present data with
the DI interaction assuming the linear proportionality between the (α, α′ ) cross section
and the relevant transition strength.
The DWBA calculations using the DI interaction were performed and fitted to themeasured cross sections by changing the amplitudes of the transition densities (α0 andδλ ) in Eqs. (5.14) and (5.16). The fitted results are shown by the solid lines in Fig. 4.9,
whereas the calculated cross section using the DD interactions with the same amplitudes
with the DI interaction are shown by the dashed lines.
The obtained transition strengths are summarized in Table. 6.1. The errors for thestrengths originate from the uncertainty of the fitting. The multipolarity λ in Table 6.1
is equal to the spin of the excited state because the spin of the ground state in the
self-conjugate A = 4n nuclei is zero.
Table 6.1: Measured transition strengths between the ground and excited states in 24 Mg,
28 Si, and 40 Ca.
Nucleus
24 Mg
24 Mg
28 Si
28 Si
28 Si
40 Ca
40 Ca

Eα
(MeV)
130
386
130
130
130
130
130

Ex
(MeV)
9.31
9.31
6.69
10.18
10.51
8.28
8.37

Jπ
0+
3
0+
3
0+
3
3−
2
2+
5
0+
3
4+
1

B(Eλ; IS)(fm2λ )a46 ± 3164 ± 48
27 ± 13(6.0 ± 0.5) × 103
12 ± 391 ± 28(2.0 ± 0.4) × 105

EWSR fraction(%)
2.3 ± 1.6
3.2 ± 2.4
0.77 ± 0.36
6.8 ± 0.5
0.60 ± 0.17
1.8 ± 0.6
1.2 ± 0.2

a The unit is fm4 for λ = 0.

The calculated cross sections fitted to the experiment reproduce the experimental
24 Mg at bothdata reasonably well over all measured angles except for the 0+
3 state in

Eα = 130 and 386 MeV. The third diﬀraction maxima of the calculated cross sections
for this state are about four times smaller than those of the measured cross sections, andthis is one reason why the errors in the deduced B(Eλ; IS) are quite large. However, the24 Mg reproduce the experiment at forward
calculated cross sections for the 0+
3 state inangles including 0◦ .
In contrast to the cases of λ = 0 and λ ≥ 2 transitions, the situation for the λ = 1

92

transitions is not simple. Since the E1 transition between isoscalar states is forbidden to
the first order, the observed E1 transitions between the isoscalar states are competitive
processes between isovector transitions allowed by the isospin symmetry breaking andhigher-order isoscalar transitions as squeezing oscillation. We can not directly compare
the absolute values of the calculated cross sections with the experimental data. because
these two types of E1 transitions have diﬀerent transition densities and the known
electromagnetic transition strengths B(E1) can not be related to the collective coupling
parameter of the transition density in Eq. (5.12).
Therefore, we determined the collective coupling parameter β1 in Eq. (5.12) to fit
the first diﬀraction maxima of the cross sections assuming that these E1 transitions are
purely higher-order isoscalar transitions. As discussed in Sec. 6.1, the diﬀraction patternof the cross sections are reproduced well by the DWBA calculations using both the DI
and DD interactions, and the DI interactions gives systematically smaller cross sections
than the DD interactions as in the other multipole transitions.
The deduced coupling parameter β1 is compared with β1EWSR given by Eq. (5.15)
which corresponds to 100% of the EWSR strengths in Table 6.2. The EWSR fractionsfor β1 are also shown.
Table 6.2: Deduced collective coupling parameters β1 and that for the transition with
100% of the EWSR strength (β1EWSR ). The experimental data and calculated
cross sections are shown in Fig. 4.8.
Nucleus
12 C
16 O
24 Mg
28 Si
28 Si
40 Ca

Eα
(MeV)
130
130
130
130
130
130

Ex
(MeV)
10.84
7.12
9.15
8.90
9.93
5.90

Jπ

β1

1−
1
1−
1
1−
1
1−
1
1−
2
1−
1

1.45 × 10−2
3.02 × 10−2
4.11 × 10−2
6.47 × 10−3
9.03 × 10−3
4.11 × 10−3

93

EWSR fraction
(%)
1
6
0.3
1
2
7

β1EWSR
1.45 × 10−1
1.27 × 10−1
7.51 × 10−1
6.55 × 10−2
6.20 × 10−2
1.56 × 10−2

Chapter 7

Summary
Strength distributions in excitation-energy spectra of atomic nuclei provide insights into
the nuclear structure because they directly reflect nuclear wave functions. Excitation
strength is fundamentally the overlap between wave functions of ground and excitedstates, and energy is an eigenvalue of a nuclear Hamiltonian associated with the wave
function. The excitation strength and energy are experimental observables, and can bedirectly compared with theoretical calculations of the nuclear structures. Therefore, thedetermination of the strength distribution is important to study the nuclear structure,for example, the cluster structures in nuclei.
The inelastic alpha scattering has selectivity to isoscalar natural-parity transitions
where transferred spin and isospin are ∆S = 0 and ∆T = 0 since both spin and isospin of
the alpha particle are zero. Therefore, the inelastic alpha scattering is useful to determine
the strength distribution in excitation-energy region where several states overlap eachother. In addition, the MDA works well to separate the strength distributions of thediﬀerent transferred angular momenta ∆L, and it enables us to obtain the strength
distribution of the isoscalar natural-parity excitations even from continuous excitationenergy spectra where many states with large widths overlap each other.
The MDA is established on the assumption that the cross sections of the inelastic alphascattering are reasonably well described by the DWBA calculation and are approximately
proportional to the relevant transition strength. However, a result contradictory to thislinear proportional relation was reported in the monopole excitation to the Hoyle statefrom the ground state in 12 C. It was claimed in Ref. [58] that this problem was due to thereaction mechanism of the inelastic alpha scattering, and the inelastic alpha scattering

94might strongly couple to the nuclear structure.
Therefore, it had been very urgent to confirm whether this puzzle of the inelastic alphascattering really exists or not. If it exists, the puzzle should be solved, otherwise thestrength distribution determined by the MDA of the inelastic alpha scattering might not
be reliable. Nevertheless, the systematic measurement of the inelastic alpha scattering
had not been performed nor examined to check the reliability of the theoretical calculation used in the MDA until now. In the present work, we systematically measured thecross sections of the inelastic alpha scattering at Eα = 130 and 386 MeV exciting lowlying discrete states in 12 C, 16 O, 20 Ne, 24 Mg, 28 Si and 40 Ca for the first time. In addition,
the comparison of the measured cross sections with the “parameter-free” DWBA calculation was carried out. All of the adjustable parameters in the DWBA calculation were
determined by the electromagnetic transition strengths and the elastic alpha scattering,
therefore there is no room for so-called “the puzzle of missing monopole strength” if theconsistency between the measured cross sections and the present DWBA calculations is
confirmed.
It was found that the DWBA calculation with the DI interaction at Eα = 386 MeV
was better than with the DD interaction and the calculation with the DI interaction at
Eα = 386 MeV was better than that at Eα = 130 MeV, and thus the inadequate density
dependence in the eﬀective αN interaction caused the “puzzle of missing monopole
strength” by overestimating the cross sections for the ∆L = 0 transitions. This puzzle
was not specific to the Hoyle state in 12 C but universally observed in all of the ∆L = 0
transitions.
We also studied ambiguities of the DWBA calculation from the distorting potentials,
phenomenological interaction, transition densities, and coupled-channel eﬀects. The
ambiguities originating from these factors in the present calculations were negligibly
small in most cases of the inelastic scattering, although the CC eﬀects sometimes gave
sizable modification to cross sections.
It was also shown that there were still some uncertainties even for the DWBA calculations with the DI interaction to determine the strength distribution. Therefore, a new
eﬀective αN interaction with a more sophisticated formula is necessary. The present
results should provide the unique and important information for theoretical studies todevelop a new reliable αN interaction.

95

ACKNOWLEDGMENTS
I would like to express a great gratitude, first and foremost, to Prof. T. Kawabata, whois my supervisor during almost all periods of my Ph.D. course at Kyoto University. He
introduced me the research field of clustering in nuclei, and the present research theme ismotivated to study the clustering in nuclei. He has been giving me continuous supportand encouragement for the experiment, the analysis, and the writing and also taking
care of my condition during these several years. Therefore, I owe what I am to him.
Both Prof. K. Imai and Prof. T. Nagae have been supporting and instructing me fromthe start of my research career as a master course student. I spent a special and precious
period under them when the new experimental facility J-PARC was just launched andthe first physics program was about to get the experimental results.
Core members of the experiment RCNP-E402 at the same laboratory with me whenthe measurement was performed, Mr. T. Baba, Mr. T. Furuno, Mr. Y. Ishii, Prof. Y.
Matsuda, Mr. M. Murata, and Ms. M. Tsumura, mainly supported the 20 Ne measurements. They helped me to prepare the experiment again and again without complaining,
and the measurements would not be carried out without their continuous and intensecooperation. The other collaborators of E402 also helped me a lot during and also after
the experiment. Especially, Prof. H. Matsubara explained and taught me how to use
the gas target at RCNP, and supported me to start the research at RCNP. Mr. K.
Inaba kindly provided me the data of the 12 C measurements. The cross sections given
from him, and his support were very important in the present work. The early works
by Mr. N. Yokota and Mr. T. Kadoya are the start point of the present work, andmany cross-section data were taken under the experiment RCNP-E308 and E369 withthem and analyzed by them. I would like to express my thankfulness and appreciation
to them. The elastic scatterings at Eα = 130 MeV at backward angles were measured
under the graduation research program by undergraduate students at Kyoto University(KADAIKENKYU P4), and these data are important to determine the αN interaction precisely. I appreciate the members of P4, the teaching assistants, and also Prof.

96

Kawabata and Prof. Kanada-En’yo.
Prof. Minomo and Prof. Funaki provided the theoretical calculations which are necessary and important for the discussion in the present work. I asked Prof. Minomo to
calculate the cross sections under many diﬀerent conditions again and again, however,
he supported me with tender care even though he must have been busy with his own
research. In addition, I had the meetings about the inelastic alpha scattering and the
puzzle of the missing monopole strength in these few years, and the discussions during
the meetings were very valuable and indispensable for this work and the paper. I would
like to thank Prof. Y. Funaki, Prof. M. Ito, Prof. M. Kimura, Prof. Y. Kanada-En’yo
Prof. K. Minomo, and Prof. M. Takashina for giving me such a special opportunity.
All of the data for the present work are taken at RCNP. I would like to thank the RCNP
cyclotron staﬀs for providing a high-quality beam for background-free measurements at
forward angles including 0◦ . Many staﬀs supported the research and my life at RCNP.
Particularly, Prof. N. Aoi and Prof. A. Tamii have been kindly supporting me andproviding a good environment for the research. I would not be able to study continuously
without their help.
Finally, I would like to express my great appreciations to my parents and brother
Hiroshi, Sanae, and Akira. They understand my situation well and always support mewith patience. I have been able to keep carrying out the present work for these year
thanks to their help.

97Appendix A

Data tables of cross sections
A.1 Elastic scattering
A.1.1 Eα = 130 MeV
Table A.1: 12 C(α, α) elastic,
Eα = 130 MeV
θc.m.
(deg)
5.0
6.0
6.2
7.3
8.3
9.4
10.5
11.5
12.6
13.7
14.7
15.8
16.9
18.0
19.0
20.1
21.1
22.2
23.3

dσ/dΩ
(mb/sr)9.74×103
6.97×103
6.40×103
3.92×103
1.81×103
7.72×102
2.06×102
6.59×101
1.02×102
1.93×102
2.48×102
2.65×102
2.17×102
1.70×102
1.10×102
7.24×101
4.82×101
4.46×101
4.79×101

Table A.1: (Continued)
θc.m.
(deg)
24.3
25.4
26.5
27.5
28.6
29.6
30.7
31.7
32.8
33.8
34.9
35.9
37.0
38.0
39.1
40.1
41.1
41.9
43.7

error
(mb/sr)5.3×102
3.9×102
3.4×1022.1×102
9.9×101
4.4×101
1.1×101
3.9
5.9
1.1×101
1.3×101
1.3×101
1.2×101
9.1
5.9
3.9
2.7
2.5
2.6

98

dσ/dΩ
(mb/sr)5.53×101
5.73×101
5.87×101
5.29×101
4.75×101
3.78×101
3.20×101
2.60×101
2.26×101
1.93×101
1.79×101
1.62×101
1.51×101
1.24×101
1.14×101
9.43
8.11
7.00
4.77

error
(mb/sr)
3.0
3.1
3.2
3.1
2.8
2.2
1.9
1.5
1.3
1.1
1.1
9.5×10−1
8.9×10−1
7.2×10−1
6.7×10−1
5.5×10−1
4.8×10−1
5.1×10−1
3.5×10−1

Table A.2: 16 O(α, α) elastic,
Eα = 130 MeV
θc.m.
(deg)
4.7
5.7
5.8
6.8
7.8
8.8
9.8
10.8
11.8
12.8
13.8
14.8
15.8
16.8
17.8
18.8
19.8
20.8
21.8
22.8
23.8
24.8
25.8
26.8
27.8
28.8
29.8

dσ/dΩ
(mb/sr)
1.53×104
1.00×104
9.82×103
5.35×103
1.91×103
6.23×102
9.41×101
8.55×101
2.34×102
3.54×102
3.68×102
3.14×102
1.95×102
1.14×102
5.38×101
3.72×101
4.31×101
6.27×101
7.37×101
8.04×101
6.97×101
6.25×101
4.61×101
4.09×101
3.47×101
3.39×101
3.26×101

