Spectroscopic factors and strength distributions for the
deeply bound orbitals in 40Ca obtained from the (~p, 2p)
reaction at 392 MeVYusuke Yasuda

Abstract
Cross sections and analyzing powers for the 40 Ca(~p, 2p) reaction were measured with a 392-MeV
polarized proton beam. Recoil momentum distributions of the cross section, which approximately reflect the Fermi momentum of nucleons in nuclei, and those of the analyzing power were measured forhole states of low-lying and deeply bound orbitals. The cross sections and the analyzing powers wereanalyzed by the distorted-wave impulse approximation (DWIA).
For the low-lying discrete states in separation energy spectra, measured recoil-momentum distributions of the cross section were well reproduced by a hole state of the 1d3/2 orbital for the peak at 8.3 MeV
and a superposition of the 2s1/2 and 1 f7/2 orbitals for the peak at 10.9 MeV respectively although theobserved spectroscopic factors for these orbitals were larger than those from the previous (e, e′ p) studies. To discuss spectroscopic factors for deeply bound orbitals in 40 Ca, the normalization factor forthe discrete states was determined as 0.53 comparing with a spectroscopic factor from the (e, e′ p) measurement. The recoil-momentum distributions of the analyzing power for the low-lying discrete states
were qualitatively reproduced by the DWIA calculation but they were overestimated in the entire recoilmomentum range. It was found that the DWIA calculation reproduces the recoil-momentum distributionof the cross section and the analyzing power qualitatively well but quantitative problems on the normalization are left.
The strength distributions for the deep-hole states were obtained by a multipole decomposition analysis and a background subtraction. They were reasonably disentangled from continuous spectrum on
the basis of characteristic behavior of the recoil-momentum distribution of the cross section dependingon the orbital angular momentum L. The centroid energies and widths of the hole-state strengths were
determined as 29.6 ± 0.5 and 48.4 ± 0.6 MeV for the 1p and 1s1/2 -hole states respectively. The normalized spectroscopic factors for the deeply bound 1 p and 1s1/2 orbitals were 49 ± 7 and 89 ± 9% of thesum-rule limits of independent-particle shell model, respectively. The reduction of the spectroscopic
factors suggests an influence of the nucleon-nucleon correlations on the spectroscopic factors.

Contents

1 Introduction
1.1

1Nuclear structure and nucleon-nucleon correlations . . . . . . . . . . . . . . . . . . .

1

1.1.1

Spectroscopic factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.1.2

Previous studies of the spectroscopic factor . . . . . . . . . . . . . . . . . . .

2

1.1.3

Previous measurements of deep-hole states . . . . . . . . . . . . . . . . . . .

4

1.1.4

Direct measurement of NN-correlated pair . . . . . . . . . . . . . . . . . . .

6

1.2

Quasi-free knock-out (p, 2p) reaction . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.3(p, 2p) experiment at 1 GeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

1.4

Purposes of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

102 Experiment

13

2.1

Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.2

Experimental conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

2.3

Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2.3.1

Beam transportation and target . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2.3.2

Dual spectrometer system . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

2.3.3

Focal-Plane detectors of GR and LAS . . . . . . . . . . . . . . . . . . . . . .

19

2.3.4

MWPC for scattering angles . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

2.3.5

Trigger and data acquisition system . . . . . . . . . . . . . . . . . . . . . . .

24

3 Data Reduction

31
iCONTENTS

ii

3.1

Polarization of proton beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

3.2

Particle Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

3.3

Subtraction of accidental coincidences . . . . . . . . . . . . . . . . . . . . . . . . . .

34

3.4

Track reconstruction of scattered particles . . . . . . . . . . . . . . . . . . . . . . . .

35

3.4.1

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

35

3.4.2

Trajectory of a charged particle . . . . . . . . . . . . . . . . . . . . . . . . .

37

3.4.3

Efficiency of the MWDC . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

3.5

Multi-wire proportional chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

3.6

Solid angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

3.7

Normalization due to the trigger efficiency of the scintillation detectors . . . . . . . . .

40

3.8

Separation energy of 40 Ca(p, 2p)39 K reaction . . . . . . . . . . . . . . . . . . . . . .

42

3.9

Cross sections and analyzing powers . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

3.10 pp-scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

4 Analysis

45

4.1

Distorted wave impulse approximation calculation . . . . . . . . . . . . . . . . . . . .

45

4.2

Multipole decomposition analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

4.3

Multipole decomposition analysis with Ay data . . . . . . . . . . . . . . . . . . . . .

49

5 Result

51

5.1

Separation-energy spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

5.2

Results for the discrete states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

5.3

Results for deeply bound states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

5.3.1

Strength distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

5.3.2

Background consideration . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

6 Discussion

79

6.1

Strength distribution of the deep-hole states . . . . . . . . . . . . . . . . . . . . . . .

79

6.2

DWIA calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

CONTENTS

iii

6.3

Spectroscopic factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

6.3.1

Comparison with experimental works . . . . . . . . . . . . . . . . . . . . . .

85

6.3.2

Comparison with theoretical works . . . . . . . . . . . . . . . . . . . . . . .

87

Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

6.4

7 Summary93

Acknowledgments

95

A Revised result

97

B Scattering angles and recoil momentum99

B.1 Scattering angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

B.2 Recoil momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

C Phase space calculation

103

C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103

C.2 Phase space for four-particle final state . . . . . . . . . . . . . . . . . . . . . . . . . .

103

C.3 Phase space for two-particle final state . . . . . . . . . . . . . . . . . . . . . . . . . .

105

C.4 Two-body kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

106

C.5 Supplement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107

D Numerical data tables

109

Chapter 1

Introduction
1.1 Nuclear structure and nucleon-nucleon correlations

A nucleus is a many body system that has been studied experimentally and theoretically for a longtime. Among different nuclear models, the independent-particle shell model (IPSM) in the nuclear
mean field describes distinctive features of nucleus like the existence of magic numbers or the spins andparities of nuclei. The IPSM assumes that the nucleons independently move in a nucleus, i.e. nucleon’s
motion in a nucleus is described by a single-particle wave function obtained by solving a Schrödinger
equation with a mean-field potential, and a nucleus is described as a kind of product of the single-particle
wave functions of nucleons. The nucleons occupy the single-particle orbitals, which are classified withprinciple quantum numbers and total angular momenta, from the deepest binding orbital to the shallow
orbitals near the Fermi surface.
In microscopic nuclear models such as the IPSM, the nuclear mean field is built on the basis of
nucleon-nucleon (NN) two-body interaction. However, some features of the NN interaction are neglected. The difference between the mean-field potential and the actual sum of the NN interactions is
known as the residual interaction or the NN correlations, and it is necessary to describe nuclear structure in detail and attracts many researchers. The short-range correlations, which are related to the strong
repulsive core of the NN two-body interaction, are the most important NN correlations. It has been aninteresting topic how the strong repulsive interaction between nucleons works in nuclei. Recently, thespin-isospin and tensor correlations have also received much attention because they play an importantrole in the binding of nucleons and affect the shell structure in exotic nuclei [1, 2].
In the theoretical point of view, in the 1950s, Jastrow proposed a method to take the influence ofthe strong two-body repulsive force into a theoretical description of the nucleus using a correlation
function [3]. This function is multiplied on the single-particle wave functions so as to take the shortrange correlation into account, and the amplitude of the modified wave functions become zero when
nucleons are in a short distance. This method was employed in the following works, and it was predictedthat high-momentum components in nucleon momentum are enhanced due to the NN correlations [4, 5,
1

2

CHAPTER 1. INTRODUCTION

6, 7, 8].

1.1.1 Spectroscopic factor
Spectroscopic factor is a useful measure to examine the NN correlation. The spectroscopic factor isdefined as a ratio of the measured cross section to the theoretical cross section calculated within the
IPSM framework. For knock-out and pick-up reactions, this factor indicates how many nucleons in
the orbital participate in the reaction channel, and approximately shows how many nucleons are in theorbital in the target nucleus. In a naive shell-model picture, the spectroscopic factor for an orbital withtotal angular momentum J is expected to be 2J + 1, since orbitals in nuclei are filled by nucleons up to
2J + 1.
In a knock-out reaction, an incident particle is scattered from a nucleon in a nucleus and the recoiled
nucleon is knocked out from the nucleus. The residual nucleus is expected to be a one-hole state because
a nucleon is independently moving in the nucleus in the view of the IPSM. When the nucleon in the
nucleus strongly correlates with another nucleon, the residual nucleus is far from one-hole state and thecorrelated nucleon is often ejected from the nucleus at the same time. Consequently, a cross section
measured under the kinematical condition that the residual nucleus is expected to remain a one-holestate decreases, that is, the existence of NN correlated pairs makes the spectroscopic factor decrease.
Therefore, the spectroscopic factor is of importance to examine the NN correlations.

1.1.2 Previous studies of the spectroscopic factor
The spectroscopic factors for orbitals in nuclei have been measured to investigate the influence of the
NN correlations on the shell structure. Early spectroscopic studies were performed by the (d, 3 He)
reaction for proton (e.g. Ref. [9]). As the progress of electron accelerators, experiments for quasifree knock-out reactions with electron beams were actively performed. At the Nationaal Instituut voor
Subatomaire Fysica, Amsterdam (NIKHEF), high resolution studies of the (e, e′ p) reaction were carriedout for nuclei in the wide mass range from 2 H to 209 Bi, as reviewed by Dieperink and Witt Huberts [10].
From the results of high resolution (e, e′ p) experiments at NIKHEF, Lapikás et al. reported the spectroscopic factors for the nucleon orbitals close to the Fermi surface in 16 O, 40 Ca, 48 Ca, 90 Zr, and 208 Pb
decrease to 60%–70% of the simple IPSM limits (2J + 1) and compared those factors with predictions
from nuclear matter calculations that include short-range and tensor correlations [11]. Figure 1.1 from
Ref. [11] shows summed spectroscopic strength as a function of the missing energy. A large reduction
is observed near the Fermi surface. This reduction of the spectroscopic factor cannot be described in
the IPSM and it is expected to be ascribed to the presence of correlations between the nucleons, comingfrom the residual nuclear interactions.
The curves in Fig. 1.1 show theoretical calculations by Benhar et al. using a microscopic nuclear
matter calculation with the correlated basis function (CBF) theory including the NN correlations and1.1. NUCLEAR STRUCTURE AND NUCLEON-NUCLEON CORRELATIONS

3

Figure 1.1: Summed spectroscopic strength observed for proton knock-out reactions from various orbitals in the closed-shell nuclei 16 O, 40 Ca, 48 Ca, 90 Zr, and 208 Pb as a function of the mean excitationenergy of the orbital relative to the Fermi energy (ǫF ). The dashed curve represents the quasi-particle
strength calculated for nuclear matter, the solid curve is derived from the nuclear matter curve by including surface effects calculated for 208 Pb [12]. This figure is taken from Ref. [11].

surface effects [12]. The quenched spectroscopic strength near the Fermi surface was explained byboth contributions of the NN correlations and the surface effects. It is worthy of notice in Fig. 1.1 thatthe hole-state strengths far below the Fermi surface, where the surface effects are not important, weresuggested to decrease to almost 80% of the IPSM limits [12].
Some theoretical calculations performed with state-dependent correlations also suggested the reduction of spectroscopic factors for deeply bound orbitals. The spectroscopic factors calculated by Fabrocini et al. for the 1s and 1p orbitals are quenched to 70% and 90% of the IPSM limits in 16 O, andto 55% and 58% in 40 Ca, respectively [13]. According to this calculation, the depletion of the spectroscopic factors by the NN correlations is 10%–15% for the valence orbitals and 30%–45% for the
deeply bound orbitals. Most of the depletion was caused by the spin-isospin and tensor components of
the NN correlations in their calculation. Bisconti et al. developed the calculation following Ref. [13] by
Fabrocini et al., and predicted the spectroscopic factors for several doubly-closed-shell nuclei to be 80%
or less of the IPSM limit for the 1s and 1p orbitals in medium and heavy nuclei [14]. It is interestingthat the spectroscopic factor for the deepest 1s orbital is suppressed owing to the NN correlations moststrongly of all the orbitals in both of the calculations [13, 14].
Since the deeply bound orbitals such as 1s and 1p orbitals are bound far below the Fermi surface inmedium and heavy nuclei, the spectroscopic factors for these orbitals will not be affected by surface
effects but will be predominantly affected by the NN correlations. Therefore, it is interesting for the

4

CHAPTER 1. INTRODUCTION

Figure 1.2: Separation energies, widths, and angular momentum assignments of the hole states obtained
from quasi-free scattering, as functions of the atomic number. This figure is taken from Ref. [15].

study of NN correlations to investigate the spectroscopic factors for the deeply bound orbitals in medium
and heavy nuclei.

1.1.3 Previous measurements of deep-hole states
The valence or low-lying orbitals, which characterize the spin state of the nucleus, have been studied
well, whereas the nature of the deeply bound orbitals in medium and heavy nuclei have not sufficiently
been investigated and the deeply bound orbitals are assumed to be fully occupied by nucleons in the
IPSM.
Figure 1.2 from Ref. [15] is a famous figure on separation energies and widths compiled by Jacoband Maris. Separation energies and widths for hole states were obtained from quasi-free reactions forthe wide mass-number range of nuclei. The separation energy is an energy to remove a nucleon froma nucleus, that is, a binding energy of the nucleon. For the heavy nuclei, the experimental knowledge
for deeply bound orbitals was far from rich and has hardly been updated. Since the hole states knocked
out from the deeply bound orbitals are thought to have short lifetimes and large widths and they overlap
each other, it is very difficult to identify deep hole states.
In the medium-mass range of nuclei, for example, 40 Ca is often used for study on nuclear structuresince it is a doubly magic nucleus. As 40 Ca has a core of closed shells with 16 nucleons (8 protons and8 neutrons), the 1s1/2 and 1p orbitals in the inner core are suitable to study correlations far below theFermi surface. Many pioneering attempts were performed so as to examine the single-particle behavior

1.1. NUCLEAR STRUCTURE AND NUCLEON-NUCLEON CORRELATIONS

5of the deep-hole states by the (e, e′ p) [16, 17, 18, 19] and (p, 2p) [20, 21] reactions with the 40 Ca target.
In the work by Mougey et al. in Ref. [16], the hole-strength distribution from the (e, e′ p) reaction was
obtained by using the momentum dependence of the spectral function as shown in Fig. 1.3. Since this
distribution is plotted in large energy bins and they tentatively gave mean removal energies of the 1sand 1p-hole states, the information on the deep-hole states were not reliable. The reported spectroscopic
factors were 75% of the IPSM limit for the 1s1/2 orbital and 95% for the 1p orbital.
Nakamura et al. also reported the result of the (e, e′ p) reaction experiment [17, 18]. The hole states
except for the 1s1/2 orbital were identified by fitting their distorted momentum distributions. The 1s1/2 hole-state distributions in Fig. 1.4 were extracted by subtracting the contribution of the upper shells
and the background due to multiple collision processes from the separation energy spectra [18]. These
distributions were not on the basis of the recoil-momentum dependence and the shapes of the 1s1/2
distributions in Fig. 1.4 are unclear. The reported spectroscopic factors for the 1s1/2 and 1p orbitals arelarger than the sum-rule limits.
Figure. 1.5 from Ref. [20] shows the strength distributions obtained from the (p, 2p) reaction by
James et al. They are better in statistics than those from the electron scattering, but low-lying states,
which correspond to the ground or first excited states of the residual nucleus 39 K, are not separated due
to insufficient separation-energy resolution of 4–5 MeV (FWHM). Although James et al. reported theratios of the measured protons to the protons occupying the nl j shell as reduction factors, it is difficult
to compare those values for the low-lying states with other measurements for checking consistency of
the analysis owing to the poor separation of the low-lying states. They separated the contributions of
the orbitals from the strength distributions by using distorted momentum distributions and identified the1s level in 40 Ca, however, they didn’t give the centroid of the distribution because of the broad natureof the level.
The reported spectroscopic factors 1s1/2 and 1p orbitals in 40 Ca are not consistent among these experiments, and their centroids of the distributions are unreliable. Thus, these values of interest are still
controversial.
After the dawn of the knockout-reaction experiment, in the 1990s, the Petersburg Nuclear Physics
Institute (PNPI) group reported that they had succeeded in identifying the 1s1/2 -hole states for mediumand heavy-mass nuclei such as 40 Ca, 90 Zr, and 208 Pb in the separation-energy spectra for the ( p, 2p) and(p, np) reactions with a proton beam at 1 GeV [22, 23]. Figure 1.6 from Ref. [22] shows separationenergy spectra for the 40 Ca(p, 2p) (upper three figures in Fig. 1.6) and 40 Ca(p, pn) (lower three figures
in Fig. 1.6) reactions, and broad bumps of the deepest 1s-hole states are observed in the spectra. There isno other experiment identifying 1s-hole states in separation energy spectra for the medium- and heavymass nuclei except the PNPI group. The high energy 1-GeV injection beam might be advantageous to
identify deep hole states. Since the absolute cross sections were not measured, the spectroscopic factors
were not be given. Nevertheless, these reports encouraged many researchers to study deep-hole statesagain.

CHAPTER 1. INTRODUCTION

6

Figure 1.3:
Hole-strength distribution
from (e, e′ p) reaction for 40 Ca obtained byMougey et al. This figure is taken from
Ref. [16].

Figure 1.4: Separation energy spectra in the
1s1/2 region for 40 Ca after subtraction of thecontributions of upper shells and backgroundsin (e, e′ p) reaction. This figure is taken from
Ref. [18].

As described above, the measurements providing the information on all of the orbitals from the surface to the inner core in a nucleus with absolute cross sections and sufficient resolution of the separationenergy were unavailable. Such a kind of measurement is necessary to compare spectroscopic factors
for low-lying states with those from the high-resolution (e, e′ p) reaction measurement and to ensure thevalues for deeply bound orbitals.

1.1.4 Direct measurement of NN-correlated pair
There are other methods that don’t employ spectroscopic factors to study the NN correlation. Several experiments were performed by knock-out reaction to directly measure the correlated nucleon pairs. Whena nucleon in a correlated pair is knocked out, the other nucleon in the pair recoils. Taking advantageof this nature, correlated nucleon pairs were measured under the back-to-back kinematical condition as
evidence of short-range correlations by the (e, e′ pp) reaction at the Thomas Jefferson National Accelerator Facility (JLab) [24, 25] and at NIKHEF [26], by the (e, e′ pn) reaction at Maintz [27, 28], and by the(p, 2pn) reaction at Brookhaven National Laboratory (BNL) [29, 30]. At JLab, Subedi et al. measured

1.2. QUASI-FREE KNOCK-OUT (P, 2P) REACTION

7

Figure 1.5: Strength distributions from the ( p, 2p) reaction obtained for various nuclei by James et al.
This figure is taken from Ref. [20].

the 12 C(e, e′ pn), 12 C(e, e′ pp), and 12 C(e, e′ p) reactions simultaneously, and they suggested from theirresult that the 80% of the nucleons in the 12 C behave as described within the shell model, and 18% ofthem are p–n correlated pairs and the rest of 2% are p–p and n–n correlated pairs.
The result of these direct measurement indicates the existence of the state in which nucleons correlate
each other. However, this kind of experiment don’t answer the question how much nucleons occupy thesingle-particle states in nuclei. The study of the spectroscopic factor can provide the information on thisquestion. Therefore, the direct measurement of NN-correlated pair and the spectroscopic study should
be complementary on the study of nuclear structure.

1.2 Quasi-free knock-out (p, 2p) reaction
Quasi-free knock-out reaction is one of the most direct methods to investigate the single-particle properties of a nucleus such as spectroscopic factors or nucleon-momentum distributions and modificationof the single-particle orbitals in nuclear medium. This is a process where an incident particle knocks
a nucleon out of a nucleus and the residual nucleus remains in a one-hole state. If the energy of theincoming particle and the momentum transfer to the nucleon in the nucleus is sufficiently large, theinfluence of the other spectator nucleons in the nucleus on the knock-out process can be neglected andthe scattering of the incident particle and the knocked-out nucleon is like a scattering in a free space.
This is the reason why this process is called quasi-free or quasi-elastic scattering. The (p, 2p) reaction

8

CHAPTER 1. INTRODUCTION

Figure 1.6: Separation-energy spectra for the 40 Ca(p, 2p) (upper three figures ) and 40 Ca(p, pn) (lower
three figures ) reactions measured with a 1 GeV proton beam at Petersburg Nuclear Physics Institute(PNPI) from Ref. [22]. The detection range of proton energy is (a) 830–870, (b) 855–887, and (c)880–915 MeV, respectively.

differs from pp scattering in a free space on the point that the knocked-out nucleon is bound and has the
Fermi momentum in a nucleus before the scattering. The cross section for the (p, 2p) reaction depends
not only on the NN-scattering amplitude but also the Fermi momentum in a nucleus. This nature of thecross section is useful to examine the momentum distribution of a nucleon in a nucleus. As quasi-freeknock-out reactions have three-body final states, measurements can be made under various kinematicalconditions to investigate a momentum distribution of a nucleon in nuclei.
The first experiments of such processes were the ( p, 2p) reaction experiments, which were performed
at Berkeley using 340 MeV incident proton beam for light nucleus and the coincident proton pairs wereobserved [31, 32]. Subsequently, the (p, 2p) reaction experiments were performed with medium-energy
proton beams at Uppsala, CERN, Liverpool, and so on, as reviewed in Refs. [33, 34, 35]. For thedescription of the (p, 2p) reaction, DWIA calculations have been developed and used. In the impulse
approximation, the proton-proton matrix element is replaced by that of free proton-proton scattering
for the kinematics of the experiment. The three wave functions of the incoming and outgoing protons
are distorted by complex optical potentials. This distortion reflects inelastic multiple scattering. The
distortion reduces the intensity of scattered protons and smears out the angular correlations expectedfrom the momentum distributions in the orbitals.
Since the 1980s, the spectroscopic study by the (p, 2p) reaction principally progressed at Indiana,
TRIUMF, Maryland, and iThemba [36, 37, 38, 39, 40, 41, 42, 43, 44]. The experiments were mostly
performed at the energy range of 100–200 MeV. The spectroscopic factors obtained with the DWIA

1.2. QUASI-FREE KNOCK-OUT (P, 2P) REACTION

9calculation were often different between the measurement conditions, that is, the absolute values of
the cross section from the DWIA calculation have some uncertainties. Nevertheless, iThemba groupobtained the spectroscopic factors for the hole states of the valence orbitals in 208 Pb and found that they
are in good agreement with the (e, e′ p) studies [44].
On the other hand, quasi-free (e, e′ p) reaction measurement has also been performed since the 1960s [33,34, 45, 46]. Only the outgoing proton suffers distortion owing to the optical potential, so that the intensity reduction and strong distortion are less severe in the (e, e′ p) scattering than in the (p, 2p) scattering.
Thus the spectroscopic factors provided by the (e, e′ p) reaction have small uncertainties [47]. Since the(e, e′ p) reaction occurs in whole of the radial range of a target nucleus [47], this reaction is effective toinvestigate whole of a bound-state wave function with respect to the radial range. However, as the crosssections are much smaller than those for the ( p, 2p) reaction owing to the electromagnetic interaction
between the incident electron and a nucleon in a nucleus, the measurement was limited to the hole statesof the low-lying orbitals.
In both of the (p, 2p) and (e, e′ p) reactions, the cross sections for the deep-hole states are muchsmaller than those for the low-lying states. Although the intensity reduction for the (p, 2p) reaction
owing to multiple collisions is more severe than that for the (e, e′ p) reaction, the (p, 2p) reaction has
an advantage to gain yields sufficiently for deep-hole states. If the (p, 2p) measurement is performed
with enough resolution of the separation energy to separate low-lying states, the deduced spectroscopic
factors can be compared with those for low-lying states from the (e, e′ p) reaction and be checked theirabsolute values. Furthermore, normalizing the spectroscopic factors to that from the (e, e′ p) reaction, it
is possible to avoid the uncertainty owing to the large intensity reduction and to determine spectroscopic
factors for deep-hole states with small uncertainty. Therefore, the (p, 2p) measurement from deep-hole
to low-lying states with sufficient resolution of the separation energy can provide reliable values ofspectroscopic factors with the aid of the result of the (e, e′ p) reaction.
The (p, 2p) reaction has an another property that the (e, e′ p) reaction doesn’t have. In the measurement of the (p, 2p) reaction with a polarized proton beam, cross sections have asymmetry depending on
the direction of the proton spin. In 1973, Jacob and Maris suggested that analyzing powers (Ay ’s) for thehole states of the j> and j< orbitals are expected to behave differently owing to the spin-orbit couplingin the nucleus [48]. Analyzing powers were measured with polarized proton beam at TRIUMF for 16 O
and 40 Ca at 200 MeV [49, 37, 38] and for 16 O at 500 MeV [39], and the J dependence of the analyzing
power were experimentally confirmed [35]. The analyzing powers for the 1p3/2 - and 1p1/2 -hole states in
the 16 O(p, 2p) reaction measured by changing a kinetic energy of an ejected proton are shown in Fig. 1.7
from Ref. [37]. Since the 1p3/2 - and 1p1/2 orbitals have a different spin-orbit coupling, the analyzing
powers for these states behave differently. This effect for the analyzing power is called Maris effect. As
a result of the extensive investigation of the (p, 2p) reaction, the analyzing power for a hole state is recognized to be due to the spin-orbit part of the optical potential for an orbital with L = 0 and to both thespin-orbit part and the effective polarization (Maris effect) for a L , 0 orbital. Therefore, the analyzing
power data can provide the information on the j> and j< orbitals and may separate contributions of j> and j< -hole states in the cross sections.

CHAPTER 1. INTRODUCTION

Analysing Power

10

40

60

80

100

120

140

Energy E1 (MeV)

Figure 1.7: Analyzing powers for the 1 p3/2 and p1/2 -hole states in the 16 O(p, 2p) reaction measured bychanging a kinetic energy of an ejected proton. They behave differently as predicted by Jacob and Marisand show the validity of Maris effect. This figure is taken from Ref. [37].

1.3 (p, 2p) experiment at 1 GeV
In 1999 and 2000, our group performed ( p, 2p) experiments with 1 GeV proton beam at PNPI to measure
the hump of the 1s-hole state in 40 Ca in a separation energy spectrum and its absolute cross section
simultaneously [50]. The magnetic spectrometers were set at an asymmetric angle condition and themeasured kinetic energies of the ejected two protons were unbalanced, following a previous work by
Volkov et al [22]. However, the 1s-hole state could not be identified in the separation energy spectrumat that time. There was no visible hump at the expected separation-energy region. Volkov et al. used amagnetic spectrometer and a time of flight detector array that consists of scintillation counters to analyze
momenta of two protons, whereas we used two magnetic spectrometers. The time of flight detector arrayhad 10 times smaller acceptance of the vertical angular acceptance than that of a magnetic spectrometer.
Therefore, it was considered that a large detected vertical angle accepted wide recoil momentum rangesof residual nuclei and possibly smeared the separation-energy spectrum.

1.4 Purposes of this work
Reliable extraction of the strength distributions of deep-hole states is of much interest. It has been
desired for long to determine the spectroscopic factors for deeply bound orbitals far below the Fermi
surface and to discuss their quenching. To clarify these matters, in the present work, we performed the
40 Ca(p, 2p) reaction experiment with a polarized proton beam at 392 MeV and with a high separation-

1.4. PURPOSES OF THIS WORK

11energy resolution of 750 keV, and measured the cross sections and the analyzing powers for the allorbitals from the inner core to the Fermi surface.
The high-resolution measurement enables us to observe the low-lying states as the discrete peaks inthe separation-energy spectrum, so these states are independently studied. The low-lying discrete states
are useful for verifying the L-dependence of the cross section and for checking the DWIA calculation.
Furthermore, it is possible to compare the measured spectroscopic factors for low-lying states with thosefrom the (e, e′ p) reaction for checking the absolute value. The uncertainty due to the strong intensity
reduction can be eliminated by comparing the spectroscopic factor for the low-lying discrete states fromthe (p, 2p) reaction with those from the (e, e′ p) reaction.
40 Ca is a suitable nucleus to study deeply bound orbitals since 40 Ca is a nucleus in the medium-mass

region and has a core of closed shells. It would be premature to use much heavier nucleus than 40 Ca. Theenergy of 392 MeV of the injection proton is higher than the energy of 200 MeV, where many (p, 2p)
experiments have been performed. Since both of the ejected protons have larger kinetic energies thanprotons in case of 200 MeV injection, the protons ought to suffer less multiple collisions. Furthermore,when a polarized proton beam is used, the J dependence of the analyzing power for the (p, 2p) reaction
might be useful for separating the contributions of the j> - and j< -hole states.
In the following chapters, the detail of the present work is presented. The experimental setup is
explained in Chap. 2. The data reduction and analysis of measured data are presented in Chap. 3 and 4.
The result is shown in Chap. 5 and discussed in Chap. 6. At last, the summary is given in Chap. 7.

12

CHAPTER 1. INTRODUCTION

Chapter 2

Experiment
The experiment was performed under Program No. E168 and E217 in the ring cyclotron facility at
RCNP, Osaka University, with a 392-MeV polarized proton beam and the dual-arm spectrometer systemin the WS beam line. A schematic layout of the RCNP cyclotron facility is shown in Fig. 2.1.

2.1 Kinematics
In the (p, 2p) reaction measurement, an injected proton interacts with a bound proton in the target
nucleus and both of the bound and injected protons go out from the nucleus. The knocked-out proton
and the scattered proton are measured in the experiment.
In the A(p, 2p)B reaction with a target nucleus A and a residual nucleus B, as illustrated in Fig. 2.2,
the separation energy Esep is given by
Esep = T 0 − T 1 − T 2 − T 3 = E x − Q,

(2.1)

where T i (i = 0, 1, 2, 3) are the kinetic energies of the incident proton (i = 0), the scattered and knockedout protons (i = 1, 2), and the residual nucleus (i = 3). The quantities E x and Q indicate the excitationenergy of the residual nucleus B and the reaction Q-value [ Q = MA − (MB + m p ) ], respectively; MA,
MB , and m p are the masses of the target nucleus, the residual nucleus, and the proton. The separationenergy is the energy required to knock out a proton from a target nucleus, corresponding to the binding
energy of the knocked-out proton.
The momenta and the scattering angles of the two ejected protons were measured in the (p, 2p)
measurement. Although the residual nucleus was not detected, its momentum (p3 ) was calculated fromthe momenta of the incident proton and two ejected protons on the basis of the momentum conservation
law. Though the residual nucleus is a spectator for quasi-free knock-out reactions in the view of the
IPSM, it recoils and has a momentum owing to the momentum conservation. The momentum ( p3 ) is
called a recoil momentum. Since the target nucleus is at rest before the scattering, the proton ejectedfrom the nucleus should have a momentum −p3 , which corresponds to the Fermi momentum, in the13CHAPTER 2. EXPERIMENT

14

LAS
WS
Grand Raiden

BLP-2

BLP-1

Ring
Cyclotron
N-BLP

0

50 m
superconducting
solenoid magnets

N
AVF
Cyclotron

Figure 2.1: Schematic view of the RCNP ring cyclotron facility.

target nucleus before the scattering
Since the nucleon-momentum distribution in a nucleus is strongly related to the orbital angular momentum L, the recoil-momentum distribution of the cross section for the (p, 2p) reaction predominantly
depends on L. Because protons in the single-particle orbitals with L , 0 cannot have zero momentum in a nucleus, the cross sections for the knock-out reaction from the single-particle orbitals with
L , 0 should have a minimum around p3 = 0. On the other hand, the cross sections of the s-holestates (L = 0) have a maximum at p3 = 0. Since the distribution of the cross section has characteristic
behavior at p3 = 0, the measurement around p3 = 0 is essential.