Table A.2: (Continued)
θc.m.
(deg)
30.7
31.7
32.7
33.7
34.7
35.7
36.6
37.6
38.6
39.3
41.0

error
(mb/sr)8.4×102
5.6×1025.2×102
2.9×102
1.0×102
3.5×101
5.5
5.0
1.4×101
2.0×101
1.9×101
1.7×101
1.1×101
6.6
3.1
2.2
2.5
3.6
4.2
4.5
3.8
3.4
2.7
2.4
2.0
2.0
1.9

99

dσ/dΩ
(mb/sr)3.35×101
3.14×101
3.05×101
2.59×101
2.36×101
1.91×101
1.62×101
1.30×101
1.11×101
9.05
7.83

error
(mb/sr)
2.0
1.9
1.8
1.5
1.4
1.1
9.7×10−1
7.8×10−1
6.7×10−1
6.8×10−1
4.9×10−1

Table A.3: 24 Mg(α, α) elastic,
Eα = 130 MeV
θc.m.
(deg)
5.4
6.3
7.3
8.2
9.1
10.1
11.0
11.9
12.9
13.8
14.8
15.7
16.6
17.6
18.5
19.4
20.3
21.3
22.2
23.1
24.1
25.0
25.9
26.9
27.8
28.7
29.6
30.6
31.5
32.4
33.3
34.2
35.2
36.1

dσ/dΩ
(mb/sr)
2.17×104
1.06×104
1.79×103
3.04×102
6.50×101
3.46×102
5.97×102
6.50×102
4.82×102
2.99×102
1.19×102
4.29×101
3.07×101
5.49×101
7.82×101
8.67×101
7.49×101
5.68×101
3.51×101
2.49×101
1.95×101
2.12×101
2.34×101
2.55×101
2.36×101
2.13×101
1.68×101
1.38×101
1.07×101
9.56
8.57
8.29
7.22
6.65

Table A.3: (Continued)
θc.m.
(deg)
36.3
37.4
38.7
39.8
42.1
44.4
46.4
48.6
51.0
52.9
55.6
57.4
59.5
61.2
64.3
67.5
70.4

error
(mb/sr)1.2×103
5.8×102
9.5×101
1.8×101
3.9
1.9×101
3.5×101
3.8×101
2.7×101
1.8×101
6.6
2.5
1.8
3.1
4.6
5.0
4.3
3.4
2.0
1.5
1.2
1.3
1.4
1.5
1.4
1.3
1.0
8.5×10−1
6.8×10−1
6.1×10−1
5.6×10−1
5.4×10−1
4.8×10−1
4.5×10−1

100

dσ/dΩ
(mb/sr)
6.26
5.17
4.36
3.69
2.51
1.51
1.13
7.17×10−1
4.99×10−1
3.40×10−1
2.37×10−1
1.57×10−1
1.02×10−1
6.77×10−2
3.53×10−2
2.08×10−2
1.11×10−2

error(mb/sr)4.4×10−1
3.6×10−1
3.1×10−1
2.6×10−1
1.8×10−1
1.1×10−1
7.9×10−2
5.0×10−2
3.5×10−2
2.4×10−2
1.7×10−2
1.1×10−2
7.1×10−3
4.7×10−3
2.5×10−3
1.5×10−3
7.8×10−4

Table A.4: 28 Si(α, α) elastic,
Eα = 130 MeV
θc.m.
(deg)
8.0
9.0
9.9
10.8
11.7
12.6
13.5
14.4
15.4
16.3
17.2
18.1
19.0
19.9
20.8
21.8
22.7
23.6
24.5
25.4
26.3
27.2
28.1
29.0
29.9
30.8
31.7
32.6
33.6
34.5
35.4
34.5
36.1
37.7
39.3
40.9

dσ/dΩ
(mb/sr)2.96×102
2.07×102
6.99×102
9.78×102
9.55×102
6.04×102
3.19×102
1.09×102
4.06×101
5.82×101
1.07×102
1.23×102
1.21×102
8.94×101
5.85×101
3.32×101
2.43×101
2.63×101
3.31×101
3.60×101
3.66×101
3.13×101
2.60×101
1.97×101
1.69×101
1.48×101
1.49×101
1.41×101
1.38×101
1.30×101
1.17×101
1.30×101
1.06×101
8.84
7.66
5.65

Table A.4: (Continued)
θc.m.
(deg)
42.5
44.1
45.6
47.2
48.8

error
(mb/sr)1.7×101
1.2×101
3.8×101
5.5×101
5.4×101
3.2×101
1.7×101
6.0
2.4
3.4
6.0
6.9
6.8
5.1
3.4
2.0
1.5
1.6
1.9
2.1
2.2
1.8
1.5
1.2
1.0
9.0×10−1
9.0×10−1
8.4×10−1
8.3×10−1
8.0×10−1
7.3×10−1
7.9×10−1
6.7×10−1
5.6×10−1
4.9×10−1
3.5×10−1

101

dσ/dΩ
(mb/sr)
3.37
2.77
2.10
1.44
9.29×10−1

error(mb/sr)2.4×10−1
2.1×10−1
1.7×10−1
1.3×10−1
9.2×10−2

Table A.5: 40 Ca(α, α) elastic,
Eα = 130 MeV
θc.m.
(deg)
4.1
5.0
5.1
6.0
6.8
7.7
8.6
9.5
10.4
11.3
12.1
13.0
13.9
14.8
15.7
16.5
17.4
18.3
19.2
20.1
20.9
21.8
22.7
23.6
24.4
25.3
26.2
27.1
27.9
28.8
29.7
30.6
31.4
32.3
33.2
34.1

dσ/dΩ
(mb/sr)
6.59×104
3.28×104
2.83×104
1.08×104
2.05×103
5.86×102
1.00×103
1.54×103
1.39×103
9.16×102
3.38×102
7.17×101
3.17×101
1.23×102
2.02×102
2.19×102
1.60×102
9.42×101
3.62×101
1.96×101
2.52×101
3.72×101
4.10×101
3.63×101
2.37×101
1.47×101
9.24
9.22
1.14×101
1.36×101
1.35×101
1.25×101
9.99
8.69
7.53
7.39

Table A.5: (Continued)
θc.m.
(deg)
34.8
36.3
37.9
39.4
41.0
42.5
44.0
45.6
47.1
48.6
50.2
51.7
53.2
54.7
56.2
57.8
59.3

error
(mb/sr)3.5×103
1.8×103
1.5×103
5.9×102
1.1×102
3.3×101
5.6×101
8.4×101
7.9×101
5.3×101
1.8×101
4.2
1.9
6.8
1.2×101
1.3×101
8.9
5.4
2.1
1.2
1.5
2.2
2.4
2.2
1.4
9.3×10−1
6.2×10−1
6.2×10−1
7.4×10−1
8.7×10−1
8.6×10−1
8.1×10−1
6.6×10−1
5.9×10−1
5.2×10−1
5.1×10−1

102

dσ/dΩ
(mb/sr)
7.00
7.37
6.29
3.87
3.45
3.17
2.76
1.72
1.26
8.94×10−1
6.69×10−1
4.27×10−1
3.76×10−1
3.02×10−1
1.97×10−1
1.65×10−1
1.05×10−1

error(mb/sr)6.6×10−1
4.7×10−1
4.2×10−1
4.3×10−1
3.2×10−1
4.0×10−1
2.7×10−1
1.4×10−1
1.1×10−1
6.5×10−2
6.0×10−2
5.4×10−2
5.4×10−2
4.8×10−2
4.3×10−2
3.9×10−2
1.4×10−2A.1.2 Eα = 386 MeV
Table A.6: 12 C(α, α) elastic,
Eα = 386 MeV
θc.m.
(deg)
2.7
3.3
3.8
4.3
4.3
4.8
5.4
5.9
6.2
6.7
7.3
7.8
8.1
8.6
9.2
9.7
9.0
9.6
10.1
10.6
11.2
11.7
12.3
12.8
13.3
13.9
14.4
15.0
15.5
16.0
16.6
17.1
17.7
18.2

dσ/dΩ
(mb/sr)2.03×104
1.73×104
1.35×104
1.02×104
1.15×104
7.52×103
5.06×103
3.05×103
2.13×103
1.19×103
6.15×102
3.69×102
3.04×102
3.02×102
3.20×102
3.26×102
3.09×102
3.25×102
3.19×102
2.89×102
2.25×102
1.83×102
1.38×102
9.83×101
5.74×101
4.04×101
2.83×101
2.06×101
1.52×101
1.37×101
1.25×101
1.13×101
9.08
7.83

Table A.6: (Continued)
θc.m.
(deg)
18.7
19.3
19.8
20.4
20.9
21.4
22.0
22.5
23.0
23.6
24.1
24.7
25.2
25.7
26.3
26.8
27.3
27.9
28.4
28.9
29.5
30.0
30.5
31.1
31.6
32.1
32.7
33.2
33.7
34.3

error
(mb/sr)1.0×103
8.7×102
6.8×102
5.1×102
5.8×102
3.8×102
2.6×102
1.5×102
1.1×102
6.0×101
3.1×101
1.9×101
9.2
9.1
9.6
9.8
9.4
9.8
9.6
8.8
6.8
5.5
4.2
3.0
1.7
1.2
8.7×10−1
6.4×10−1
4.7×10−1
4.2×10−1
3.9×10−1
3.5×10−1
2.8×10−1
2.4×10−1

103

dσ/dΩ
(mb/sr)
6.45
4.98
3.00
2.26
1.64
1.18
7.36×10−1
5.80×10−1
4.63×10−1
3.85×10−1
2.99×10−1
2.59×10−1
2.16×10−1
1.79×10−1
1.26×10−1
1.04×10−1
7.93×10−2
6.10×10−2
3.96×10−2
3.20×10−2
2.50×10−2
1.99×10−2
1.47×10−2
1.25×10−2
1.01×10−2
8.38×10−3
5.81×10−3
4.82×10−3
3.99×10−3
3.01×10−3

error(mb/sr)2.0×10−1
1.5×10−1
9.1×10−2
6.8×10−2
5.0×10−2
3.6×10−2
2.3×10−2
1.8×10−2
1.4×10−2
1.2×10−2
9.2×10−3
8.0×10−3
6.7×10−3
5.6×10−3
3.9×10−3
3.2×10−3
2.5×10−3
1.9×10−3
1.3×10−3
1.0×10−3
8.2×10−4
6.7×10−4
5.2×10−4
4.6×10−4
3.8×10−4
3.3×10−4
2.2×10−4
1.9×10−4
1.7×10−4
1.3×10−4

Table A.7: 20 Ne(α, α) elastic,
Eα = 386 MeV
θc.m.
(deg)
5.1
5.6
7.3
7.8
8.3
8.8
9.0
9.5
10.0
10.5
10.7
11.2
11.7
12.2
12.4
12.9
13.4
13.9
14.1
14.6
15.1
15.6
15.8
16.3
16.8
17.3
17.5
18.0
18.5
19.0
19.2
19.7
20.2
20.7
21.0
21.4
21.9
22.3

dσ/dΩ
(mb/sr)9.44×102
3.57×102
5.19×102
5.35×102
4.77×102
3.84×102
3.40×102
2.39×102
1.56×102
9.72×101
6.75×101
4.73×101
3.98×101
3.47×101
3.50×101
3.24×101
2.84×101
2.36×101
2.10×101
1.64×101
1.18×101
8.18
6.55
4.56
3.30
2.46
2.06
1.66
1.43
1.24
1.09
8.48×10−1
7.01×10−1
5.18×10−1
4.47×10−1
3.34×10−1
2.43×10−1
1.71×10−1

Table A.7: (Continued)
θc.m.
(deg)
22.6
23.1
23.6
24.0
24.3
24.8
25.2
25.7
26.0
26.4
26.9
27.4

error
(mb/sr)5.6×101
2.7×101
2.5×101
2.6×101
2.3×101
1.9×101
1.6×101
1.1×101
7.7
5.0
3.4
2.4
2.1
1.8
1.7
1.6
1.4
1.2
1.1
8.5×10−1
6.3×10−1
4.6×10−1
3.8×10−1
2.8×10−1
2.1×10−1
1.7×10−1
1.2×10−1
1.0×10−1
9.0×10−2
8.0×10−2
6.5×10−2
5.2×10−2
4.5×10−2
3.5×10−2
3.0×10−2
2.4×10−2
1.9×10−2
1.4×10−2

104

dσ/dΩ
(mb/sr)1.50×10−1
1.05×10−1
8.39×10−2
7.11×10−2
6.90×10−2
4.73×10−2
4.06×10−2
2.86×10−2
3.31×10−2
2.13×10−2
1.87×10−2
1.38×10−2

error(mb/sr)1.3×10−2
9.8×10−3
8.3×10−3
7.4×10−3
7.3×10−3
5.6×10−3
5.1×10−3
4.0×10−3
3.4×10−3
2.5×10−3
2.3×10−3
1.9×10−3

Table A.8: 24 Mg(α, α) elastic,
Eα = 386 MeV
θc.m.
(deg)
5.3
5.7
6.2
6.7
7.0
7.5
8.0
8.4
9.4
9.8
10.3
10.8
11.7
12.2
12.7
13.1
14.1
14.5
15.0
15.5
16.4
16.9
17.4
17.8
18.8
19.2
19.7
20.2
21.1
21.6
22.1
22.5
23.5
23.9
24.4
24.9
25.8
26.3

dσ/dΩ
(mb/sr)8.78×102
5.00×102
6.96×102
9.11×102
1.09×103
1.10×103
9.65×102
7.40×102
3.28×102
1.91×102
1.22×102
8.48×101
7.60×101
7.69×101
7.10×101
5.74×101
3.11×101
2.18×101
1.44×101
9.56
5.58
4.80
4.05
3.56
2.20
1.67
1.14
8.5×10−1
4.49×10−1
3.67×10−1
2.37×10−1
1.96×10−1
1.28×10−1
1.04×10−1
7.85×10−2
6.18×10−2
2.84×10−2
2.52×10−2

Table A.8: (Continued)
θc.m.
(deg)
26.7
27.2
28.8

error
(mb/sr)5.3×101
3.9×101
4.6×101
5.4×101
4.5×101
4.4×101
4.1×101
3.7×101
1.4×101
1.1×101
8.4
7.2
8.5
8.4
8.1
7.4
1.4
1.1
9.3×10−1
7.6×10−1
6.6×10−1
6.1×10−1
5.7×10−1
5.2×10−1
2.8×10−1
2.4×10−1
2.0×10−1
1.7×10−1
9.8×10−2
8.4×10−2
6.9×10−2
6.3×10−2
3.1×10−2
2.8×10−2
2.4×10−2
2.2×10−2
1.3×10−2
1.2×10−2

105

dσ/dΩ
(mb/sr)
1.68×10−2
1.17×10−2
2.7×10−3

error(mb/sr)8.9×10−3
7.7×10−3
1.3×10−3

A.2 Inelastic scattering
A.2.1 Eα = 130 MeV
Table A.9: 12 C(α, α′ ) inelastic,
Ex = 4.44 MeV (2+1 ),Eα = 130 MeV
θc.m.
(deg)
0.5
2.8
3.9
5.0
6.1
6.2
7.3
8.4
9.4
10.5
11.6
12.7
13.8
14.8
15.9
17.0
18.1

dσ/dΩ
(mb/sr)1.86×101
2.33×101
1.91×101
2.08×101
3.20×101
3.39×101
4.94×101
6.20×101
7.32×101
6.61×101
6.14×101
4.49×101
3.37×101
2.18×101
1.73×101
1.47×101
1.73×101

Table A.10: 12 C(α, α′ ) inelastic,
Ex = 7.65 MeV (0+2 ),Eα = 130 MeV

error(mb/sr)9.7×10−1
1.4
1.1
1.2
1.8
1.9
2.9
3.3
3.8
3.5
3.2
2.4
1.8
1.2
9.4×10−1
8.0×10−1
9.3×10−1

θc.m.
(deg)
0.5
2.9
3.9
5.0
6.1
6.2
7.3
8.4
9.5
10.6
11.7
12.7
13.8
14.9
16.0
17.1
18.1

106

dσ/dΩ
(mb/sr)2.76×101
1.89×101
1.01×101
3.69
1.33
1.19
2.16
4.20
5.97
6.30
5.78
3.84
2.35
1.35
1.25
1.45
1.89

error
(mb/sr)1.4
1.1
6.3n×10−1
2.4×10−1
1.0×10−1
8.8×10−2
1.5×10−1
2.5×10−1
4.1×10−1
3.7×10−1
3.4×10−1
2.3×10−1
1.5×10−1
8.9×10−2
8.3×10−2
9.1×10−2
1.2×10−1

Table A.11: 12 C(α, α′ ) inelastic,
Ex = 9.64 MeV (3−1 ),Eα = 130 MeV
θc.m.
(deg)
0.5
2.9
3.9
5.0
6.1
6.3
7.3
8.4
9.5
10.6
11.7
12.8
13.9
14.9
16.0
17.1
18.2

dσ/dΩ
(mb/sr)
1.88
2.71
3.06
3.64
4.40
4.47
5.32
7.19
9.49
1.06×101
1.13×101
1.01×101
9.20
6.73
4.98
3.18
2.76

Table A.12: 12 C(α, α′ ) inelastic,
Ex = 10.84 MeV (1−1 ),Eα = 130 MeV

error(mb/sr)1.5×10−1
2.8×10−1
3.1×10−1
3.1×10−1
3.5×10−1
3.4×10−1
4.0×10−1
4.8×10−1
6.1×10−1
6.8×10−1
7.1×10−1
6.5×10−1
6.0×10−1
4.4×10−1
3.4×10−1
2.2×10−1
2.0×10−1

θc.m.
(deg)
2.9
4.0
6.3
7.4
8.4
9.5
10.6
11.7
12.8
13.9
15.0
16.1
17.1
18.2

107

dσ/dΩ
(mb/sr)9.81×10−1
8.72×10−1
4.94×10−1
2.54×10−1
1.13×10−1
1.61×10−1
3.44×10−1
5.32×10−1
6.23×10−1
6.51×10−1
5.72×10−1
3.64×10−1
2.02×10−1
1.12×10−1

error(mb/sr)1.2×10−1
1.1×10−1
5.8×10−2
4.1×10−2
2.5×10−2
2.9×10−2
4.7×10−2
5.5×10−2
6.2×10−2
6.4×10−2
5.6×10−2
4.3×10−2
2.8×10−2
2.1×10−2

Table A.13: 16 O(α, α′ ) inelastic,
Ex = 6.05 MeV (0+2 ),Eα = 130 MeV
θc.m.
(deg)
0.5
2.7

dσ/dΩ
(mb/sr)3.33
2.38

Table A.15: 16 O(α, α′ ) inelastic,
Ex = 6.92 MeV (2+1 ),Eα = 130 MeV

error(mb/sr)6.6×10−1
9.9×10−1

θc.m.
(deg)
0.5
2.7
3.7
4.7
5.7
5.8
6.8
7.8
8.9
9.9
10.9
11.9
12.9
13.9
14.9
15.9
16.9

Table A.14: 16 O(α, α′ ) inelastic,
Ex = 6.13 MeV (3−1 ),Eα = 130 MeV
θc.m.
(deg)
0.5
2.7
3.7
4.7
5.7
5.8
6.8
7.8
8.9
9.9
10.9
11.9
12.9
13.9
14.9
15.9
16.9

dσ/dΩ
(mb/sr)6.8
1.13×101
1.40×101
1.52×101
1.77×101
1.85×101
2.26×101
2.57×101
3.33×101
3.56×101
3.89×101
3.47×101
3.08×101
2.18×101
1.55×101
9.39
7.47

error
(mb/sr)2.5
7.9×10−1
9.5×10−1
9.2×10−1
1.1
1.0
1.3
1.5
1.8
1.9
2.1
1.9
1.7
1.2
8.6×10−1
5.25×10−1
4.23×10−1

108

dσ/dΩ
(mb/sr)1.10×101
1.30×101
1.27×101
1.54×101
2.21×101
2.25×101
2.91×101
3.08×101
3.00×101
2.06×101
1.39×101
6.99
4.20
3.58
6.02
7.76
9.53

error
(mb/sr)
6.4×10−1
8.9×10−1
8.7×10−1
9.3×10−1
1.3
1.3
1.6
1.7
1.6
1.1
7.7×10−1
4.0×10−1
2.5×10−1
2.6×10−1
4.1×10−1
4.9×10−1
5.9×10−1

Table A.16: 16 O(α, α′ ) inelastic,
Ex = 7.12 MeV (1−1 ),Eα = 130 MeV
θc.m.
(deg)
0.5
2.7
3.7
4.7
5.7
5.8
6.8
7.9
8.9
9.9
10.9
11.9
12.9
13.9
14.9
15.9
16.9

dσ/dΩ
(mb/sr)
2.10
9.15
1.17×101
1.07×101
8.31
8.16
4.63
1.45
4.19×10−1
1.21
2.48
3.62
4.05
3.62
2.87
1.83
1.25

Table A.17: 16 O(α, α′ ) inelastic,
Ex = 9.84 MeV (2+2 ),Eα = 130 MeV

error(mb/sr)1.4×10−1
8.6×10−1
1.0
8.0×10−1
6.5×10−1
5.6×10−1
3.5×10−1
1.3×10−1
6.9×10−2
1.1×10−1
1.9×10−1
2.6×10−1
2.9×10−1
2.6×10−1
2.2×10−1
1.4×10−1
1.1×10−1

θc.m.
(deg)
0.5
2.7
3.7
4.7
5.7
5.8
6.9
7.9
8.9
9.9
10.9
11.9
12.9
14.0
15.0
16.0
17.0

109

dσ/dΩ
(mb/sr)6.70×10−1
5.7×10−1
3.66×10−1
1.83×10−1
1.96×10−1
1.43×10−1
3.14×10−1
4.4×10−1
5.46×10−1
8.39×10−1
6.74×10−1
7.25×10−1
5.76×10−1
4.36×10−1
3.70×10−1
3.34×10−1
3.66×10−1

error(mb/sr)8.9×10−2
1.8×10−1
6.9×10−2
6.5×10−2
4.7×10−2
2.4×10−2
6.1×10−2
1.3×10−1
5.2×10−2
8.7×10−2
6.4×10−2
7.9×10−2
8.6×10−2
4.7×10−2
4.2×10−2
3.8×10−2
4.1×10−2

Table A.18: 16 O(α, α′ ) inelastic,
Ex = 10.36 MeV (4+1 ),Eα = 130 MeV
θc.m.
(deg)
0.5
2.7
3.7
4.7
5.7
5.8
6.9
7.9
8.9
9.9
10.9
11.9
12.9
14.0
15.0
16.0
17.0

dσ/dΩ
(mb/sr)9.33×10−1
4.14×10−1
1.91×10−1
1.09×10−1
1.50×10−1
1.82×10−1
2.64×10−1
3.39×10−1
4.67×10−1
6.02×10−1
6.56×10−1
6.24×10−1
5.57×10−1
4.19×10−1
3.28×10−1
2.60×10−1
3.29×10−1

Table A.19: 16 O(α, α′ ) inelastic,
Ex = 11.52 MeV (2+3 ),Eα = 130 MeV

error(mb/sr)5.8×10−2
7.0×10−2
4.0×10−2
2.3×10−2
2.8×10−2
2.9×10−2
3.8×10−2
4.0×10−2
5.1×10−2
7.9×10−2
6.9×10−2
6.0×10−2
5.5×10−2
4.7×10−2
4.1×10−2
3.4×10−2
3.9×10−2

θc.m.
(deg)
0.5
2.7
3.7
4.7
5.7
5.9
6.9
7.9
8.9
9.9
10.9
12.0
13.0
14.0
15.0
16.0
17.0

110

dσ/dΩ
(mb/sr)
8.4
7.22
7.21
8.50
1.06×101
1.09×101
1.30×101
1.23×101
1.23×101
9.40
6.58
3.43
2.29
2.12
2.78
3.28
4.16

error
(mb/sr)1.7
5.6×10−1
5.6×10−1
5.8×10−1
7.1×10−1
7.1×10−1
8.3×10−1
7.6×10−1
7.6×10−1
5.9×10−1
4.4×10−1
2.4×10−1
1.7×10−1
1.7×10−1
2.1×10−1
2.4×10−1
2.9×10−1

Table A.20: 16 O(α, α′ ) inelastic,
Ex = 12.05 MeV (0+3 ),Eα = 130 MeV
θc.m.
(deg)
0.5
2.7
3.7
4.7
5.7
5.9
6.9
7.9
8.9
9.9
10.9
12.0
13.0
14.0
15.0
16.0
17.0

dσ/dΩ
(mb/sr)
8.9
4.75
2.42
7.35×10−1
2.06×10−1
1.92×10−1
5.79×10−1
1.18
1.49
1.54
1.41
1.04
6.29×10−1
1.51×10−1
3.13×10−1
7.80×10−1
8.84×10−1

Table A.21: 24 Mg(α, α′ ) inelastic,
Ex = 1.37 MeV (2+1 ),Eα = 130 MeV

error(mb/sr)1.4
3.9×10−1
2.4×10−1
8.9×10−2
3.9×10−2
3.8×10−2
7.2×10−2
1.1×10−1
1.3×10−1
1.3×10−1
1.3×10−1
9.2×10−2
6.5×10−2
2.9×10−2
4.2×10−2
7.6×10−2
8.4×10−2

θc.m.
(deg)
0.5
2.5
3.4
4.3
5.3
5.4
6.3
7.3
8.2
9.1
10.1
11.0
12.0
12.9
13.8
14.8
15.7

111

dσ/dΩ
(mb/sr)1.20×1028.80×101
6.71×101
9.20×101
1.43×102
1.49×102
1.97×102
1.93×102
1.74×102
1.08×102
5.81×101
1.90×101
1.19×101
2.46×101
4.46×101
5.48×101
5.70×101error(mb/sr)
6.4
4.8
4.1
4.9
7.6
7.9
1.0×101
1.0×101
9.0
5.6
3.1
1.1
6.7
1.3
2.4
2.9
3.0

Table A.22: 24 Mg(α, α′ ) inelastic,
Ex = 4.12 MeV (4+1 ),Eα = 130 MeV
θc.m.
(deg)
7.3
8.2
9.2
10.1
11.0
12.0
12.9
13.9
14.8
15.7

dσ/dΩ
(mb/sr)5.0×10−1
7.00×10−1
8.2×10−1
1.12
1.21
1.32
1.43
1.29
1.20
1.13

Table A.24: 24 Mg(α, α′ ) inelastic,
Ex = 6.01 MeV (4+2 ),Eα = 130 MeV

error(mb/sr)2.1×10−1
7.9×10−2
2.5×10−1
2.7×10−1
1.2×10−1
9.1×10−2
1.3×10−1
1.2×10−1
1.2×10−1
1.2×10−1

θc.m.
(deg)
0.5
2.5
3.4
4.4
5.3
5.4
6.4
7.3
8.2
9.2
10.1
11.1
12.0
12.9
13.9
14.8
15.7