2.2. EXPERIMENTAL CONDITIONS

15

Figure 2.2: Notation for the kinematics of the A(p, 2p)B reaction in the laboratory system. The incident
energy of the proton is T 0 = 392 MeV in the present study. The recoil momentum of the residual nucleus
B is indicated by p3 .

2.2 Experimental conditions

Some types of experiments have been performed until now. One of them is a symmetric experiment
where the two outgoing protons have equal emerged angles and kinetic energies. Although this simple
measurement condition is advantageous for the theoretical calculation, only a part of events measured by
the counters satisfy this condition. When the kinetic energies of two ejected protons are kept fixed andthe direction of an ejected proton is changed to measure the recoil momentum distribution, distorting
potentials for the outgoing protons can be fixed and the uncertainty from the distortion potentials due
to change of kinetic energies of protons can be decreased. In a so-called energy-sharing experiment,
the directions of two ejected protons are kept fixed and the energies of them are varied. A large partof measured events can be used and it is easier to vary the recoil momentum by changing energies ofejected protons than changing the directions of protons.
Since the cross sections of deep-hole states are expected to be small, the experiments were performed
under the energy-sharing condition in this work. The magnetic fields of the spectrometers and the angleof the LAS were varied, while the sum of the kinetic energies of the two measured protons T 1 + T 2 was
kept constant at each separation energy. The directions of the ejected protons were set at asymmetriccondition following the experiments at PNPI where the 1s-hole states were identified in separation energy spectra by Volkov et al. The angle of the GR was fixed at 25.5◦ . In order to separate the hole states,
the recoil-momentum distributions of the cross section and the analyzing power were measured in theregion of 0–200 MeV/c in the separation energy region of 0–89 MeV. The experimental parameters arelisted in Table 2.1. The kinematical sets are grouped according to the range of the measured separationenergies.

CHAPTER 2. EXPERIMENT

16

Table 2.1: Measured kinematical sets (central values). The angle of the GR was fixed at 25.5◦ . θLAS
indicates the angle of the LAS which was set according to the measured separation-energy region.
Esep (MeV)
T 1 (MeV) θLAS (deg) T 2 (MeV)
Set 1
0–17
a
290.49
56.41
95.00
b
270.2056.41
115.29c251.0756.41
134.42d233.22
56.41
152.27
Set 2

6–37

a
b
c
d

270.20251.07233.22
218.25

52.06
52.06
52.06
52.06

95.00
114.13131.98146.95

Set 3

25– 54abcd251.07233.22
218.25203.70

48.00
48.00
48.00
48.00

95.00
112.85127.82142.37

Set 4

40–75

a
b
c
d

233.22
218.25203.70189.00

44.26
44.26
44.26
44.26

95.00
109.97124.52
139.22

Set 5

56–89

a
b
c
d

218.25203.70189.00175.00

41.18
41.18
41.18
41.18

95.00
109.55124.25138.25

2.3 Experimental setup
2.3.1 Beam transportation and target
A polarized proton beam was provided by a high intensity polarized ion source (HIPIS) [51] and was
injected to the K = 120 MeV AVF (Azimuthally Varying Field) cyclotron. At the same time, thepolarization axis was adjusted to the vertical direction by bending the beam with both an electrostatic
deflector and a bending magnet. The direction of the beam polarization was reversed every secondby switching the strong and weak transition units of the HIPIS, alternatively. The proton beam wasaccelerated to the energy of 64.2 MeV by AVF cyclotron and further accelerated to 392 MeV by the K =400 MeV ring cyclotron [52]. The proton beam extracted from the ring cyclotron was achromaticallytransported to the target in the scattering chamber through the WS beamline [53]. The beam spot sizeat the target point was typically 1 mm in diameter. After passing through the target, the beam wastransported into a Faraday cup in the shielding wall. The beam current collected in the Faraday cup was2.3. EXPERIMENTAL SETUP

17

monitored with a current digitizer (model 1000C) from Brookhaven Instruments Corporation.
The beam polarization was continuously monitored using a beam-line polarimeter (BLP) systemusing a polyethylene target. In the polarimeter system, kinematical coincidence was used to selectp-H scattering from (CH2 )n foil. A pair of protons scattered to the opposite directions in the centerof mass system are detected in coincidence by a pair of the scintillation detectors. Two pairs of thescintillation detectors (L–L’ and R–R’ pairs ) were placed in the horizontal plane to measure left/rightscattering asymmetry as shown in Fig. 2.3. The other pairs were arranged in the vertical plane tomeasure up/down scattering asymmetry. The scintillation detectors were placed at the laboratory anglesof 17.0◦ and 69.7◦ where the value of the analyzing power for pp-scattering is nearly maximum at thisinjection energy. Delayed coincidence events between different beam bunches were also measured toestimate the numbers of accidental coincidence protons. Beam polarization of 60–70% was achieved in
the experiment.
The calcium targets were made from a natural calcium block. Pieces of the calcium block were rolled
thin and they were cut into the rectangular shapes with 20 mm × 30 mm. As calcium is an easily
oxidizable metal, the pieces of calcium block were soaked in liquid paraffin when they were rolled andstored. The targets used in the measurements were two sheets of calcium foil with thicknesses of 53
and 24 mg/cm2 . The thickness of 24 mg/cm2 was normalized to the thicker target with the thickness of
53 mg/cm2 by comparing the cross sections in some kinematical regions. The thickness of 24 mg/cm2
was thinner than the value of 27 mg/cm2 estimated from its area and weight during the preparation.
The uncertainty of the target thickness was probably due to matter of its uniformity. Assuming that the
thicker target had the same amount of uncertainty due to the matter of its uniformity, the target witha thickness of 53 mg/cm2 has the uncertainty of 6%. The uncertainty of the thickness of 24 mg/cm2
was estimated as 6% combining the uncertainty of the normalization factor. The targets used in the

z

y

R

L

17.0°

Plastic
Scintillator

69.7°

L'

R'

x

PMT
Polystyrene
TargetBeam

Figure 2.3: Layout of BLP.

CHAPTER 2. EXPERIMENT

18

Table 2.2: Targets used in the measurements.
NumberExperiment

thickness

1
2

E168
E217

53 mg/cm2
27 mg/cm2

thickness(Normalized)—
24 mg/cm2

uncertainty
6%
6%

measurements were summarized in Table 2.2. The oxygen contamination was estimated from elastic
scattering and was less than 1% relative to calcium in weight.

2.3.2 Dual spectrometer system
Scattered protons were analyzed with the dual-spectrometer system, the Grand Raiden spectrometer(GR) [54, 55] and the large-acceptance spectrometer (LAS) [56, 57]. A schematic view of the system isshown in Fig. 2.4.
The GR was designed and constructed for high-resolution measurements with a momentum resolutionp/δp = 37000. The design specifications are listed in Table 2.3. The GR consists of three dipole
magnets (D1, D2, and DSR), two quadrupole magnets (Q1 and Q2), a sextupole magnet (SX), and amultipole magnet (MP) as shown in Fig. 2.4. The second order ion-optical properties like the tilting
angle of the focal plane are adjusted by the SX magnet, and higher-order aberrations are minimized by
the MP magnet and the curvatures of the pole edges at the entrance and exit of the dipole magnets. The

Grand Raiden (GR)To Faraday cup

MP

D2

D1

Large AcceptanceSpectrometer (LAS)
D

Q2
SX
Q1
DSR
Focal Plane
Detectors

TargetPolarized
proton beam

Q
Focal Plane
Detectors

03m

Figure 2.4: Schematic view of the dual-arm spectrometer system at RCNP.

2.3. EXPERIMENTAL SETUP

19

Table 2.3: Design specification of the Grand Raiden spectrometer (GR).
Mean orbit radius
Total deflection angle
Range of the setting angle
Momentum range
Momentum dispersion
Momentum resolution (p/∆p)
Tilting angle of focal plane
Focal plane length
Maximum magnet rigidity
Maximum field strength (D1, D2)
Maximum field gradient (Q1)
Maximum field gradient (Q2)
Horizontal magnification (x|x)
Vertical magnification (y|y)
Horizontal angle acceptance
Vertical angle acceptance
Maximum solid angle
Flight path of the central ray
a

3m
162◦
−4◦ to 90◦
5%15.45 m37,000 a
45◦
120 cm
5.4 T·m
1.8 T
0.13 T/cm
0.033 T/cm
-0.417
5.98±20 mrad±70 mrad4.3 msr
20 m

The source width is assumed to be 1 mm.

third dipole magnet (DSR) required for in-plane polarization transfer measurements was not used in thepresent experiment. The LAS is the second arm spectrometer with a resolution p/δp = 5, 000. It was
designed to have a large solid angle (≈20 msr) and a wide momentum acceptance (30%). The design
specifications are listed in Table 2.4. The LAS consists of a quadrupole (Q) and a dipole magnet (D).

2.3.3 Focal-Plane detectors of GR and LAS
The two scattered protons were detected with the focal plane detectors at each spectrometer. Each focal
plane detector consists of two plastic scintillation counters and two vertical-drift-type multiwire-drift
chambers (MWDCs).
The focal plane detector system of the GR consists of two MWDCs and two planes of plastic scintillators, as illustrated in Fig. 2.5. The type of any plane of the MWDCs is, so-called, a vertical drift
chamber (VDC), in which electrons and ions drift perpendicularly to the anode plane [58]. Specificationof the GR-MWDCs are summarized in Table 2.5. Each MWDC has two anode wire planes (X and U),
sandwiched between three cathode planes. Anode planes include sense wires and potential wires. The
structure of an X-wire plane is schematically illustrated in Fig. 2.6. The wire configurations of the X andU planes of the GR-MWDC are shown in Fig. 2.7. It should be noted that the spacing of sense wires aredifferent between X-plane (6 mm) and U-plane (4 mm) for the GR-MWDCs. The potential wires serve
to make a uniform electric field between the cathode plane and the anode plane. High voltages of -300 V

CHAPTER 2. EXPERIMENT

20

Table 2.4: Design specification of the Large Acceptance Spectrometer (LAS).
Mean orbit radius
Total deflection angle
Range of the setting angle
Momentum range
Momentum resolution (p/∆p)
Tilting angle of focal plane
Focal plane length
Maximum magnet rigidity
Maximum field strength (D)Maximum field gradient (Q)(Sextupole component)(Octupole component)
Horizontal magnification (x|x)
Vertical magnification (y|y)
Horizontal angle acceptance
Vertical angle acceptance
Maximum solid angle
Flight path of the central ray
The value is given for the high momentum end (δ = +15% ).

M M
PS PS W W
2 1 DC DC
2 1a1.75 m70◦
0◦ to 130◦
30%
5,000
∼ 57◦ a170 cm3.22 T·m
1.6 T-74 mT/cm
0.465 T/cm2 m
0.029 T/cm3 m
-0.40
-7.3
±60 mrad±100 mrad∼20 msr
6.2 m

45°

0

1m

Figure 2.5: Focal plane detectors of the GR.

were applied to the potential wires in both planes and -5.6 kV to the cathode planes of the MWDCs.
The sense wires were grounded (0 V). The gas multiplications by avalanche processes are only occurred
near the sense wires. Mixture gas of argon (71.4%), iso-butan (28.6%), and iso-propyl-alcohol wasused. The iso-propyl-alcohol in the vapor pressure at 2 ◦ C was mixed in the argon gas in order to reducethe deterioration due to the aging effect like the polymerizations of gas at the wire surface. Signals from

2.3. EXPERIMENTAL SETUP

21

the anode wires were pre-amplified and discriminated by LeCroy 2735DC cards, which were directly
connected on the printed bases of the MWDCs without cables. Output ECL signals of the 2735DC
cards were transferred to LeCroy 3377 TDCs, in which information on the hit timing of each wire wasdigitized.

Cathod Plane
2mm

10mm

Anode Wires
Potential Wire
10mm

Sense Wire

Cathod Plane
Figure 2.6: Structure of each plane of MWDC.

y
48.2°

6mm

MWDC-X4mm

xzz = beam direction

MWDC-U

Figure 2.7: Wire configurations of the X-plane and U-plane of the GR-MWDC.

The GR drift chambers were backed by two plastic scintillation counters, the size of which is 1200W mm
× 120H mm × 10t mm. The scintillation light was detected by photo-multiplier tubes (HAMAMATSU
H1161) at both sides of the scintillators. Signals from these scintillators were used to generate a trigger signal of the GR event. An aluminum plate with a thickness of 10 mm was placed between two
scintillators in order to prevent the secondary electrons produced in one scintillator from hitting anotherscintillator.
The focal plane detector system of the LAS consists of two MWDCs and two planes of plastic scintillators. The detector layout is shown in Fig. 2.8. In order to cover the vertically broad focal plane of
the LAS, both scintillator planes consist of three (up, middle, and down) scintillation counters, whichare 2000W mm × 150H mm with a thickness of 6mm respectively. Fast photo-multiplier tubes (HAMAMATSU H1949) were used on both sides of each scintillator. Aluminum plates with a thickness of 3 mm
were placed between the scintillator planes for the same purpose as the GR side.

CHAPTER 2. EXPERIMENT

22

The LAS-MWDCs are similar to those for the GR, except for the size and wire configuration. Although each MWDC consists of three anode planes (X, U, and V), the V-plane has not been used owing
to the lack of the readout electronics. The wire configurations of the X and U planes of the LAS-MWDC
are shown in Fig. 2.9. A high voltage of -5.4 kV was supplied to the cathode plane, and -300 V to thepotential wires. The specification of the LAS-MWDCs is summarized in Table 2.6.

MWDC1
MWDC2
PS1
PS2
0m

1m

Scintillator
Hodoscope

(U)
(M)
(D)

(U)
(M)
(D)

Sideview of the trigger scintillators .

Figure 2.8: Focal plane detectors of the LAS.

2.3.4 MWPC for scattering angles
Two multiwire proportional chambers (MWPCs) were newly installed at the entrances of the GR and
LAS to acquire the vertical-scattering-angle information which cannot be precisely determined frommeasurements in the focal planes of the spectrometers owing to their ion-optical properties. The precise
measurement of the scattering angles is important in calculating the recoil momentum of the residual
nucleus. The layout around the scattering chamber and the geometrical relation of anode planes andthe center of the scattering chamber are illustrated in Figs. 2.10 and 2.11. The specification of the
MWPCs are summarized in Table. 2.7. The structure of an anode-wire plane is schematically illustrated
in Fig. 2.12. The MWPCs consist of two horizontal wire planes (X, X’) and two vertical wire planes(Y, Y’) whose wire pitch is 2.02 mm. Typically, one or two wires were hit per plane for one trajectoryof a charged particle. The counter gas was a mixture of argon (66%), iso-butane (33%), freon (0.3%),
and iso-propyl-alcohol (vapor pressure at 2 ◦ C). High voltages of -4.9 kV and -4.7 kV were applied tothe cathode planes of the MWPC for the GR and LAS, respectively. Signals from the anode wires werepre-amplified and discriminated by Nanometric N277-C3 cards or preamp-cards by REPIC. The wirehit pattern was converted into the central position and the number of the hit wires for every cluster by
the LeCroy PCOS III system.

2.3. EXPERIMENTAL SETUP

23

Table 2.5: Specifications of the MWDCs for the GR.
Wire configuration
Active area
Number of sense wires
Cathode-anode gap
Anode wire spacing
Sense wire spacing
Sense wires
Potential wires
CathodeCathode voltage
Potential-Wire Voltage
Gas mixture
Entrance and exit window
Distance between two MWDCs
Pre-amplifier

X (0◦ =vertical), U (48.2◦ )1150W mm × 120H mm
192 (X), 208 (U)
10 mm
2 mm
6 mm (X), 4 mm (U)20 µmφ gold-plated tungsten wire
50 µmφ beryllium-copper wire
10 µm-thick carbon-aramid film
-5.6 kV-300 V
Argon (71%)+Iso-butane (28.6%)+Iso-propyl-alcohol (2◦ C vapor pressure)12.5 µm-thick aramid film
250 mm
LeCroy 2735DC

Table 2.6: Specifications of the MWDCs for the LAS.
Wire configuration
Active area
Number of sense wires
Cathode-anode gap
Anode wire spacing
Sense wire spacing
Sense wires
Potential wires
CathodeCathode voltage
Potential-Wire Voltage
Gas mixture
Entrance and exit window
Distance between two MWDCs
Pre-amplifier

X (0◦ =vertical), U (31◦ ), V (−31◦ )1700W mm × 350H mm
272 (X), 256 (U), 256 (V)
10 mm
2 mm
6 mm
20 µmφ gold-plated tungsten wire
50 µmφ beryllium-copper wire
10 µm-thick carbon-aramid film
-5.4 kV-300 V
Argon (71%)+Iso-butane (28.6%)+Iso-propyl-alcohol (2◦ C vapor pressure)
25 µm-thick aramid film
164 mm
LeCroy 2735DC

CHAPTER 2. EXPERIMENT

24

31°

6mm

m
6m

MWDC-X

MWDC-U
y

-31°

x
6m
m

z

z = beam direction

MWDC-V
Figure 2.9: Wire configurations of the X-plane, U-plane, and V-plane of the LAS-MWDC.

The collimators made of lead were placed in front of the MWPCs for both of the spectrometers.
Table 2.7: Specifications of the MWPCs for the GR and LAS.

Wire configuration
Active area
Number of sense wires
Cathode-anode gap
Anode wire spacing
Sense wires
Guard wires
CathodeCathode voltage
Gas mixture
Entrance and exit window
Pre-amplifier

GR
LAS
X (0◦ =vertical), Y (90◦ =horizontal)
30W mm × 62H mm
94W mm × 158H mm
16 (X), 32 (Y)
48 (X), 80 (Y)
6.4 mm
2.02 mm
20 µmφ gold-plated tungsten wire
50 µmφ Beryllium-Copper wire
6 µm-thick carbon-aramid film
-4.9 kV-4.7 kV
Argon (66%)+Iso-butane (33%)+Freon (0.3%)+Iso-propyl-alcohol (2◦ C vapor pressure)
50 µm-thick aramid film

2.3.5 Trigger and data acquisition system
The data acquisition was initiated by the trigger signals from the GR and LAS scintillators. The readout
electronics and trigger systems of the focal plane scintillators for the GR and LAS are illustrated in2.3. EXPERIMENTAL SETUP

25

Figure 2.10: Schematic view of the layout around the scattering chamber. The proton beam is injected
into the reaction target at the center of the scattering chamber and is transported to the Faraday cup
placed about 25 m downstream of the target. The MWPCs and the lead collimators are installed in frontof the quadrupole magnets.

Fig. 2.13 and Fig. 2.14, and were placed near the focal planes of the GR and LAS, respectively. Any
output of photomultiplier tube (PMT) was first divided into two signals and one was discriminated bya constant fraction discriminator (CFD) and the other was sent to a FERA (Fast Encoding Readout
ADC (analog-to-digital converter);LeCroy 4300B) module. One of the CFD outputs was transmitted
to the TDC (time-to-digital converter) system that consists of TFCs (Time to FERA Converter;LeCroy4303) and FERAs. A coincidence signal of two PMT-outputs on both sides of the same scintillator was
generated by a Mean-Timer circuit, in which the times of two signals were averaged. Thus, the position
dependence of output timing caused by the difference of the propagation time in the long scintillator
was minimized.
The trigger system was constructed with LeCroy 2366 universal logic modules (ULM) with field
programmable gate-array (FPGA) chips. As shown in Figs. 2.13 and 2.14, the trigger system received
signals from the outputs of Mean Timers and generated the GR and LAS trigger under the condition thatsignals from PS1 and PS2 coincide internally. In the LAS, PS1 or PS2 was generated when there wasat least one signal of three Mean Timer outputs corresponding to up, middle, and down scintillators.
The GR trigger gave the gate signals of ADC modules and start signals of TDC modules for the GR
focal plane detectors, while the LAS trigger was used as the ADC gate signals and the TDC start signals
for the LAS focal plane detectors. The coincidence trigger of the GR and LAS was also generated in
the 2366 module at the GR side, where the output timing was determined by the LAS trigger. The
logic diagram for coincidence event of the GR and LAS is shown in Fig. 2.15. The (p, 2p) event was

CHAPTER 2. EXPERIMENT

26

12.8mmCenter of
Scattering Chamber

To the GR
(LAS)

Y2

X2

Y1

X1

DGR(LAS)

D GR / LAS = 542.3 mm / 551.3 mm ( E168 )D GR / LAS = 544.1 mm / 546.0 mm ( E217 )Figure 2.11: Geometrical relation of anode planes of the MWPC and the center of the scattering chamber.

6.4 mm

6.4 mmcharged particle

sense wirecathode foil

2.02mmcathode foil

Figure 2.12: Structure of each wire plane of the MWPCs.

measured by GR+LAS coincidence mode, and a GR single event and a LAS single event were alsomeasured under the sampling condition. The main trigger output started the data acquisition system.
A schematic diagram of the data acquisition (DAQ) system [59, 60] is shown in Fig. 2.16. The
digitized data of the TDCs and ADCs for the detectors were transferred in parallel via the ECL bus tothe high speed memory modules (HSM) in the VME crate. The flow controlling event tagger module(FCET) [61], that was installed in each CAMAC crate for TDCs and ADCs, attaches the event header,
event number, and input register words to the data from the LeCroy FERA and FERET system for thesubsequent event reconstruction, and transfers the data via the ECL bus. Similarly, the rapid data transfer
module (RDTM) [62] manages the data from the LeCroy PCOS III system. The CAMAC actions are
excluded in this data transfer process. In the present measurements, the DAQ system treated the datafrom the GR-MWDCs, the GR-scintillators, the LAS-MWDCs, the LAS-scintillators, and the MWPCs
for the GR and LAS, in parallel. Each transfer line has two HSMs, which work as a double-buffer andreduce dead time in transferring buffered data.
The stored data in the HSMs were moved to a reflective memory module of RM5576 through the2.3. EXPERIMENTAL SETUP

27

VME bus by an MC68040 based CPU board, and the data in the RM5576 was automatically copied
to another RM5576 module in the counting room through the link of optical fiber cables. A SUN
workstation read the data from the RM5576 in the counting room and transferred them to an IBM
RS/60000 workstation via an FDDI network. This data transfer method with reflective memory modules
and optical fiber cables was used in the beamtime of E168. In the beamtime of E217, new data transfer
method, which directly transfers the data to the workstation by a gigabit Ethernet, was installed andused. The data was stored in the large hard disk connected to the workstation. The event reconstruction
and online data analysis were also performed on this computer.
Beam current was adjusted so that the live time of the DAQ system was kept at almost 80–90%.
PS1-L

CFD

PS1-R

CFD

PS2-L

CFD

PS2-R

CFD

PS1

Mean
Timer

GR trigger

delay

PS2

Mean
Timer

35

delay

17

PS1
PS2
t (ns)FERAT F CTiming Chart

FERA

delay

delay

Figure 2.13: Logic diagram for the GR focal plane detectors.

CHAPTER 2. EXPERIMENT

28

PS1U-L

CFD

PS1U-R

CFD

PS1U-L

CFD

PS1U-R

CFD

PS1D-L

CFD

PS1D-R

CFD

PS2U-L

CFD

PS2U-R

CFD

PS2U-L

CFD

PS2U-R

CFD

Mean
Timer
delay

PS1
Mean
Timer

Mean
Timer

LAS trigger

Mean
Timer
delay

PS2
Mean
Timer

50
15

PS2D-L

CFD

PS2D-R

CFD

PS1
PS2

Mean
Timer

t (ns)

FERA

T F C

.....

Timing Chart

FERA

.....

delay

delay

Figure 2.14: Logic diagram for the LAS focal plane detectors.

GR trigger

Sampling
1/n

LAS trigger

Sampling

1/m

GR
sample trigger~500

GR+LAS
coincidence trigger

LAS
GR

LAS
sample triggert (ns)
Timing Chartfor GR+LAS coincidence

Figure 2.15: Logic diagram for coincidence event of the GR and LAS.

Scaler29

Gigabit Ethernet(E217)

Fiber-Optic Link
(E168)
200m

K3922

FCET
FERET
FERA

FERA DRV

2.3. EXPERIMENTAL SETUPCAMAC Crate

GR Scintillators

IBM RS/6000 SP(kasuga)

FCET

SFVME

HSM
HSM
HSM
HSM
HSM
HSM
K2917
RM5576

CPU
K3922

Scaler

FCET
FERET
FERA

FERA DRVLAS Scintillators

HSM
HSM
HSM
HSM

LeCroy 3377 System

RM5576VME
GR-MWDCs

FDDI

CAMAC Crate

Event Buildsun

RecordingFCET

LAS-MWDCs

SUN/SS20

LeCroy 3377 System

RDTModule
GR/LAS-MWPCs
LeCroy PCOS-III System

Figure 2.16: Schematic view of the data acquisition system.

On-line Analysis

30

CHAPTER 2. EXPERIMENTChapter 3

Data Reduction
A program code ’Yosoi analyzer’ has been developed for analyzing experimental data obtained with the
GR and/or LAS spectrometer system at RCNP. The analyzed results were stored in a HBOOK [63] fileand graphically displayed using a program PAW [64]. The data analysis was mainly carried out by using
the central computer system at RCNP, that is, IBM RS/6000SP system.

3.1 Polarization of proton beam
The beam polarization was measured by using the beam-line polarimeter 1 (BLP1) located at the firststraight beam-line section in the west experimental hall. The BLP2 was used for monitoring the transmission and polarization of the beam.
Yield (NL (NR )) in coincidence by the pair of scintillators of L(R) and L’(R’) in Fig. 2.3 for spin-up (↑)
and spin-down (↓) modes are described asp↑

NL↑ = NL − NLa↑ = σ0 (θL )Nt Nb↑ ǫL ∆ΩL (1 + Ay (θL )p↑y ),p↑NR↑ = NR − NRa↑ = σ0 (θR )Nt Nb↑ ǫR ∆ΩR (1 − Ay (θR )p↑y ),NL↓ = NLp↓ − NLa↓ = σ0 (θL )Nt Nb↓ ǫL ∆ΩL (1 − Ay (θL )p↓y ),p↓NR↓ = NR − NRa↓ = σ0 (θR )Nt Nb↓ ǫR ∆ΩR (1 + Ay (θR )p↓y ).

(3.1a)(3.1b)(3.1c)(3.1d)The superscripts of ↑ and ↓ represent the quantities in spin-up (↑) and spin-down (↓) modes. N p andN a are the numbers of prompt and accidental coincidence events, σ0 (θ) and Aeffy (θ) are the unpolarizedcross section and the analyzing power for p + p scattering. Nt and Nb are the numbers of the target
and beam particles. py , ǫ and ∆Ω are the beam-polarization in the vertical direction, the efficiency,
and the solid angle of each scintillation detector, respectively. The accidental coincidence event N a
was estimated using the number of forward counter L(R) event coincident with the event of backward
counter L’(R’) in the next beam bunch.
The angular acceptances of the polarimeter were determined by collimating the backward protons.
31CHAPTER 3. DATA REDUCTION32

If there is no instrumental asymmetry, namely ǫL ∆ΩL = ǫR ∆ΩR = ǫ∆Ω, the beam polarization can be
expressed as follows;
1 2Q − (L + R)
Ay (θ)L−R
2/Q − L−1 + R−11.
p↓ =Ay (θ)
R−1 − L−1

p↑ =

L=

NL↓
NL↑

,

R=

NR↓
NR↑

,

Q=

Nb↓
Nb↑

(3.2a)(3.2b)(3.3)

The scattering angles of forward and backward protons for a 392 MeV proton beam were set at θlab =17◦ and θlab = 69.7◦ , respectively, The effective analyzing power of Ay (17◦ ) = 0.44 ± 0.01 for pp scattering from the polyethylene target was used in Eq. (3.2) to determine the beam polarization. This valueof Ay (17◦ ) for 392 MeV was previously calibrated in comparison with two polarization values measured
before and after the ring cyclotron for a vertically polarized proton beam. The beam polarization before the acceleration by the ring cyclotron, which was measured by BLP-N between the AVF and ring
cyclotrons, was determined from the asymmetry for the 12 C(p, p) elastic scattering with the analyzing
power data of the 12 C(p, p) elastic scattering measured at RCNP [65]. The beam polarization of 60–70%
was achieved in the experiments.

3.2 Particle Identification
Combination of the time of flight (TOF) through the spectrometer and the energy deposition to the plastic
scintillator (∆E) provided the particle identification at the focal plane of the spectrometers. Photons fromthe scintillators were detected by PMTs attached on both the left and right sides. The photon number
is attenuated owing to the absorption by the scintillator material during the transmission. The photon
number I can be described as a function of the path length x,
 x
(3.4)
I(x) = I0 exp − ,lwhere I0 is the initial photon numbers and l is the attenuation length of the scintillator material. Suppose
the distances between the emitting point of the photons and the left/right PMTs are xL and xR , thegeometrical mean I¯ of the photon numbers at both sides is
r
 x 
 x 
 x +x 
 L
L
R
L
R
· I0 exp −= I0 exp −= I0 exp −.
(3.5)
I¯ = I0 exp −
l
l
2l
2l
where L = xL + xR is the length of the scintillator. Eq. (3.5) shows that I¯ is independent on a position
where a particle hits and becomes a good measure of energy deposition to the scintillator. Since theenergy loss of the charged particles in the scintillator material is described by the well-known BetheBloch formula, the I¯ spectra are useful for the particle identification. Figure.3.1 shows the ∆E signal
from PS1 of the GR. The peaks corresponding to protons and deuterons are recognized.

Counts

3.2. PARTICLE IDENTIFICATION

10 6
10 5
10 4
10 3
10 2
10
1

33

ProtonDeuteron0

200 400 600 800 1000 1200 1400 1600
∆E1 (ch.)Figure 3.1: Energy loss spectrum for the PS1 of the GR, which is the mean of the pulse-height signalsof the left and right PMTs.

Figure 3.2 (a) shows the TDC spectrum that indicates the time difference between the mean-time
signal from PS1 of the GR and the RF signal from the ring cyclotron. The RF signal, which was filteredat the half rate with a rate divider module, stopped the TDC and provided a periodic spectrum. Bothof the two prominent peaks at 550 ch and 1050 ch in Fig 3.2 (a) indicate the spectra for proton trigger
events, and the difference between the two peaks in TDC channel corresponds an interval of 60-ns
period between beam bunches. The influence of the momentum acceptance of the spectrometer in the
TDC spectrum was corrected by using the trajectory angle and the position at the focal plane, as seenin Fig. 3.2 (b). The time difference between the mean-time signal of a scintillator and the RF signal
from the ring cyclotron reflects the time that the scattered particle takes to reach the scintillator from thetarget through the spectrometer. Since the particles have different velocities that depend on their masses
under the same momentum condition, the TDC spectra for this time difference provides information on
the masses of the particles.
Protons were identified by setting gates on the TDC spectra because the ∆E peak of proton has a longtail at higher ∆E region. The ∆E spectra for the PS1 scintillator was also used for the particle identification to eliminate the γ-ray contribution. These gates identifying proton are schematically displayedin the two-dimensional scatter plot of the corrected TDC and ∆E of the GR in Fig. 3.3. The proton
identification of the LAS was performed in the identical way.
In the analysis of the set1a and a part of the set2a in Table 2.1, however, proton events were identified
only with the ∆E spectra because the TDC data was not available owing to a trouble. In the part of the
set2a where the TDC information was available, both of the particle identification were compared, andthe yield identified with only the ∆E spectra was 0.5% smaller than that with the TDC and ∆E spectra.
Since the ∆E spectra for the PS1 at the set1a shows a similar shape to that at the set2a, the influence ofthe lack of the TDC data for the particle identification at the set1a is expected to be as small as that at

200

600

Deuteron(a)

800 1000 1200 1400

0

200

600

Proton400

DeuteronProton

TDC (ch.)

10 8
10 7
10 6
10 5
10 4
10 3
2
10
10
1

400

ProtonDeuteron0

Deuteron

CountsCounts

10 8
10 7
10 6
10 5
10 4
10 3
10 2
10
1

ProtonCHAPTER 3. DATA REDUCTION

34

(b)

800 1000 1200 1400
TDCcorr. (ch.)

Figure 3.2: (a): TDC spectrum that indicates the time difference between the mean-time signal fromthe PS1 of the GR and the RF signal from the AVF cyclotron. The RF signal was filtered at the halfrate. (b): TDC spectrum corrected by the trajectory angle and the position at the focal plane so that the
influence of the momentum acceptance of the spectrometer on the TDC is eliminated.

set2a.

3.3 Subtraction of accidental coincidences
The proton beam from the cyclotron has a time structure of approximately 60-ns period between bunches.
To estimate the yield of accidental events, coincidence between the signals of the GR and LAS fromadjacent beam bunches was allowed by increase in the width of the trigger signal of the GR. The yield of
the true (p, 2p) events, which must originate from the same beam bunch, was estimated by subtracting
the yield of the accidental events from the yield of the coincident events in the same beam bunch.
A TDC spectrum for the time difference between the trigger signals of the GR and LAS is shown in
Fig. 3.4. Each peak corresponds to one beam bunch. A prompt peak consists of the true and accidental

3.4. TRACK RECONSTRUCTION OF SCATTERED PARTICLES

35

1500

600

ProtonProton400

Deuteron

Deuteron∆E1 (ch.)

1000

500

0

0

200

800 1000 1200 1400

TDCcorr. (ch.)
Figure 3.3: Two-dimensional scatter plot of the corrected TDC and ∆E1 of the GR. The events enclosedby the solid lines are recognized as the proton events.

coincidence events, while the other peaks consist only of the accidental coincidence events. Assuming
the beam has no micro-structures, that is, the same number of protons are included in all beam bunches,the yield of true coincidence events can be extracted by subtracting the events of one of the accidental
bunch from those of the true bunch.

3.4 Track reconstruction of scattered particles
3.4.1 Multi-wire drift chambers
The trajectories of charged particles entering the focal planes of both spectrometers were determinedwith the GR and LAS-MWDCs. As shown in Fig. 3.5, the position p of an incident charged particle
at an anode plane of the MWDC is determined from the drift lengths di−1 , di of at least more than two
wires in the same cluster. A cluster means that it has at least two adjacent hit wires. Since the X- andγ-rays mostly hit one wire only, background events by photons can be almost excluded. When |di | is theCHAPTER 3. DATA REDUCTION

36

x 10 2

Counts

4000
3000
2000
1000
0

600

800

1000

1200

1400

TDC (ch.)Figure 3.4: TDC spectrum for the time difference between the GR and LAS trigger signals in a measurement of the 40 Ca(p, 2p) reaction.

ChargedParticle
Cathod Planepd i-1
Anode Wires
di

d i+1

Sense Wire

Cathod Plane

Figure 3.5: Position in a plane of MWDC.

minimum drift length in a cluster with three hit wires, the position p is simply calculated as
p = pi + lWS

di−1 + di+1,
di−1 − di+1

(di−1 > 0, di+1 < 0)(3.6)

where pi is the position of i-th wire, lWS is the sense wire spacing, and a negative value is taken for di+1
because electrons moving to (i − 1)-th wire and (i + 1)-th wire drift in the opposite direction. In thestandard setting of both the GR and LAS-MWDCs, particles with correct trajectories usually hit morethan three sense wires. The incident angle (θ) are also roughly estimated by tan θ = (di−1 − di+1 )/2lWS
with the angular resolution of about 2◦ . The drift velocity is almost constant but it considerably deviates
near the sense wires owing to the steep gradient of the electric field. Since the TDC value only gives
the drift time of each wire, one must convert this to the drift length. The so-called x − t calibration wasmade for each wire plane using the real data taken in the present measurements. Typical drift velocityof the uniform region is about 48 µm/ns.

3.4. TRACK RECONSTRUCTION OF SCATTERED PARTICLES

37

X
z’

x’

ΘVDC
z

y, y’
Incident particle
X

e
an
pl
1-

e
an
pl
1-

e
an
pl
2U
e
an
pl
2-

U

X

(x’,y’)

Figure 3.6: Coordinate systems for the ray-tracing with two MWDCs.

The residual distribution defined as

Residual =di−1 + di+1
− di
2

(3.7)

was used for the estimation of the resolution, and for all planes of the GR and LAS- MWDCs, theresolutions were less than 400 µm (FWHM). The position resolution δp depends on the incident angle√
and is mostly better than the residual resolution because the intrinsic resolution δdi of each wire is 6/3of the residual distribution, which is deduced from Eq. (3.7).

3.4.2 Trajectory of a charged particle
Two sets of X and U positions of anode planes can completely determine the three dimensional trajectoryof the charged particle. The wire configurations of the X-planes and U-planes of the GR and LASMWDCs are shown in Figs. 2.7 and 2.9.
We define two coordinate systems: the central-ray coordinate in which the z-axis is the momentum
direction of the central ray and the focal-plane coordinate in which z′ -axis is perpendicular to the anodeplanes of the MWDCs, as shown in Fig. 3.6. The z − x and z′ − x′ planes are the median plane of thespectrometer. In both coordinate systems, the center of the X1-plane is taken as the origin.
′

In the focal-plane coordinate system, the horizontal and vertical positions (x′ ,y′ ) and angles (θ′x = dx
dz′ ,
dy′
′
θy = dz′ ) of an incident particle are calculated from px1 , pu1 , px2 , and pu2 , that are obtained using

CHAPTER 3. DATA REDUCTION

38

Eq. (3.6).
px2 − px1,
LDC
pu2 − pu1,=
LDC
tan θ′x tan θu′
−
=
tan ψ
tan ψ
= px1

tan θ′x =(3.8)

tan θu′(3.9)

tan θy′x′0

u0 = pu1 − z′u1 · tan θu′
u′
x′0
− 0
y′0 =
tan ψ tan ψ(3.10)(3.11)(3.12)(3.13)

where LDC = z′x2 − z′x1 = z′u2 − z′u1 is the distance of two MWDCs, ψ is the tilting angle of U-planes, andx′0 , y′0 are the horizontal and vertical positions at the z′0 plane. LDC was 250 mm for the GR and 164 mm
for the LAS. When the position resolution δp is 300 µm and the multiple scattering effect is neglected,√
the horizontal angular resolution is given as (0.3 2/250) · cos2 θGR ∼ 0.85 mrad for the GR (θGR ∼ 45◦ )√and (0.3 2/164)·cos2 θLAS ∼ 0.89 mrad for the LAS (θLAS ∼ 54◦ ) at FWHM, respectively. The vertical
position and angular resolutions are about two times worse than those in the horizontal direction. The
position resolution of 300 µm corresponds to about 10 keV for the GR and 10–20 keV for the LAS
depending on the magnetic field, and they are smaller than the energy spread of the beam (∼ 200 keV)in the present experiments.
In the central-ray coordinate, the horizontal and vertical angles are converted asθ x = θ′x − ΘV DC
tan θy =tan θ′x cos ΘV DC(3.14)(3.15)

where ΘV DC is the tilting angle of the MWDC for the GR (45◦ ) or LAS (54◦ ). Using the ion-opticalmatrix, one can trace these angles back to the scattering angles on the target. The focal plane of the
GR almost agreed with the X-plane of MWDC1, and small aberrations were empirically corrected bylooking at some two dimensional plots like an x − θ correlation spectrum. In the case of the LAS, themomentum deviation and scattering angle relative to the central ray was obtained from the trajectory at
the focal plane by calculating the 4th-order matrix.

3.4.3 Efficiency of the MWDC
Efficiency of the MWDC was estimated by using (p, 2p) events that a proton is detected with both of the
GR and LAS. Estimation of efficiency of the MWDC needs events that a proton surely goes through the
MWDC. (p, 2p) events were identified by the following condition in case of the estimation of tracking
efficiency of the GR-MWDCs,
I True coincidence between the GR and LAS ( TDC between the GR and LAS )3.5. MULTI-WIRE PROPORTIONAL CHAMBERS

39

II Identification of a proton for the LAS ( ∆E and TOF )III Success in a track reconstruction with the LAS-MWDCs

IV Identification of a proton for the GR (∆E and TOF)
TOF is raw TDC data which is not corrected by information from the GR-MWDCs.
V Central region of the GR-MWDC
Position information obtained by a trigger scintillation counter was used.
We used the events that satisfy all the conditions I–V as a sample. For this sample, the number of theevents that a track reconstruction was succeeded with the GR-MWDCs was counted; the efficiency wasevaluated as the ratio of this number to the number of the sample. The efficiency of the GR-MWDCs
was about 95%. The efficiency of the LAS-MWDC was estimated in the same way by replacing a roleof the GR and LAS, and was 80–85%.

3.5 Multi-wire proportional chambers
Each MWPC at the entrance of the spectrometer was used to determine the scattering angle of thecharged particle that comes to the focal plane. The LeCroy PCOS III system provides hit-wire information by the address of the cluster centroid and the width of the cluster. The cluster centroid with the odd
hit wires is given as the wire position at the center of the hit wires, and that with the even hit wires isgiven as the position between two wires with a half bit address. The address of the cluster centroid isobtained in the accuracy of half pitch of the sense wire, that is, it is determined in the width of 1.01 mm∼ ±0.505 mm. The scattering angle of the particle was determined from the horizontal and vertical
positions at the MWPC, assuming that the particle comes from the center of the target. The events with
only one hit wire in a cluster was predominant in the measurement. In the present analysis, each anodeplane was required to have only one cluster to determine the position of charged particle in the anodeplane. When the angular resolution is estimated with the wire spacing of 2.02 mm, it is approximately
s
!2
!2
2.02
1
+
∼ 0.22◦ ∼ ±0.11◦ ,(3.16)
550
550
where the source width is assumed to be 1 mm in diameter and 550 mm is a approximate distance fromthe center of the scattering chamber to an anode plane.
For an estimation of the efficiency of the MWPC, we used the events that a track reconstruction with
the MWDC succeeds for a proton at the focal plane. For this sample, the number of the events that ascattering angle was obtained was counted; the efficiency was evaluated as the ratio of this number tothe number of the sample. The efficiency of the GR-MWPC was about 90% and that of the LAS-MWPC40

CHAPTER 3. DATA REDUCTION

was about 40–60%. The multi-cluster events decreased the efficiency of the LAS-MWPC because the
LAS-MWPC has a large active area.

3.6 Solid angles
The horizontal scattering angle obtained with the MWDCs at the focal plane was used for particles
coming to the GR, whereas, the horizontal scattering angle obtained with the MWPC at the entrance of
the LAS was used for particles coming to the LAS because the resolution of the horizontal stattering
angle obtained with the MWDCs at the LAS was worse than that with the MWPC owing to the multiple
scattering. The vertical scattering angles obtained with the MWPCs were used for particles coming to
the GR and LAS.
Figure 3.7 shows the two dimensional scatter plots of the horizontal (dx/dz) and vertical (dy/dz)
scattering angles for protons analyzed with the GR and LAS.
The solid angle for the GR was determined by the lead collimator as 2.4 msr. Although the leadcollimator was installed at the entrance of the LAS to determine the solid angle in a rectangle shape, thescattering angle for the LAS shown in Fig. 3.7 (b) doesn’t show a rectangle but a hexagonal shape. The
chipped corners in Fig. 3.7 (b) indicates that a part of protons which pass through the collimator stopped
in the LAS. From the shape of the corners, the vacuum chamber in the quadrupole magnet probably
reduced the acceptance. The solid angle for the LAS eliminated the chipped corners was estimated as
19.1 msr, which is a little smaller than 20.0 msr determined by the lead collimator.
When the LAS was set at 56.41◦ , pp scattering event also measured at large scattering angle part in
the solid angle for the LAS. As the cross section for pp scattering is much larger than ( p, 2p) reaction,
just a tiny part of the protons from pp scattering that lost its energy in the spectrometer possibly overlap
the spectrum for the (p, 2p) reaction. Therefore, only at this LAS angle setting (set1a–c), the part ofthe solid angle that is larger than 58.41◦ for the horizontal scattering angle was eliminated, and its solid
angle 16.4 msr was used.

3.7 Normalization due to the trigger efficiency of the scintillation detectors
For a charged particle, a passage through the two plastic scintillation counters and an aluminum plate,which is placed between the scintillators, provides a trigger signal at each focal plane. The detection
efficiency of the scintillation counter system for the GR was previously measured with a proton beam
at 300 MeV changing the thickness of the aluminum plate, and the measured results show that theefficiency decreases as the thickness of the aluminum plate increases [66]. The inefficiency must be proportional to the total reaction cross section between proton and materials. The proton-27 Al total reaction
cross section is almost constant within a 10% fluctuation for the proton energy of below 400 MeV [67].

3.7. NORMALIZATION DUE TO THE TRIGGER EFFICIENCY OF THE SCINTILLATION
DETECTORS

10

4

(b)

(a)
5
dy/dz (deg)dy/dz (deg)

2

0

0

-5

-2

-4

41

-2

0
dx/dz (deg)2-10-5 -2.5 0

2.5

5dx/dz (deg)Figure 3.7: Two dimensional scatter plot of the horizontal and vertical scattering angles of protons
analyzed with the GR (a) and LAS (b).

Therefore, the proton-energy dependence of the inefficiency was considered to be constant in the energy
region of the present (p, 2p) measurement. The inefficiency of the detection for the GR system, whichused the aluminum plates with a thickness of 10 mm, was estimated as ǭGR = 2.2 ± 0.2% including
the uncertainty of 10% from the energy dependence of the total reaction cross section. The inefficiencyof the scintillator-detection system for the LAS, which used the aluminum plates with a thickness of
3 mm, was estimated as ǭLAS = 1.4 ± 0.1% by interpolation from the measurement results for the GR.
Combined both the inefficiencies, the detection efficiency by the scintillation detectors for the (p, 2p)
measurement was estimated byǫGR×LAS = (1 − ǭGR ) × (1 − ǭLAS ) = 96.4 ± 0.3 %.

(3.17)

This efficiency is due to only the scintillation detectors, which provide a trigger for DAQ system, but
it doesn’t include the wire-counter efficiencies. The correction using this efficiency was applied on theyield for the (p, 2p) measurement.

CHAPTER 3. DATA REDUCTION

42

3.8 Separation energy of 40Ca(p, 2p)39K reaction
In each of the spectrometers, the horizontal position at the focal plane corresponds to the momentum
deviation from the central momentum in the spectrometer owing to the ion-optics property. The momentum of the particle at the focal plane was obtained from the horizontal position and Bρ value for thecentral ray of the spectrometer. The energy loss for protons in the air between the scattering chamber
and the spectrometer and in the MWPC was corrected using Bethe-Bloch formula. The momentumof the residual nucleus was calculated using the momentum conservation law. The separation energy
was obtained via Eq. (2.1). The two dimensional scatter plot of the energies of coincidence two protons measured with the GR and LAS and the separation-energy spectrum for the 40 Ca(p, 2p) reaction at
T 0 = 392 MeV are shown in Fig. 3.8(a) and (b) The thick loci that correspond to the transitions to theground and lower excited states in 39 K can be seen in Fig. 3.8(a). Several discrete states, such as theground state (3/2+ ) and the first excited state (2.52 MeV 1/2+ ), are observed in Fig. 3.8(b).

3.9 Cross sections and analyzing powers
The differential cross section of the (p, 2p) reaction in the laboratory system is written as
Y
d4 σ
=
,
dΩ1 dΩ2 dT 1 dT 2 QAǫη∆Ω1 ∆Ω2 ∆T 1 ∆T 2

(3.18)

where Y is the (p, 2p) coincidence yield, Q is the total number of protons in the incident beam, A is
the number of target particles per unit area, ǫ is the detection efficiency, and η is the live time ratio ofthe data acquisition system. ∆Ω1,2 are the solid angles of the spectrometers of the GR (1) and LAS (2),and ∆T 1 and ∆T 2 are the energy range of the proton detected with the GR and LAS. ∆T 1 = 8 MeV and∆T 2 = 0.1 MeV were used for the separation-energy spectra in Chap. 5.
For a certain separation energy Esep of the (p, 2p) reaction, the differential cross section of
Y
d3 σ
=
,
dΩ1 dΩ2 dT 1 QAǫη∆Ω1 ∆Ω2 ∆T 1

(3.19)

is used to show its recoil momentum distribution. ∆T 1 = 4 MeV was used for the recoil momentum
distributions.
For polarized proton beam, the yields for spin-up (↑) and spin-down (↓) modes are described as

Y↑ =
Y↓ =

d3 σ
(1 + p↑ )Q↑ Aǫ ↑ η↑ ∆Ω1 ∆Ω2 ∆T 1 ,dΩ1 dΩ2 dT 1
d3 σ
(1 − p↓ )Q↓ Aǫ ↓ η↓ ∆Ω1 ∆Ω2 ∆T 1 .
dΩ1 dΩ2 dT 1

(3.20)(3.21)

3.9. CROSS SECTIONS AND ANALYZING POWERS

160

43

4000

(a)

39

K (g.s.)
3/2 +

39

g.s

.) 3

3000

/2 +

CountsT2 (MeV)

140

K(

(b)

120

2000
2.52
+
1/2

100
1000
80
240

250

0

260

0

T1 (MeV)2040

Esep (MeV)

Figure 3.8: (a): Two dimensional scatter plot of the energies of coincidence two protons measured with
the GR and LAS in the 40 Ca(p, 2p) reaction at T 0 = 392 MeV. (b): Separation-energy spectrum for the
40 Ca(p, 2p) reaction.

The superscripts of ↑ and ↓ represent the quantities in spin-up (↑) and spin-down (↓) modes. The
differential cross section and the analyzing power are written asdσdΩ1 dΩ2 dT 1

=

Ay =

1
p↓ N ↑ + p↑ N ↓,↑↓A∆Ω1 ∆Ω2 ∆T 1
p +pN↑ − N↓,p↓ N ↑ + p↑ N ↓(3.22)(3.23)

where
N↑ =

Y↑
,
Q↑ ǫ ↑ η↑

N↓ =

Y↓
.
Q↓ ǫ ↓ η↓(3.24)

CHAPTER 3. DATA REDUCTION

44

3.10

pp-scattering

To examine the absolute value of the cross section measured with the system that consists of the WS
beamline, the GR and LAS spectrometers, and the focal plane detectors, pp scattering was measured at
25.5◦ for the GR and at 60.0◦ for the LAS with 9.2 mg/cm2 polyethylene targets. The uncertainties of theangle settings for the GR and LAS are quite smaller than 0.1◦ . Since the solid angle and the momentum
acceptance of the LAS are larger than those of the GR, the acceptance of pp scattering measurement
was determined by the GR.
The separation energy defined with only the energies of two scattered protons in the final state as′ = T − T − T was used to identify pp scattering event. E ′ must be 0 in case of pp scattering.
Esep
0
1
2
sep

The analyzing power was obtained in the same way as that for the (p, 2p) reaction.

The measured cross section of pp scattering was 15.52 ± 0.28 mb/sr, which differs by 2.3% fromthe SAID [68, 69] calculation value of 15.89 mb/sr. The value of 15.89 mb/sr was evaluated by folding
over the solid-angle acceptance, as was same as the following calculation for the analyzing power. The
difference of 2.3% between the measured cross section and the SAID calculation value is as small as
the uncertainty of the thickness of polyethylene target (2%). The measured analyzing power Ay was
0.336 ± 0.005. The difference between the measured value and the SAID calculation value of Ay = 0.365
is 0.029 and it is small enough for the following discussion.

Chapter 4

Analysis
We utilized the spectrum at the set1 for the analysis of the discrete peaks, and those at the set2–5 forthe analysis of the broad spectrum region. To deduce spectroscopic factors and to compare the recoilmomentum distribution of the cross section and the analyzing power with theoretical calculation, DWIA
calculation was carried out.

4.1 Distorted wave impulse approximation calculation
The cross sections and the analyzing powers were calculated for the 40 Ca(p, 2p)39 K reaction using a
DWIA code THREEDEE [70].
The cross section for the quasi-elastic scattering is calculated assuming that the quasi-elastic scattering is a N–N scattering process in a target nucleus. The program code THREEDEE includes spin-orbit
effects in distorting potentials and makes the factorization approximation in separating the matrix elements of the two-body transition operator outside the distorted wave integral. When we consider aquasi-free reaction A(a, cd)B where A = B + b, the cross section can be written as [70]d3 σdΩc Ωd dEc
=

X
1
2π
ωBC 2 S~v
′ ′′ (2J + 1) (2S a + 1)
ρa ρc ρdJM

2X

σa σ′c σ′′
d
Λσb

′ ′′
(2L + 1)1/2 (LΛS b σb |JM) T σLΛ
′ ′′ hσc σd |t|σa σb i ,
aσ σ
c d

ρa ρ′c ρ′′d(4.1)
where S a = S b = 21 are the spins of the particles a and b, v is the incident velocity, ωB is the phase spacefactor, and C 2 S is the spectroscopic factor. J, M are the total angular momentum quantum numbers ofparticle b. The quantities L and Λ are the orbital angular momentum carried by b and its projection.
The two-body transition operator t is separated outside the distorted wave integral by the factorization
approximation. The spin matrices are given in the different coordinates where the directions of propagation of particles a, c, and d are ẑ, zˆ′ , and zˆ′′ axes. Spin projections are denoted as ρi . The quantity45

CHAPTER 4. ANALYSIS

46

Table 4.1: Bound-state parameters of the Woods-Saxon potential for 40 Ca nucleus.
Nucleus
Orbital r0 [fm]a [fm] Vso [MeV]Elton and Swift [73] 2s
1.30
0.60
17/2
1.30
0.60
8.3
1d
1.30
0.60
12
1p
1.30
0.60
30
1s
1.30
0.60
Kramer et al. [74]

1d3/2
2s1/2
1 f7/2

1.295(47)1.276(59)1.348(67)

0.65
0.65
0.65

7.15 a8.45 baThis Vso value is calculated with Thomas spin-orbit parameter λ = 25 and depth of the potential-well V0 = 51.6 MeV in
Ref. [74].
bThis Vso value is calculated with Thomas spin-orbit parameter λ = 25 and depth of the potential-well V0 = 61.0 MeV in
Ref. [74].

T LΛ is defined by
−1/2
T σLΛ
′ ′′ = (2 + 1)
a σc σdρa ρ′c ρ′′
d

Z

 
 


χσ(−)′ ρ′ ∗ ~r ′ χσ(−)′′ ρ′′ ∗ ~r ′′ φLΛ ~r χσ(+)a ρa γ~r d~r,
c c

(4.2)

d d

where γ = B/A, the χ± are distorted waves which also convert the spin states from σ to ρ, and φLΛ
is the spatial part of the bound state wave function of particle b. The inclusion of spin-orbit distortion
increases the complexity of the description of the distorted waves. For an incident polarized beam, we
can omit the average on the spin state of the particle a in the description of the cross section in Eq. (4.1).
In the present calculations, the NN scattering t matrix with the on-shell amplitude was approximately
employed in the final energy prescription, although the two-body t matrix used for the (p, 2p) reaction
should be evaluated in the nuclear medium including off-shell properties. The t matrix was taken fromthe solution of the phase shift analysis by Arndt et al [71].
The distorted waves were calculated using the Schrödinger equivalent potential from the Dirac global
optical model parameters obtained by Cooper et al. [72]. The EDAD-1 potential parameter set was usedin the calculations. The wave functions for the bound nucleons were calculated by adjusting the depthof the Woods-Saxon potential to reproduce the empirical separation energies. The parameters provided
by Elton and Swift [73] were employed for the Woods-Saxon potential and a Coulomb potential for a
uniformly charged sphere with a radius of 1.25 M 1/3 fm, where M is a mass number of the nucleus. The
parameters are listed in Table 4.1.
Since the radius of the bound-state potential is critical for calculating the cross section, Elton’s parameter choice was confirmed by evaluating the root-mean-square radius (RMSR) of the potential expectedfrom the point nucleon distribution in the nucleus as follows.

4.2. MULTIPOLE DECOMPOSITION ANALYSIS

47

2 i can be written as
The mean-square radius (MSR) of the charge distribution of the nucleus hrch
22hrch
i = hr2p i + hrch
ip +

N 2
hr in
Z ch

(4.3)

2 i , and hr2 i denote the MSRs of the point proton distribution in the nucleus, the charge
where hr2p i, hrchpch ndistribution of the proton, and the charge distribution of the neutron, respectively. Z and N indicate thenumber of protons and neutrons in the nucleus. The RMSRs of the charge distribution of the proton
2 i1/2 = 0.8791 fm [75] and hr2 i1/2 = 3.4764 fm [75], respectively. The
and of the 40 Ca nucleus are hrchpch Ca
2 i is −0.1149 fm2 [75]. Note that the negative value of hr2 i
MSR of the neutron charge distribution hrchnch n
comes from the negative charge distribution in the larger-radius region of the neutron. From Eq. (4.3),
the RMSR of the proton distribution in the 40 Ca nucleus is estimated as

hr2p i1/2 =



2
2
hrch
iCa − hrch
ip −= 3.38 fm.

N 2 1/2
hr in
Z ch

(4.4)

The MSR of the potential in the nucleus can be considered as the sum of the MSR of the nucleon
distribution in the nucleus and that of NN interaction range. Assuming the effective interaction range
2 i = 5.78 fm2 calculated from the mass-number-dependent formula hr2 i = (0.132 ± 0.013)A2/3 +of hrint
int(4.24 ± 0.24) fm2 [76], we evaluated the RMSR of the 40 Ca potential as
1/2

2 1/2
2
i
= 4.15 fm.
hrpot
iCa = hr2p i + hrint(4.5)

The MSR of the proton distribution is used instead of the MSR of the nucleon distribution because
40 Ca has the same number of protons and neutrons. It was confirmed that the RMSR of 4.14 fm for
the Woods-Saxon potential with the Elton parameter is in good agreement with the estimated RMSR of
4.15 fm for the 40 Ca potential expected from the point nucleon distribution.
To compare the measured and calculated cross sections, the cross sections calculated with the DWIA
code were averaged over the angular and momentum acceptances of the spectrometers.

4.2 Multipole decomposition analysis
Since the deep-hole states induced by nucleon knockout reactions from deeply bound orbitals stay far
above the particle-decay threshold and have short lifetimes, these states have large widths and overlap
each other. If we assume that the cross sections for different hole states do not coherently interfere witheach other, the total cross section can be described by the superposition of the cross sections for different
hole states by the (p, 2p) reaction. Since the recoil-momentum distribution of the ( p, 2p) reaction crosssection predominantly depends on L, a superposition of the cross sections for the ( p, 2p) reaction fromdifferent L orbitals was used for a fitting to the experimental recoil-momentum distribution data. The
single-particle-hole strength for each orbital was obtained by the fitting. This technique is known asl-decomposition [77] or multipole decomposition analysis (MDA).

CHAPTER 4. ANALYSIS

48

1s1/2 (× 0.5)
1p 3/2
1p 1/2
1d 5/2

0.06

0.04

3

2

d σ/d Ω 1dΩ 2dE1 [ µ b/Sr MeV]

0.08

0.02

0
-300

-200

-100

0

100

200

300

P3 [ MeV/c ]Figure 4.1: Recoil-momentum distributions of the cross sections for the 1s1/2 -, 1p3/2 -, 1p1/2 -,and 1d5/2 -hole states by DWIA calculations. They were calculated with spectroscopic factorsof 1 at a separation energy 15 MeV under a certain kinematical condition.

Figure 4.1 shows recoil momentum distributions of the cross sections for some single hole states
calculated by the DWIA code. The cross section shows characteristic recoil-momentum distribution
depending on the angular momentum (L) of the hole state. As the angular momentum L increases, themaximum position of the cross section moves to larger recoil-momentum region. The MDA utilizes thisL-dependence of the cross sections.
The momentum distribution of the cross section at a separation energy Esep is written in terms of the(Esep , T 1 ) for the single-particle-hole state obtained from a DWIA calcula(p, 2p) cross section σDWIA
αtion as follows:X
σcalc (Esep , T 1 ) =S α (Esep )σDWIA(Esep , T 1 ),
(4.6)
α
α

where α indicates an occupied single-particle orbital in the nucleus, for example α = 2s1/2 , 1d5/2 , etc.
S α (Esep ) is the hole-state strength for the orbital α. The quantity σ(Esep , T 1 ) denotes the triple-differential
cross section,
d3 σ
,
(4.7)
σ(Esep , T 1 ) =dΩ1 dΩ2 dT 1
where Ω1 , Ω2 , and T 1 represent the solid angles of the two spectrometers and the kinetic energy of oneof the scattered protons, respectively.

4.3. MULTIPOLE DECOMPOSITION ANALYSIS WITH AY DATA

49

The χ2 for fitting to the cross-section data is defined by:
X  σexp (Esep , T 1i ) − σcalc (Esep , T 1i ) 2
2
2

 ,
χ = χcs =
∆σ(E
,
T
)
sep
1i
i

(4.8)

where σexp (Esep , T 1 ) and ∆σ(Esep , T i ) are the measured cross section and its uncertainty including thestatistical and the systematic uncertainties. From the analysis of pp scattering in Sec. 3.10, the systematic uncertainty of 2–3% was estimated, and the 6% uncertainty arose from the thicknesses of thecalcium targets. Therefore, we added these two kinds of uncertainties in quadrature and included thesystematic uncertainty of 7% in the uncertainties for the fittings. The summation index i runs to thenumber of the data and the S α s were searched to minimize the χ2 at each separation energy. The uncertainties in the S α s correspond to a change of 1 in χ2 from the minimum value. The spectroscopic factor
for the orbital α was given by summing the strengths as
XSα =S α (Esep ).
(4.9)
Esep

4.3 Multipole decomposition analysis with Ay data
The cross sections of each single-particle state for spin-up and spin-down states of the proton beamwith the polarization p are written by using the cross section σαDW IA (Esep , T 1 ) and the analyzing powerIA
AyDW
α (E sep , T 1 ) as


X
DW IA
DW IA
(E
,
T
)
=
S
σ
(E
,
T
)
1
+
pA
(E
,
T
)
(4.10)
σcalc
sep 1
α α
sep 1
sep 1 ,
yα
↑
α

σcalc
↓ (E sep , T 1 ) =

X
α





IA
S α σαDW IA (Esep , T 1 ) 1 − pAyDWα (E sep , T 1 )where the ↑ and ↓ show the spin states of the proton beam.