Table A.23: 24 Mg(α, α′ ) inelastic,
Ex = 4.24 MeV (2+2 ),Eα = 130 MeV
θc.m.
(deg)
0.5
2.5
3.4
4.4
5.3
5.4
6.3
7.3
8.2
9.2
10.1
11.0
12.0
12.9
13.9
14.8
15.7

dσ/dΩ
(mb/sr)1.27×101
1.76×101
1.18×101
9.69
1.14×101
1.16×101
1.49×101
1.50×101
1.50×101
1.05×101
6.57
2.71
1.58
1.87
3.90
5.31
6.59

error
(mb/sr)7.6×10−1
1.1
7.3×10−1
5.93×10−1
6.9×10−1
8.3×10−1
1.1
8.8×10−1
8.7×10−1
6.3×10−1
4.2×10−1
1.9×10−1
1.2×10−1
1.4×10−1
2.7×10−1
3.5×10−1
4.4×10−1

112

dσ/dΩ
(mb/sr)5.99×10−1
7.65×10−1
1.04
1.18
1.64
1.62
2.16
2.98
4.30
5.48
7.12
7.50
7.69
6.11
4.56
2.60
1.60

error
(mb/sr)5.8×10−2
7.4×10−2
9.3×10−2
9.6×10−2
1.3×10−1
1.3×10−1
1.6×10−1
2.6×10−1
2.6×10−1
3.3×10−1
4.2×10−1
4.4×10−1
4.5×10−1
3.6×10−1
2.8×10−1
1.7×10−1
1.1×10−1

Table A.25: 24 Mg(α, α′ ) inelastic,
Ex = 6.43 MeV (0+2 ),Eα = 130 MeV
θc.m.
(deg)
0.5
2.5
3.4
4.4
5.3
5.4
6.4
7.3
8.2
9.2
10.1
11.1
12.0
12.9
13.9
14.8
15.8

dσ/dΩ
(mb/sr)2.87×101
1.63×101
6.57
1.28
5.46×10−1
7.38×10−1
2.81
4.44
4.92
3.54
2.53
1.31
1.31
1.43
2.45
2.59
2.39

Table A.26: 24 Mg(α, α′ ) inelastic,
Ex = 7.35 MeV (2+3 ),Eα = 130 MeV

error(mb/sr)1.6
9.8×10−1
4.3×10−1
1.0×10−1
5.3×10−2
6.7×10−2
2.0×10−1
3.4×10−1
3.5×10−1
2.7×10−1
1.6×10−1
9.1×10−2
9.1×10−2
1.3×10−1
1.6×10−1
1.7×10−1
1.6×10−1

θc.m.
(deg)
0.5
2.5
3.4
4.4
5.3
5.4
6.4
7.3
8.2
9.2
10.1
11.1
12.0
12.9
13.9
14.8
15.8

113

dσ/dΩ
(mb/sr)
1.72
2.45
2.79
3.96
5.76
5.85
6.91
5.52
4.11
1.96
9.20×10−1
8.25×10−1
1.66
2.28
2.74
2.18
1.69

error
(mb/sr)1.3×10−1
1.8×10−1
2.1×10−1
2.7×10−1
3.7×10−1
3.8×10−1
4.4×10−1
3.9×10−1
3.2×10−1
1.3×10−1
6.7×10−2
6.1×10−2
1.1×10−1
1.5×10−1
1.7×10−1
1.4×10−1
1.2×10−1

Table A.27: 24 Mg(α, α′ ) inelastic,
Ex = 7.62 MeV (3−1 ),Eα = 130 MeV
θc.m.
(deg)
0.5
2.5
3.4
4.4
5.3
5.4
6.4
7.3
8.2
9.2
10.1
11.1
12.0
12.9
13.9
14.8
15.8

dσ/dΩ
(mb/sr)
3.33
7.23
8.32
8.54
8.85
8.60
9.06
9.55
1.16×101
1.17×101
1.12×101
9.92
7.83
4.82
3.32
2.39
2.69

Table A.28: 24 Mg(α, α′ ) inelastic,
Ex = 8.36 MeV (3−2 ),Eα = 130 MeV

error(mb/sr)2.3×10−1
4.7×10−1
5.3×10−1
5.3×10−1
5.5×10−1
5.3×10−1
5.6×10−1
6.2×10−1
7.5×10−1
6.6×10−1
6.4×10−1
5.7×10−1
4.5×10−1
2.9×10−1
2.1×10−1
1.6×10−1
1.7×10−1

θc.m.
(deg)
0.5
2.5
3.4
4.4
5.3
5.4
6.4
7.3
8.3
9.2
10.1
11.1
12.0
13.0
13.9
14.8
15.8

114

dσ/dΩ
(mb/sr)
3.95
4.60
6.27
6.67
9.3
9.0
1.23×101
1.49×101
1.85×101
1.81×101
1.59×101
1.09×101
6.97
3.24
2.16
2.53
3.66

error
(mb/sr)9.3×10−1
5.8×10−1
7.5×10−1
7.5×10−1
1.5
1.0
1.5
9.4×10−1
1.0
9.6×10−1
1.8
1.3
9.4×10−1
4.7×10−1
1.8×10−1
2.9×10−1
3.7×10−1

Table A.29: 24 Mg(α, α′ ) inelastic,
Ex = 9.00 MeV (2+5 ),Eα = 130 MeV
θc.m.
(deg)
2.5
3.4
4.4
5.3
5.4
6.4
7.3
8.3
9.2
10.1
11.1
12.0
13.0
13.9
14.8
15.8

dσ/dΩ
(mb/sr)6.78×10−1
6.68×10−1
8.64×10−1
1.18
1.07
1.19
9.74×10−1
5.00×10−1
3.10×10−1
1.98×10−1
1.90×10−1
2.87×10−1
3.32×10−1
2.88×10−1
2.25×10−1
1.93×10−1

Table A.30: 24 Mg(α, α′ ) inelastic,
Ex = 9.15 MeV (1−1 ),Eα = 130 MeV

error(mb/sr)
6.7×10−2
6.6×10−2
7.5×10−2
9.6×10−2
9.0×10−2
9.8×10−2
7.1×10−2
7.0×10−2
6.4×10−2
2.0×10−2
1.9×10−2
2.6×10−2
3.0×10−2
4.1×10−2
2.2×10−2
2.0×10−2

θc.m.
(deg)
2.5
3.4
4.4
5.3
5.4
6.4
7.3
8.3
9.2
10.1
11.1
12.0
13.0
13.9
14.8
15.8

115

dσ/dΩ
(mb/sr)6.50×10−1
7.68×10−1
7.99×10−1
8.32×10−1
8.0×10−1
6.23×10−1
4.80×10−1
2.93×10−1
2.39×10−1
1.73×10−1
2.11×10−1
2.82×10−1
3.32×10−1
3.65×10−1
3.32×10−1
2.70×10−1

error(mb/sr)
6.5×10−2
7.4×10−2
7.0×10−2
7.3×10−2
1.1×10−1
9.2×10−2
3.9×10−2
4.9×10−2
6.1×10−2
1.8×10−2
2.1×10−2
4.0×10−2
3.0×10−2
4.9×10−2
3.0×10−2
2.6×10−2

Table A.31: 24 Mg(α, α′ ) inelastic,
Ex = 9.31 MeV (0+3 ),Eα = 130 MeV
θc.m.
(deg)
0.5
2.5
3.4
4.4
5.3
5.4
6.4
7.3
8.3
9.2
10.1
11.1
12.0
13.0
13.9
14.8
15.8

dσ/dΩ
(mb/sr)
8.36
4.71
1.60
4.57×10−1
3.62×10−1
4.19×10−1
9.52×10−1
1.36
1.13
7.66×10−1
4.62×10−1
3.60×10−1
6.42×10−1
1.06
1.18
1.32
1.16

Table A.32: 24 Mg(α, α′ ) inelastic,
Ex = 10.36 MeV (2+7 ),Eα = 130 MeV

error(mb/sr)5.2×10−1
3.2×10−1
1.3×10−1
4.6×10−2
3.9×10−2
4.4×10−2
8.2×10−2
9.4×10−2
1.1×10−1
9.7×10−2
3.8×10−2
3.1×10−2
5.0×10−2
7.6×10−2
1.1×10−1
9.3×10−2
8.3×10−2

θc.m.
(deg)
0.5
2.5
3.4
4.4
5.3
5.4
6.4
7.3
8.3
9.2
10.1

116

dσ/dΩ
(mb/sr)
1.18
1.60
1.52
1.76
2.10
2.13
2.32
2.34
2.13
1.71
1.08

error
(mb/sr)9.8×10−2
1.3×10−1
1.3×10−1
1.3×10−1
1.5×10−1
1.6×10−1
1.7×10−1
1.5×10−1
1.4×10−1
1.7×10−1
1.1×10−1

Table A.33: 28 Si(α, α′ ) inelastic,
Ex = 1.78 MeV (2+1 ),Eα = 130 MeV
θc.m.
(deg)
0.5
2.4
3.3
4.3
5.2
5.3
6.2
7.1
8.0
9.0
9.9
10.8
11.7
12.6
13.5
14.5
15.4

dσ/dΩ
(mb/sr)
1.28×102
1.30×102
9.08×101
9.01×101
1.32×102
1.38×102
1.82×102
1.82×102
1.61×102
1.01×102
5.20×101
1.58×101
1.09×101
2.46×101
4.47×101
5.45×101
5.62×101

Table A.34: 28 Si(α, α′ ) inelastic,
Ex = 4.62 MeV (4+1 ),Eα = 130 MeV

error(mb/sr)
6.7
7.4
5.3
5.0
7.2
7.3
9.5
9.4
8.4
5.3
2.8
8.9×10−1
6.3×10−1
1.4
2.4
2.9
3.0

θc.m.
(deg)
2.4
3.3
4.3
5.2
5.3
6.2
7.1
8.1
9.0
9.9
10.8
11.7
12.6
13.6
14.5
15.4

117

dσ/dΩ
(mb/sr)5.94×10−1
7.4×10−1
7.80×10−1
1.07
1.01
1.55
2.06
3.19
3.96
5.09
4.98
4.79
3.37
2.44
1.47
1.24

error
(mb/sr)9.3×10−2
1.1×10−1
8.4×10−2
1.1×10−1
8.3×10−2
1.7×10−1
1.4×10−1
2.0×10−1
2.5×10−1
3.1×10−1
3.0×10−1
2.9×10−1
2.2×10−1
1.6×10−1
1.0×10−1
8.7×10−2

Table A.35: 28 Si(α, α′ ) inelastic,
Ex = 4.98 MeV (0+2 ),Eα = 130 MeV
θc.m.
(deg)
0.5
2.4
3.3
4.3
5.2
5.3
6.2
7.1
8.1
9.0
9.9
10.8
11.7
12.7
13.6
14.5
15.4

dσ/dΩ
(mb/sr)3.18×101
2.55×101
1.17×101
2.68
9.88×10−1
1.21
4.04
5.97
6.23
4.02
2.21
1.28
2.28
3.76
5.42
5.33
4.81

Table A.36: 28 Si(α, α′ ) inelastic,
Ex = 6.69 MeV (0+3 ),Eα = 130 MeV

error(mb/sr)5.6
1.7
8.7×10−1
2.2×10−1
9.9×10−2
9.6×10−2
3.4×10−1
3.6×10−1
3.7×10−1
2.5×10−1
1.5×10−1
9.2×10−2
1.5×10−1
2.4×10−1
3.3×10−1
3.2×10−1
2.9×10−1

θc.m.
(deg)
0.5
2.4
3.3
4.3
5.2
5.3
7.1
8.1
9.0
9.9
10.8
12.7
13.6
14.5
15.4

118

dσ/dΩ
(mb/sr)
3.18
2.56
1.39
6.5×10−1
3.6×10−1
3.84×10−1
9.42×10−1
1.13
8.4×10−1
6.5×10−1
2.56×10−1
1.51×10−1
1.83×10−1
2.14×10−1
1.94×10−1

error(mb/sr)
4.9×10−1
2.6×10−1
1.7×10−1
1.5×10−1
1.4×10−1
3.9×10−2
7.1×10−2
2.6×10−1
1.5×10−1
2.1×10−1
2.5×10−2
1.7×10−2
2.0×10−2
2.1×10−2
5.1×10−2

Table A.37: 28 Si(α, α′ ) inelastic,
Ex = 6.88 MeV (3−1 ),Eα = 130 MeV
θc.m.
(deg)
0.5
2.4
3.3
4.3
5.2
5.3
6.2
7.1
8.1
9.0
9.9
10.8
11.7
12.7
13.6
14.5
15.4

dσ/dΩ
(mb/sr)4.8
1.17×101
1.56×101
1.94×101
2.41×101
2.48×101
3.25×101
4.08×101
4.97×101
4.94×101
4.63×101
3.29×101
2.29×101
1.23×101
8.84
8.69
1.19×101

Table A.38: 28 Si(α, α′ ) inelastic,
Ex = 7.93 MeV (2+2 ),Eα = 130 MeV

error(mb/sr)1.5
8.7×10−1
1.1
1.2
1.5
1.4
1.8
2.2
2.7
2.7
2.5
1.8
1.3
7.0×10−1
5.2×10−1
5.0×10−1
6.7×10−1

θc.m.
(deg)
0.5
2.4
3.3
4.3
5.2
5.3
6.2
7.1
8.1
9.0
9.9
10.8
11.8
12.7
13.6
14.5
15.4

119

dσ/dΩ
(mb/sr)
3.4
3.30
2.60
2.89
3.81
3.83
4.38
3.82
3.37
1.68
8.44×10−1
3.81×10−1
5.50×10−1
8.56×10−1
1.25
1.10
9.69×10−1

error(mb/sr)1.4
5.4×10−1
2.6×10−1
2.3×10−1
2.9×10−1
2.6×10−1
2.9×10−1
2.4×10−1
2.2×10−1
1.2×10−1
6.6×10−2
3.4×10−2
4.6×10−2
6.6×10−2
9.1×10−2
7.3×10−2
7.1×10−2