,(4.11)

The analyzing power at Esep is written as
Acalc
y (E sep , T 1 )
=
=

calc
calc
1 σ↑ (Esep , T 1 ) − σ↓ (Esep , T 1 )calc
p σcalc↑ (E sep , T 1 ) + σ↓ (E sep , T 1 )P
DW IA (E , T ) A DW IA (E , T )
sep 1
sep 1
α S α (E sep ) σα
yα
.
P
DW
IA
(Esep , T 1 )α S α (E sep ) σα

In the MDA with the analyzing power data, the χ2 for the Ay data is defined as

2
calc
X  Aexpy (E sep , T i ) − Ay (E sep , T i ) 2χ Ay = ,∆Ay (Esep , T i )i(4.12)(4.13)

where Aexpy (E sep , T 1 ) and ∆Ay (E sep , T i ) are a measured analyzing power and the uncertainty. In the caseof MDA including the cross section and the analyzing power, χ2Ay is also included and the total χ2 isdefined as χ2 = χ2cs +χ2Ay .

50

CHAPTER 4. ANALYSIS

Chapter 5

Result5.1 Separation-energy spectra
Separation-energy spectra at the measured kinematical set1–5 are displayed in Figs. 5.1–5.5. Yields
of some parts of the spectra are reduced owing to the limits of the momentum acceptance of the spectrometers. Vertical dotted lines in Fig. 5.3–5.5 show boundaries where the momentum acceptance isguaranteed. Parts of the yields at the outside of the dotted lines are lost owing to the momentum acceptance. The typical separation-energy resolution is 750 keV at full width at half maximum (FWHM).
The spectra display the cross section as a function of the separation energy and show the distributionof hole states. The transitions to the ground and lower excited states in 39 K are seen as the discrete peaks
in Fig. 5.1 and 5.2. In Fig 5.3, a hump is seen around 30 MeV, where the hole state of the 1p orbitalis expected from previous experiments [18, 19, 22]. Over 40 MeV, the cross section decreases as theseparation energy increases, and no peak or hump cannot be seen.
In these measurements, the vertical acceptances of the spectrometers were able to be decreased inoffline analysis by using the information of the installed MWPCs in front of the spectrometers. Following a previous work by Volkov et al [22], separation energy spectra with small vertical acceptancegates were prepared, however, any hump structure was not able to be recognized as same as the spectra
without the vertical gates.
Two prominent discrete peaks at 8.3 and 10.9 MeV in the separation energy spectra were analyzed at
the set1 independently because they are well isolated, and the data of the broad spectrum region at the
separation energies above 12 MeV was analyzed by the MDA at the set2–5 to separate the contribution
from some hole states.

51

52

CHAPTER 5. RESULT

Figure 5.1: Separation-energy spectra at the measured kinematical set1 in Table 2.1.

5.1. SEPARATION-ENERGY SPECTRA

Figure 5.2: Separation-energy spectra at the measured kinematical set2 in Table 2.1.

53

54

CHAPTER 5. RESULT

Figure 5.3: Separation-energy spectra at the measured kinematical set3 in Table 2.1. Vertical dottedlines show boundaries where the momentum acceptance is guaranteed. Yields at the outside of thevertical dotted lines decrease due to the finite momentum acceptance of the spectrometers.

5.1. SEPARATION-ENERGY SPECTRA

55

Figure 5.4: Separation-energy spectra at the measured kinematical set4 in Table 2.1. Vertical dottedlines show boundaries where the momentum acceptance is guaranteed. Yields at the outside of thevertical dotted lines decrease due to the finite momentum acceptance of the spectrometers.

56

CHAPTER 5. RESULT

Figure 5.5: Separation-energy spectra at the measured kinematical set5 in Table 2.1. Vertical dottedlines show boundaries where the momentum acceptance is guaranteed. Yields at the outside of thevertical dotted lines decrease due to the finite momentum acceptance of the spectrometers.

5.2. RESULTS FOR THE DISCRETE STATES

57

5.2 Results for the discrete states

10.9

The separation-energy spectrum up to 20 MeV at the set1 is shown in Fig. 5.6. Two prominent peaks areobserved at 8.3 and 10.9 MeV. The recoil-momentum distributions of the cross section and the analyzing
power for these peaks are displayed in Figs. 5.7 and 5.8, respectively. The error bars that represent thestatistical uncertainties are smaller than the symbols.

4000

1d 3/2

and

1f7/2
12.0

8.3

Count s

6000

2s1/2

2000
0

0

5
10
15
20
Separation Energy [MeV]

Figure 5.6: Separation-energy spectrum for the 40 Ca(p, 2p) reaction up to Esep =20 MeV at the kinematical set1 (Table 2.1). Two prominent peaks are observed
at 8.3 MeV and 10.9 MeV. The MDA was performed at separation energies above
12 MeV.

In Fig. 5.7, the cross section of the first hole state has a minimum near p3 = 0 MeV/c in the recoilmomentum distribution. This suggests that the peak consists of a hole state of a single-particle orbitalwith L , 0, as mentioned in Sec. 2.2.
In 40 Ca, the orbital with L , 0 near the Fermi surface is 1d orbital. The 1d orbital is split into thej< and j> orbitals by the spin-orbit interaction and the j> orbital is bound more deeply than the j<orbital. The j< orbital, that is, the 1d3/2 orbital ought to contribute a hole state of the first peak. The
solid curves in Fig. 5.7 show the recoil-momentum distributions for the hole state of the 1d3/2 orbital
obtained from the DWIA calculation, which are normalized to the measured cross section. The DWIA
calculation reproduces the measured cross section reasonably well. The calculation roughly reproduces
the dependence of the recoil momentum on the analyzing power, but the value of the analyzing poweris systematically overestimated in the entire recoil-momentum range. The spectroscopic factor deduced
as the normalization factor is listed in Table 5.1 and is larger than the 2J + 1 value of 4.
In Fig. 5.8, the recoil-momentum distribution of the cross section for the peak at 10.9 MeV exhibits
a maximum around 0 MeV/c, which is characteristic for a hole state of a single-particle orbital with
L = 0. Therefore, the hole state of the 2s1/2 orbital is expected to be dominant in the peak at 10.9 MeV.
Comparing with the level structure of 39 K [78], we see that the hole states of the 1 f7/2 and 2p3/2 orbitals

58

CHAPTER 5. RESULT

Figure 5.7: Cross section (a) and analyzing power (b) of the 40 Ca(p, 2p) reaction for the peak at 8.3 MeV
as functions of the recoil momentum p3 . The solid line represents the DWIA calculations fitted by thespectroscopic factor for the 1d3/2 orbital. The discrepancy in the Ay (b) between the data and the DWIA
calculation is discussed in the text (Sec.5.2A.).

will be included in the peak as well. Since the contribution of the 2p3/2 -hole state is negligibly small
compared with those of the 2s1/2 - and 1 f7/2 -hole states [9], the peak at 10.9 MeV was analyzed as
the sum of the 2s1/2 - and 1 f7/2 -hole states. We searched the best set of normalization factors for the
calculated cross sections for the 2s1/2 - and 1 f7/2 -hole states to reproduce the measured cross sections.
The result is shown in Fig. 5.8 and the obtained spectroscopic factors are listed in Table 5.1. The fitted
thick solid line for the cross section agrees well with the experimental data, but the calculated analyzing
power is larger than the measurement, which is similar to the results for the 1d3/2 -hole state.
The spectroscopic factors for the valence orbitals were reported from the previous ( p, 2p) [40, 79, 80](e, e′ p) [47] and (d,3 He) [9, 47, 81] reactions as listed in Table 5.1. The values from the (p, 2p) reaction
contradict because its measurement angles and used distorting potentials are different. This implies there
is uncertainties in spectroscopic factors for the ( p, 2p) reaction owing to the distorting potential and thekinematical condition. The spectroscopic factors reported by Doll et al. from the (d,3 He) reaction
are systematically larger than those from the (e, e′ p) reaction. However, Kramer et al. reanalyzed thesame (d,3 He) data in Ref. [9] and showed that the spectroscopic factors for the 1d3/2 and 2s1/2 orbitals
obtained from the (e, e′ p) and (d,3 He) reactions are consistent [47]. As the obtained spectroscopic
factors in the present work are larger than those consistent values from the (e, e′ p) and (d,3 He) reaction
and that for the 1d3/2 -hole state is over the 2J + 1, the obtained spectroscopic factors are overestimated.
This leads that the DWIA calculation underestimated the absolute cross sections.
The phenomena that the measured analyzing powers were smaller than the calculation was previously
observed and discussed by Noro et al. [82], Miller et al. [83], and Hatanaka et al. [84]. Furthermore,a similar reduction in the polarization P was observed by Miklukho et al. [85] and Andreev et al. [86].
These authors suggested that the reduction might be caused by the influence of the effective mean

5.2. RESULTS FOR THE DISCRETE STATES

59

Figure 5.8: Results of the MDA for the peak at 10.9 MeV. The cross section (a) and the analyzing
power (b) are shown as functions of the recoil momentum p3 . The thick solid line shows the fitted
results. The thin solid line and the dotted line show the contributions of the 2s1/2 and 1 f7/2 states,
respectively. The discrepancy in the Ay (b) between the data and the DWIA calculation is discussed in
the text (Sec.5.2A.).

density on the NN interaction. The systematic discrepancies between the experimental data and the
DWIA calculation in Figs. 5.7 (b) and 5.8 (b) possibly imply the modification of the NN interaction in
the nucleus. Although various extensive theoretical studies have been performed, no theoretical model
has succeeded in reproducing the experimental analyzing power quantitatively.
The measured recoil-momentum distributions of the cross section for these two discrete peaks arewell reproduced by fitting with the distributions calculated with the DWIA. The absolute values of thecross section might have uncertainties because the obtained spectroscopic factors are larger than thosefrom other measurements. However, it was confirmed for the L = 0 and L = 2 orbitals that the DWIA
calculation reproduces characteristic behavior of the recoil-momentum distribution depending on theorbital angular momentum L. The MDA for the deep-hole states uses the calculated recoil-momentum
distributions of the cross section from different L orbitals, and first requires the reliability of them in
reproducing a one-hole state for respective L orbitals. The result of these two discrete peaks assuresvalidity for practicing the MDA for deep-hole states.
On the other hand, the recoil-momentum distribution of the analyzing power for the 1d3/2 -hole state
was not reproduced by the DWIA calculation even though it is a pure one-hole state. Therefore, the
DWIA calculation of the analyzing power is not satisfactorily reliable, and we didn’t accept the MDA
with the analyzing power data and abandoned to distinguish the hole states of the j> and j< orbitals byusing the analyzing power data. The results shown in the following sections are obtained from the MDA
only with the cross-section data.

CHAPTER 5. RESULT

60

Table 5.1: Spectroscopic factors for the discrete peaks in the 40 Ca measurement. Kramer et
al. reanalyzed the previous (d,3 He) data of Doll et al. in Ref. [9] in their study [47]. The
present result is shown with the statistical uncertainty obtained in the fitting or MDA.
Esep [MeV] Orbital Spectroscopic factor(p, 2p)E p = 392 MeV, RCNP, Osaka
Present work
8.3
1d3/2
4.87 ± 0.09
10.9
2s1/2
1.61 ± 0.04
10.9
1 f7/2
1.12 ± 0.06
E p = 45.0–148.2 MeV, Maryland
Samanta et al. [40]

8.33
10.85

1d3/2
2s1/2

3.9–6.0
1.0–1.6

E p = 200 MeV, TRIUMF, Antonuk et al. [38]Kudo and Miyazaki [79]
8.33

1d3/2

1.46–3.37

8.33

1d3/2

2.44–2.80

8.328
10.850

1d3/2
2s1/2

2.58 ± 0.19
1.03 ± 0.07

8.33
10.85
11.15

1d3/2
2s1/2
1 f7/2

3.70
1.65
0.58

8.328
10.850

1d3/2
2s1/2

2.30
1.03

8.33
10.86
11.15

1d3/2
2s1/2
1 f7/2

4.23
1.62
0.46

Mano and Kudo [80](e, e′ p)
NIKHEF
Kramer et al. [47](d,3 He)
Ed = 52 MeV, Karlsruhe
Doll et al. [9]Kramer et al. [47] (reanalysis)

Ed = 34.4 MeV, Oak Ridge
Hiebert et al. [81]5.3 Results for deeply bound states
The MDA was performed for each 2-MeV bin in the separation energy region of 12–84 MeV with thecross section data. The hole states of the 1s1/2 , 2s1/2 , 1p and 1d orbitals were taken into account in
the MDA since they are expected to dominate in the broad spectrum region from the naive shell-model
picture. As hole states of j> and j< orbitals with the same L cannot be distinguished in the present MDA
without the analyzing power, the single-particle-hole states were treated under the following assumptions.

5.3. RESULTS FOR DEEPLY BOUND STATES

61

Since the 1d orbital is split into the 1d3/2 and 1d5/2 orbitals and the 1d5/2 orbital is bound moredeeply than 1d3/2 orbital, the hole state of the 1d5/2 orbital is expected to dominate the d-state strength
at separation energies above 12 MeV. The relative strengths for the 1p1/2 - and 1p3/2 -hole states were
fixed in the ratio of 2:4 as suggested by the naive shell-model picture. Since the recoil-momentum
distributions of the cross sections for the 1s1/2 - and 2s1/2 -hole states are quite similar, the strengths
for the 1s1/2 - and 2s1/2 -hole states cannot be distinguished by the MDA. Therefore, the strengths ofthe 2s1/2 - and 1s1/2 -hole states were assumed to be the s-hole-state strengths in the lower and higher
separation-energy regions, respectively.
In Figs. 5.9–5.18, the recoil-momentum dependencies of the cross section and the analyzing powerat the kinematical set2–5 are displayed. The error bars for the experimental data represent the statisticaluncertainties. The thick solid lines in the figures of the cross section and the analyzing power indicate
the fitted results by the MDA, and the other lines represent the contribution of the component hole states.
Note that the analyzing power data was not used for the MDA. The lines in the figures of the analyzing
power were drawn by using the results of the MDA with the cross section data.
The MDA was carried out for the data at the set2 with both the 2s1/2 - and 1s1/2 -hole states as a s-holestate because the border of the 2s1/2 -and 1s1/2 -hole states were unknown. Figures 5.9 and 5.10 are
the results with the 2s1/2 -hole state as a s-hole state while Figs. 5.11 and 5.12 are the results with the1s1/2 -hole state. The contributions from the 1d5/2 - and 1p-hole states are almost identical between in
Figs. 5.9 and 5.11. The differences between the contributions from the 2s1/2 - and 1s1/2 -hole states are
insignificant. The MDA for the data at the set3–5 was carried out with the 1s1/2 -hole states as a s-holestate.
Figure 5.19 shows the typical examples of the fitted recoil-momentum distributions of the cross section and the analyzing power at separation energies of 17, 33, and 53 MeV. The fitted recoil-momentum
distributions of the cross section reasonably well reproduce the measured cross-section data at eachenergy. It is found that the hole states of the 1d5/2 , 1p, and 1s1/2 orbitals are dominant in the recoilmomentum distributions of the cross section at 17, 33, and 53 MeV, respectively. However, the calculated analyzing power shown in Figs. 5.19(d)–5.19(f) overestimates the experimental data at all of theseenergies.

5.3.1 Strength distributions
After the MDA was performed, the strength distributions obtained for the 1s1/2 -, 2s1/2 -, 1p-, and 1d5/2 hole states are shown in Fig. 5.20. The differences of the strengths of the 2s1/2 - and 1s1/2 -hole states are
due to the differences of the magnitude of the calculated cross section. The strength distributions from
the MDA results with the 2s1/2 - and 1s1/2 -hole states are in good agreement for the 1 p, and 1d5/2 -holestates. Since the strength distribution obtained for the s-hole state has a minimum at 24 MeV, the s-holestate strengths were divided into the 2s1/2 and 1s1/2 strengths at 24 MeV.
The strength distributions obtained for the 1s1/2 - and 1p-hole states in Fig. 5.20 have hump structures

CHAPTER 5. RESULT

62

around 50 and 30 MeV, respectively. The hump of the 1p-hole state around 30 MeV is consistent withthe hump structure in separation energy spectra in Fig. 5.3.

5.3.2 Background consideration
The continuum physical backgrounds are also observed in the strength distributions. Although the continuum strengths are appreciable at separation energies larger than 80 MeV in the strength distributions
for the 1s1/2 and 1p-hole states, it is improbable that a single-particle-hole state has such a high separation energy. These strengths are considered to be the contributions of many-body processes such as(p, 3p) or (p, 2pn) reactions or multistep processes that follow the knock-out reaction.
In the (p, 3p) and (p, 2pn) reactions, the final states are four-body systems with the configurationsof 3p+38 Ar and 2p + n+38 K, respectively. The phase-space volumes of the four-body final states wereconsidered under the condition that two protons in the final state were detected by the GR and LAS
but the other particles in the final states were not detected. The combined phase-space volume was obtained by averaging the phase-space volumes of the (p, 3p) and (p, 2pn) final states with equal weights.
Since the 40 Ca(p, 3p)38 Ar and 40 Ca(p, 2pn)38 K reaction channels open at 14.7 and 21.4 MeV in theseparation-energy spectrum for the 40 Ca(p, 2p)39 K reaction, respectively, the combined phase-spacevolume increases from 14.7 MeV.
For the multistep processes, Cowley et al. studied the contribution of the rescattering processes that
follow the knock-out reaction to the coincidence measurement of the 40 Ca(p, 2p) reaction [87]. The
theoretical cross sections for the rescattering processes as an incoherent sum of the (p, p′ p′′ ) and (p, 2p)
reactions well reproduced their experimental coincidence spectra. The contributions of the rescattering
processes have shapes similar to the four-body phase space estimated here. Both of them gradually increase from the lower separation energy region. Thus, we have used the estimated four-body phase spaceas the background-shape model including the four-body and the rescattering processes; the adequacy ofthe model will be discussed later.
An asymmetric Lorentzian shape was suggested by Sartor and Mahaux to describe the spectral function of a hole state near the particle threshold [88]. It can be written asf (Esep ) = N

1 + (Esep − Ec )A(Esep − Ec )2 + 41 Γ2(5.1)

where Ec and Γ are the centroid energy and width of the peak, and N and A are a normalization parameterand an asymmetric parameter. When the parameter A becomes small, the function becomes symmetric.
We fitted a combination of the asymmetric Lorentzian function and the estimated phase-space volume tothe strength distributions for the 1s1/2 , 1p, and 1d5/2 -hole states. The dotted lines in Fig. 5.21 show thephase-space volumes that seem to give reasonable background shapes. The background for the 2s1/2 hole state was determined to be of the same magnitude in the cross section as that for the 1s1/2 -holestate.
The reduced χ2 of χ2ν = 1.0 and 0.9 were obtained in the fit for the 1s1/2 and 1p-hole states. The5.3. RESULTS FOR DEEPLY BOUND STATES

63

Table 5.2: Centroid energies and widths (FWHM) of the strength distributions for the hole
states of the 1p and 1s1/2 orbitals in 40 Ca. For the present result, the uncertainties wereobtained as the region corresponding to a change of 1 in χ2 from the minimum value in thefitting of the asymmetric Lorentzian and the background.
Centroid
Width( MeV )( MeV )
Present work(p, 2p)1p29.6 ± 0.5
11.4 ± 1.2
1s1/2
48.4 ± 0.6
23.4 ± 1.1
Mougey et al.
1p
41(e, e′ p) [16]1s1/256Amaldi Jr. et al. a1p
32 ± 418 ± 5′(e, e p) [19]1s1/2
77 ± 1446 ± 24
Nakamura et al. b
1p
35 ± 1
21 ± 3′(e, e p) [18]1s1/2 (A)58.7 ± 1.2
36 ± 11s1/2 (B)
58.4 ± 1.1
32 ± 1cVolkov et al.
1p1/2
29.8 ± 0.1
8.5 ± 1.1
(p, 2p) [22]1p3/234.7 ± 0.3
9.4 ± 1.2
1s1/2
53.6 ± 0.7
18.8 ± 14a

The values were obtained by fitting Maxwellian curves to the separation-energy spectra. We evaluated thelisted widths from the values given in Ref. [19] (where the authors show the distances between the zero and the
maximum of the Maxwellian curve) by regarding them as a 3σ in Gaussian function.
b
Two results (A) and (B) were presented for the 1s1/2 orbital.
c
Volkov et al. disentangled the 1p1/2 and 1p3/2 -hole states by fitting the separation-energy spectra with some
Gaussian functions.

reduced χ2 is defined as χ2ν = χ2 /ν, where ν is the number of degrees of freedom in the fit. In the χ2 fit
for the 1d5/2 -hole state, χ2ν value of 2.3 was obtained, that is, it was larger than 1. If the model in the fit
were perfect, χ2ν would be 1. The larger χ2ν is due to an inadequate model used in the fit. To compensate
this inadequacy of the model, the uncertainties in the parameters were estimated in the 1d5/2 -hole state
case, following the method of Terashima et al. in Ref. [89]. The uncertainties in the strength data wereartificially increased by multiplying all the strength uncertainties by a constant factor so that χ2ν became
1, and so increased uncertainties in the parameters were obtained. The increased uncertainties involvethe model uncertainties that originate in the inadequacy of the model, and they were defined as the total
uncertainties. The uncertainties estimated without increase in the uncertainties in the strength data were
defined as the statistical uncertainties.
The centroid energies (Ec ) and widths (Γ) of the peaks for the 1s1/2 - and 1p-hole states were obtained
as the parameters of the asymmetric Lorentzian function in the fit; they are listed in Table 5.2 with theresults from the previous experiments. The total uncertainties in the centroid energies and widths were
estimated as mentioned before. The centroid energies of the 1p and 1s1/2 -hole states are deduced to be
29.6 ± 0.5 and 48.4 ± 0.6 MeV, respectively.
The spectroscopic factors obtained by subtraction of the fitted four-body background are listed in

CHAPTER 5. RESULT

64

Table 5.3: Spectroscopic factors relative to the IPSM limits for the orbitals in 40 Ca. They
are obtained after the background subtraction in the present study. The second uncertainty,
in parentheses, is the statistical uncertainty estimated by the propagation of the strength uncertainties at each 2-MeV bin and the uncertainty of the contribution of the background. The
first one is the total uncertainty that includes also the model uncertainty, which is mentioned
in the text. The spectroscopic factors deduced by the DWIA calculation with the nonlocality
correction are also shown. The detail discussion is given in Chap. 6.
Spectroscopic factor
IPSM limit
1d3/2
1 f7/2
2s1/2
1d5/2
1p
1s1/2

4
8
2
6
6
2

with nonlocality correction

1.22 ± 0.02(±0.02)
0.14 ± 0.01(±0.01)
1.00 ± 0.03(±0.03)
1.60 ± 0.12(±0.11)
0.93 ± 0.12(±0.12)
1.69 ± 0.10(±0.10)

0.92 ± 0.02(±0.02)
0.10 ± 0.01(±0.01)
0.75 ± 0.02(±0.02)
1.19 ± 0.09(±0.08)
0.71 ± 0.08(±0.08)
1.36 ± 0.12(±0.10)

Table 5.4: Spectroscopic factors relative to the IPSM limits for the orbitals in 40 Ca withoutbackground subtraction. The uncertainty is the statistical uncertainty estimated by the propagation of the strength uncertainties at each 2-MeV bin.
Spectroscopic factor
IPSM limit
2s1/2
1d5/2
1p
1s1/2

2
6
6
2

with nonlocality correction

1.02 ± 0.03
2.23 ± 0.11
1.87 ± 0.07
2.53 ± 0.05

0.77 ± 0.02
1.73 ± 0.08
1.32 ± 0.05
1.89 ± 0.04

Table 5.3, together with the total and statistical uncertainties. The spectroscopic factors without background subtraction are listed in Table 5.4 with only statistical uncertainties.
The spectroscopic factors of the 2s1/2 and 1p orbitals are consistent with 100% within the uncertainty,
whereas the spectroscopic factors of 1d3/2 , 1d5/2 and 1s1/2 -hole states are over 100% of the 2J+1 values.
The spectroscopic factors of over 100% implies the uncertainty of the DWIA calculation, as mentioned
in Sec. 5.2. The absolute value of the spectroscopic factor is discussed in Chap. 6.

5.3. RESULTS FOR DEEPLY BOUND STATES

65

Figure 5.9: Cross section for each 2-MeV bin in the separation energy at the set2 as a function of therecoil momentum p3 . The thick solid line shows the MDA results including 2s1/2 -hole state as a s-holestate. The dotted lines, the short-dashed lines, and the dash-dotted lines show the contributions of 2s1/2 -,1p-, and 1d5/2 -hole states, respectively.

66

CHAPTER 5. RESULT

Figure 5.10: Analyzing power for each 2-MeV bin in the separation energy at the set2 as a function of therecoil momentum p3 . The thick solid line shows the MDA results including 2s1/2 -hole state as a s-holestate. The analyzing power data was not included in the MDA. The dotted lines, the short-dashed lines,
and the dash-dotted lines show the contributions of 2s1/2 -, 1p-, and 1d5/2 -hole states, respectively.

5.3. RESULTS FOR DEEPLY BOUND STATES

67

Figure 5.11: Cross section for each 2-MeV bin in the separation energy at the set2 as a function of therecoil momentum p3 . The thick solid line shows the MDA results including 1s1/2 -hole state as a s-holestate. The dotted lines, the short-dashed lines, and the dash-dotted lines show the contributions of 1s1/2 -,1p-, and 1d5/2 -hole states, respectively.

68

CHAPTER 5. RESULT

Figure 5.12: Analyzing power for each 2-MeV bin in the separation energy at the set2 as a function of therecoil momentum p3 . The thick solid line shows the MDA results including 1s1/2 -hole state as a s-holestate. The analyzing power data was not included in the MDA. The dotted lines, the short-dashed lines,
and the dash-dotted lines show the contributions of 1s1/2 -, 1p-, and 1d5/2 -hole states, respectively.

5.3. RESULTS FOR DEEPLY BOUND STATES

69

Figure 5.13: Cross section for each 2-MeV bin in the separation energy at the set3 as a function of
the recoil momentum p3 . The thick solid line shows the MDA results. The dotted lines, the shortdashed lines, and the dash-dotted lines show the contributions of 1s1/2 -, 1p-, and 1d5/2 -hole states,
respectively.

70

CHAPTER 5. RESULT

Figure 5.14: Analyzing power for each 2-MeV bin in the separation energy at the set3 as a function of
the recoil momentum p3 . The analyzing power data was not included in the MDA. The dotted lines,
the short-dashed lines, and the dash-dotted lines show the contributions of 1s1/2 -, 1p-, and 1d5/2 -holestates, respectively.

5.3. RESULTS FOR DEEPLY BOUND STATES

71

Figure 5.15: Cross section for each 2-MeV bin in the separation energy at the set4 as a function of
the recoil momentum p3 . The thick solid line shows the MDA results. The dotted lines, the shortdashed lines, and the dash-dotted lines show the contributions of 1s1/2 -, 1p-, and 1d5/2 -hole states,
respectively.

72

CHAPTER 5. RESULT

Figure 5.16: Analyzing power for each 2-MeV bin in the separation energy at the set4 as a function of
the recoil momentum p3 . The analyzing power data was not included in the MDA. The dotted lines,
the short-dashed lines, and the dash-dotted lines show the contributions of 1s1/2 -, 1p-, and 1d5/2 -holestates, respectively.

5.3. RESULTS FOR DEEPLY BOUND STATES

73

Figure 5.17: Cross section for each 2-MeV bin in the separation energy at the set5 as a function of
the recoil momentum p3 . The thick solid line shows the MDA results. The dotted lines, the shortdashed lines, and the dash-dotted lines show the contributions of 1s1/2 -, 1p-, and 1d5/2 -hole states,
respectively.

74

CHAPTER 5. RESULT

Figure 5.18: Analyzing power for each 2-MeV bin in the separation energy at the set5 as a function of
the recoil momentum p3 . The analyzing power data was not included in the MDA. The dotted lines,
the short-dashed lines, and the dash-dotted lines show the contributions of 1s1/2 -, 1p-, and 1d5/2 -holestates, respectively.

5.3. RESULTS FOR DEEPLY BOUND STATES

Figure 5.19: Cross section and analyzing power as a function of the recoil momentum p3 forthe selected separation-energy bins at 17 [(a) and (d)], 33 [(b) and (e)] and 53 MeV [(c) and(f)]. The thick solid lines show the MDA results. The thin solid lines, the dotted lines, theshort-dashed lines, and the dash-dotted lines show 1p-, and 1d5/2 -hole states, respectively.

75

76

CHAPTER 5. RESULT

Figure 5.20: Strength distributions obtained in the separation-energy region of 12–84 MeV. The solidlines are shown to guide the eyes. The vertical dotted line at 24 MeV in (a) shows the border where
the s-hole state contribution was divided into the contribution of the 2s1/2 -hole state and that of the1s1/2 -hole state.

5.3. RESULTS FOR DEEPLY BOUND STATES

77

Figure 5.21: Strength distributions for the hole states of the (a) 1s1/2 , (b) 1p, (c) 2s1/2 , and (d) 1d5/2
orbitals in the separation energy region of 12–84 MeV. The solid lines and the dash-dotted lines in
the figures for the 1s1/2 -, 1p- and 1d5/2 -hole states show the fitted curves and the fitted asymmetric
Lorentzian functions, respectively. The short-dashed lines show the estimated contribution of the fourbody background.

78

CHAPTER 5. RESULTChapter 6

Discussion
6.1 Strength distribution of the deep-hole states

The strengths for the orbital states for each 2-MeV bin were independently obtained by the MDA ateach 2-MeV bin, that is, the strength distributions in Fig. 5.20 were obtained without any assumption
on the shape of the distribution. This is different from the previous analyses by Volkov et al. [22] andby Nakamura et al. [18]. The obtained strength distribution shows clearer shapes than those from theprevious experiments, which are shown in Figs. 1.3, 1.4, 1.5, and 1.6 in Chap. 1. It was useful to discuss
the strength distributions and the contribution of the background.
In Fig. 6.1, the centroids and widths of the strength distributions for the 1p- and 1s1/2 -hole states
are compared with those obtained from the previous experiments listed in Table 5.2. Vertical lines inFig. 6.1 indicate uncertainties of the centroids as the standard deviation. The centroid energies for the
1p- and 1s1/2 -hole states were determined with small uncertainties in the present analysis. The obtained
centroid energies and widths of the 1p and 1s1/2 -hole states are close to the results from the ( p, 2p)
experiment at PNPI [22]. The centroid values from the present work are the smallest among the listed
values.
Since the centroid of the obtained distribution for the 1 p-hole state corresponds to the peak position
at 30 MeV in the separation energy spectra in Fig. 5.3, the peak at 30 MeV in the separation energy
spectra should be the peak of 1p-hole state. This correspondence assures that the MDA reliably worked.
Furthermore, it shows that a one-hole state can be observed even for such a deeply bound orbital.
The success in obtaining the strength distributions relies on the good reproduction of the recoilmomentum distribution of the cross section for a one-hole state, which is confirmed by analyzing two
low-lying discrete peaks.
79

CHAPTER 6. DISCUSSION

80

Figure 6.1: Centroids (symbols) and FWHM widths (solid lines) of the strength
distributions for the 1p- and 1s1/2 -hole states with those obtained from the previous
experiments, from Table 5.2. Vertical lines indicate uncertainties of the centroids
as standard deviation. Mougey et al. did not give the uncertainties.