Table A.39: 28 Si(α, α′ ) inelastic,
Ex = 8.26 MeV (2+3 ),Eα = 130 MeV
θc.m.
(deg)
2.4
4.3
5.2
5.3
6.2
7.2
8.1
9.0
9.9
10.8
11.8
12.7
13.6
14.5
15.4

dσ/dΩ
(mb/sr)4.1×10−1
3.05×10−1
2.83×10−1
2.77×10−1
2.69×10−1
4.13×10−1
4.99×10−1
5.49×10−1
5.77×10−1
5.07×10−1
3.74×10−1
2.35×10−1
1.86×10−1
1.99×10−1
3.47×10−1

Table A.40: 28 Si(α, α′ ) inelastic,
Ex = 8.90 MeV (1−1 ),Eα = 130 MeV

error(mb/sr)
1.6×10−1
4.3×10−2
4.1×10−2
3.1×10−2
3.0×10−2
3.7×10−2
6.5×10−2
4.7×10−2
4.9×10−2
4.3×10−2
3.4×10−2
2.4×10−2
2.0×10−2
2.0×10−2
3.1×10−2

θc.m.
(deg)
0.5
2.4
3.3
4.3
5.2
5.3
6.2
7.2
8.1
9.0
9.9
10.8
11.8
12.7
13.6
14.5
15.4

120

dσ/dΩ
(mb/sr)
1.61
3.27
3.31
2.21
1.05
9.08×10−1
2.91×10−1
5.07×10−1
1.40
2.04
2.14
1.68
1.24
8.12×10−1
9.69×10−1
1.27
1.57

error
(mb/sr)2.2×10−1
2.6×10−1
2.6×10−1
1.6×10−1
8.4×10−2
7.4×10−2
3.1×10−2
4.2×10−2
9.8×10−2
1.5×10−1
1.5×10−1
1.2×10−1
9.0×10−2
5.9×10−2
6.8×10−2
8.9×10−2
1.1×10−1

Table A.41: 28 Si(α, α′ ) inelastic,
Ex = 9.48 MeV (2+4 ),Eα = 130 MeV
θc.m.
(deg)
0.5
2.4
3.4
4.3
5.2
5.3
6.2
7.2
8.1
9.0
9.9
10.8
11.8
12.7
13.6
14.5
15.4

dσ/dΩ
(mb/sr)9.08×10−1
1.04
1.15
1.32
1.73
1.70
1.97
1.74
1.51
1.16
6.29×10−1
4.56×10−1
4.39×10−1
6.64×10−1
7.80×10−1
6.57×10−1
5.42×10−1

Table A.42: 28 Si(α, α′ ) inelastic,
Ex = 9.93 MeV (1−2 ),Eα = 130 MeV

error(mb/sr)7.1×10−2
1.4×10−1
1.5×10−1
1.2×10−1
1.5×10−1
1.3×10−1
1.4×10−1
1.2×10−1
1.1×10−1
8.6×10−2
5.2×10−2
4.0×10−2
3.8×10−2
5.4×10−2
6.1×10−2
5.1×10−2
4.4×10−2

θc.m.
(deg)
0.5
2.4
3.4
4.3
5.2
5.3
6.2
7.2
8.1
9.0
9.9
10.8
11.8
12.7
13.6
14.5
15.4

121

dσ/dΩ
(mb/sr)
1.48
5.61
7.27
6.69
4.69
4.08
2.01
7.26×10−1
6.27×10−1
1.47
1.85
1.83
1.47
7.75×10−1
3.24×10−1
2.30×10−1
4.02×10−1

error(mb/sr)2.2×10−1
4.0×10−1
5.0×10−1
5.0×10−1
3.7×10−1
3.3×10−1
1.9×10−1
5.6×10−2
5.0×10−2
1.1×10−1
1.4×10−1
1.2×10−1
1.0×10−1
5.6×10−2
2.8×10−2
2.3×10−2
3.5×10−2

Table A.43: 28 Si(α, α′ ) inelastic,
Ex = 10.18 MeV (3−2 ),Eα = 130 MeV
θc.m.
(deg)
0.5
2.4
3.4
4.3
5.2
5.3
6.2
7.2
8.1
9.0
9.9
10.8
11.8
12.7
13.6
14.5
15.5

dσ/dΩ
(mb/sr)
4.63
6.78
7.91
8.38
8.97
8.59
9.64
1.01×101
1.16×101
1.15×101
1.10×101
7.83
6.08
3.33
2.25
2.06
2.56

Table A.44: 28 Si(α, α′ ) inelastic,
Ex = 10.51 MeV (2+5 ),Eα = 130 MeV

error(mb/sr)3.1×10−1
6.1×10−1
6.8×10−1
6.0×10−1
6.4×10−1
6.1×10−1
6.6×10−1
6.5×10−1
7.4×10−1
7.9×10−1
7.6×10−1
5.2×10−1
4.2×10−1
2.4×10−1
1.7×10−1
1.7×10−1
2.0×10−1

θc.m.
(deg)
0.5
2.4
3.4
4.3
5.2
5.3
6.2
7.2
8.1
9.0
9.9
10.8
11.8
12.7
13.6
14.5
15.5

122

dσ/dΩ
(mb/sr)7.7×10−1
3.76×10−1
4.20×10−1
8.2×10−1
1.17
1.27
1.35
1.22
8.20×10−1
4.30×10−1
1.83×10−1
8.9×10−2
2.30×10−1
2.69×10−1
3.57×10−1
3.60×10−1
2.67×10−1

error(mb/sr)4.4×10−1
8.5×10−2
9.3×10−2
1.0×10−1
1.3×10−1
1.4×10−1
1.4×10−1
1.1×10−1
8.7×10−2
7.1×10−2
4.2×10−2
2.3×10−2
3.6×10−2
3.8×10−2
4.4×10−2
4.7×10−2
4.0×10−2

Table A.45: 40 Ca(α, α′ ) inelastic,
Ex = 3.35 MeV (0+2 ),Eα = 130 MeV
θc.m.
(deg)
0.4
4.1
6.0
6.9
7.7
9.5
10.4
11.3
12.1
13.0
13.9
14.8

dσ/dΩ
(mb/sr)2.8×10−1
1.80×10−1
2.18×10−1
1.29×10−1
6.6×10−2
3.45×10−2
6.2×10−2
7.2×10−2
5.65×10−2
2.34×10−2
3.39×10−2
7.9×10−2

Table A.47: 40 Ca(α, α′ ) inelastic,
Ex = 3.90 MeV (2+1 ),Eα = 130 MeV

error(mb/sr)1.5×10−1
2.6×10−2
5.4×10−2
1.6×10−2
2.4×10−2
6.6×10−3
1.4×10−2
1.6×10−2
7.9×10−3
6.8×10−3
6.6×10−3
1.2×10−2

θc.m.
(deg)
0.4
2.3
3.2
4.1
5.0
5.1
6.0
6.9
7.7
8.6
9.5
10.4
11.3
12.2
13.0
13.9
14.8

Table A.46: 40 Ca(α, α′ ) inelastic,
Ex = 3.74 MeV (3−1 ),Eα = 130 MeV
θc.m.
(deg)
0.4
2.3
3.2
4.1
5.0
5.1
6.0
6.9
7.7
9.5
10.4
11.3
12.2
13.0
13.9
14.8

dσ/dΩ
(mb/sr)5.10
3.02×101
3.97×101
4.22×101
4.60×101
4.68×101
5.29×101
6.43×101
8.02×101
6.76×101
3.96×101
2.08×101
7.77
8.39
1.41×101
2.17×101error(mb/sr)3.1×10−1
1.7
2.3
2.4
3.0
3.0
3.3
3.4
4.2
3.6
2.1
1.2
4.5×10−1
4.8×10−1
8.1×10−1
1.2

123

dσ/dΩ
(mb/sr)6.36
1.01×101
7.89
1.19×101
1.53×101
1.67×101
2.10×101
1.94×101
1.52×101
6.08
1.74
1.46
3.88
5.97
6.72
4.90
2.81

error
(mb/sr)3.8×10−1
6.4×10−1
5.1×10−1
7.3×10−1
1.2
1.3
1.5
1.1
8.6×10−1
3.7×10−1
1.2×10−1
1.0×10−1
2.4×10−1
3.5×10−1
3.9×10−1
3.0×10−1
1.9×10−1

Table A.48: 40 Ca(α, α′ ) inelastic,
Ex = 4.49 MeV (5−1 ),Eα = 130 MeV
θc.m.
(deg)
0.4
4.1
5.0
5.1
6.0
6.9
7.7
8.6
9.5
10.4
11.3
12.2
13.0
13.9
14.8

dσ/dΩ
(mb/sr)
4.54×10−1
8.71×10−1
9.18×10−1
8.4×10−1
1.01
1.46
2.16
2.95
4.34
5.11
5.97
5.95
5.63
4.02
2.79

Table A.49: 40 Ca(α, α′ ) inelastic,
Ex = 5.90 MeV (1−1 ),Eα = 130 MeV

error(mb/sr)3.8×10−2
8.0×10−2
8.3×10−2
1.4×10−1
3.0×10−1
1.1×10−1
1.5×10−1
1.9×10−1
2.7×10−1
3.1×10−1
3.6×10−1
3.5×10−1
3.3×10−1
2.6×10−1
1.8×10−1

θc.m.
(deg)
0.4
2.3
3.2
4.1
5.0
5.1
6.0
6.9
7.7
8.6
9.5
10.4
11.3
12.2
13.0
13.9
14.8

124

dσ/dΩ
(mb/sr)
3.49×10−1
3.61
3.65
3.11
1.84
1.60
4.91×10−1
2.5×10−1
5.43×10−1
9.13×10−1
1.10
7.86×10−1
4.93×10−1
1.64×10−1
1.49×10−1
4.17×10−1
4.40×10−1

error(mb/sr)3.1×10−2
2.6×10−1
2.6×10−1
2.2×10−1
1.5×10−1
1.3×10−1
9.5×10−2
1.3×10−1
4.7×10−2
7.1×10−2
8.3×10−2
6.1×10−2
4.2×10−2
1.7×10−2
1.6×10−2
3.8×10−2
4.0×10−2

Table A.50: 40 Ca(α, α′ ) inelastic,
Ex = 6.29 MeV (3−2 ),Eα = 130 MeV
θc.m.
(deg)
0.4
2.3
3.2
4.1
5.0
5.1
6.0
6.9
7.7
8.6
9.5
10.4
11.3
12.2
13.0
13.9
14.8

dσ/dΩ
(mb/sr)
1.20
5.53
6.57
8.72
1.00×101
1.02×101
1.25×101
1.19×101
1.27×101
1.18×101
9.54
5.17
2.36
1.12
1.64
2.70
3.58

Table A.51: 40 Ca(α, α′ ) inelastic,
Ex = 8.09 MeV (2+2 ),Eα = 130 MeV

error(mb/sr)8.5×10−2
3.8×10−1
4.4×10−1
5.5×10−1
6.2×10−1
6.4×10−1
7.7×10−1
8.7×10−1
8.3×10−1
6.8×10−1
5.6×10−1
3.2×10−1
1.6×10−1
7.9×10−2
1.1×10−1
1.8×10−1
2.3×10−1

θc.m.
(deg)
0.4
2.3
3.2
4.1
5.0
5.1
6.0
6.9
7.8
8.6
9.5
10.4
11.3
12.2
13.1
13.9
14.8

125

dσ/dΩ
(mb/sr)
2.57
3.01
3.15
5.11
6.27
6.58
6.81
5.00
2.55
1.37
6.61×10−1
1.03
1.79
2.10
1.92
1.33
9.41×10−1

error(mb/sr)1.7×10−1
3.0×10−1
3.2×10−1
3.4×10−1
4.1×10−1
4.3×10−1
4.4×10−1
3.1×10−1
3.0×10−1
9.9×10−2
5.4×10−2
7.7×10−2
1.2×10−1
1.4×10−1
1.3×10−1
9.7×10−2
7.3×10−2

Table A.52: 40 Ca(α, α′ ) inelastic,
Ex = 8.28 MeV (0+3 ),Eα = 130 MeV
θc.m.
(deg)
0.4
2.3
3.2
4.1
5.0
5.1
6.0
6.9
7.8
8.6
9.5
10.4
11.3
12.2
13.1
13.9
14.8

dσ/dΩ
(mb/sr)
9.54
4.52
1.21
1.62×10−1
4.75×10−1
4.62×10−1
1.07
1.19
9.4×10−1
6.33×10−1
3.37×10−1
1.52×10−1
4.24×10−1
5.23×10−1
5.24×10−1
3.20×10−1
1.59×10−1

Table A.53: 40 Ca(α, α′ ) inelastic,
Ex = 8.37 MeV (4+1 ),Eα = 130 MeV

error(mb/sr)5.5×10−1
3.2×10−1
1.1×10−1
2.4×10−2
5.1×10−2
5.0×10−2
9.5×10−2
8.8×10−2
2.0×10−1
7.3×10−2
6.0×10−2
1.7×10−2
3.7×10−2
4.2×10−2
4.2×10−2
3.1×10−2
1.9×10−2

θc.m.
(deg)
0.4
2.3
3.2
4.1
5.0
5.1
6.0
6.9
7.8
8.6
9.5
10.4
11.3
12.2
13.1
13.9
14.8

126

dσ/dΩ
(mb/sr)1.03
7.75×10−1
5.84×10−1
8.12×10−1
1.05
1.08
1.55
1.85
2.25
2.53
3.13
2.74
2.37
1.60
9.78×10−1
4.75×10−1
3.94×10−1

error(mb/sr)3.9×10−1
7.6×10−2
6.1×10−2
7.6×10−2
9.3×10−2
9.5×10−2
1.3×10−1
1.3×10−1
2.9×10−1
2.2×10−1
2.0×10−1
1.8×10−1
1.6×10−1
1.1×10−1
7.0×10−2
4.2×10−2
3.7×10−2

Table A.54: 40 Ca(α, α′ ) inelastic,
Ex = 8.58 MeV (2+3 ),Eα = 130 MeV
θc.m.
(deg)
0.4
2.3
3.2
4.1
5.0
5.1
6.0
6.9
7.8
8.6
9.5
10.4
11.3
12.2
13.1
13.9
14.8

dσ/dΩ
(mb/sr)
1.23
1.68
2.11
2.70
3.54
3.74
4.26
3.44
2.42
1.03
4.97×10−1
4.80×10−1
7.67×10−1
1.09
1.20
9.24×10−1
5.21×10−1

error(mb/sr)8.7×10−2
1.4×10−1
1.7×10−1
2.0×10−1
2.5×10−1
2.6×10−1
3.0×10−1
2.2×10−1
2.9×10−1
1.2×10−1
4.3×10−2
4.1×10−2
6.0×10−2
7.7×10−2
8.4×10−2
7.2×10−2
4.5×10−2127A.2.2 Eα = 386 MeV
Table A.55: 12 C(α, α′ ) inelastic,
Ex = 4.44 MeV (2+1 ),Eα = 386 MeV
θc.m.
(deg)
0.4
1.0
1.8
2.5
3.0
3.5
4.0
4.9
5.4
5.9
6.2
6.7
7.3
7.8
8.6
9.1
9.6
10.2
11.0
11.5
12.1
12.6
13.4
14.0
14.5

dσ/dΩ
(mb/sr)7.55×101
7.76×101
7.31×101
6.43×101
6.13×101
6.47×101
7.41×101
1.00×102
1.04×102
1.06×102
1.12×102
1.06×1028.80×101
6.75×101
4.55×101
3.17×101
2.12×101
1.60×101
1.39×101
1.64×101
1.92×101
2.04×101
2.16×101
1.98×101
1.72×101

Table A.55: (Continued)
θc.m.
(deg)
15.0
15.8
16.4
16.9
17.5
18.3
18.8
19.3
19.9
20.7
21.2
21.8
22.3
22.5
23.1
23.6
24.2
24.7
25.2
25.8
26.3
26.8
27.4
27.9
28.5
29.0
29.5

error
(mb/sr)
8.5
5.1
4.1
1.0
1.0
1.0
1.1
5.0
5.2
5.3
5.6
5.3
4.4
3.4
2.3
1.6
1.1
8.0×10−1
7.0×10−1
8.2×10−1
9.6×10−1
1.0
1.1
9.9×10−1
8.6×10−1

128

dσ/dΩ
(mb/sr)
1.36×101
1.02×101
7.27
5.07
3.46
1.98
1.71
1.60
1.52
1.44
1.29
1.10
9.26×10−1
8.20×10−1
6.68×10−1
5.26×10−1
3.71×10−1
2.95×10−1
2.34×10−1
1.85×10−1
1.32×10−1
1.15×10−1
9.95×10−2
8.46×10−2
6.45×10−2
5.72×10−2
4.80×10−2

error(mb/sr)
6.8×10−1
5.1×10−1
3.7×10−1
2.6×10−1
1.7×10−1
1.0×10−1
8.7×10−2
8.2×10−2
7.8×10−2
8.0×10−2
7.3×10−2
6.3×10−2
5.4×10−2
2.5×10−2
2.1×10−2
1.6×10−2
1.1×10−2
9.1×10−3
7.2×10−3
5.8×10−3
4.0×10−3
3.5×10−3
3.1×10−3
2.6×10−3
2.0×10−3
1.8×10−3
1.5×10−3

Table A.56: 12 C(α, α′ ) inelastic,
Ex = 7.65 MeV (0+2 ),Eα = 386 MeV
θc.m.
(deg)
0.4
1.0
1.8
2.5
3.0
3.5
4.1
4.3
4.9
5.4
5.9
6.2
6.7
7.3
7.8
8.6
9.1
9.6
10.2
11.0

dσ/dΩ
(mb/sr)8.27×101
7.32×101
5.14×101
3.27×101
1.96×101
9.10
2.80
9.01×10−1
1.07
2.78
4.46
6.01
6.82
5.93
4.49
2.51
1.44
1.01
1.02
1.50

Table A.56: (Continued)
θc.m.
(deg)
11.5
12.1
12.6
13.4
14.0
14.5
15.0
15.9
16.4
16.9
17.5
18.3
18.8
19.4
19.9
20.7
21.3
21.8
22.3

error
(mb/sr)1.6
9.0×10−1
6.0×10−1
7.0×10−1
6.0×10−1
4.0×10−1
2.0×10−1
5.8×10−2
6.7×10−2
1.5×10−1
2.4×10−1
3.2×10−1
3.6×10−1
3.1×10−1
2.4×10−1
1.3×10−1
7.3×10−2
5.2×10−2
5.2×10−2
7.6×10−2

129

dσ/dΩ
(mb/sr)
1.90
2.08
1.99
1.71
1.38
9.59×10−1
6.04×10−1
3.45×10−1
2.43×10−1
1.93×10−1
2.11×10−1
2.03×10−1
1.92×10−1
1.77×10−1
1.56×10−1
1.24×10−1
7.80×10−2
6.62×10−2
3.91×10−2

error(mb/sr)9.6×10−2
1.1×10−1
1.0×10−1
8.6×10−2
7.0×10−2
4.9×10−2
3.1×10−2
1.9×10−2
1.4×10−2
1.1×10−2
1.2×10−2
1.2×10−2
1.1×10−2
1.0×10−2
9.1×10−3
1.2×10−2
9.0×10−3
8.2×10−3
6.1×10−3

Table A.57: 12 C(α, α′ ) inelastic,
Ex = 9.64 MeV (3−1 ),Eα = 386 MeV
θc.m.
(deg)
0.4
1.0
1.5
2.1
2.7
3.3
3.8
4.3
4.3
4.9
5.4
5.9
6.1
6.7
7.2
7.8
8.5
9.0
9.6
10.1
10.9
11.5
12.0
12.5
13.4

dσ/dΩ
(mb/sr)
2.55
3.24
2.86
4.04
5.50
6.80
8.55
1.07×101
1.01×101
1.28×101
1.50×101
1.69×101
2.05×101
2.06×101
2.12×101
1.94×101
1.59×101
1.29×101
9.43
6.71
3.77
3.04
2.83
2.84
3.27

Table A.57: (Continued)
θc.m.
(deg)
13.9
14.4
15.0
15.8
16.3
16.9
17.4
18.2
18.8
19.3
19.9
20.7
21.2
21.8
22.3
23.1
23.7
24.2
24.7
25.5
26.1
26.6
27.2

error
(mb/sr)3.4×10−1
2.5×10−1
2.6×10−1
2.6×10−1
7.4×10−2
7.7×10−2
8.2×10−2
8.9×10−2
1.1×10−1
1.2×10−1
1.3×10−1
1.4×10−1
3.9×10−1
3.9×10−1
3.9×10−1
3.7×10−1
3.9×10−2
3.5×10−2
3.0×10−2
2.6×10−2
1.9×10−2
1.7×10−2
1.7×10−2
1.6×10−2
1.7×10−2

130

dσ/dΩ
(mb/sr)
3.40
3.26
2.96
2.20
1.73
1.29
8.74×10−1
4.26×10−1
3.19×10−1
2.41×10−1
2.01×10−1
1.38×10−1
1.24×10−1
1.21×10−1
9.24×10−2
9.54×10−2
7.60×10−2
6.36×10−2
4.53×10−2
2.66×10−2
1.85×10−2
1.59×10−2
1.30×10−2

error(mb/sr)1.8×10−2
1.7×10−2
1.6×10−2
1.5×10−2
1.3×10−2
1.1×10−2
9.2×10−3
6.8×10−3
5.9×10−3
5.2×10−3
4.8×10−3
1.0×10−2
9.9×10−3
9.7×10−3
8.7×10−3
8.0×10−3
7.1×10−3
6.5×10−3
5.5×10−3
3.5×10−3
3.0×10−3
2.7×10−3
2.5×10−3

Table A.58: 16 O(α, α′ ) inelastic,
Ex = 6.13 MeV (3−1 ),Eα = 386 MeV
θc.m.
(deg)
2.4
2.9
3.4
3.9
4.0
4.4
4.9
5.4
5.6
6.1
6.6
7.1

dσ/dΩ
(mb/sr)1.91×101
2.15×101
2.40×101
2.86×101
2.91×101
3.85×101
4.72×101
5.31×101
5.43×101
6.08×101
6.14×101
6.06×101error(mb/sr)
1.2
1.3
1.5
1.7
1.8
2.3
2.7
3.1
3.1
3.5
3.5
3.4

Table A.59: 16 O(α, α′ ) inelastic,
Ex = 6.92 MeV (2+1 ),Eα = 386 MeV
θc.m.
(deg)
2.4
2.9
3.4
3.9
4.0
4.4
4.9
5.4
5.6
6.1
6.6
7.1

dσ/dΩ
(mb/sr)
2.49×101
3.24×101
4.18×101
5.40×101
5.11×101
5.98×101
6.75×101
6.44×101
6.21×101
5.52×101
3.95×101
2.84×101

Table A.60: 16 O(α, α′ ) inelastic,
Ex = 11.52 MeV (2+3 ),Eα = 386 MeV
θc.m.
(deg)
1.0
2.4
2.9
3.4
3.9
4.0
4.9
5.4
5.6
6.1
6.6
7.1

dσ/dΩ
(mb/sr)1.25×101
1.56×101
1.91×101
2.43×101
3.00×101
3.39×101
3.45×101
3.26×101
3.19×101
2.73×101
2.37×101
1.57×101error(mb/sr)
1.8
1.0
1.2
1.5
1.8
2.0
2.1
2.0
1.9
1.6
1.4
9.9×10−1

Table A.61: 16 O(α, α′ ) inelastic,
Ex = 12.05 MeV (0+3 ),Eα = 386 MeV

error(mb/sr)
1.5
2.0
2.5
3.1
3.0
3.4
3.8
3.7
3.5
3.2
2.3
1.7

θc.m.
(deg)
1.0
2.4
2.9
3.4
3.9
4.4
4.9
5.4
5.6
6.1
6.6
7.1

131

dσ/dΩ
(mb/sr)1.63×101
4.84
1.81
2.80×10−1
2.80×10−1
8.6×10−1
1.75
2.12
2.24
2.25
2.13
1.02

error
(mb/sr)1.0
4.2×10−1
2.0×10−1
5.4×10−2
5.4×10−2
1.1×10−1
2.0×10−1
2.3×10−1
2.2×10−1
2.2×10−1
2.2×10−1
1.3×10−1

Table A.62: 20 Ne(α, α′ ) inelastic,
Ex = 1.63 MeV (2+1 ),Eα = 386 MeV
θc.m.
(deg)
2.9
3.4
3.9
4.1
4.6
5.1
5.6
7.3
7.8
8.3
8.8
9.0
9.5
10.0

dσ/dΩ
(mb/sr)1.94×102
2.51×102
2.86×102
3.25×102
3.25×102
2.93×102
2.29×102
3.14×101
2.89×101
3.41×101
4.18×101
4.27×101
4.58×101
4.14×101

Table A.63: 20 Ne(α, α′ ) inelastic,
Ex = 4.25 MeV (4+1 ),Eα = 386 MeV

error(mb/sr)9.7
1.2×101
1.4×101
1.6×101
1.6×101
1.4×101
1.1×101
2.3
2.1
2.4
2.9
2.6
2.7
2.5

θc.m.
(deg)
4.6
5.1
5.6
7.3
7.8
8.3
8.8
9.0
9.5
10.0
10.5
10.7
11.2
11.7
12.2
12.4
12.9
13.4
13.9

132

dσ/dΩ
(mb/sr)
3.47
5.73
7.04
8.76
7.63
5.53
3.72
3.18
2.24
2.03
2.29
2.61
3.39
3.57
3.55
3.50
3.19
2.58
1.87

error
(mb/sr)2.8×10−1
4.1×10−1
4.8×10−1
8.6×10−1
7.7×10−1
6.1×10−1
4.7×10−1
3.2×10−1
2.6×10−1
2.4×10−1
2.5×10−1
2.3×10−1
2.8×10−1
2.9×10−1
2.9×10−1
2.4×10−1
2.2×10−1
1.8×10−1
1.4×10−1

Table A.64: 20 Ne(α, α′ ) inelastic,
Ex = 5.62 MeV (3−1 ),Eα = 386 MeV
θc.m.
(deg)
2.4
2.9
5.1
5.6
7.3
7.8
12.4
12.9
13.4
13.9

dσ/dΩ
(mb/sr)
1.33×101
1.54×101
2.45×101
2.56×101
1.99×101
1.36×101
5.01
4.25
3.07
2.38

Table A.66: 24 Mg(α, α′ ) inelastic,
Ex = 1.37 MeV (2+1 ),Eα = 386 MeV

error(mb/sr)8.6×10−1
9.6×10−1
1.4
1.4
1.6
1.2
3.3×10−1
2.9×10−1
2.2×10−1
1.8×10−1

θc.m.
(deg)
2.2
2.7
3.2
3.6
3.7
4.1
4.6
5.1
5.1
5.6
6.1
6.5
6.9
7.4
7.8
8.3
8.6
9.1
9.6
10.0
10.4
10.8
11.3
11.8