6.2 DWIA calculation
For the first two peaks at 8.3 and 10.9 MeV in the separation energy spectra, the cross sections and theanalyzing powers are reproduced by the DWIA calculation qualitatively well as shown in Figs. 5.7 and
5.8, and the peaks seem to involve scarcely any background. Nevertheless the spectroscopic factor forthe 1d3/2 -hole state exceeds 100% of the 2J + 1 value, and that of the 2s1/2 -hole state is much larger thanthe experimental values from the (e, e′ p) and (d,3 He) reactions as shown in Table 5.1. Here the validityof the DWIA calculation is discussed for the hole states of the valence orbitals in the following paragraphs. For the 1d3/2 , 2s1/2 and 1 f7/2 -hole states, the DWIA calculation was performed with different
parameters to examine uncertainties in the obtained spectroscopic factors.
First, the results of the DWIA calculation with various distorting potentials are displayed in Figs. 6.2
and 6.3. The several parameter sets of Cooper et al. [72] and Hama et al. [90] were used as the distorting
potential. The used potentials are listed in Table 6.1; the obtained spectroscopic factors for the respective
potentials are shown in Fig 6.4. The calculations by the plane-wave-impulse approximation (PWIA) arealso displayed by thick solid lines in Figs. 6.2 and 6.3 to investigate the contribution of the distortion.
The contribution from the 2s1/2 -hole state is displayed but that from the 1 f7/2 -hole state is not in Fig. 6.3
for simplicity.
In Figs. 6.2 and 6.3, all the results of the DWIA calculation with various potentials qualitatively
reproduced the experimental data as well as the result with EDAD1 used in the previous chapter, andthe recoil-momentum distributions of the cross section and the analyzing power for the used distorting
potentials are almost indistinguishable. Comparing the DWIA calculation with the PWIA calculation in6.2. DWIA CALCULATION

81

Table 6.1: Global optical potentials used for the distortion in the DWIA calculation.
Potential
EDAD1
EDAD2
EDAD3
EDAICA
DP1
DP2

Data for fit
12 C, 16 O, 40 Ca, 90 Zr, 208 Pb(p, p) 20-1040 MeV
12 C, 16 O, 40 Ca, 90 Zr, 208 Pb(p, p) 20-1040 MeV
12 C, 16 O, 40 Ca, 90 Zr, 208 Pb(p, p) 20-1040 MeV
40 Ca(p, p) 20-1040 MeV
40 Ca, 48 Ca, 56 Fe, 60 Ni, 90 Zr, 208 Pb(p, p) 65-1040 MeV

200

1

150

0.5

Reference
[72]
[72]
[72]
[72]
[90]
[90]

3

100
50

Ay

2

d σ/d Ω 1dΩ 2dE1 [µ b/sr MeV]

40 Ca, 48 Ca, 56 Fe, 60 Ni, 90 Zr, 208 Pb(p, p) 65-1040 MeV

Num. of parameters
106
154
176
70
80
84

0

-0.5
PWIA

EDAD1
EDAD2
EDAD3
EDAICA
DP1
DP2

0
-1
-200 -1000100-200 -1000100Recoil momentum p 3 [MeV/c] Recoil momentum p 3 [MeV/c]

Figure 6.2: DWIA calculations of the 40 Ca(p, 2p) reaction for the 1d3/2 -hole state
at 8.3 MeV with different optical potentials. The parameter sets of Cooper et al.
(EDAD2, EDAD3, EDAICA) [72] and of Hama et al. (DP1, DP2) [90] were used
for the distorting potential. The strengths of the 1d3/2 -hole state were obtained byfitting to the experimental data. The thick solid line indicates the PWIA calculation.

Figs. 6.2 and 6.3, it is found that the distorting potential plays an important role to reproduce the recoilmomentum distribution of the cross section and the analyzing power. The qualitative reproduction of
the cross section and the analyzing power shows that the DWIA works well to reproduce the data forthe 392-MeV proton. The obtained spectroscopic factors in Fig 6.4 shows that the cross sections by the
DWIA calculation are reduced to about 20–30% of the PWIA values and the relative strength between
these three orbitals are nearly invariable. However, all of the spectroscopic factors are much larger than
the results from the (e, e′ p) and (d,3 He) reactions.
Second, the sensitivity of the spectroscopic factors to various bound state parameters in the DWIA
calculation is investigated. The recoil-momentum distribution of the cross section and the analyzing
power with the different radii of the Woods-Saxon potential for the 1d3/2 - and 2s1/2 -hole states are
displayed in Figs. 6.5 and 6.6. A change of the diffuseness parameter slightly modifies the distributions

CHAPTER 6. DISCUSSION

3

600
500
400
300
200
100

1
0.5

Ay

2

d σ/d Ω 1dΩ 2dE1 [µ b/sr MeV]

82

0

-0.5
PWIA

EDAD1
EDAD2
EDAD3
EDAICA
DP1
DP2

0
-1
-200 -1000100-200 -1000100Recoil momentum p 3 [MeV/c] Recoil momentum p 3 [MeV/c]

Figure 6.3: DWIA calculations of the 40 Ca(p, 2p) reaction for the 2s1/2 -hole state
at 10.9 MeV with different optical potentials. The used optical potentials were sameas those for the 1s3/2 -hole state in Fig. 6.2. The strength of the 2s1/2 -hole state wasdetermined by the MDA for the peak. The only contribution of 2s1/2 -hole state aredisplayed for simplicity. The thick solid line indicates the PWIA calculation.

of the cross section and the analyzing power. The relation between the obtained spectroscopic factors
and the radius parameter is shown in Fig. 6.7. A change of 0.01 fm in the radius will change thespectroscopic factors by about 3, 2, and 5% for the 1d3/2 -, 2s1/2 -, and 1 f7/2 -hole states respectively,
relative to the values with 1.30 fm by Elton and Swift. Next, the diffuseness parameter was changed
within a range. The relation between the obtained spectroscopic factors and the diffuseness is shown in
Fig. 6.8. A change of the diffuseness parameter doesn’t modify distributions of the cross section andthe analyzing power so much as the radius does, and the change of 0.01 fm in the diffuseness parameter
will change the spectroscopic factors by about 1% for all of the 1d3/2 -, 2s1/2 -, and 1 f7/2 -hole states
relative to the values with 0.60 fm by Elton and Swift. The strength of the spin–orbit term in the opticalpotential also influences on the spectroscopic factors as shown in Fig. 6.9. The spectroscopic factor
gradually changes about 2% by 1 MeV for the 1d3/2 -hole state.
The nonlocality corrections also give sizable influence on the calculation. Although the present analysis shown in Chap. 5 doesn’t includes the nonlocality corrections, some recent analyses of the knockout
reactions take them into account, and Kramer et al. also uses it for their analyses of the (e, e′ p) and(d,3 He) reactions. In order to investigate the influence of the nonlocality corrections on the spectroscopic factors, the DWIA calculation for the present work was performed with the nonlocality corrections with a nonlocality range of β = 0.85 fm by Perey and Buck [91] in the bound state wave function
and in the reaction. Using the calculated cross sections for the MDA, the obtained spectroscopic factors
were shown in Fig. 6.10 with the result obtained in the previous chapter. It is found in Fig. 6.10 that thenonlocality corrections almost uniformly decrease the spectroscopic factors for all of the orbitals thougha couple of the spectroscopic factors are still larger than the shell-model limits of the 2J + 1 values.

6.2. DWIA CALCULATION

83

Figure 6.4: Spectroscopic factors obtained with the different distorting potentials for the 1d3/2 -, 2s1/2 -, and 1 f7/2 -holestates.

As seen in the previous paragraphs, the bound state parameters give some influence on the spectroscopic factors. The difference of the spectroscopic factors between from the present analysis and fromthe (e, e′ p) reaction by Kramer et al. in the reference [47] might be attributed to the difference in thebound state parameters used in the DWIA calculation. The bound-state parameters used by Kramer et
al. are listed in Table 4.1. Kramer et al. performed their analysis with the same bound state wave
function for the (e, e′ p) and (d,3 He) reactions and obtained consistent spectroscopic factors [47]. The
parameters in Table 4.1 were determined by the analysis for the (e, e′ p) reaction on the grounds that the(e, e′ p) reaction is sensitive to the whole of the bound-state wave function. It should be noted that theradius and diffuseness parameters by Kramer et al. are optimized for each orbital although those values
by Elton and Swift in Table 4.1, which are used in the present analysis, are same for all the orbitals in
40 Ca. The analysis by Kramer et al. includes the nonlocality corrections with a nonlocality range of
β = 0.85 fm by Perey and Buck [91] in the bound state wave function and in the reaction. By using
the same bound-state parameters with Kramer et al., the cross sections for the 40 Ca(p, 2p) reaction arecalculated and the spectroscopic factors of 3.23 for the 1d3/2 and 1.19 for the 2s1/2 are obtained. Even if
the same bound-state parameters and the nonlocality-correction parameters are used for the ( p, 2p) and(e, e′ p) reactions, the spectroscopic factors from the ( p, 2p) reaction are still larger than those from the(e, e′ p) reaction by 20% or more. The differences of the spectroscopic factors from the ( p, 2p) in thepresent analysis and the previous result by Kramer et al. are not explained by the bound-state parameters
only. Therefore, other problems still remain in the DWIA calculation.
It should be noted that the spectroscopic factors from the ( p, 2p) reaction are not settled even at
the proton-injection energy of 200 MeV, where many ( p, 2p) experiments have vigorously been performed [40, 44, 79, 80]. It is seen in Table 5.1 that the spectroscopic factors for the valence orbitals in
40 Ca change by a factor of about 2 at the different measured angles.

CHAPTER 6. DISCUSSION

84

Figure 6.5: DWIA calculations of the 40 Ca(p, 2p) reaction for the 1d3/2 -hole statewith the different radius parameters of the Woods-Saxon potential. The strengthsof the 1d3/2 -hole state were obtained by fitting to the experimental data.

Table 6.2: Normalized spectroscopic factors relative to the IPSM limits for the orbitals in
40 Ca. The description of the uncertainty is same as Table 5.3 except that these total uncertainties include also the uncertainty of the normalization.
Spectroscopic factor
IPSM limit
1d3/2
1 f7/2
2s1/2
1d5/2
1p
1s1/2

4
8
2
6
6
2

with nonlocality correction

0.65 ± 0.05(±0.01)
0.074 ± 0.007(±0.004)
0.53 ± 0.04(±0.01)
0.85 ± 0.09(±0.06)
0.49 ± 0.07(±0.06)
0.89 ± 0.09(±0.06)

0.65 ± 0.05(±0.01)
0.069 ± 0.007(±0.04)
0.53 ± 0.04(±0.01)
0.83 ± 0.09(±0.05)
0.49 ± 0.07(±0.06)
0.95 ± 0.11(±0.07)

The aim of this work is to investigate the spectroscopic factors for the deeply bound orbitals. In order
to discuss them in spite of the unsolved problems of the DWIA calculation, the spectroscopic factors
were normalized by using the value from the (e, e′ p) reaction. Since there was no way to optimize
bound-state parameters for the overlapped deeply bound orbitals, the common bound state parameters
by Elton and Swift were used. As the hole state of the 1d3/2 orbital is well separated from the other
orbital contributions in the separation energy spectra, the spectroscopic factor of the 1d3/2 orbital wasused as the normalization reference. The ratio of the spectroscopic factor for the 1d3/2 orbital from the(e, e′ p) reaction by Kramer et al. to that obtained from the present analysis was 0.53. Thus, the deduced
spectroscopic factors are normalized by the factor of 0.53 as listed in Table 6.2. The uncertainty of 8%
from the normalization is added in quadrature for the total uncertainty.

6.3. SPECTROSCOPIC FACTORS

85

Figure 6.6: DWIA calculations of the 40 Ca(p, 2p) reaction for the 2s1/2 -hole statewith the different radius parameters of the Woods-Saxon potential. The strength of
the 2s1/2 -hole state was determined by the MDA for the peak. The only contribution of 2s1/2 -hole state are displayed for simplicity.

6.3 Spectroscopic factors
6.3.1 Comparison with experimental works
The spectroscopic factors from the present analysis are compared with those obtained from the previous(e, e′ p) experiments in Table 6.3, and they are illustrated as a function of the separation energy in
Fig. 6.11. They are shown as ratios to the IPSM limits. The experimental uncertainties are not given forthe spectroscopic factors obtained by Mougey et al. [16].
The spectroscopic factors obtained from the (e, e′ p) experiment by Nakamura et al. are larger than
the sum-rule limits except for the 1d5/2 orbital [18]. Their large spectroscopic factors for the 1 p and1s1/2 orbitals suggest the underestimation of the cross section in the calculation or the inclusion of thecontinuum background in the higher-separation-energy region.
The spectroscopic factors of 0.65 and 0.75 obtained by Mougey et al. [16] for the 2s1/2 and 1s1/2
orbitals are as low as the present results of 0.53 and 0.89, whereas the spectroscopic factor of 0.95 forthe 1p orbital is much larger than the present result of 0.49. The centroid energy of 41 MeV obtained byMougey et al. for the 1p orbital is much higher than the present result of 29.6 MeV as seen in Fig. 6.1.
As can be seen in Fig. 6.1, both the centroid energies of the 1p and 1s orbitals by Mougey et al. arehigher than those from the present result. The larger spectroscopic factor and the higher centroid energy
for the 1p orbital can be explained by inclusion of the continuum background in the 1p-hole strengths
in the higher-separation-energy region or by an insufficient separation of the 1p- and 1s-hole states. It
is also plausible that the extremely high centroid energy of 77 MeV for the 1s1/2 -hole state reported by
Amaldi et al. [19] is due to the continuum background at higher separation energy.

86

Figure 6.7: Spectroscopic factors obtainedwith the different radius parameters of the
Woods-Saxon potential for the 1d3/2 -, 2s1/2 and 1 f7/2 -hole states.

CHAPTER 6. DISCUSSION

Figure 6.8: Spectroscopic factors obtainedwith the different diffuseness parameters of the
Woods-Saxon potential for the 1d3/2 -, 2s1/2 -,and 1 f7/2 -hole states.

The present analysis also depends on the background treatment. The spectroscopic factors obtained
without background subtraction for the 1s1/2 , 1p, and 1d5/2 orbitals are 1.5–2.0 times larger than thosewith backgrounds subtraction, as seen in Tables 5.3 and 5.4. This indicates that the spectroscopic factors
are greatly affected by the background estimation.
The background from the four-body final states in the high-missing-energy region in the 12 C(e, e′ p)reaction was investigated by Fissum et al. at JLab [92]. The measured separation-energy spectra werecompared with calculation of the (e, e′ pp) and (e, e′ pn) reactions included meson-exchange currents,
isobar currents, central short-range correlations, and tensor medium-range correlations. However, thecalculated cross section was smaller than the measured cross section by about 50% and did not clarify
the total background components.
The contribution of the rescattering processes was studied by Cowley et al. [87]. It is remarkable thatthe calculated rescattering processes well reproduced the experimental coincidence spectra especiallyin the kinematical region in which the quasi-free events were hardly expected. The rescattering process
cannot be ignored in some kinematical regions in the coincidence measurement.
Although the background components are still controversial, in the present study, we used the fourbody phase space as the background-shape model, taking account of the four-body and rescattering
processes. To compensate the inadequacy of the model, the uncertainties in the deduced observables
were estimated to include the model uncertainties, as previously mentioned.
A large reduction in the single-particle strength has also been observed for the strongly bound valence
neutrons near the Fermi surface in proton-rich unstable nuclei [93, 94]. This reduction has been suggested to be due to the strong p–n interactions [95]. Although the nucleons in the 1p and 1s1/2 orbitals
in 40 Ca are also strongly bound, these orbitals are far below the Fermi surface. Therefore, the mechanisms for reducing the spectroscopic factors for the 1 p and 1s1/2 orbitals in 40 Ca are expected to bedifferent. We need further experimental study on the spectroscopic factors in other nuclei to clarify the6.3. SPECTROSCOPIC FACTORS

87

Figure 6.9: Correlation between the spectroscopic factors for the 1d3/2 -hole state
and the strength of the spin–orbit term for the bound state.

NN correlations that contribute to the reduction of the spectroscopic factors for deeply bound orbitals.

6.3.2 Comparison with theoretical works
The spectroscopic factors from the present result are compared with those from theoretical studies inTable 6.4, and they are illustrated as a function of the separation energy in Fig. 6.12. They are shown asratios to the IPSM limits.
The calculations reported by Fabrocini et al. [13] and by Bisconti et al. [14] used state dependent
correlations with central and tensor components, so-called f6 correlation. Fabrocini et al. ignored the
Coulomb interaction in the mean-field potential and used the ls coupling scheme of the single-particle
wave function basis. Bisconti et al. followed the calculation in Ref. [13] by Fabrocini et al., andconsidered the presence of the antiparallel spin terms and distinguished between proton and neutron
contributions in the j j coupling scheme for sake of completeness. Both calculations suggested that the
spectroscopic factor decreases as the binding energy increases, as can be seen in Fig. 6.12. This is contrary to the prediction by the nuclear matter calculation by Benhar et al.[12]. Furthermore, Bisconti etal. insisted from the results of 12 C, 16 O, 40 Ca, 48 Ca, and 208 Pb that the spectroscopic factors increase
when the principle quantum number n and the l j values increase. Although the values predicted by
Bisconti et al. show a rather moderate reduction, the quantitative difference between these calculations
was not discussed. The role played by the various correlations have been unclear.
The spectroscopic factors of 0.86 and 0.87 for the 2s1/2 orbital predicted by Fabrocini et al. [13] andBisconti et al. [14] considerably exceed the present result of 0.53. This discrepancy could be explainedby surface effects, which are not taken into account in their calculations. It is theoretically known thatthe surface effects reduce the spectroscopic factors for the orbitals near the Fermi level.
Though the spectroscopic factor of 0.85 for the 1d5/2 orbital is consistent with the values predicted

88

CHAPTER 6. DISCUSSION

Figure 6.10: Spectroscopic factors relative to the IPSM limits for the orbitals in 40 Ca as a function of theseparation energy. The solid line and dashed line are shown to guide the eyes. The strengths shown by
the squares are obtained by the MDA with the DWIA calculation including the nonlocality correctionin the bound state wave function and in the reaction. The nonlocality corrections almost uniformly
decrease the spectroscopic factors for all of the orbitals.

by Fabrocini et al. and by Bisconti et al., those for the 1s1/2 and 1p orbitals show a different tendency
from the predictions. The spectroscopic factor of 0.89 for the 1s1/2 orbital is slightly smaller than unity,
but the reduction of the spectroscopic factor is not significant because the experimental uncertainty
is as large as 0.09. On the other hand, the spectroscopic factor of 0.49 for the 1p orbital is largely
suppressed and is smaller than the predicted value of 0.58 by Fabrocini et al. The large reduction of thespectroscopic factor for the 1 p orbital indicates a strong influence of NN correlations in the inner corefar below the Fermi surface.
The spectroscopic factors for the 1d5/2 , 1p and 1s1/2 don’t show monotonous decrease or increase
with respect to binding energy as predicted. The present result possibly suggests other dependence of
the NN correlations, e.g., l dependence, and a clue to disentangle the role played by the correlations.
Therefore, it is interesting to investigate the spectroscopic factor for the 1p orbital in detail. The spectroscopic factor for the 1p orbital was obtained on the assumption that the relative strengths for the 1p1/2
and 1p3/2 -hole states are in the ratio of 2:4. It is important to deduce the spectroscopic factors for the
1p1/2 and 1p3/2 -hole states separately to obtain further information on the reduction of the spectroscopic
factor for the deeply bound orbitals.

6.4 Perspectives
The 1p1/2 - and 1p3/2 -hole states could not be separated in the present study because the DWIA calculation of the analyzing power was not fully reliable. It is important to separate the hole states of the 1p1/2

6.4. PERSPECTIVES

89

Table 6.3: Spectroscopic factors relative to the IPSM limits for the orbitals in 40 Ca obtained fromprevious experiments. For the present result, the description of the uncertainty is same as Table 6.2.
IPSM limit

Present work

Mougey et al. [16] a

Nakamura et al.[18] b

2s1/2

2

0.65

1.0 ± 0.1

1d3/2
1d5/2

10
4
6

0.53 ± 0.04(±0.01)

1d

1p

a

0.49 ± 0.07(±0.06)

6
1s1/2

b

0.65 ± 0.05(±0.01)
0.85 ± 0.09(±0.06)

0.89 ± 0.09(±0.06)

2

0.77

0.95
0.75

1.1 ± 0.4
0.78 ± 0.27

1.70 ± 0.15

2.60 ± 0.15 (A)1.9 ± 0.1 (B)The uncertainties were not given for the spectroscopic factors by Mougey et al..
Two results ((A) and (B)) were presented for 1s1/2 orbital.

Table 6.4: Spectroscopic factors relative to the IPSM limits for the orbitals in 40 Ca obtained fromtheoretical studies. For the present result, the description of the uncertainty is same as Table 6.2.
IPSM limit

Present work

Fabrocini et al. [13]Bisconti et al. [14]

2s1/2

2

0.86

0.87

1d3/2
1d5/2

10
4
6

0.53 ± 0.04(±0.01)

0.49 ± 0.07(±0.06)

0.58

1p1/2
1p3/2

6
2
4

1s1/2

2

0.89 ± 0.09(±0.06)

0.55

1d

1p

0.65 ± 0.05(±0.01)
0.85 ± 0.09(±0.06)

0.87
0.85
0.86
0.81
0.82
0.78

and 1p3/2 orbitals, that is, the hole states of the j> and j< orbitals, by use of the analyzing power data,
to obtain further information on the deeply bound orbitals and to understand NN correlations in atomic
nuclei. Therefore, progress in the reaction theory is strongly desired because it enables us to use theanalyzing power data in the MDA.
Following the present work, another ( p, 2p) experiment with a polarized proton beam was done at
200 MeV [96] at RCNP. The analyzing power data was reproduced by the DWIA calculation better thanthat in the present work, and it encourages us to separate the hole states of the j> and j< orbitals. Since
that measurement was performed under the symmetric angle condition for ejected protons, the symmetric angle condition seems to be advantageous to the analyzing power measurement. Understanding why
the DWIA calculation reproduced the analyzing power data in the lower injection-energy experimentremains to be done. Although background contribution might interfere the analyzing power for deeply
bound orbitals, the analyzing power measurement at 200 MeV under the symmetric angle condition

90

CHAPTER 6. DISCUSSION

Figure 6.11: Spectroscopic factors relative to the IPSM limits for the orbitals in 40 Ca as a function of
the separation energy. The dashed lines are shown to guide the eyes.

might be useful to separate the 1p1/2 - and 1p3/2 -hole states.
We also need a systematic measurement of the spectroscopic factors for the orbitals far below theFermi surface in various nuclei. It should be highly helpful for us to clarify the NN correlation in theatomic nuclei.

6.4. PERSPECTIVES

91

Figure 6.12: Spectroscopic factors relative to the IPSM limits for the orbitals in 40 Ca as a function of
the separation energy with the predictions by Fabrocini et al. [13] and Bisconti et al. [14]. For the
separation energies of the predictions, the values of the present result are used since no predicted values
are given in their papers. The separation energies for the 1 p1/2 and 1p3/2 orbitals are plotted deviating±2 MeV from 29.6 MeV respectively. The solid line (present result), dashed line (Fabrocini et al.), anddash-dotted line (Bisconti et al.) are shown to guide the eyes.

92

CHAPTER 6. DISCUSSION

Chapter 7

Summary
A 40 Ca(~p, 2p) experiment was performed with a 392-MeV polarized proton beam to measure the recoilmomentum distributions of the cross section and the analyzing power in the separation-energy region of
0–89 MeV.
Although a hump structure is seen around 30 MeV in the measured separation energy spectra, no peakor hump cannot be seen above 40 MeV, where 1s-hole state is expected to be seen.
Two prominent peaks were observed at 8.3 and 10.9 MeV in the separation energy spectra, and theirrecoil-momentum distributions of the cross section show characteristic behavior for a hole state of asingle-particle orbital with L , 0 and L = 0, respectively. The DWIA calculation reproduces thedependence of the recoil momentum on the measured cross section qualitatively well. This assured
the validity of the MDA with the cross-section data. Although the DWIA calculation reproduces thedependence of the recoil momentum on the measured analyzing power qualitatively, it systematicallyoverestimates the analyzing power for 1d3/2 and 2s1/2 -hole states. The obtained spectroscopic factors
for the 1d3/2 and 2s1/2 are larger than those from the (e, e′ p) and (d,3 He) reactions. The differences ofthe spectroscopic factors for the low-lying orbitals between from the (p, 2p) in the present work andfrom the (e, e′ p) reactions are due not only to the bound-state parameters but also to the treatment of the
distortion in the (p, 2p) reaction. It is found that there are some uncertainties in the absolute values of
the cross section and the analyzing power from the DWIA calculation under the present experimental
condition.
The strength distributions for the deep-hole states were obtained by the MDA for each 2-MeV binin the separation energy without any assumption on the shape of the distribution and were successfully
separated from the continuum background by subtraction of a four-body background. The centroid
energies and widths of the distributions were deduced to be 29.6 ± 0.5 and 48.4 ± 0.6 MeV for the 1p
and 1s1/2 -hole states, respectively. The hump of the 1p-hole state around 30 MeV is consistent with the
hump structure in the separation energy spectra.
Taking the uncertainty of the DWIA calculation into the consideration, the spectroscopic factors
were normalized by using the value for the 1d3/2 orbital from the (e, e′ p) reaction and discussed. The
93

94

CHAPTER 7. SUMMARY

normalized spectroscopic factors for the 1 p and 1s1/2 orbitals were obtained as 49 ± 7% and 89 ± 9% of
the IPSM limits, respectively. Although the small quenching for the 1s1/2 orbital doesn’t give a decisive
evidence of the influence of the NN correlations, the large quenching for the 1 p orbital suggests that the
spectroscopic factor for the orbital in the inner core is reduced owing to the NN correlations. Further
development of the DWIA calculation of the analyzing power is needed to separate the hole states of the
1p1/2 and 1p3/2 orbitals and clarify the influence of the NN correlations on the spectroscopic factors.

Acknowledgements
I would like to express my gratitude to Professor H. Sakaguchi, who was my supervisor at the graduate
course in Kyoto University. He suggested the study of deep hole states in the medium-mass region by(p, 2p) reaction and have patiently been discussed the subject on physics with me. I have continuously
learned experimental technique from him.
I would like to thank Professor T. Kawabata for his helpful discussion and guidance on the present
work. I appreciate his critical advice on a draft of a paper.
I acknowledge Professor M. Yosoi, who is kind enough to provide his analysis software and to givethe detail of making the MWPCs. As he is one of the experimentalists who have a detailed knowledgeof an experiment with the Grand Raiden spectrometer, I have gotten lots of advices from him not onlyin the experiments but also in the analysis of the data.
I am deeply grateful to Professor T. Noro, from whom I have learned much of (p, 2p) experiment. He
gave me chances to join the experiments at the Petersburg Nuclear Physics Institute (PNPI), Gatchina,
Russia. We started the study of deep hole states by (p, 2p) reaction since these experiments.
I would like to thank Professor T. Nagae for his thoughtful suggestions for this thesis.
I gratefully thank the scientific and technical staff and cyclotron crew at RCNP, especially Professor
K. Hatanaka and Dr. S. Ninomiya, for their support for providing a stable beam during the experimentand for the beam-time management.
I would address my thank to Dr. H. Takeda, who had developed the RDTM module and supported its
operation during the beamtime. I am grateful to Dr. M. Itoh for his hard work in the preparation andin the beamtime. I am thankful to Dr. M. Uchida and Dr. H. P. Yoshida for their great contribution to
the experiment, especially to the data acquisition. I wish to thank Dr. S. Terashima, Mr. S. Kishi, and
Dr. J. Zenihiro, who worked in whole of the experiment and took much of midnight shifts.
I would like to thank Professor T. Wakasa for his support and advices on a draft of a paper.
I would like to thank all other members of the RCNP-E168/217 experiments; Mr. S. Asaji, Dr. K. Fujita, Mr. Y. Hagihara, Dr. T. Ishida, Professor Y. Sakemi, Dr. Y. Shimizu, Dr. Y. Tameshige, and
Mr. T. Yonemura for their great contributions.

95

I spent 7 years as a member of the experimental nuclear hadronic physics (NH) group at Kyoto
University. I would like to express many thanks Professor K. Imai for devoted direction. I would like tothank Dr. T. Murakami, from whom I learned a lot of things on physics. I also thank all members of thegroup, especially friendly coworkers, Dr. M. Kitaguchi and Dr. M. Miyabe.
I would like to thank Professor. A. Ozawa, Professor. H. Okamura, and Professor. A. Tamii for theirsupport and giving me chances to continue the research on nuclear physics at University of Tsukuba andat RCNP, Osaka University.
I would like to express my appreciation to my parents, who have supported and encouraged me. I
also thank Dr. K. Yasuda, who is a biologist and one of my relatives. She gave some advice and cheered
me up.
Finally, I thank my wife Sachiko for her patience and encouragement during all these years.

Appendix A

Revised result
We had already published the result of this work in Ref. [97] in 2010. After the publication, a mistakein the estimation of the cross section was found during the preparation of this thesis. The results inthis thesis were obtained after correcting the mistake and improving the inadequate estimation of the
MWPC efficiency. The revised result in this thesis is compared with the published result in Ref. [97] inTables A.1 –A.4. The errata based on the revised result will be published.
In the reanalysis, at first, the spectroscopic factors were evaluated in the same way as that in Ref. [97],which are listed as “Raw” in Tables A.1, A.3, and A.4. However, the obtained spectroscopic factors are
considerably larger than the published values, and some of them exceed 100% of the 2J + 1 value. It was
concluded that there are some uncertainties in the absolute values of the cross section from the DWIA
calculation under the present experimental condition. Taking the uncertainty of the DWIA calculation
into the consideration, the spectroscopic factors were normalized by using the value for the 1d3/2 orbitalfrom the (e, e′ p) reaction by Kramer et al. in Ref. [47], which are listed as “Normalized” in Tables A.1,
A.3, and A.4.
Table A.1: Spectroscopic factors for the discrete peaks in the 40 Ca measurement. The first uncertainty is the total uncertainty and the second, in parentheses, is the statistical uncertainty includedin the total. In the published results, the uncertainty from the DWIA calculation is included in the
total uncertainty. In the reanalyzed results, the uncertainty from the normalization reference [47]is included in the total uncertainty of the normalized results.
Esep [MeV] Orbital
Spectroscopic factor
Published [97]
8.3
10.9
10.9

1d3/2
2s1/2
1 f7/2

Reanalyzed
Raw
Normalized
4.87 ± 0.09(±0.09) 2.58 ± 0.20(±0.05)
1.61 ± 0.04(±0.04) 0.85 ± 0.07(±0.02)
1.12 ± 0.06(±0.06) 0.59 ± 0.06(±0.03)

3.12 ± 0.53(±0.06)
1.01 ± 0.17(±0.03)
0.78 ± 0.14(±0.04)

97

APPENDIX A. REVISED RESULT

98

Table A.2: Centroid energies and widths (FWHM) of the strength distributions for the hole statesof the 1p and 1s1/2 orbitals in 40 Ca. The first uncertainty is the total uncertainty, which includes thestatistical and the model uncertainties, and the second, in parentheses, is the statistical uncertainty
included in the total.
Orbital1p
1s1/2

Published [97]Centroid
Width( MeV )( MeV )
30.0 ± 0.4(±0.3) 10.3 ± 1.1(±0.9)49.6 ± 0.6(±0.6) 21.3 ± 0.9(±0.9)

Reanalyzed
Centroid
Width( MeV )( MeV )
29.6 ± 0.5(±0.5) 11.4 ± 1.2(±1.2)
48.4 ± 0.6(±0.6) 23.4 ± 1.1(±1.1)

Table A.3: Spectroscopic factors relative to the IPSM limits for the orbitals in 40 Ca. They are
obtained after subtraction of the background. The first uncertainty is the total uncertainty and thesecond, in parentheses, is the statistical uncertainty included in the total. The model uncertainty is
included in the total uncertainty. The uncertainty from the DWIA calculation are included in the total uncertainty of the published results, while the uncertainty from the normalization reference [47]is included in the total uncertainty of the normalized results.
Orbital IPSM limit
Spectroscopic factor
Published [97]1d3/2
1 f7/2
2s1/2
1d5/2
1p
1s1/2

0.78 ± 0.13(±0.01)
0.097 ± 0.017(±0.005)
0.60 ± 0.10(±0.02)
0.94 ± 0.17(±0.06)
0.49 ± 0.10(±0.06)
0.78 ± 0.14(±0.05)

4
8
2
6
6
2

Reanalyzed
Raw
Normalized
1.22 ± 0.02(±0.02) 0.65 ± 0.05(±0.01)
0.14 ± 0.01(±0.01) 0.074 ± 0.007(±0.004)
1.00 ± 0.03(±0.03) 0.53 ± 0.04(±0.01)
1.60 ± 0.12(±0.11) 0.85 ± 0.09(±0.06)
0.93 ± 0.12(±0.12) 0.49 ± 0.07(±0.06)
1.69 ± 0.10(±0.10) 0.89 ± 0.09(±0.06)

Table A.4: Spectroscopic factors relative to the IPSM limits for the orbitals in 40 Ca without background subtraction. The first uncertainty is the total uncertainty and the second, in parentheses,
is the statistical uncertainty included in the total. The uncertainty from the DWIA calculation are
included in the total uncertainty of the published results, while the uncertainty from the normalization reference [47] is included in the total uncertainty of the normalized results.
Orbital IPSM limit
Spectroscopic factor
Published [97]
2s1/2
1d5/2
1p
1s1/2

2
6
6
2

0.61 ± 0.11(±0.02)
1.33 ± 0.23(±0.05)
1.41 ± 0.24(±0.04)
1.12 ± 0.19(±0.03)

Reanalyzed
Raw
Normalized
1.02 ± 0.03(±0.03) 0.54 ± 0.04(±0.01)
2.23 ± 0.11(±0.11) 1.18 ± 0.11(±0.06)
1.87 ± 0.07(±0.07) 0.99 ± 0.08(±0.04)
2.53 ± 0.05(±0.05) 1.34 ± 0.11(±0.03)

Appendix B

Scattering angles and recoil momentum
B.1 Scattering angle
Scattering angles for the emitted protons from the target were obtained with the MWPCs in front of thespectrometers and with the VDCs at the focal planes. Angular resolution by the MWPC was determinedby the wire spacing, a beam-spot size at the target, and the distance between the target and the anodeplane. It is estimated at approximately 0.11◦ , as mentioned in Sec. 3.5.
Horizontal angular resolutions by the MWDC were evaluated by using pp scattering event for reference. The event of pp scattering was measured at 25.5◦ for the GR and at 60.0◦ for the LAS. For pp
scattering, the measured kinetic energy and horizontal scattering angle of the emitted proton strongly
correlate to each other, and they are one-to-one correspondence in the laboratory system since pp scattering event is two body reaction and two particles are identical. Figure B.1 (A) shows two dimensional
plot of the horizontal scattering angle (θGR ) and the position at the focal plane (XGR ), which corresponds
to the kinetic energy. The width of the locus line indicates the resolution of the scattering angle andthe kinetic energy. The locus was corrected so that the kinetic-energy dependence of the horizontal′ )scattering angle was canceled, as shown in Fig. B.1 (B). In −50 ≤ XGR ≤ 50, the corrected angle (θGR
was projected on the horizontal angle axis , which is shown in Fig. B.1 (C), and fitted with Gaussianfunction. The estimated angular resolution was 0.08◦ for the GR. It is smaller than that by the MWPC
for the GR. The horizontal angular resolution by the MWDC for the LAS was also estimated in the same
way as the GR case and was 0.23◦ . It is larger than that by the MWPC for the LAS. The worse angularresolution for the LAS is due to the low proton energy and thick film at the exit of the LAS spectrometer.

B.2 Recoil momentum→
Recoil momentum (−p3 ) of the residual nucleus in the (p, 2p) reaction was calculated on the basis of the
momentum conservation law as
−
→
→
→
→
p =−
p −−
p −−
p
(B.1)
3

0

1

99

2

100

APPENDIX B. SCATTERING ANGLES AND RECOIL MOMENTUM

Figure B.1: (A) Two dimensional plot of the horizontal scattering angle (θGR ) and the position at the
focal plane (XGR ), which corresponds to the kinetic energy, for the GR. (B) Two dimensional plot of thehorizontal scattering angle corrected so that the kinetic-energy dependence of the horizontal scattering′ ) and X . (C) Corrected angle (θ′ ) projected on the horizontal angle axis inangle was canceled (θGR
GR
GR
−50 ≤ XGR ≤ 50.
−
where →pi (i = 0, 1, 2) are the momentum of the incident proton (i = 0), the scattered and knockedout protons (i = 1, 2), respectively. In pp scattering event, the recoil momentum doesn’t exist since the→residual particle doesn’t exist. The momentum −p3 calculated by Eq. B.1 should be zero for pp scattering−→
event. It is useful to estimate p3 and the deviation from zero in pp scattering event for the accuracy ofmomentum reconstruction.
The information from the MWPCs was used for the vertical scattering angle for particles coming
to the GR and the horizontal and vertical scattering angles for particles coming to the LAS. Here,→in estimating the momentum −p3 in pp scattering event, the scattering angles from the MWPCs wererandomly deviated in the angular resolution by Monte Carlo technique for smoothing. The obtained→
result for |−p3 | in pp scattering event is shown in Fig. B.2. The average deviation from µ defined byh|xi − µ|i =Z|x − µ| f (x) dx(B.2)

B.2. RECOIL MOMENTUM

101

Figure B.2: Momentum |p3 | estimated in pp scattering event. The scattering angles from the MWPCs
were randomly deviated in the angular resolution by Monte Carlo technique for smoothing.

→
was estimated for x = |−p3 |, where f (x) is a distribution function and µ=0. The obtained average deviation
was 4.4 MeV/c. If we didn’t have any information on the vertical scattering angles and ignored them,that is, the vertical scattering angles were taken as zero, the average deviation would be 2.9 MeV/c andbe a little underestimate.

102

APPENDIX B. SCATTERING ANGLES AND RECOIL MOMENTUM

Appendix C

Phase space calculation
C.1

Introduction

The (p, 3p) and (p, 2pn) reactions are possible components of the background in the (p, 2p) reaction,
while the dynamical aspects in the (p, 3p) and (p, 2pn) reactions are unclear. In order to estimate thecontribution of those reactions in the (p, 2p) reaction measurement without the dynamical details of the(p, 3p) and (p, 2pn) reactions, the phase space for the 4-particle final state was calculated. Phase spacedescribes a kinematical aspect of the process. In this chapter, the four-body phase space is derived fromits definition.
As the DWIA calculation in the present study employs the noninvariant form, the noninvariant phasespace is employed here. The momentum p j and energy E j of the j-th particle satisfy the following
relation
E 2j − p2j = m2j ,(C.1)
where m j is the mass of the j-th particle. As is defined in Werbrouck’s textbook [98], the probability
dPn of a n-particle final state with a total momentum P and total energy E is defined as
d3n Pn = d3n Rn H,(C.2)

where H is the square of the noninvariant matrix element for the process and contains the dynamical
aspects of the process, and Rn is pure kinematical and the Lorentz noninvariant phase space defined by
3n

d Rn =

C.2

(Y
n
j=1

3

)

d p j δ3

n
X
j=1

n
 X

pj − P δ
Ej − E .

(C.3)

j=1

Phase space for four-particle final state

The processes that we take account of are the 40 Ca(p, 3p)38 Ar and 40 Ca(p, 2pn)38 K reactions. As the
final states have the configuration of 3p+38 Ar or of 2p + n+38 K, respectively, the four-body phase space
103

APPENDIX C. PHASE SPACE CALCULATION

104

R4 is discussed here. The calculation employs the condition that the two protons in the final states aredetected by GR and LAS but the other particles in the final states are not detected, as is same as theexperiment.
Let particles 1 and 2 correspond to protons detected by GR and LAS. The particles 3 and 4 are the
combination of p+38 Ar or n+38 K. The phase space for the four-body final states is given from Eq. (C.3)
as
(Y
) X
4
4
4
 X

4·3
3
d R4 =
d p j δ3pj − P δ
Ej − E .
(C.4)
j=1

j=1

j=1

From Eq. (C.4), we have the four-body phase space in the integral form

Z


 

d3 p1 d3 p2 d3 p3 d3 p4 δ3 p1 + p2 + p3 + p4 − P δ E1 + E2 + E3 + E4 − E
Z
Z

 

3
3
=
d p1 d p2 d3 p3 d3 p4 δ3 p3 + p4 − P′ δ E3 + E4 − E ′ ,(C.5)


R4 E, P =where E ′ and P′ are defined by E ′ = E − E1 − E2 and P′ = P − p1 − p2 , respectively. The last integral in
Eq. (C.5) corresponds to the two-body phase space R2 E ′ , P′ with total momentum P′ and total energy
E′.
Z

 

′
′
d3 p3 d3 p4 δ3 p3 + p4 − P′ δ E3 + E4 − E ′R2 E , P =(C.6)

Therefore, we haveR4 E, P =Zd3 p1 d3 p2 R2 E ′ , P′ .

(C.7)

Under the condition that only two protons in the final states are detected, the state of the rest particles is indefinite. The 4-particle final states could contribute in the separation energy spectrum for the
40 Ca(p, 2p)39 K reaction. Excluded the reaction Q-value, the separation energy calculated from the energies of the injected and detected protons corresponds to the excitation energy of the residual nucleus
in the 3-particle final states, while it corresponds to the effective mass of the particles 3 and 4 in the
4-particle final states. The effective mass M34 of the particles 3 and 4 is defined by


2
M34
= E3 + E4 2 − p3 + p4 2 .

(C.8)

In order to estimate the contribution of the 4 particle final states in the separation energy spectrum forthe 40 Ca(p, 2p)39 K reaction, the mass distribution of the four-body phase space
dR4
dM34

(C.9)

is needed. By using the conservation laws of momentum and energy, the effective mass M34 is expressed
as
2
M34
=

=



E − E1 − E2 2 − P − p1 − p2 2


M 2 + m21 + m22 − 2EE1 + 2P · p1 2 − 2EE2 + 2E1 E2 − 2P · p1 − 2p1 · p2 .

(C.10)

C.3. PHASE SPACE FOR TWO-PARTICLE FINAL STATE105

Let p j = |p j |, and differentiating Eq. (C.10) with respect to p2 gives
dM34
dp2
dE

= −

!
 p2
1 
E − E1+ P cos θ2 − p1 cos θ12 .
M34
E2(C.11)

p

The use of d p jj = E jj gives the other forms of d3 p j
d3 p j = p2j dp j dΩ j = p j E j dE j dΩ j .

(C.12)

Combining Eq. (C.7), Eq. (C.11), and Eq. (C.12) yields
R4 E, PZ==

p1 E1 dE1 dΩ1 p22 dp2 dΩ2 R2 E ′ , P′Z

p1 E1 dE1 dΩ1 p22 dΩ2 dM34Z

dE1 dΩ1 dΩ2 p1 E1 p22




dp2
R2 E ′ , P′ .
dM34

(C.13)

Therefore, Eq. (C.9) is obtained as

dR4
dM34

=


dp2
R2 E ′ , P ′ .
dM34(C.14)The phase space for the four-body final state is calculated under the condition that two protons froma final state are detected by the GR and LAS. Therefore, the parameters of particle 1 and 2 are given by
the measurement condition. The integral range of these parameters are determined by the acceptancesof the GR and LAS.

C.3

Phase space for two-particle final state

When the parameters of the particles 1 and 2 are given, the energy E ′ and momentum P’ of the rest
two-body system are determined.
At the integration of Eq. (C.6) over p4 , the integration rules for a δ-function gives′

R2 E , P

′

Z

=

Z

=



d3 p3 d3 p4 δ3 p3 + p4 − P′ δ E3 + E4 − E ′" ∂Ad p3 ψ p4
∂ p4
3

−1 #

,

p4 = p4 ∗where A = p4 − P′ − p3 . Since the solution p4 ∗ of A = 0 is p4 ∗ = P′ − p3 and ∂∂A
p = 1,
4

R2 E ′ , P ′



=

Z

d3 p3 δq

p23 + m23 +q
P′ − p3 2 + m24 − E ′ .

(C.15)

APPENDIX C. PHASE SPACE CALCULATION106

The following integration over p3 yields
Z
Z
q

q

′
′
R2 E , P=dΩ3P′ − p3 2 + m24 − E ′p23 dp3 δ
p23 + m23 +Z=

where B =Sinceq

p23 + m23 +

q

dΩ3

p23

−1 #

∂B
∂p3

′

=

p3 − P′ cos θ3
+ q
,
2
2
2
′
2
p3 + m3
P − p3 + m4p3

q#p∗3 p∗3 − P′ cos θ3 −1
+
=
dΩ3
E3∗
E4∗
#
"
Z
p∗3 p∗3 − P′ cos θ3 −1∗2+= 2π d cos θ3 p3
E3∗
E4∗
"

Z

′

R2 E , PE3∗ =q(C.17)

p∗2
3

where

C.4

(C.16)

p3 =p∗3

P′ − p23 2 + m24 − E ′ and p3 ∗ is the solution of B = 0.
∂B
∂p3


R2 E ′ , P′ is derived as

"

2
p∗2
3 + m3 ,

E4∗ =

q


P′ − p∗3 2 + m24 .

(C.18)(C.19)

Two-body kinematics

As defined P′ = |P′ |, B = 0 givesq

p23 + m23 +q

P′ − p23 2 + m24 − E ′ = 0 .

This equation is simplified in a straightforward way as

a p23 + 2 b p3 + c = 0(C.20)where
a = 4 E ′2 − P′2 cos2 θ3 ,b = −2 P′ cos θ3 α ,c = 4 E ′2 m23 − α2 ,α = E ′2 + m23 − m24 − P′2 .
The solution of Eq. (C.20) is
p3 =

−b ±

√

b2 − ac.
a(C.21)

C.5. SUPPLEMENT107

The physical solution p∗3 of the B = 0 is the p3 solutions that are real and positive. This solution of p∗3
makes the two-body phase space of Eq. (C.18) definite, therefore, the mass distribution of the four-bodyphase space of Eq. (C.14) is also given.
Here, the condition under which the physical solution exists is discussed at a different viewpoint. In
the center of mass system of the particles 3 and 4, the momentum of the particle 3 is given as
q
 2
  2

M34 − m3 + m4 2 M34− m3 − m4 2
pc3 =.
(C.22)
2M34
pc′

The relative velocities βc3 = E3∗ and β = EP′ are defined, and γ3∗ = q 1 c2 , γ = √1 . There are 3
3

1−β3

1−β

cases depending on the relative velocities. Let θ3 be the angle between the directions of p3 and P’. The
number of the physical solution of p3 and the maximum of the angle θ3 are classified by 3 cases.
1. β < βc3There is one physical solution for p3 and θ3 max = 180◦ .
2. β = βc3There is one physical solution for p3 and θ3 max = 90◦ .

3. β > βc3
There are two physical solutions for p3 and θ3 max is given as follows.
cos θ3 max =


C.5

s

!
β∗3 γ3∗ 2

1−

βγ(C.23)

Supplement

The probability of the reaction for the n-particle final state Pn is defined in Lorentz invariant form as
d4n Pn = d4n Rn H.

(C.24)

H is the square of the invariant matrix element for the process and Rn is the invariant phase space defined
by
n
n
n
Y
X
Y

d4n Rn =d4 q j δ4
qj − Q(C.25)
δ q2j − m2j .
j=1

j=1

j=1

q j represents a four-momentum of the j-th particle that has an energy E j and a momentum p j ,It satisfies the following relation
q j = E j, pj .

(C.26)

q2j = E 2j − p2j = m2j .

(C.27)

APPENDIX C. PHASE SPACE CALCULATION108

The total four-momentum Q of the system has the components

Q = E, P .

(C.28)

By using a property of the Dirac δ-function, the following relation is proved.

Zδ q2j − m2j d q j 0

=

"
"


∂
 q2j − m2j∂ qj 0−1 #

q j 0 =E j∂  2=
 q j 0 − p2j − m2j∂ qj 0#"1=
2 q j 0 q j  =E j

−1 #



q j 0=q

p2j +m2j

0

=

1
2E j

(C.29)

This relation simplify the phase space as
3nd Rn =nY
d3 p j
j=1

2E j

3

δ

n
X
j=1

n
 X

pj − P δ
Ej − E .

(C.30)

j=1

The probability in the invariant form is equivalent to that in the noninvariant form provided that
H=

n
Y
1
H.
2E
j
j=1

(C.31)

Appendix D

Numerical data tables
Table D.1: Data table for the 40 Ca(~p, 2p) reaction for the peaks at 8.3 and 10.9 MeV in the separationenergy at the set1 in Table 2.1.
Esep
(MeV)
8.3
8.3
8.3
8.3
8.3
8.3
8.3
8.3
8.3
8.3
8.3
8.3
8.3
8.3
8.3
10.9
10.9
10.9
10.9
10.9
10.9
10.9
10.9
10.9
10.9
10.9
10.9
10.9
10.9
10.9

E1
(MeV)
230.0
234.0
238.0
246.0
250.0
254.0
258.0
266.0
270.0
274.0
278.0
286.0
290.0
294.0
298.0
230.0
234.0
238.0
246.0
250.0
254.0
258.0
266.0
270.0
274.0
278.0
286.0
290.0
294.0
298.0

p3
(MeV/c)−164.3−154.9−145.4−126.4−116.7−107.1
−97.4
−77.7
−67.8
−57.9
−47.9
−28.0
−18.6
−11.4
12.0
−160.3−150.9−141.3−122.1−112.4−102.6
−92.8
−73.0
−62.9
−52.8
−42.6
−22.2
−12.5
6.9
13.3

dσ
dΩ1 dΩ2 dE1
2(µb/sr MeV)
59.23 ± 0.62
73.63 ± 0.70
86.84 ± 0.76
116.27 ± 0.86
124.91 ± 0.91
130.78 ± 0.92
131.42 ± 0.93
121.58 ± 0.90
104.50 ± 0.83
88.61 ± 0.78
74.03 ± 0.72
49.89 ± 0.41
43.50 ± 0.55
40.51 ± 0.55
41.09 ± 0.57
34.77 ± 0.48
35.93 ± 1.14
36.34 ± 0.50
32.42 ± 0.49
28.14 ± 0.46
27.89 ± 0.47
32.34 ± 0.50
72.82 ± 0.75
116.95 ± 0.96
172.49 ± 1.20
239.18 ± 1.45
382.27 ± 1.43
422.91 ± 2.14
437.52 ± 2.11
416.79 ± 1.98

109

Ay
0.215 ± 0.015
0.244 ± 0.013
0.231 ± 0.012
0.202 ± 0.010
0.233 ± 0.010
0.209 ± 0.010
0.207 ± 0.010
0.154 ± 0.010
0.097 ± 0.011
0.086 ± 0.012
0.059 ± 0.014
0.130 ± 0.013
0.220 ± 0.020
0.353 ± 0.021
0.415 ± 0.021
0.002 ± 0.020
0.031 ± 0.020
0.015 ± 0.020−0.085 ± 0.021−0.053 ± 0.023
0.062 ± 0.024
0.242 ± 0.022
0.432 ± 0.014
0.429 ± 0.011
0.409 ± 0.010
0.387 ± 0.008
0.326 ± 0.006
0.310 ± 0.008
0.282 ± 0.007
0.263 ± 0.007

E1
(MeV)
214.0
218.0
222.0
226.0
230.0
234.0
238.0
246.0
250.0
254.0
258.0
266.0
270.0
274.0
278.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0
246.0
250.0
254.0
258.0
266.0
270.0
274.0
278.0

p3
(MeV/c)−169.3−160.6−151.9−143.1−134.4−125.6−116.9
−99.3
−90.6
−81.9
−73.3
−56.5
−48.6
−41.4
−35.4
−166.1−157.3−148.5−139.7−130.9−122.1−113.2
−95.5
−86.7
−77.8
−69.1
−52.1
−44.1
−36.9
−31.1

dσ
dΩ1 dΩ2 dE1
2(µb/sr MeV)
19.58 ± 0.37
19.07 ± 0.37
19.54 ± 0.37
23.07 ± 0.40
31.58 ± 0.45
28.20 ± 0.43
27.05 ± 0.43
44.81 ± 0.54
36.81 ± 1.07
32.90 ± 1.35
32.41 ± 1.36
36.02 ± 0.42
32.77 ± 0.40
33.86 ± 0.41
35.18 ± 0.42
38.10 ± 0.80
46.08 ± 0.53
53.07 ± 0.57
56.73 ± 0.59
56.41 ± 0.59
63.88 ± 0.62
68.06 ± 0.64
56.86 ± 0.60
59.41 ± 1.22
60.35 ± 1.54
56.77 ± 1.23
48.26 ± 0.46
47.33 ± 0.47
44.67 ± 0.46
41.20 ± 0.45

Ay
0.071 ± 0.026
0.078 ± 0.027
0.076 ± 0.027
0.078 ± 0.025
0.039 ± 0.020
0.062 ± 0.022
0.087 ± 0.022
0.044 ± 0.017
0.069 ± 0.019
0.111 ± 0.021
0.117 ± 0.021
0.231 ± 0.018
0.236 ± 0.019
0.260 ± 0.019
0.265 ± 0.018
0.110 ± 0.018
0.057 ± 0.016
0.044 ± 0.015
0.053 ± 0.015
0.060 ± 0.015
0.064 ± 0.014
0.028 ± 0.013
0.072 ± 0.015
0.088 ± 0.014
0.083 ± 0.014
0.121 ± 0.015
0.148 ± 0.015
0.204 ± 0.015
0.194 ± 0.016
0.217 ± 0.017

Esep
(MeV)
17.0
17.0
17.0
17.0
17.0
17.0
17.0
17.0
17.0
17.0
17.0
17.0
17.0
17.0
17.0
19.0
19.0
19.0
19.0
19.0
19.0
19.0
19.0
19.0
19.0
19.0
19.0
19.0
19.0
19.0

E1
(MeV)
214.0
218.0
222.0
226.0
230.0
234.0
238.0
246.0
250.0
254.0
258.0
266.0
270.0
274.0
278.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0
246.0
250.0
254.0
258.0
266.0
270.0
274.0
278.0

Table D.2 (Continued)dσp3dΩ1 dΩ2 dE1(MeV/c) (µb/sr2 MeV)
−162.9 27.93 ± 0.43
−154.1 32.04 ± 0.46
−145.2 35.90 ± 0.48
−136.3 39.01 ± 0.50
−127.5 40.84 ± 0.51
−118.5 45.33 ± 0.53
−109.6 46.61 ± 0.54
−91.7 42.66 ± 1.00
−82.7 42.46 ± 1.04
−73.8 42.54 ± 0.54
−64.9 38.81 ± 0.52
−47.7 34.27 ± 0.40
−39.6 31.55 ± 0.40
−32.4 28.95 ± 0.39
−26.8 26.97 ± 0.38
−159.7 13.08 ± 0.32
−150.8 15.05 ± 0.34
−141.9 16.14 ± 0.35
−133.0 17.46 ± 0.36
−124.0 18.41 ± 0.37
−115.0 20.30 ± 0.39
−106.0 21.38 ± 0.40
−87.9 20.35 ± 0.40
−78.8 21.21 ± 0.41
−69.7 22.01 ± 0.42
−60.7 20.81 ± 0.41
−43.2 19.35 ± 0.33
−35.0 19.14 ± 0.34
−27.8 17.89 ± 0.34
−22.7 16.64 ± 0.33

Ay
0.122 ± 0.022
0.046 ± 0.020
0.068 ± 0.019
0.053 ± 0.018
0.049 ± 0.017
0.079 ± 0.017
0.035 ± 0.016
0.030 ± 0.017
0.011 ± 0.018
0.081 ± 0.018
0.113 ± 0.019
0.154 ± 0.018
0.166 ± 0.019
0.162 ± 0.021
0.163 ± 0.022
0.145 ± 0.034
0.129 ± 0.032
0.047 ± 0.031
0.083 ± 0.029
0.110 ± 0.028
0.100 ± 0.027
0.060 ± 0.026
0.117 ± 0.028
0.059 ± 0.027
0.068 ± 0.027
0.135 ± 0.028
0.078 ± 0.027
0.111 ± 0.027
0.106 ± 0.029
0.076 ± 0.031

APPENDIX D. NUMERICAL DATA TABLES

Esep
(MeV)
13.0
13.0
13.0
13.0
13.0
13.0
13.0
13.0
13.0
13.0
13.0
13.0
13.0
13.0
13.0
15.0
15.0
15.0
15.0
15.0
15.0
15.0
15.0
15.0
15.0
15.0
15.0
15.0
15.0
15.0

110

Table D.2: Data table for the 40 Ca(~p, 2p) reaction at the set2 in Table 2.1.
∆Esep = 2 MeV.

Esep
(MeV)
21.0
21.0
21.0
21.0
21.0
21.0
21.0
21.0
21.0
21.0
21.0
21.0
21.0
21.0
21.0
23.0
23.0
23.0
23.0
23.0
23.0
23.0
23.0
23.0
23.0
23.0
23.0
23.0
23.0
23.0

E1
(MeV)
214.0
218.0
222.0
226.0
230.0
234.0
238.0
246.0
250.0
254.0
258.0
266.0
270.0
274.0
278.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0
246.0
250.0
254.0
258.0
266.0
270.0
274.0
278.0

Table D.2 (Continued)dσp3dΩ1 dΩ2 dE1(MeV/c) (µb/sr2 MeV)
−156.6
8.49 ± 0.28
−147.69.33 ± 0.29
−138.6 10.11 ± 0.30
−129.6 11.67 ± 0.32
−120.6 11.72 ± 0.32
−111.5 12.68 ± 0.33
−102.4 13.22 ± 0.34
−84.1 13.96 ± 0.36
−74.9 13.18 ± 0.36
−65.7 13.82 ± 0.36
−56.5 12.70 ± 0.36
−38.7 12.66 ± 0.30
−30.4 12.00 ± 0.30
−23.2 11.47 ± 0.30
−18.9 11.10 ± 0.30
−153.48.62 ± 0.28
−144.49.18 ± 0.29
−135.4 10.86 ± 0.31
−126.3 11.34 ± 0.32
−117.1 11.97 ± 0.32
−108.0 13.24 ± 0.33
−98.8 13.78 ± 0.34
−80.3 14.26 ± 0.36
−71.0 14.75 ± 0.37
−61.7 14.18 ± 0.37
−52.4 14.59 ± 0.38
−34.1 13.66 ± 0.31
−25.7 12.91 ± 0.31
−18.7 12.41 ± 0.31
15.6 11.80 ± 0.31

Ay
−0.015 ± 0.046
0.109 ± 0.044
0.101 ± 0.042
0.067 ± 0.038
0.051 ± 0.038
0.102 ± 0.036
0.068 ± 0.036
0.081 ± 0.036
0.086 ± 0.038
0.127 ± 0.037−0.025 ± 0.040
0.041 ± 0.037
0.109 ± 0.039
0.060 ± 0.041
0.160 ± 0.042
0.098 ± 0.046
0.157 ± 0.044
0.139 ± 0.040
0.140 ± 0.039
0.097 ± 0.038
0.120 ± 0.035
0.016 ± 0.035
0.094 ± 0.035
0.090 ± 0.035
0.102 ± 0.036
0.155 ± 0.036
0.161 ± 0.035
0.105 ± 0.037
0.147 ± 0.038
0.228 ± 0.040

Esep
(MeV)
25.0
25.0
25.0
25.0
25.0
25.0
25.0
25.0
25.0
25.0
25.0
25.0
25.0
25.0
25.0
27.0
27.0
27.0
27.0
27.0
27.0
27.0
27.0
27.0
27.0
27.0
27.0
27.0
27.0
27.0

E1
(MeV)
214.0
218.0
222.0
226.0
230.0
234.0
238.0
246.0
250.0
254.0
258.0
266.0
270.0
274.0
278.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0
246.0
250.0
254.0
258.0
266.0
270.0
274.0
278.0

Table D.2 (Continued)dσp3dΩ1 dΩ2 dE1(MeV/c) (µb/sr2 MeV)
−150.38.64 ± 0.28
−141.29.72 ± 0.30
−132.1 11.29 ± 0.31
−122.9 12.32 ± 0.33
−113.7 13.40 ± 0.33
−104.5 14.77 ± 0.35
−95.2 15.66 ± 0.36
−76.5 17.15 ± 0.38
−67.1 17.11 ± 0.39
−57.7 16.53 ± 0.39
−48.2 17.55 ± 0.40
−29.6 15.59 ± 0.32
−21.0 14.45 ± 0.32
−14.3 13.18 ± 0.31
13.3 12.91 ± 0.31
−147.2 10.80 ± 0.30
−138.0 12.12 ± 0.32
−128.9 13.62 ± 0.33
−119.6 15.39 ± 0.35
−110.4 16.93 ± 0.36
−101.1 18.44 ± 0.38
−91.7 19.57 ± 0.39
−72.8 21.32 ± 0.41
−63.2 20.88 ± 0.41
−53.7 21.69 ± 0.42
−44.1 21.45 ± 0.42
−25.0 18.49 ± 0.33
−16.2 16.62 ± 0.33
−10.2 15.71 ± 0.32
12.6 14.62 ± 0.32

Ay
0.113 ± 0.046
0.122 ± 0.043
0.079 ± 0.039
0.059 ± 0.037
0.138 ± 0.035
0.075 ± 0.033
0.071 ± 0.032
0.082 ± 0.031
0.119 ± 0.032
0.081 ± 0.033
0.117 ± 0.032
0.105 ± 0.032
0.103 ± 0.034
0.123 ± 0.037
0.063 ± 0.037
0.134 ± 0.039
0.140 ± 0.037
0.079 ± 0.034
0.121 ± 0.032
0.080 ± 0.030
0.087 ± 0.029
0.116 ± 0.028
0.110 ± 0.027
0.099 ± 0.028
0.124 ± 0.027
0.087 ± 0.028
0.058 ± 0.028
0.089 ± 0.030
0.076 ± 0.032
0.071 ± 0.034