Table A.65: 20 Ne(α, α′ ) inelastic,
Ex = 6.73 MeV (0+2 ),Eα = 386 MeV
θc.m.
(deg)
0.5
2.4
2.9
3.4

dσ/dΩ
(mb/sr)8.02×101
3.30
7.9×10−1
1.05

error
(mb/sr)5.3
2.7×10−1
1.1×10−1
1.5×10−1

133

dσ/dΩ
(mb/sr)2.27×102
2.09×102
3.94×102
5.23×102
5.35×102
6.19×102
6.45×102
5.96×102
5.47×102
4.37×102
3.25×102
2.00×102
1.23×102
8.09×101
7.48×101
9.35×101
1.18×102
1.42×102
1.52×102
1.41×102
1.20×102
9.78×101
6.72×101
4.60×101error(mb/sr)1.2×101
1.2×101
2.1×101
2.7×101
3.2×101
3.6×101
3.8×101
3.5×101
2.9×101
2.3×101
1.7×101
1.1×101
6.8
4.6
4.2
5.2
6.5
7.7
8.3
7.7
6.6
5.4
3.8
2.6

Table A.67: 24 Mg(α, α′ ) inelastic,
Ex = 6.43 MeV (0+2 ),Eα = 386 MeV
θc.m.
(deg)
2.2
2.7
3.2
3.6
3.7
4.1
4.6
5.1
5.1
5.6
6.1
6.5
6.9
7.4
7.8
8.3
8.6
9.1
9.6
10.0
10.4
10.8
11.3
11.8

dσ/dΩ
(mb/sr)1.83×101
6.84
1.62
1.41
1.76
4.24
8.17
8.27
8.50
7.51
5.52
3.17
1.74
1.07
1.04
1.55
1.90
2.11
1.93
1.46
1.09
6.70×10−1
3.60×10−1
3.00×10−1

Table A.68: 24 Mg(α, α′ ) inelastic,
Ex = 7.35 MeV (2+3 ),Eα = 386 MeV

error(mb/sr)9.3×10−1
3.5×10−1
8.7×10−2
7.7×10−2
1.1×10−1
2.7×10−1
4.9×10−1
4.7×10−1
4.4×10−1
3.9×10−1
2.9×10−1
1.7×10−1
9.3×10−2
5.9×10−2
5.7×10−2
8.4×10−2
1.0×10−1
1.1×10−1
1.0×10−1
7.9×10−2
5.9×10−2
3.8×10−2
2.1×10−2
1.7×10−2

θc.m.
(deg)
0.7
1.0
2.4
2.8
3.3
3.8
3.5
4.0
4.5
4.9
5.3
5.7
6.2
6.7
7.0
7.5
8.0
8.4
8.8
9.3
9.7
10.2
10.6
11.0
11.5
12.0

134

dσ/dΩ
(mb/sr)
7.18
6.54
5.19
7.85
8.76
1.01×101
1.07×101
1.02×101
9.53
7.42
6.51
3.33
1.14
7.31×10−1
4.95×10−1
1.16
1.50
1.95
1.96
1.43
9.40×10−1
5.18×10−1
3.14×10−1
2.40×10−1
3.52×10−1
3.29×10−1

error(mb/sr)2.5×10−1
2.6×10−1
2.4×10−1
2.4×10−1
2.5×10−1
2.7×10−1
3.4×10−1
3.8×10−1
4.3×10−1
5.5×10−1
4.0×10−1
5.1×10−1
1.3×10−1
9.7×10−2
7.0×10−2
8.8×10−2
2.0×10−1
1.1×10−1
4.7×10−2
3.7×10−2
3.0×10−2
2.2×10−2
1.8×10−2
1.6×10−2
1.9×10−2
2.1×10−2

Table A.69: 24 Mg(α, α′ ) inelastic,
Ex = 7.62 MeV (3−1 ),Eα = 386 MeV
θc.m.
(deg)
0.7
1.0
2.4
2.8
3.3
3.8
3.5
4.0
4.5
4.9
5.3
5.7
6.2
6.7
7.0
7.5
8.0
8.4
8.8
9.3
9.7
10.2
10.6
11.0
11.5
12.0

dσ/dΩ
(mb/sr)
4.90
6.76
9.91
1.06×101
1.13×101
1.18×101
1.22×101
1.32×101
1.59×101
1.91×101
2.06×101
2.03×101
1.89×101
1.53×101
1.30×101
9.79
6.35
3.95
2.16
1.77
1.98
2.63
3.29
3.70
3.77
3.65

Table A.70: 24 Mg(α, α′ ) inelastic,
Ex = 8.36 MeV (3−2 ),Eα = 386 MeV

error(mb/sr)2.9×10−1
3.3×10−1
3.3×10−1
2.8×10−1
2.8×10−1
2.8×10−1
3.5×10−1
4.4×10−1
5.1×10−1
6.5×10−1
3.3×10−1
3.3×10−1
3.2×10−1
2.9×10−1
2.1×10−1
1.9×10−1
1.6×10−1
1.4×10−1
8.7×10−2
6.4×10−2
2.1×10−1
8.9×10−2
8.9×10−2
8.1×10−2
8.3×10−2
7.7×10−2

θc.m.
(deg)
0.7
1.0
2.4
2.8
3.3
3.8
3.5
4.0
4.5
4.9
5.3
5.7
6.2
6.7
7.0
7.5
8.0
8.4
8.8
9.3
9.7
10.2
10.6
11.0
11.5
12.0

135

dσ/dΩ
(mb/sr)
4.57
5.47
7.10
8.94
1.18×101
1.53×101
1.51×101
2.18×101
2.65×101
3.01×101
3.29×101
3.12×101
2.89×101
2.14×101
1.86×101
1.30×101
7.46
5.01
3.67
3.73
4.30
4.79
4.67
4.47
3.62
2.82

error
(mb/sr)2.3×10−1
2.8×10−1
3.1×10−1
3.0×10−1
3.3×10−1
3.7×10−1
5.0×10−1
7.0×10−1
7.4×10−1
9.0×10−1
5.6×10−1
5.3×10−1
5.2×10−1
5.4×10−1
3.2×10−1
2.5×10−1
1.9×10−1
1.6×10−1
7.0×10−2
6.7×10−2
7.3×10−2
7.7×10−2
8.9×10−2
8.5×10−2
7.6×10−2
6.8×10−2

Table A.71: 24 Mg(α, α′ ) inelastic,
Ex = 9.00 MeV (2+5 ),Eα = 386 MeV
θc.m.
(deg)
0.8
1.0
2.4
2.8
3.3
3.8
3.5
4.0
4.5
4.9
5.3
5.7
6.2
6.7
7.0
7.5
8.0
8.4
8.8
9.3
9.7
10.2
10.6
11.0
11.5
12.0

dσ/dΩ
(mb/sr)9.2×10−1
1.30
1.84
2.39
3.20
3.37
3.88
4.24
3.77
2.78
2.24
1.57
8.57×10−1
7.34×10−1
6.13×10−1
5.72×10−1
5.80×10−1
6.14×10−1
5.85×10−1
4.67×10−1
3.77×10−1
2.15×10−1
2.06×10−1
1.68×10−1
1.07×10−1
1.17×10−1

Table A.72: 24 Mg(α, α′ ) inelastic,
Ex = 9.15 MeV (1−1 ),Eα = 386 MeV

error(mb/sr)1.2×10−1
1.2×10−1
1.4×10−1
1.3×10−1
1.6×10−1
1.6×10−1
2.1×10−1
2.6×10−1
2.4×10−1
2.7×10−1
2.4×10−1
1.0×10−1
7.8×10−2
7.6×10−2
5.0×10−2
4.8×10−2
4.8×10−2
4.9×10−2
2.6×10−2
2.2×10−2
2.0×10−2
1.5×10−2
1.6×10−2
1.4×10−2
1.2×10−2
1.2×10−2

θc.m.
(deg)
0.8
1.0
2.4
2.8
3.3
3.8
3.5
4.0
4.5
4.9
5.3
5.7
6.2
6.7
7.0
7.5
8.0
8.4
8.8
9.3
9.7
10.2
10.6
11.0
11.5
12.0

136

dσ/dΩ
(mb/sr)5.6×10−1
5.78×10−1
8.3×10−1
9.96×10−1
9.63×10−1
1.17
1.18
7.9×10−1
6.1×10−1
1.1×10−1
1.3×10−1
2.07×10−1
1.78×10−1
1.31×10−1
9.9×10−2
1.20×10−1
1.87×10−1
2.60×10−1
2.11×10−1
1.32×10−1
1.08×10−1
1.30×10−1
3.03×10−2
3.58×10−2
6.70×10−2
7.8×10−2

error(mb/sr)1.1×10−1
9.4×10−2
1.0×10−1
9.7×10−2
8.9×10−2
1.0×10−1
1.3×10−1
1.4×10−1
1.3×10−1
1.5×10−1
2.1×10−1
6.1×10−2
5.1×10−2
4.5×10−2
2.8×10−2
2.9×10−2
3.3×10−2
3.7×10−2
1.8×10−2
1.3×10−2
1.2×10−2
1.3×10−2
8.9×10−3
8.4×10−3
9.8×10−3
1.1×10−2

Table A.73: 24 Mg(α, α′ ) inelastic,
Ex = 9.31 MeV (0+3 ),Eα = 386 MeV
θc.m.
(deg)
0.8
1.0
2.4
2.8
3.3
3.8
3.5
4.0
4.5
4.9
5.3
5.7
6.2
6.7
7.0
7.5
8.0
8.4
8.8
9.3
9.7
10.2
10.6
11.0
11.5
12.0

dσ/dΩ
(mb/sr)1.84×101
1.69×101
3.09
9.20×10−1
4.74×10−1
1.29
7.8×10−1
1.88
2.75
2.67
2.20
1.92
8.98×10−1
5.23×10−1
5.26×10−1
6.22×10−1
8.71×10−1
1.12
1.44
1.33
1.09
7.42×10−1
5.81×10−1
4.32×10−1
3.48×10−1
3.95×10−1

error(mb/sr)3.9×10−1
3.7×10−1
1.7×10−1
8.5×10−2
6.3×10−2
1.0×10−1
1.0×10−1
1.6×10−1
2.0×10−1
2.1×10−1
2.4×10−1
1.1×10−1
7.9×10−2
6.4×10−2
4.6×10−2
5.0×10−2
5.8×10−2
6.6×10−2
3.9×10−2
3.6×10−2
3.3×10−2
2.7×10−2
2.5×10−2
2.2×10−2
1.9×10−2
2.1×10−2

Table A.74: 28 Si(α, α′ ) inelastic,
Ex = 1.78 MeV (2+1 ),Eα = 386 MeV
θc.m.
(deg)
2.2
2.7
3.1
3.5
3.6
4.0
4.5
4.9
5.1
5.6
6.0
6.5

dσ/dΩ
(mb/sr)1.78×102
1.82×102
2.24×102
2.64×102
2.85×102
3.27×102
3.24×102
2.93×102
2.88×102
2.19×102
1.41×102
8.30×101error(mb/sr)9.7
9.9
1.2×101
1.4×101
1.5×101
1.7×101
1.7×101
1.6×101
1.5×101
1.2×101
7.8
4.8

Table A.75: 28 Si(α, α′ ) inelastic,
Ex = 4.98 MeV (0+2 ),Eα = 386 MeV
θc.m.
(deg)
2.2
2.7
3.1
3.5
3.6
4.0
4.5
4.9
5.1
5.6
6.0
6.5

137

dσ/dΩ
(mb/sr)1.77×101
6.60
1.80
2.40
2.70
5.90
9.40
1.03×101
1.02×101
9.20
5.90
2.60

error
(mb/sr)1.0
4.2×10−1
1.4×10−1
1.9×10−1
2.1×10−1
3.9×10−1
5.7×10−1
6.2×10−1
6.2×10−1
5.6×10−1
3.8×10−1
1.8×10−1

Table A.76: 40 Ca(α, α′ ) inelastic,
Ex = 3.74 MeV (3−1 ),Eα = 386 MeV
θc.m.
(deg)
2.1
2.6
3.0
3.3
3.4
3.8
4.3
4.7
4.9
5.3
5.8
6.2

dσ/dΩ
(mb/sr)
7.51×101
8.55×101
8.56×101
1.09×102
1.12×102
1.36×102
1.56×102
1.78×102
1.84×102
1.79×102
1.60×102
1.12×102

Table A.77: 40 Ca(α, α′ ) inelastic,
Ex = 3.90 MeV (2+1 ),Eα = 386 MeV

error(mb/sr)
4.4
5.0
5.1
6.4
6.5
7.8
8.9
1.0×101
1.0×101
1.0×101
9.1
6.5

θc.m.
(deg)
2.1
2.6
3.0
3.3
3.4
3.8
4.3
4.7
4.9
5.3
5.8
6.2

138

dσ/dΩ
(mb/sr)2.50×101
3.29×101
4.08×101
5.35×101
5.97×101
6.38×101
6.36×101
4.46×101
4.59×101
2.66×101
1.50×101
8.38

error
(mb/sr)
1.6
2.1
2.6
3.4
3.7
3.9
3.9
2.8
2.9
1.8
1.1
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147

List of Tables
2.1

Thicknesses and isotope enrichment of the targets used in the present
measurements with the 4 He2+ beams at Eα = 130 and 386 MeV. All thefoil targets are self-supporting. . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2

Design specification of the magnetic spectrometer Grand Raiden . . . . . 30

2.3

Design specification of the MWDCs for GR placed at the focal plane.
Supplied voltages are for the 20 Ne measurements. . . . . . . . . . . . . . . 31

2.4

Angular ranges for the cross section measurements of the elastic alphascattering at Eα = 130 and 386 MeV. . . . . . . . . . . . . . . . . . . . . 37

2.5

Angular ranges for the cross section measurements of the inelastic alphascattering at Eα = 130 and 386 MeV. . . . . . . . . . . . . . . . . . . . . 37

4.1

List of the excited states identified in the present work. . . . . . . . . . . 53

5.1

Parameterization for the density-independent (DI) and density-dependent(DD) αN interaction at Eα = 130 MeV. . . . . . . . . . . . . . . . . . . . 67

5.2

Parameterization for the density-independent (DI) and density-dependent(DD) αN interaction at Eα = 386 MeV. . . . . . . . . . . . . . . . . . . . 68

5.3

Electromagnetic transition strengths B(E0) [47, 72–76] and the deformation parameters α0 for the λ = 0 transitions. α0 is determined toreproduce B(E0) for each transition. . . . . . . . . . . . . . . . . . . . . . 73

5.4

Electromagnetic transition strengths B(E2) [47, 72–76] and the deformation lengths δ2 for the λ = 2 transitions. δ2 is determined to reproduce
B(E2) for each transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.5Electromagnetic transition strengths B(E3) [47, 72–76] and the deformation lengths δ3 for the λ = 3 transitions. δ3 is determined to reproduce
B(E3) for each transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

148

5.6

Electromagnetic transition strengths B(E4) [47, 72–76] and the deformation lengths δ4 for the λ = 4 transitions. δ4 is determined to reproduce
B(E4) for each transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.1

Measured transition strengths between the ground and excited states in
24 Mg, 28 Si, and 40 Ca.