111

112

E1
(MeV)
214.0
218.0
222.0
226.0
230.0
234.0
238.0
246.0
250.0
254.0
258.0
266.0
270.0
274.0
278.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0
246.0
250.0
254.0
258.0
266.0
270.0
274.0
278.0

Ay
0.116 ± 0.039
0.108 ± 0.035
0.107 ± 0.032
0.158 ± 0.031
0.101 ± 0.028
0.077 ± 0.028
0.098 ± 0.026
0.050 ± 0.026
0.077 ± 0.027
0.069 ± 0.027
0.064 ± 0.028
0.059 ± 0.027
0.058 ± 0.030
0.061 ± 0.032
0.116 ± 0.035
0.088 ± 0.040
0.173 ± 0.036
0.104 ± 0.033
0.080 ± 0.030
0.076 ± 0.029
0.072 ± 0.028
0.056 ± 0.027
0.101 ± 0.027
0.098 ± 0.027
0.009 ± 0.028
0.092 ± 0.029
0.048 ± 0.029
0.028 ± 0.031
0.080 ± 0.035
0.051 ± 0.036

Esep
(MeV)
33.0
33.0
33.0
33.0
33.0
33.0
33.0
33.0
33.0
33.0
33.0
33.0
33.0
33.0
33.0
35.0
35.0
35.0
35.0
35.0
35.0
35.0
35.0
35.0
35.0
35.0
35.0
35.0
35.0
35.0

E1
(MeV)
214.0
218.0
222.0
226.0
230.0
234.0
238.0
246.0
250.0
254.0
258.0
266.0
270.0
274.0
278.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0
246.0
250.0
254.0
258.0
266.0
270.0
274.0
278.0

Table D.2 (Continued)dσp3dΩ1 dΩ2 dE1(MeV/c) (µb/sr2 MeV)
−138.0 10.28 ± 0.30
−128.7 13.24 ± 0.33
−119.3 13.64 ± 0.34
−109.9 15.27 ± 0.35
−100.4 16.53 ± 0.36
−90.9 18.05 ± 0.38
−81.3 19.03 ± 0.38
−61.9 20.78 ± 0.41
−52.1 20.41 ± 0.41
−42.1 18.32 ± 0.40
−32.1 17.08 ± 0.39
−11.7 15.86 ± 0.32
−1.7 14.61 ± 0.32
9.5 13.71 ± 0.31
20.2 11.32 ± 0.30
−135.0 10.07 ± 0.30
−125.6 11.71 ± 0.32
−116.2 12.86 ± 0.33
−106.7 14.15 ± 0.34
−97.2 14.80 ± 0.35
−87.6 16.41 ± 0.36
−78.0 16.86 ± 0.37
−58.4 17.56 ± 0.39
−48.5 17.18 ± 0.39
−38.5 16.43 ± 0.39
−28.4 15.88 ± 0.38
−7.9 15.13 ± 0.32
3.3 13.38 ± 0.31
13.6 11.74 ± 0.30
24.4 10.56 ± 0.29

Ay
0.092 ± 0.042
0.146 ± 0.035
0.109 ± 0.035
0.113 ± 0.032
0.101 ± 0.031
0.073 ± 0.029
0.077 ± 0.028
0.036 ± 0.028
0.083 ± 0.028
0.051 ± 0.030−0.020 ± 0.032
0.048 ± 0.032
0.050 ± 0.034
0.126 ± 0.036
0.146 ± 0.041
0.090 ± 0.042
0.156 ± 0.038
0.052 ± 0.036
0.074 ± 0.034
0.138 ± 0.033
0.071 ± 0.031
0.016 ± 0.031
0.049 ± 0.031
0.009 ± 0.032
0.043 ± 0.033
0.057 ± 0.034
0.061 ± 0.033
0.106 ± 0.036
0.082 ± 0.040
0.044 ± 0.043

APPENDIX D. NUMERICAL DATA TABLES

Esep
(MeV)
29.0
29.0
29.0
29.0
29.0
29.0
29.0
29.0
29.0
29.0
29.0
29.0
29.0
29.0
29.0
31.0
31.0
31.0
31.0
31.0
31.0
31.0
31.0
31.0
31.0
31.0
31.0
31.0
31.0
31.0

Table D.2 (Continued)dσp3dΩ1 dΩ2 dE1(MeV/c) (µb/sr2 MeV)
−144.1 11.09 ± 0.31
−134.9 12.96 ± 0.32
−125.7 14.96 ± 0.35
−116.4 16.33 ± 0.36
−107.0 18.36 ± 0.37
−97.6 18.97 ± 0.38
−88.2 21.63 ± 0.40
−69.1 22.30 ± 0.42
−59.4 22.24 ± 0.42
−49.7 21.76 ± 0.42
−40.0 20.98 ± 0.42
−20.5 19.66 ± 0.34
−11.4 16.71 ± 0.33
7.1 15.66 ± 0.32
13.8 13.95 ± 0.32
−141.0 10.80 ± 0.31
−131.8 12.74 ± 0.33
−122.5 14.44 ± 0.34
−113.1 17.28 ± 0.37
−103.7 18.20 ± 0.37
−94.3 19.28 ± 0.38
−84.7 19.99 ± 0.39
−65.5 21.69 ± 0.42
−55.7 21.88 ± 0.42
−45.9 21.11 ± 0.42
−36.0 19.84 ± 0.41
−16.0 17.75 ± 0.33
−6.6 16.08 ± 0.32
6.7 14.20 ± 0.32
16.5 13.25 ± 0.31

Table D.3: Data table for the 40 Ca(~p, 2p) reaction at the set3 in Table 2.1.
∆Esep = 2 MeV.
Esep
(MeV)
29.0
29.0
29.0
29.0
29.0
29.0
29.0
29.0
29.0
29.0
29.0
29.0
29.0
29.0
31.0
31.0
31.0
31.0
31.0
31.0
31.0
31.0
31.0
31.0
31.0
31.0
31.0
31.0

E1
(MeV)
202.0
206.0
210.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0
246.0
250.0
254.0
258.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0
246.0
250.0
254.0
258.0

p3
(MeV/c)−147.6−139.3−130.9−122.5−114.2−105.8
−97.5
−89.2
−80.9
−72.8
−57.0
−49.5
−42.7
−37.0
−144.4−136.0−127.5−119.1−110.6−102.2
−93.7
−85.3
−77.0
−68.7
−52.6
−45.1
−38.3
−32.6

dσ
dΩ1 dΩ2 dE1
2(µb/sr MeV)
9.30 ± 0.31
11.25 ± 0.33
12.39 ± 0.34
13.30 ± 0.36
15.23 ± 0.38
17.02 ± 0.39
18.52 ± 0.41
19.17 ± 0.41
20.77 ± 0.43
20.41 ± 0.43
20.00 ± 0.31
19.27 ± 0.31
18.89 ± 0.31
18.70 ± 0.31
9.14 ± 0.31
10.90 ± 0.33
12.89 ± 0.35
13.83 ± 0.36
14.34 ± 0.37
16.30 ± 0.39
18.05 ± 0.41
18.75 ± 0.41
19.61 ± 0.42
20.23 ± 0.43
18.70 ± 0.30
18.48 ± 0.30
18.38 ± 0.31
17.12 ± 0.30

Ay
0.106 ± 0.045
0.198 ± 0.040
0.147 ± 0.038−0.009 ± 0.037
0.079 ± 0.034
0.070 ± 0.032
0.088 ± 0.030
0.122 ± 0.029
0.086 ± 0.028
0.081 ± 0.028
0.061 ± 0.024
0.025 ± 0.025
0.046 ± 0.026
0.036 ± 0.026
0.041 ± 0.046
0.149 ± 0.041
0.197 ± 0.037
0.062 ± 0.036
0.019 ± 0.036
0.092 ± 0.033−0.008 ± 0.031
0.079 ± 0.029
0.021 ± 0.029
0.094 ± 0.028
0.039 ± 0.026
0.030 ± 0.026
0.030 ± 0.026
0.032 ± 0.028

Esep
(MeV)
33.0
33.0
33.0
33.0
33.0
33.0
33.0
33.0
33.0
33.0
33.0
33.0
33.0
33.0
35.0
35.0
35.0
35.0
35.0
35.0
35.0
35.0
35.0
35.0
35.0
35.0
35.0
35.0

E1
(MeV)
202.0
206.0
210.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0
246.0
250.0
254.0
258.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0
246.0
250.0
254.0
258.0

Table D.3 (Continued)dσp3dΩ1 dΩ2 dE1(MeV/c) (µb/sr2 MeV)
−141.2
9.69 ± 0.31
−132.7 11.04 ± 0.32
−124.2 11.85 ± 0.34
−115.6 13.56 ± 0.36
−107.1 14.92 ± 0.37
−98.6 15.63 ± 0.39
−90.0 17.68 ± 0.40
−81.5 17.64 ± 0.40
−73.0 18.24 ± 0.41
−64.6 18.61 ± 0.42
−48.2 17.35 ± 0.30
−40.6 16.99 ± 0.30
−33.7 16.38 ± 0.30
−28.3 16.27 ± 0.30
−138.09.46 ± 0.31
−129.4 10.40 ± 0.32
−120.8 11.72 ± 0.34
−112.2 12.18 ± 0.35
−103.6 13.90 ± 0.37
−94.9 15.13 ± 0.38
−86.3 15.69 ± 0.39
−77.7 16.11 ± 0.39
−69.1 16.66 ± 0.40
−60.5 17.34 ± 0.41
−43.9 16.14 ± 0.29
−36.1 14.74 ± 0.29
−29.2 15.08 ± 0.29
−24.1 14.50 ± 0.29

Ay
0.083 ± 0.044
0.104 ± 0.040
0.101 ± 0.040
0.085 ± 0.037
0.040 ± 0.035
0.168 ± 0.034
0.150 ± 0.031
0.037 ± 0.031
0.025 ± 0.030−0.023 ± 0.030
0.040 ± 0.027
0.073 ± 0.028
0.110 ± 0.029
0.089 ± 0.029
0.063 ± 0.045
0.031 ± 0.043
0.088 ± 0.040
0.064 ± 0.040
0.087 ± 0.036
0.021 ± 0.035
0.060 ± 0.034
0.039 ± 0.033
0.028 ± 0.032
0.052 ± 0.032
0.054 ± 0.029
0.063 ± 0.031
0.035 ± 0.031
0.079 ± 0.032

113

114

E1
(MeV)
202.0
206.0
210.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0
246.0
250.0
254.0
258.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0
246.0
250.0
254.0
258.0

Ay
0.061 ± 0.049
0.082 ± 0.045
0.067 ± 0.043
0.041 ± 0.042
0.065 ± 0.039
0.041 ± 0.038
0.020 ± 0.037−0.008 ± 0.035
0.085 ± 0.035
0.026 ± 0.034
0.058 ± 0.032
0.054 ± 0.032
0.052 ± 0.032
0.084 ± 0.036
0.120 ± 0.053
0.089 ± 0.052
0.091 ± 0.045−0.040 ± 0.046
0.018 ± 0.044
0.067 ± 0.042
0.120 ± 0.042
0.030 ± 0.040
0.100 ± 0.039
0.056 ± 0.038
0.006 ± 0.035−0.007 ± 0.036
0.109 ± 0.037
0.059 ± 0.038

Esep
(MeV)
41.0
41.0
41.0
41.0
41.0
41.0
41.0
41.0
41.0
41.0
41.0
41.0
41.0
41.0
43.0
43.0
43.0
43.0
43.0
43.0
43.0
43.0
43.0
43.0
43.0
43.0
43.0
43.0

E1
(MeV)
202.0
206.0
210.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0
246.0
250.0
254.0
258.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0
246.0
250.0
254.0
258.0

Table D.3 (Continued)dσp3dΩ1 dΩ2 dE1(MeV/c) (µb/sr2 MeV)
−128.6
7.06 ± 0.29
−119.87.20 ± 0.29
−110.98.64 ± 0.31
−102.1
9.23 ± 0.33
−93.2 10.01 ± 0.33
−84.3 10.59 ± 0.34
−75.3 10.53 ± 0.35
−66.3 11.57 ± 0.35
−57.3 11.69 ± 0.36
−48.3 11.90 ± 0.37
−30.6 11.60 ± 0.27
−22.3 11.01 ± 0.27
−15.6 10.81 ± 0.27
13.3
9.96 ± 0.27
−125.5
6.81 ± 0.29
−116.6
7.22 ± 0.29
−107.77.48 ± 0.30
−98.8
7.97 ± 0.31
−89.8
8.74 ± 0.32
−80.8
8.75 ± 0.33
−71.7
9.90 ± 0.34
−62.6 10.70 ± 0.35
−53.5 10.85 ± 0.36
−44.3 10.70 ± 0.36
−26.2 10.99 ± 0.27
−17.7 10.19 ± 0.27
−11.2 10.23 ± 0.27
11.5
9.94 ± 0.27

Ay
0.070 ± 0.057
0.042 ± 0.056
0.088 ± 0.049
0.079 ± 0.049
0.059 ± 0.046
0.017 ± 0.045
0.051 ± 0.046
0.080 ± 0.041
0.032 ± 0.041
0.020 ± 0.041
0.035 ± 0.037
0.008 ± 0.039
0.037 ± 0.040
0.083 ± 0.043
0.036 ± 0.058
0.139 ± 0.056
0.052 ± 0.055
0.055 ± 0.054
0.026 ± 0.051
0.035 ± 0.051
0.067 ± 0.047
0.037 ± 0.044−0.002 ± 0.044
0.064 ± 0.045
0.063 ± 0.039
0.067 ± 0.041
0.109 ± 0.042
0.068 ± 0.043

APPENDIX D. NUMERICAL DATA TABLES

Esep
(MeV)
37.0
37.0
37.0
37.0
37.0
37.0
37.0
37.0
37.0
37.0
37.0
37.0
37.0
37.0
39.0
39.0
39.0
39.0
39.0
39.0
39.0
39.0
39.0
39.0
39.0
39.0
39.0
39.0

Table D.3 (Continued)dσp3dΩ1 dΩ2 dE1(MeV/c) (µb/sr2 MeV)
−134.88.56 ± 0.30
−126.29.73 ± 0.32
−117.5 10.47 ± 0.32
−108.8 11.46 ± 0.35
−100.1 12.46 ± 0.35
−91.4 13.40 ± 0.37
−82.6 13.74 ± 0.37
−73.9 14.39 ± 0.38
−65.1 15.10 ± 0.39
−56.4 15.75 ± 0.39
−39.5 14.05 ± 0.28
−31.5 14.19 ± 0.28
−24.7 14.03 ± 0.29
−20.0 12.50 ± 0.28
−131.77.80 ± 0.30
−123.08.04 ± 0.30
−114.29.72 ± 0.32
−105.4 10.06 ± 0.33
−96.6 10.86 ± 0.34
−87.8 11.54 ± 0.35
−78.9 11.58 ± 0.35
−70.1 12.05 ± 0.36
−61.2 12.76 ± 0.37
−52.3 13.51 ± 0.38
−35.0 12.27 ± 0.27
−27.0 12.16 ± 0.27
−20.1 11.66 ± 0.27
−16.3 11.46 ± 0.28

Esep
(MeV)
45.0
45.0
45.0
45.0
45.0
45.0
45.0
45.0
45.0
45.0
45.0
45.0
45.0
45.0
47.0
47.0
47.0
47.0
47.0
47.0
47.0
47.0
47.0
47.0
47.0
47.0
47.0
47.0

E1
(MeV)
202.0
206.0
210.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0
246.0
250.0
254.0
258.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0
246.0
250.0
254.0
258.0

Table D.3 (Continued)dσp3dΩ1 dΩ2 dE1(MeV/c) (µb/sr2 MeV)
−122.45.95 ± 0.28
−113.5
6.28 ± 0.29
−104.5
7.25 ± 0.30
−95.5
8.09 ± 0.31
−86.4
7.84 ± 0.32
−77.3
8.33 ± 0.33
−68.2
8.59 ± 0.33
−58.9
9.66 ± 0.34
−49.7 10.21 ± 0.35
−40.3
9.93 ± 0.35
−21.7 10.63 ± 0.27
−13.0 10.10 ± 0.26
−7.3 10.17 ± 0.27
11.6 10.01 ± 0.27
−119.46.00 ± 0.28
−110.4
6.34 ± 0.29
−101.4
6.49 ± 0.29
−92.3
6.88 ± 0.30
−83.1
7.16 ± 0.31
−73.9
7.98 ± 0.32
−64.7
9.45 ± 0.34
−55.4 10.02 ± 0.35
−46.0 10.54 ± 0.35
−36.5
9.78 ± 0.35
−17.4 10.47 ± 0.26
−8.2 10.91 ± 0.27
5.1 10.35 ± 0.27
13.6
9.62 ± 0.27

Ay
0.003 ± 0.065
0.133 ± 0.063
0.110 ± 0.057
0.055 ± 0.053
0.073 ± 0.055
0.148 ± 0.054
0.077 ± 0.053
0.019 ± 0.047
0.012 ± 0.046
0.097 ± 0.048
0.015 ± 0.040
0.069 ± 0.042
0.084 ± 0.042
0.114 ± 0.043
0.087 ± 0.064
0.087 ± 0.062
0.062 ± 0.061
0.040 ± 0.061
0.120 ± 0.059
0.046 ± 0.055
0.118 ± 0.049
0.094 ± 0.046
0.176 ± 0.045
0.019 ± 0.048
0.072 ± 0.040
0.085 ± 0.039
0.024 ± 0.041
0.073 ± 0.044

Esep
(MeV)
49.0
49.0
49.0
49.0
49.0
49.0
49.0
49.0
49.0
49.0
49.0
49.0
49.0
49.0
51.0
51.0
51.0
51.0
51.0
51.0
51.0
51.0
51.0
51.0
51.0
51.0
51.0
51.0

E1
(MeV)
202.0
206.0
210.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0
246.0
250.0
254.0
258.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0
246.0
250.0
254.0
258.0

Table D.3 (Continued)dσp3dΩ1 dΩ2 dE1(MeV/c) (µb/sr2 MeV)
−116.45.15 ± 0.27
−107.36.30 ± 0.28
−98.2
6.33 ± 0.29
−89.1
7.57 ± 0.31
−79.9
7.30 ± 0.31
−70.6
8.37 ± 0.32
−61.3
8.11 ± 0.32
−51.9
9.58 ± 0.34
−42.4
9.66 ± 0.34
−32.8
9.59 ± 0.35
−13.3 10.21 ± 0.26
−3.5 10.08 ± 0.27
6.8 10.02 ± 0.27
17.0 10.31 ± 0.27
−113.45.07 ± 0.27
−104.35.38 ± 0.28
−95.2
6.19 ± 0.28
−86.0
6.51 ± 0.30
−76.7
7.03 ± 0.31
−67.4
7.20 ± 0.31
−58.0
7.87 ± 0.32
−48.5
8.82 ± 0.34
−38.9
9.39 ± 0.34
−29.3
9.33 ± 0.35
−9.7
9.82 ± 0.26
1.9
9.44 ± 0.26
10.8
9.57 ± 0.26
21.0
9.42 ± 0.26

Ay
0.056 ± 0.073
0.080 ± 0.062
0.000 ± 0.062
0.183 ± 0.056
0.048 ± 0.059−0.020 ± 0.053
0.072 ± 0.055
0.046 ± 0.047
0.072 ± 0.048
0.082 ± 0.048
0.074 ± 0.041
0.014 ± 0.042
0.075 ± 0.042
0.063 ± 0.041−0.008 ± 0.074
0.108 ± 0.071
0.009 ± 0.063
0.050 ± 0.064
0.084 ± 0.060
0.091 ± 0.060
0.063 ± 0.056−0.010 ± 0.051
0.031 ± 0.049−0.021 ± 0.050
0.032 ± 0.042
0.044 ± 0.044
0.100 ± 0.044
0.069 ± 0.045

115

116
Table D.4: Data table for the 40 Ca(~p, 2p) reaction at the set4 in Table 2.1.
∆Esep = 2 MeV.

E1
(MeV)
202.0
206.0
210.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0
246.0
250.0
254.0
258.0

Ay
0.094 ± 0.074
0.135 ± 0.073−0.058 ± 0.070
0.095 ± 0.067
0.070 ± 0.065
0.083 ± 0.064
0.085 ± 0.056
0.110 ± 0.053
0.032 ± 0.053
0.047 ± 0.051
0.018 ± 0.047
0.065 ± 0.044
0.059 ± 0.045
0.034 ± 0.047

E1
(MeV)
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0

p3
(MeV/c)−140.9−133.0−125.1−109.3−101.5
−93.7
−86.0
−78.3
−70.9
−63.6
−56.6
−50.1
−44.3
−137.8−129.7−121.7−105.8
−97.9
−90.0
−82.1
−74.4
−66.8
−59.4
−52.3
−45.7
−39.9

dσ
dΩ1 dΩ2 dE1
2(µb/sr MeV)
5.57 ± 0.36
6.41 ± 0.37
7.55 ± 0.39
7.55 ± 0.32
8.64 ± 0.33
8.83 ± 0.34
9.23 ± 0.32
9.74 ± 0.33
9.15 ± 0.32
10.75 ± 0.34
9.30 ± 0.26
9.78 ± 0.27
9.93 ± 0.27
5.20 ± 0.35
5.57 ± 0.36
6.25 ± 0.37
7.35 ± 0.32
7.09 ± 0.32
8.59 ± 0.34
8.48 ± 0.31
8.76 ± 0.32
9.69 ± 0.33
9.23 ± 0.33
9.28 ± 0.26
8.67 ± 0.26
8.75 ± 0.27

Ay
0.088 ± 0.096
0.051 ± 0.087
0.104 ± 0.078−0.004 ± 0.063−0.043 ± 0.057−0.079 ± 0.057
0.031 ± 0.049
0.127 ± 0.048
0.090 ± 0.051−0.046 ± 0.045−0.017 ± 0.045
0.066 ± 0.043
0.040 ± 0.043
0.156 ± 0.101
0.051 ± 0.098−0.054 ± 0.090
0.117 ± 0.063
0.056 ± 0.065
0.032 ± 0.058
0.086 ± 0.052−0.007 ± 0.052
0.078 ± 0.048
0.050 ± 0.051
0.094 ± 0.045
0.000 ± 0.048
0.056 ± 0.048

APPENDIX D. NUMERICAL DATA TABLES

Esep
(MeV)
53.0
53.0
53.0
53.0
53.0
53.0
53.0
53.0
53.0
53.0
53.0
53.0
53.0
53.0

Table D.3 (Continued)dσp3dΩ1 dΩ2 dE1(MeV/c) (µb/sr2 MeV)
−110.65.03 ± 0.27
−101.45.13 ± 0.27
−92.2
5.56 ± 0.28
−83.0
6.11 ± 0.30
−73.7
6.36 ± 0.30
−64.3
6.56 ± 0.30
−54.8
7.81 ± 0.32
−45.3
8.30 ± 0.33
−35.7
8.43 ± 0.33
−26.1
9.08 ± 0.34
−7.6
8.80 ± 0.26
6.5
9.44 ± 0.26
15.4
9.15 ± 0.26
25.5
8.80 ± 0.26

Esep
(MeV)
41.0
41.0
41.0
41.0
41.0
41.0
41.0
41.0
41.0
41.0
41.0
41.0
41.0
43.0
43.0
43.0
43.0
43.0
43.0
43.0
43.0
43.0
43.0
43.0
43.0
43.0

Esep
(MeV)
45.0
45.0
45.0
45.0
45.0
45.0
45.0
45.0
45.0
45.0
45.0
45.0
45.0
47.0
47.0
47.0
47.0
47.0
47.0
47.0
47.0
47.0
47.0
47.0
47.0
47.0

E1
(MeV)
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0

Table D.4 (Continued)dσp3dΩ1 dΩ2 dE1(MeV/c) (µb/sr2 MeV)
−134.6
4.84 ± 0.35
−126.54.95 ± 0.35
−118.46.70 ± 0.37
−102.3
6.52 ± 0.31
−94.3
7.16 ± 0.32
−86.3
7.44 ± 0.32
−78.3
8.03 ± 0.31
−70.4
8.51 ± 0.31
−62.7
9.02 ± 0.32
−55.2
8.94 ± 0.33
−48.0
9.34 ± 0.26
−41.3
8.82 ± 0.26
−35.5
9.35 ± 0.27
−131.54.89 ± 0.35
−123.35.04 ± 0.35
−115.15.93 ± 0.37
−98.8
6.30 ± 0.31
−90.7
6.72 ± 0.31
−82.6
7.77 ± 0.33
−74.5
8.11 ± 0.31
−66.5
7.63 ± 0.31
−58.6
8.95 ± 0.32
−50.9
9.31 ± 0.33
−43.6
8.81 ± 0.26
−36.8
9.02 ± 0.27
−31.1
8.72 ± 0.26

Ay
0.296 ± 0.109
0.195 ± 0.105
0.147 ± 0.084
0.017 ± 0.069
0.083 ± 0.066−0.024 ± 0.064
0.038 ± 0.055
0.038 ± 0.053
0.003 ± 0.051
0.056 ± 0.052
0.019 ± 0.044
0.052 ± 0.047
0.020 ± 0.045
0.116 ± 0.108−0.049 ± 0.106
0.099 ± 0.094
0.267 ± 0.072−0.034 ± 0.069
0.075 ± 0.062
0.022 ± 0.054−0.092 ± 0.058
0.059 ± 0.051−0.032 ± 0.051−0.018 ± 0.047−0.009 ± 0.047
0.084 ± 0.048

Esep
(MeV)
49.0
49.0
49.0
49.0
49.0
49.0
49.0
49.0
49.0
49.0
49.0
49.0
49.0
51.0
51.0
51.0
51.0
51.0
51.0
51.0
51.0
51.0
51.0
51.0
51.0
51.0

E1
(MeV)
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0

Table D.4 (Continued)dσp3dΩ1 dΩ2 dE1(MeV/c) (µb/sr2 MeV)
−128.35.44 ± 0.36
−120.14.80 ± 0.35
−111.95.47 ± 0.36
−95.4
6.27 ± 0.31
−87.1
7.08 ± 0.32
−78.9
6.85 ± 0.32
−70.7
6.94 ± 0.30
−62.6
8.34 ± 0.31
−54.5
8.36 ± 0.32
−46.7
8.86 ± 0.33
−39.2
8.56 ± 0.26
−32.4
8.59 ± 0.26
−26.7
8.72 ± 0.27
−125.34.30 ± 0.34
−116.94.98 ± 0.35
−108.6
5.34 ± 0.36
−91.9
5.95 ± 0.30
−83.6
6.71 ± 0.32
−75.3
6.77 ± 0.32
−66.9
7.20 ± 0.30
−58.7
8.29 ± 0.32
−50.5
8.26 ± 0.32
−42.4
9.08 ± 0.33
−34.8
8.03 ± 0.26
−27.9
8.58 ± 0.26
−22.4
8.39 ± 0.27

Ay
0.092 ± 0.098
0.102 ± 0.110
0.023 ± 0.100
0.107 ± 0.071
0.092 ± 0.066−0.002 ± 0.069
0.092 ± 0.062
0.066 ± 0.054
0.058 ± 0.054
0.046 ± 0.053
0.110 ± 0.048
0.096 ± 0.048
0.096 ± 0.048−0.056 ± 0.120−0.098 ± 0.107
0.132 ± 0.101−0.038 ± 0.075
0.011 ± 0.069
0.124 ± 0.069
0.117 ± 0.059
0.093 ± 0.054
0.051 ± 0.055
0.126 ± 0.052
0.019 ± 0.051−0.031 ± 0.049
0.102 ± 0.050

117

118

E1
(MeV)
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0

Ay
0.015 ± 0.111−0.001 ± 0.113−0.075 ± 0.114
0.034 ± 0.086
0.031 ± 0.076
0.074 ± 0.073
0.044 ± 0.058−0.014 ± 0.059−0.024 ± 0.056
0.020 ± 0.056
0.038 ± 0.051
0.063 ± 0.049
0.016 ± 0.053
0.183 ± 0.125−0.070 ± 0.113−0.038 ± 0.110
0.083 ± 0.086−0.002 ± 0.088
0.044 ± 0.071
0.070 ± 0.061−0.005 ± 0.059
0.033 ± 0.060
0.087 ± 0.056−0.080 ± 0.053
0.122 ± 0.053
0.016 ± 0.054

Esep
(MeV)
57.0
57.0
57.0
57.0
57.0
57.0
57.0
57.0
57.0
57.0
57.0
57.0
57.0
59.0
59.0
59.0
59.0
59.0
59.0
59.0
59.0
59.0
59.0
59.0
59.0
59.0

E1
(MeV)
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0

Table D.4 (Continued)dσp3dΩ1 dΩ2 dE1(MeV/c) (µb/sr2 MeV)
−116.23.74 ± 0.33
−107.6
4.34 ± 0.34
−99.1
4.43 ± 0.35
−81.9
5.08 ± 0.30
−73.2
5.48 ± 0.31
−64.5
6.15 ± 0.32
−55.8
5.90 ± 0.29
−47.1
7.23 ± 0.30
−38.4
7.03 ± 0.31
−29.7
7.64 ± 0.32
−21.4
6.60 ± 0.25
−14.1
7.06 ± 0.26
10.5
7.39 ± 0.26
−113.23.90 ± 0.34
−104.64.05 ± 0.34
−96.0
4.21 ± 0.34
−78.6
4.17 ± 0.29
−69.9
5.10 ± 0.30
−61.1
5.38 ± 0.31
−52.2
6.02 ± 0.29
−43.3
6.68 ± 0.30
−34.4
6.41 ± 0.30
−25.6
7.04 ± 0.31
−17.0
6.59 ± 0.25
−9.5
6.88 ± 0.25
8.2
6.93 ± 0.26

Ay
−0.038 ± 0.135−0.134 ± 0.120
0.125 ± 0.119
0.237 ± 0.085
0.068 ± 0.082
0.050 ± 0.075
0.016 ± 0.070−0.103 ± 0.061
0.029 ± 0.063
0.076 ± 0.059
0.041 ± 0.060
0.032 ± 0.058
0.081 ± 0.056
0.007 ± 0.131−0.011 ± 0.127
0.083 ± 0.123
0.138 ± 0.102
0.071 ± 0.086−0.097 ± 0.084−0.028 ± 0.070
0.050 ± 0.064−0.094 ± 0.068−0.044 ± 0.063−0.032 ± 0.060−0.032 ± 0.059
0.079 ± 0.059

APPENDIX D. NUMERICAL DATA TABLES

Esep
(MeV)
53.0
53.0
53.0
53.0
53.0
53.0
53.0
53.0
53.0
53.0
53.0
53.0
53.0
55.0
55.0
55.0
55.0
55.0
55.0
55.0
55.0
55.0
55.0
55.0
55.0
55.0