6.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Deduced collective coupling parameters β1 and that for the transition
with 100% of the EWSR strength (β1EWSR ). The experimental data andcalculated cross sections are shown in Fig. 4.8. . . . . . . . . . . . . . . . 93

A.1

12 C(α, α) elastic, E

α = 130 MeV. . . . . . . . . . . . . . . . . . . . . . . 98

A.1 (Continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
A.2

16 O(α, α) elastic, E = 130 MeV
α. . . . . . . . . . . . . . . . . . . . . . . 99

A.2 (Continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
A.3

24 Mg(α, α) elastic, E = 130 MeV
α. . . . . . . . . . . . . . . . . . . . . . 100

A.3 (Continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
A.4

28 Si(α, α) elastic, E = 130 MeV
α. . . . . . . . . . . . . . . . . . . . . . . 101

A.4 (Continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
A.5

40 Ca(α, α) elastic, E = 130 MeV . . . . . . . . . . . . . . . . . . . . . . . 102
αA.5 (Continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
A.6

12 C(α, α) elastic, E

α = 386 MeV. . . . . . . . . . . . . . . . . . . . . . . 103

A.6 (Continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
A.7

20 Ne(α, α) elastic, E

α = 386 MeV . . . . . . . . . . . . . . . . . . . . . . . 104

A.7 (Continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
A.8

24 Mg(α, α) elastic, E = 386 MeV
α. . . . . . . . . . . . . . . . . . . . . . 105

A.8 (Continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
12 C(α, α′ ) inelastic, E = 4.44 MeV (2+ ), E = 130 MeV . . . . . . . . . 106
x
α
1
+
12
′
A.10 C(α, α ) inelastic, Ex = 7.65 MeV (02 ), Eα = 130 MeV . . . . . . . . . 106
A.11 12 C(α, α′ ) inelastic, Ex = 9.64 MeV (3−1 ), Eα = 130 MeV . . . . . . . . . 107
12′
A.12 C(α, α ) inelastic, Ex = 10.84 MeV (1−1 ), Eα = 130 MeV . . . . . . . . . 107+
A.13 16 O(α, α′ ) inelastic, Ex = 6.05 MeV (02 ), Eα = 130 MeV . . . . . . . . . 108
A.14 16 O(α, α′ ) inelastic, Ex = 6.13 MeV (3−1 ), Eα = 130 MeV . . . . . . . . . 108
16′
A.15 O(α, α ) inelastic, Ex = 6.92 MeV (2+1 ), Eα = 130 MeV . . . . . . . . . 108
A.16 16 O(α, α′ ) inelastic, Ex = 7.12 MeV (1−1 ), Eα = 130 MeV . . . . . . . . . 109

A.9

149

A.17 16 O(α, α′ ) inelastic, Ex = 9.84 MeV (2+2 ), Eα = 130 MeV . . . . . . . . . 109
A.18 16 O(α, α′ ) inelastic, Ex = 10.36 MeV (4+1 ), Eα = 130 MeV . . . . . . . . . 110
A.19 16 O(α, α′ ) inelastic, Ex = 11.52 MeV (2+3 ), Eα = 130 MeV . . . . . . . . . 110
A.20 16 O(α, α′ ) inelastic, Ex = 12.05 MeV (0+3 ), Eα = 130 MeV . . . . . . . . . 111
A.21 24 Mg(α, α′ ) inelastic, Ex = 1.37 MeV (2+1 ), Eα = 130 MeV . . . . . . . . . 111
A.22 24 Mg(α, α′ ) inelastic, Ex = 4.12 MeV (4+1 ), Eα = 130 MeV . . . . . . . . . 112
A.23 24 Mg(α, α′ ) inelastic, Ex = 4.24 MeV (2+2 ), Eα = 130 MeV . . . . . . . . . 112
A.24 24 Mg(α, α′ ) inelastic, Ex = 6.01 MeV (4+2 ), Eα = 130 MeV . . . . . . . . . 112
A.25 24 Mg(α, α′ ) inelastic, Ex = 6.43 MeV (0+2 ), Eα = 130 MeV . . . . . . . . . 113
A.26 24 Mg(α, α′ ) inelastic, Ex = 7.35 MeV (2+3 ), Eα = 130 MeV . . . . . . . . . 113
A.27 24 Mg(α, α′ ) inelastic, Ex = 7.62 MeV (3−1 ), Eα = 130 MeV . . . . . . . . . 114
A.28 24 Mg(α, α′ ) inelastic, Ex = 8.36 MeV (3−2 ), Eα = 130 MeV . . . . . . . . . 114
A.29 24 Mg(α, α′ ) inelastic, Ex = 9.00 MeV (2+5 ), Eα = 130 MeV . . . . . . . . . 115
A.30 24 Mg(α, α′ ) inelastic, Ex = 9.15 MeV (1−1 ), Eα = 130 MeV . . . . . . . . . 115
A.31 24 Mg(α, α′ ) inelastic, Ex = 9.31 MeV (0+3 ), Eα = 130 MeV . . . . . . . . . 116
A.32 24 Mg(α, α′ ) inelastic, Ex = 10.36 MeV (2+7 ), Eα = 130 MeV . . . . . . . . 116
A.33 28 Si(α, α′ ) inelastic, Ex = 1.78 MeV (2+1 ), Eα = 130 MeV . . . . . . . . . 117
A.34 28 Si(α, α′ ) inelastic, Ex = 4.62 MeV (4+1 ), Eα = 130 MeV . . . . . . . . . 117
A.35 28 Si(α, α′ ) inelastic, Ex = 4.98 MeV (0+2 ), Eα = 130 MeV . . . . . . . . . 118
A.36 28 Si(α, α′ ) inelastic, Ex = 6.69 MeV (0+3 ), Eα = 130 MeV . . . . . . . . . 118
A.37 28 Si(α, α′ ) inelastic, Ex = 6.88 MeV (3−1 ), Eα = 130 MeV . . . . . . . . . 119
A.38 28 Si(α, α′ ) inelastic, Ex = 7.93 MeV (2+2 ), Eα = 130 MeV . . . . . . . . . 119
A.39 28 Si(α, α′ ) inelastic, Ex = 8.26 MeV (2+3 ), Eα = 130 MeV . . . . . . . . . 120
A.40 28 Si(α, α′ ) inelastic, Ex = 8.90 MeV (1−1 ), Eα = 130 MeV . . . . . . . . . 120
A.41 28 Si(α, α′ ) inelastic, Ex = 9.48 MeV (2+4 ), Eα = 130 MeV . . . . . . . . . 121
A.42 28 Si(α, α′ ) inelastic, Ex = 9.93 MeV (1−2 ), Eα = 130 MeV . . . . . . . . . 121
A.43 28 Si(α, α′ ) inelastic, Ex = 10.18 MeV (3−2 ), Eα = 130 MeV . . . . . . . . . 122
A.44 28 Si(α, α′ ) inelastic, Ex = 10.51 MeV (2+5 ), Eα = 130 MeV . . . . . . . . . 122
A.45 40 Ca(α, α′ ) inelastic, Ex = 3.35 MeV (0+2 ), Eα = 130 MeV . . . . . . . . . 123
A.46 40 Ca(α, α′ ) inelastic, Ex = 3.74 MeV (3−1 ), Eα = 130 MeV . . . . . . . . . 123
A.47 40 Ca(α, α′ ) inelastic, Ex = 3.90 MeV (2+1 ), Eα = 130 MeV . . . . . . . . . 123
A.48 40 Ca(α, α′ ) inelastic, Ex = 4.49 MeV (5−1 ), Eα = 130 MeV . . . . . . . . . 124
A.49 40 Ca(α, α′ ) inelastic, Ex = 5.90 MeV (1−1 ), Eα = 130 MeV . . . . . . . . . 124
A.50 40 Ca(α, α′ ) inelastic, Ex = 6.29 MeV (3−2 ), Eα = 130 MeV . . . . . . . . . 125

150

A.51 40 Ca(α, α′ ) inelastic, Ex = 8.09 MeV (2+2 ), Eα = 130 MeV . . . . . . . . . 125
A.52 40 Ca(α, α′ ) inelastic, Ex = 8.28 MeV (0+3 ), Eα = 130 MeV . . . . . . . . . 126
A.53 40 Ca(α, α′ ) inelastic, Ex = 8.37 MeV (4+1 ), Eα = 130 MeV . . . . . . . . . 126
A.54 40 Ca(α, α′ ) inelastic, Ex = 8.58 MeV (2+3 ), Eα = 130 MeV . . . . . . . . . 127
A.55 12 C(α, α′ ) inelastic, Ex = 4.44 MeV (2+1 ), Eα = 386 MeV . . . . . . . . . 128
A.55 (Continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
A.56 12 C(α, α′ ) inelastic, Ex = 7.65 MeV (0+2 ), Eα = 386 MeV . . . . . . . . . 129
A.56 (Continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
A.57 12 C(α, α′ ) inelastic, Ex = 9.64 MeV (3−1 ), Eα = 386 MeV . . . . . . . . . 130
A.57 (Continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
A.58 16 O(α, α′ ) inelastic, Ex = 6.13 MeV (3−1 ), Eα = 386 MeV . . . . . . . . . 131
A.59 16 O(α, α′ ) inelastic, Ex = 6.92 MeV (2+1 ), Eα = 386 MeV . . . . . . . . . 131
A.60 16 O(α, α′ ) inelastic, Ex = 11.52 MeV (2+3 ), Eα = 386 MeV . . . . . . . . . 131
A.61 16 O(α, α′ ) inelastic, Ex = 12.05 MeV (0+3 ), Eα = 386 MeV . . . . . . . . . 131
A.62 20 Ne(α, α′ ) inelastic, Ex = 1.63 MeV (2+1 ), Eα = 386 MeV . . . . . . . . . 132
A.63 20 Ne(α, α′ ) inelastic, Ex = 4.25 MeV (4+1 ), Eα = 386 MeV . . . . . . . . . 132
A.64 20 Ne(α, α′ ) inelastic, Ex = 5.62 MeV (3−1 ), Eα = 386 MeV . . . . . . . . . 133
A.65 20 Ne(α, α′ ) inelastic, Ex = 6.73 MeV (0+2 ), Eα = 386 MeV . . . . . . . . . 133
A.66 24 Mg(α, α′ ) inelastic, Ex = 1.37 MeV (2+1 ), Eα = 386 MeV . . . . . . . . . 133
A.67 24 Mg(α, α′ ) inelastic, Ex = 6.43 MeV (0+2 ), Eα = 386 MeV . . . . . . . . . 134
A.68 24 Mg(α, α′ ) inelastic, Ex = 7.35 MeV (2+3 ), Eα = 386 MeV . . . . . . . . . 134
A.69 24 Mg(α, α′ ) inelastic, Ex = 7.62 MeV (3−1 ), Eα = 386 MeV . . . . . . . . . 135
A.70 24 Mg(α, α′ ) inelastic, Ex = 8.36 MeV (3−2 ), Eα = 386 MeV . . . . . . . . . 135
A.71 24 Mg(α, α′ ) inelastic, Ex = 9.00 MeV (2+5 ), Eα = 386 MeV . . . . . . . . . 136
A.72 24 Mg(α, α′ ) inelastic, Ex = 9.15 MeV (1−1 ), Eα = 386 MeV . . . . . . . . . 136
A.73 24 Mg(α, α′ ) inelastic, Ex = 9.31 MeV (0+3 ), Eα = 386 MeV . . . . . . . . . 137
A.74 28 Si(α, α′ ) inelastic, Ex = 1.78 MeV (2+1 ), Eα = 386 MeV . . . . . . . . . 137
A.75 28 Si(α, α′ ) inelastic, Ex = 4.98 MeV (0+2 ), Eα = 386 MeV . . . . . . . . . 137
A.76 40 Ca(α, α′ ) inelastic, Ex = 3.74 MeV (3−1 ), Eα = 386 MeV . . . . . . . . . 138
A.77 40 Ca(α, α′ ) inelastic, Ex = 3.90 MeV (2+1 ), Eα = 386 MeV . . . . . . . . . 138

151