Table D.4 (Continued)dσp3dΩ1 dΩ2 dE1(MeV/c) (µb/sr2 MeV)
−122.2
4.82 ± 0.35
−113.84.70 ± 0.35
−105.44.63 ± 0.35
−88.5
5.08 ± 0.30
−80.1
6.07 ± 0.31
−71.6
6.35 ± 0.32
−63.2
7.45 ± 0.30
−54.8
7.42 ± 0.30
−46.4
8.00 ± 0.32
−38.2
8.26 ± 0.32
−30.4
7.91 ± 0.26
−23.3
8.51 ± 0.26
−18.1
7.95 ± 0.26
−119.24.08 ± 0.34
−110.7
4.66 ± 0.35
−102.24.77 ± 0.35
−85.2
5.12 ± 0.30
−76.6
4.99 ± 0.30
−68.1
6.60 ± 0.32
−59.5
7.01 ± 0.30
−50.9
7.47 ± 0.31
−42.4
7.37 ± 0.31
−34.0
8.08 ± 0.32
−25.9
7.70 ± 0.26
−18.7
7.79 ± 0.26
−14.1
7.79 ± 0.26

Esep
(MeV)
61.0
61.0
61.0
61.0
61.0
61.0
61.0
61.0
61.0
61.0
61.0
61.0
61.0
63.0
63.0
63.0
63.0
63.0
63.0
63.0
63.0
63.0
63.0
63.0
63.0
63.0

E1
(MeV)
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0

Table D.4 (Continued)dσp3dΩ1 dΩ2 dE1(MeV/c) (µb/sr2 MeV)
−110.33.99 ± 0.34
−101.7
4.39 ± 0.35
−93.0
4.66 ± 0.35
−75.4
5.32 ± 0.30
−66.6
5.95 ± 0.31
−57.7
5.33 ± 0.30
−48.7
6.01 ± 0.29
−39.7
6.12 ± 0.30
−30.6
6.21 ± 0.30
−21.5
6.71 ± 0.31
−12.6
6.03 ± 0.25
−4.9
6.27 ± 0.25
8.4
5.97 ± 0.25
−107.53.71 ± 0.34
−98.8
4.09 ± 0.34
−90.0
4.44 ± 0.35
−72.3
3.99 ± 0.29
−63.4
5.11 ± 0.30
−54.4
5.24 ± 0.30
−45.4
5.73 ± 0.29
−36.2
5.15 ± 0.29
−27.0
5.88 ± 0.30
−17.7
5.99 ± 0.30
−8.4
5.98 ± 0.25
1.7
5.51 ± 0.25
10.9
5.61 ± 0.25

Ay
0.033 ± 0.130
0.138 ± 0.118
0.085 ± 0.114
0.110 ± 0.082
0.146 ± 0.075
0.023 ± 0.084
0.014 ± 0.070
0.108 ± 0.069
0.053 ± 0.069
0.075 ± 0.065−0.072 ± 0.065
0.049 ± 0.063
0.045 ± 0.067−0.241 ± 0.139
0.033 ± 0.126
0.107 ± 0.118−0.161 ± 0.107−0.026 ± 0.086−0.009 ± 0.085−0.037 ± 0.072−0.021 ± 0.079
0.112 ± 0.072
0.081 ± 0.072−0.002 ± 0.066
0.075 ± 0.071−0.020 ± 0.072

Esep
(MeV)
65.0
65.0
65.0
65.0
65.0
65.0
65.0
65.0
65.0
65.0
65.0
65.0
65.0
67.0
67.0
67.0
67.0
67.0
67.0
67.0
67.0
67.0
67.0
67.0
67.0
67.0

E1
(MeV)
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0

Table D.4 (Continued)dσp3dΩ1 dΩ2 dE1(MeV/c) (µb/sr2 MeV)
−104.7
3.78 ± 0.33
−95.9
3.92 ± 0.34
−87.1
4.31 ± 0.35
−69.3
4.42 ± 0.29
−60.3
4.74 ± 0.30
−51.3
4.27 ± 0.29
−42.2
5.61 ± 0.29
−33.0
5.52 ± 0.29
−23.7
5.82 ± 0.29
−14.4
5.70 ± 0.30
−5.1
4.86 ± 0.24
5.3
4.72 ± 0.24
14.7
4.50 ± 0.24
−102.03.40 ± 0.33
−93.1
3.35 ± 0.33
−84.3
3.99 ± 0.34
−66.4
3.89 ± 0.28
−57.4
4.05 ± 0.29
−48.3
4.47 ± 0.30
−39.2
5.14 ± 0.28
−30.0
4.68 ± 0.29
−20.8
5.69 ± 0.29
−11.8
5.04 ± 0.29
−5.2
4.58 ± 0.24
10.0
4.80 ± 0.24
19.1
4.27 ± 0.24

Ay
0.085 ± 0.133
0.124 ± 0.130
0.039 ± 0.121−0.038 ± 0.096
0.151 ± 0.092
0.109 ± 0.101−0.033 ± 0.074
0.201 ± 0.075
0.084 ± 0.072
0.077 ± 0.074−0.097 ± 0.079−0.055 ± 0.081
0.084 ± 0.086
0.244 ± 0.148
0.181 ± 0.150−0.097 ± 0.129−0.048 ± 0.107−0.039 ± 0.105
0.092 ± 0.097−0.029 ± 0.079
0.004 ± 0.087
0.125 ± 0.073
0.124 ± 0.082−0.079 ± 0.083
0.045 ± 0.080−0.022 ± 0.090

119

120

E1
(MeV)
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0

Ay
0.358 ± 0.159
0.389 ± 0.162
0.160 ± 0.146−0.037 ± 0.113
0.021 ± 0.106
0.039 ± 0.099
0.132 ± 0.084
0.057 ± 0.086−0.137 ± 0.086
0.066 ± 0.081−0.102 ± 0.089−0.021 ± 0.091−0.058 ± 0.094
0.012 ± 0.138
0.083 ± 0.159−0.132 ± 0.142
0.013 ± 0.108
0.116 ± 0.111−0.039 ± 0.116−0.029 ± 0.091−0.117 ± 0.092−0.017 ± 0.084
0.034 ± 0.099
0.073 ± 0.089−0.102 ± 0.092
0.017 ± 0.104

Esep
(MeV)
73.0
73.0
73.0
73.0
73.0
73.0
73.0
73.0
73.0
73.0
73.0
73.0
73.0

E1
(MeV)
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0
230.0
234.0
238.0

Table D.4 (Continued)dσp3dΩ1 dΩ2 dE1(MeV/c) (µb/sr2 MeV)
−94.3
2.75 ± 0.32
−85.4
3.07 ± 0.33
−76.5
3.47 ± 0.33
−58.6
2.90 ± 0.28
−49.7
3.59 ± 0.28
−40.8
4.15 ± 0.29
−32.2
3.96 ± 0.27
−24.1
4.10 ± 0.28
−17.4
3.88 ± 0.28
−14.5
3.86 ± 0.28
17.6
3.68 ± 0.23
24.7
3.33 ± 0.23
33.5
3.08 ± 0.23

Ay
−0.129 ± 0.174
0.306 ± 0.160
0.089 ± 0.145−0.064 ± 0.140
0.180 ± 0.116−0.076 ± 0.102
0.023 ± 0.099
0.182 ± 0.097−0.031 ± 0.103
0.122 ± 0.102−0.052 ± 0.101
0.262 ± 0.111−0.134 ± 0.121

APPENDIX D. NUMERICAL DATA TABLES

Esep
(MeV)
69.0
69.0
69.0
69.0
69.0
69.0
69.0
69.0
69.0
69.0
69.0
69.0
69.0
71.0
71.0
71.0
71.0
71.0
71.0
71.0
71.0
71.0
71.0
71.0
71.0
71.0

Table D.4 (Continued)dσp3dΩ1 dΩ2 dE1(MeV/c) (µb/sr2 MeV)
−99.3
3.19 ± 0.34
−90.5
3.08 ± 0.33
−81.6
3.44 ± 0.33
−63.6
3.71 ± 0.29
−54.6
4.02 ± 0.29
−45.5
4.28 ± 0.29
−36.5
4.72 ± 0.28
−27.4
4.73 ± 0.28
−18.6
4.80 ± 0.29
−10.9
5.13 ± 0.29
8.6
4.27 ± 0.24
14.8
4.12 ± 0.24
23.8
4.09 ± 0.24
−96.8
3.64 ± 0.33
−87.9
3.11 ± 0.33
−79.0
3.46 ± 0.33
−61.0
3.85 ± 0.28
−52.0
3.76 ± 0.29
−43.0
3.62 ± 0.29
−34.1
4.41 ± 0.28
−25.4
4.32 ± 0.28
−17.4
4.81 ± 0.28
−11.9
4.01 ± 0.28
13.0
4.18 ± 0.24
19.8
4.15 ± 0.24
28.6
3.55 ± 0.23

Table D.5: Data table for the 40 Ca(~p, 2p) reaction at the set4 in Table 2.1.
∆Esep = 2 MeV.
Esep
(MeV)
59.0
59.0
59.0
59.0
59.0
59.0
59.0
59.0
59.0
59.0
59.0
59.0
61.0
61.0
61.0
61.0
61.0
61.0
61.0
61.0
61.0
61.0
61.0
61.0

E1
(MeV)
174.0
178.0
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0
174.0
178.0
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0

p3
(MeV/c)−119.0−111.1
−95.3
−87.5
−79.7
−64.2
−56.6
−49.2
−42.2
−35.7
−30.2
−26.4
−115.9−108.0
−92.0
−84.1
−76.1
−60.4
−52.7
−45.1
−37.9
−31.3
−25.8
−22.5

dσ
dΩ1 dΩ2 dE1
2(µb/sr MeV)
3.36 ± 0.27
3.74 ± 0.27
4.19 ± 0.29
4.35 ± 0.29
4.52 ± 0.29
4.82 ± 0.28
4.97 ± 0.29
5.98 ± 0.30
4.76 ± 0.18
5.18 ± 0.18
5.01 ± 0.18
5.60 ± 0.19
3.48 ± 0.27
3.67 ± 0.27
3.94 ± 0.28
3.82 ± 0.28
4.07 ± 0.29
4.54 ± 0.28
4.94 ± 0.29
4.84 ± 0.29
4.52 ± 0.18
5.00 ± 0.18
5.07 ± 0.18
5.21 ± 0.19

Ay
0.165 ± 0.114
0.074 ± 0.105
0.094 ± 0.098−0.092 ± 0.095
0.034 ± 0.093
0.085 ± 0.086
0.146 ± 0.085−0.005 ± 0.074
0.061 ± 0.134
0.134 ± 0.057
0.076 ± 0.059−0.039 ± 0.055
0.039 ± 0.111−0.116 ± 0.106
0.112 ± 0.102−0.086 ± 0.106
0.108 ± 0.101
0.102 ± 0.090−0.041 ± 0.085−0.036 ± 0.088−0.028 ± 0.063−0.020 ± 0.058
0.022 ± 0.058−0.072 ± 0.058

Esep
(MeV)
63.0
63.0
63.0
63.0
63.0
63.0
63.0
63.0
63.0
63.0
63.0
63.0
65.0
65.0
65.0
65.0
65.0
65.0
65.0
65.0
65.0
65.0
65.0
65.0

E1
(MeV)
174.0
178.0
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0
174.0
178.0
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0

Table D.5 (Continued)dσp3dΩ1 dΩ2 dE1(MeV/c) (µb/sr2 MeV)
−112.93.23 ± 0.27
−104.93.73 ± 0.27
−88.7
4.24 ± 0.29
−80.7
4.19 ± 0.28
−72.6
4.54 ± 0.30
−56.6
4.11 ± 0.28
−48.7
4.84 ± 0.29
−41.0
5.05 ± 0.29
−33.6
4.70 ± 0.18
−26.8
4.53 ± 0.18
−21.5
5.09 ± 0.18
18.9
4.77 ± 0.18
−110.03.13 ± 0.27
−101.83.01 ± 0.26
−85.5
3.32 ± 0.28
−77.3
4.01 ± 0.28
−69.1
4.55 ± 0.29
−52.8
4.06 ± 0.27
−44.8
4.65 ± 0.29
−36.9
4.99 ± 0.29
−29.2
3.97 ± 0.17
−22.4
4.42 ± 0.18
−17.2
4.23 ± 0.18
15.7
4.49 ± 0.18

Ay
0.049 ± 0.118
0.001 ± 0.103−0.032 ± 0.097
0.169 ± 0.097
0.072 ± 0.093−0.160 ± 0.099
0.081 ± 0.086−0.012 ± 0.085−0.064 ± 0.061
0.052 ± 0.063
0.071 ± 0.058−0.059 ± 0.063
0.082 ± 0.121
0.043 ± 0.125−0.027 ± 0.120−0.021 ± 0.102
0.141 ± 0.091
0.088 ± 0.097−0.003 ± 0.090−0.111 ± 0.086
0.033 ± 0.070
0.004 ± 0.065−0.022 ± 0.069
0.029 ± 0.066

121

122

E1
(MeV)
174.0
178.0
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0
174.0
178.0
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0

Ay
0.081 ± 0.143−0.043 ± 0.128−0.064 ± 0.108−0.184 ± 0.099−0.102 ± 0.100−0.084 ± 0.098
0.090 ± 0.097
0.152 ± 0.108
0.087 ± 0.074
0.006 ± 0.066
0.068 ± 0.068−0.002 ± 0.075−0.223 ± 0.138
0.066 ± 0.119
0.014 ± 0.107−0.129 ± 0.135
0.025 ± 0.107−0.007 ± 0.099−0.046 ± 0.097−0.017 ± 0.101
0.062 ± 0.077−0.109 ± 0.077
0.009 ± 0.078
0.066 ± 0.077

Esep
(MeV)
71.0
71.0
71.0
71.0
71.0
71.0
71.0
71.0
71.0
71.0
71.0
71.0
73.0
73.0
73.0
73.0
73.0
73.0
73.0
73.0
73.0
73.0
73.0
73.0

E1
(MeV)
174.0
178.0
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0
174.0
178.0
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0

Table D.5 (Continued)dσp3dΩ1 dΩ2 dE1(MeV/c) (µb/sr2 MeV)
−101.32.89 ± 0.27
−92.9
3.12 ± 0.27
−76.1
3.31 ± 0.27
−67.6
3.20 ± 0.28
−59.1
3.89 ± 0.28
−42.0
4.50 ± 0.28
−33.4
3.68 ± 0.28
−24.8
3.34 ± 0.27
−16.3
3.46 ± 0.17
−8.7
3.39 ± 0.17
6.7
3.38 ± 0.17
13.4
3.54 ± 0.18
−98.5
2.71 ± 0.26
−90.1
3.13 ± 0.27
−73.1
3.36 ± 0.28
−64.6
3.67 ± 0.28
−56.0
3.03 ± 0.27
−38.6
3.69 ± 0.27
−29.9
3.78 ± 0.28
−21.1
3.19 ± 0.28
−12.3
3.36 ± 0.17
−4.0
3.05 ± 0.17
6.9
3.21 ± 0.17
15.8
3.20 ± 0.17

Ay
−0.043 ± 0.132
0.272 ± 0.124
0.042 ± 0.118−0.104 ± 0.123−0.068 ± 0.105−0.021 ± 0.091
0.002 ± 0.109−0.183 ± 0.120−0.103 ± 0.079−0.079 ± 0.081
0.065 ± 0.082
0.042 ± 0.080
0.198 ± 0.139
0.117 ± 0.122
0.018 ± 0.121−0.077 ± 0.111−0.097 ± 0.129−0.047 ± 0.106
0.054 ± 0.108−0.241 ± 0.128−0.168 ± 0.081
0.144 ± 0.089
0.031 ± 0.086−0.047 ± 0.088

APPENDIX D. NUMERICAL DATA TABLES

Esep
(MeV)
67.0
67.0
67.0
67.0
67.0
67.0
67.0
67.0
67.0
67.0
67.0
67.0
69.0
69.0
69.0
69.0
69.0
69.0
69.0
69.0
69.0
69.0
69.0
69.0

Table D.5 (Continued)dσp3dΩ1 dΩ2 dE1(MeV/c) (µb/sr2 MeV)
−107.02.61 ± 0.26
−98.8
3.00 ± 0.27
−82.3
3.74 ± 0.28
−74.0
4.16 ± 0.29
−65.7
4.19 ± 0.29
−49.1
4.19 ± 0.28
−40.9
4.30 ± 0.29
−32.8
3.82 ± 0.28
−24.9
3.72 ± 0.17
−17.8
4.30 ± 0.18
−13.0
4.26 ± 0.18
13.4
3.86 ± 0.18
−104.12.73 ± 0.26
−95.8
3.17 ± 0.26
−79.2
3.77 ± 0.28
−70.8
2.89 ± 0.27
−62.4
3.75 ± 0.28
−45.5
4.17 ± 0.28
−37.1
4.27 ± 0.29
−28.7
4.14 ± 0.29
−20.6
3.53 ± 0.17
−13.3
3.58 ± 0.17
9.3
3.60 ± 0.17
12.6
3.76 ± 0.18

Esep
(MeV)
75.0
75.0
75.0
75.0
75.0
75.0
75.0
75.0
75.0
75.0
75.0
75.0
77.0
77.0
77.0
77.0
77.0
77.0
77.0
77.0
77.0
77.0
77.0
77.0

E1
(MeV)
174.0
178.0
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0
174.0
178.0
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0

Table D.5 (Continued)dσp3dΩ1 dΩ2 dE1(MeV/c) (µb/sr2 MeV)
−95.9
2.25 ± 0.25
−87.4
2.68 ± 0.26
−70.3
3.39 ± 0.28
−61.6
3.26 ± 0.27
−53.0
3.33 ± 0.27
−35.4
2.72 ± 0.27
−26.6
3.36 ± 0.28
−17.6
3.36 ± 0.28
−8.6
3.10 ± 0.17
0.7
3.11 ± 0.17
9.8
2.74 ± 0.17
19.1
3.13 ± 0.17
−93.2
2.51 ± 0.26
−84.7
2.56 ± 0.26
−67.5
2.96 ± 0.27
−58.8
2.90 ± 0.27
−50.1
3.20 ± 0.28
−32.5
3.15 ± 0.27
−23.7
2.78 ± 0.27
−14.8
2.97 ± 0.27
−6.4
2.89 ± 0.17
5.4
2.76 ± 0.17
13.8
2.84 ± 0.17
23.1
2.77 ± 0.17

Ay
−0.200 ± 0.162−0.042 ± 0.138
0.037 ± 0.117
0.136 ± 0.120
0.182 ± 0.118−0.077 ± 0.141−0.192 ± 0.121
0.075 ± 0.119
0.027 ± 0.087−0.045 ± 0.088−0.044 ± 0.099
0.018 ± 0.089−0.085 ± 0.147
0.001 ± 0.143
0.026 ± 0.130−0.202 ± 0.136−0.001 ± 0.124
0.035 ± 0.125
0.006 ± 0.142
0.086 ± 0.135−0.129 ± 0.093−0.021 ± 0.097
0.023 ± 0.095−0.060 ± 0.101

Esep
(MeV)
79.0
79.0
79.0
79.0
79.0
79.0
79.0
79.0
79.0
79.0
79.0
79.0
81.0
81.0
81.0
81.0
81.0
81.0
81.0
81.0
81.0
81.0
81.0
81.0

E1
(MeV)
174.0
178.0
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0
174.0
178.0
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0

Table D.5 (Continued)dσp3dΩ1 dΩ2 dE1(MeV/c) (µb/sr2 MeV)
−90.7
2.31 ± 0.26
−82.1
3.13 ± 0.26
−64.9
2.87 ± 0.27
−56.2
2.85 ± 0.27
−47.4
3.01 ± 0.27
−29.9
2.78 ± 0.27
−21.3
2.59 ± 0.27
−13.0
2.79 ± 0.27
−7.1
2.69 ± 0.16
10.1
2.64 ± 0.16
18.3
2.57 ± 0.17
27.4
2.63 ± 0.17
−88.3
2.78 ± 0.26
−79.7
2.51 ± 0.26
−62.4
2.55 ± 0.27
−53.7
2.76 ± 0.27
−45.0
2.51 ± 0.27
−27.8
2.51 ± 0.27
−19.6
2.52 ± 0.27
−12.6
2.37 ± 0.27
10.2
2.45 ± 0.16
14.9
2.47 ± 0.17
22.9
2.31 ± 0.16
31.9
2.76 ± 0.17

Ay
0.103 ± 0.159−0.043 ± 0.120
0.001 ± 0.135
0.051 ± 0.136
0.315 ± 0.131
0.058 ± 0.142
0.070 ± 0.149
0.127 ± 0.143−0.082 ± 0.099−0.057 ± 0.101−0.056 ± 0.105
0.022 ± 0.103
0.109 ± 0.135−0.012 ± 0.148
0.055 ± 0.151
0.137 ± 0.142−0.069 ± 0.153
0.122 ± 0.155
0.327 ± 0.154
0.219 ± 0.158−0.037 ± 0.106−0.014 ± 0.108−0.138 ± 0.116−0.002 ± 0.099

123

124

E1
(MeV)
174.0
178.0
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0
174.0
178.0
186.0
190.0
194.0
202.0
206.0
210.0
214.0
218.0
222.0
226.0

Ay
0.092 ± 0.190
0.007 ± 0.157−0.007 ± 0.118−0.150 ± 0.135−0.103 ± 0.140
0.205 ± 0.153−0.054 ± 0.149
0.007 ± 0.127
0.058 ± 0.114
0.213 ± 0.115
0.070 ± 0.119
0.015 ± 0.114
0.034 ± 0.134
0.108 ± 0.153
0.034 ± 0.153−0.008 ± 0.120−0.047 ± 0.135
0.101 ± 0.136−0.079 ± 0.155
0.343 ± 0.203
0.203 ± 0.122
0.099 ± 0.109
0.212 ± 0.129
0.231 ± 0.128

APPENDIX D. NUMERICAL DATA TABLES

Esep
(MeV)
83.0
83.0
83.0
83.0
83.0
83.0
83.0
83.0
83.0
83.0
83.0
83.0
85.0
85.0
85.0
85.0
85.0
85.0
85.0
85.0
85.0
85.0
85.0
85.0

Table D.5 (Continued)dσp3dΩ1 dΩ2 dE1(MeV/c) (µb/sr2 MeV)
−86.0
1.90 ± 0.25
−77.4
2.34 ± 0.26
−60.1
3.36 ± 0.28
−51.5
2.89 ± 0.27
−42.9
2.76 ± 0.27
−26.3
2.40 ± 0.26
−19.0
2.56 ± 0.27
−14.0
3.04 ± 0.27
14.3
2.27 ± 0.16
19.8
2.30 ± 0.16
27.7
2.24 ± 0.16
36.6
2.34 ± 0.16
−83.8
2.79 ± 0.26
−75.2
2.37 ± 0.25
−58.0
2.53 ± 0.27
−49.5
3.26 ± 0.27
−41.1
2.86 ± 0.27
−25.5
2.84 ± 0.27
−19.4
2.44 ± 0.26
−16.6
1.94 ± 0.26
18.7
2.11 ± 0.16
24.7
2.41 ± 0.16
32.6
2.03 ± 0.16
41.4
2.05 ± 0.16

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BIBLIOGRAPHY

List of Tables
Measured kinematical sets (central values). The angle of the GR was fixed at 25.5◦ .
θLAS indicates the angle of the LAS which was set according to the measured separationenergy region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2.2

Targets used in the measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

2.3

Design specification of the Grand Raiden spectrometer (GR). . . . . . . . . . . . . . .

19

2.4

Design specification of the Large Acceptance Spectrometer (LAS). . . . . . . . . . . .

20

2.5

Specifications of the MWDCs for the GR. . . . . . . . . . . . . . . . . . . . . . . . .

23

2.6

Specifications of the MWDCs for the LAS. . . . . . . . . . . . . . . . . . . . . . . .

23

2.7

Specifications of the MWPCs for the GR and LAS. . . . . . . . . . . . . . . . . . . .

24

4.1

Bound-state parameters of the Woods-Saxon potential for 40 Ca nucleus. . . . . . . . .

46

5.1

Spectroscopic factors for the discrete peaks in the 40 Ca measurement. Kramer et al.
reanalyzed the previous (d,3 He) data of Doll et al. in Ref. [9] in their study [47]. The
present result is shown with the statistical uncertainty obtained in the fitting or MDA. .

60

Centroid energies and widths (FWHM) of the strength distributions for the hole states of
the 1p and 1s1/2 orbitals in 40 Ca. For the present result, the uncertainties were obtained
as the region corresponding to a change of 1 in χ2 from the minimum value in the fittingof the asymmetric Lorentzian and the background. . . . . . . . . . . . . . . . . . . .

63

Spectroscopic factors relative to the IPSM limits for the orbitals in 40 Ca. They are obtained after the background subtraction in the present study. The second uncertainty, inparentheses, is the statistical uncertainty estimated by the propagation of the strength uncertainties at each 2-MeV bin and the uncertainty of the contribution of the background.
The first one is the total uncertainty that includes also the model uncertainty, which is
mentioned in the text. The spectroscopic factors deduced by the DWIA calculation with
the nonlocality correction are also shown. The detail discussion is given in Chap. 6. .

64

2.1

5.2

5.3

133

LIST OF TABLES

134

Spectroscopic factors relative to the IPSM limits for the orbitals in 40 Ca without background subtraction. The uncertainty is the statistical uncertainty estimated by the propagation of the strength uncertainties at each 2-MeV bin. . . . . . . . . . . . . . . . .

64

6.1

Global optical potentials used for the distortion in the DWIA calculation.

. . . . . . .

81

6.2

Normalized spectroscopic factors relative to the IPSM limits for the orbitals in 40 Ca. The
description of the uncertainty is same as Table 5.3 except that these total uncertainties
include also the uncertainty of the normalization. . . . . . . . . . . . . . . . . . . . .

84

Spectroscopic factors relative to the IPSM limits for the orbitals in 40 Ca obtained fromprevious experiments. For the present result, the description of the uncertainty is sameas Table 6.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

Spectroscopic factors relative to the IPSM limits for the orbitals in 40 Ca obtained fromtheoretical studies. For the present result, the description of the uncertainty is same as
Table 6.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

A.1 Spectroscopic factors for the discrete peaks in the 40 Ca measurement. The first uncertainty is the total uncertainty and the second, in parentheses, is the statistical uncertainty
included in the total. In the published results, the uncertainty from the DWIA calculation is included in the total uncertainty. In the reanalyzed results, the uncertainty fromthe normalization reference [47] is included in the total uncertainty of the normalized
results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

A.2 Centroid energies and widths (FWHM) of the strength distributions for the hole statesof the 1p and 1s1/2 orbitals in 40 Ca. The first uncertainty is the total uncertainty, which
includes the statistical and the model uncertainties, and the second, in parentheses, isthe statistical uncertainty included in the total. . . . . . . . . . . . . . . . . . . . . .

98

A.3 Spectroscopic factors relative to the IPSM limits for the orbitals in 40 Ca. They are obtained after subtraction of the background. The first uncertainty is the total uncertainty
and the second, in parentheses, is the statistical uncertainty included in the total. The
model uncertainty is included in the total uncertainty. The uncertainty from the DWIA
calculation are included in the total uncertainty of the published results, while the uncertainty from the normalization reference [47] is included in the total uncertainty of thenormalized results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

5.4

6.3

6.4

LIST OF TABLES

135

A.4 Spectroscopic factors relative to the IPSM limits for the orbitals in 40 Ca without background subtraction. The first uncertainty is the total uncertainty and the second, inparentheses, is the statistical uncertainty included in the total. The uncertainty from the
DWIA calculation are included in the total uncertainty of the published results, while
the uncertainty from the normalization reference [47] is included in the total uncertaintyof the normalized results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

D.1 Data table for the 40 Ca(~p, 2p) reaction for the peaks at 8.3 and 10.9 MeV in the separation energy at the set1 in Table 2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . .

109

D.2 Data table for the 40 Ca(~p, 2p) reaction at the set2 in Table 2.1. ∆Esep = 2 MeV. . . . .

110

D.3 Data table for the 40 Ca(~p, 2p) reaction at the set3 in Table 2.1. ∆Esep = 2 MeV. . . . .

113

D.4 Data table for the 40 Ca(~p, 2p) reaction at the set4 in Table 2.1. ∆Esep = 2 MeV. . . . .

116

D.5 Data table for the 40 Ca(~p, 2p) reaction at the set4 in Table 2.1. ∆Esep = 2 MeV. . . . .

121

136

LIST OF TABLES

List of Figures

1.1

1.2

1.3

1.4

1.5

1.6

1.7

2.1

Summed spectroscopic strength observed for proton knock-out reactions from various
orbitals in the closed-shell nuclei 16 O, 40 Ca, 48 Ca, 90 Zr, and 208 Pb as a function of themean excitation energy of the orbital relative to the Fermi energy (ǫF ). The dashedcurve represents the quasi-particle strength calculated for nuclear matter, the solid curveis derived from the nuclear matter curve by including surface effects calculated for
208 Pb [12]. This figure is taken from Ref. [11]. . . . . . . . . . . . . . . . . . . . . .

3

Separation energies, widths, and angular momentum assignments of the hole states obtained from quasi-free scattering, as functions of the atomic number. This figure is takenfrom Ref. [15]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4Hole-strength distribution from (e, e′ p) reaction for 40 Ca obtained by Mougey et al. This
figure is taken from Ref. [16]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6Separation energy spectra in the 1s1/2 region for 40 Ca after subtraction of the contributions of upper shells and backgrounds in (e, e′ p) reaction. This figure is taken from
Ref. [18]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

Strength distributions from the ( p, 2p) reaction obtained for various nuclei by James et
al. This figure is taken from Ref. [20]. . . . . . . . . . . . . . . . . . . . . . . . . . .

7

Separation-energy spectra for the 40 Ca(p, 2p) (upper three figures ) and 40 Ca(p, pn)(lower three figures ) reactions measured with a 1 GeV proton beam at Petersburg Nuclear Physics Institute (PNPI) from Ref. [22]. The detection range of proton energy is(a) 830–870, (b) 855–887, and (c) 880–915 MeV, respectively. . . . . . . . . . . . . .

8Analyzing powers for the 1p3/2 and p1/2 -hole states in the 16 O(p, 2p) reaction measured
by changing a kinetic energy of an ejected proton. They behave differently as predicted
by Jacob and Maris and show the validity of Maris effect. This figure is taken from
Ref. [37]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

Schematic view of the RCNP ring cyclotron facility.

14

137

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

LIST OF FIGURES

138

2.2

Notation for the kinematics of the A(p, 2p)B reaction in the laboratory system. The incident energy of the proton is T 0 = 392 MeV in the present study. The recoil momentumof the residual nucleus B is indicated by p3 . . . . . . . . . . . . . . . . . . . . . . . .

15

2.3

Layout of BLP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.4

Schematic view of the dual-arm spectrometer system at RCNP. . . . . . . . . . . . . .

18

2.5

Focal plane detectors of the GR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.6

Structure of each plane of MWDC. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.7

Wire configurations of the X-plane and U-plane of the GR-MWDC. . . . . . . . . . .

21

2.8

Focal plane detectors of the LAS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

2.9

Wire configurations of the X-plane, U-plane, and V-plane of the LAS-MWDC. . . . .

24

2.10 Schematic view of the layout around the scattering chamber. The proton beam is injected
into the reaction target at the center of the scattering chamber and is transported to the
Faraday cup placed about 25 m downstream of the target. The MWPCs and the leadcollimators are installed in front of the quadrupole magnets. . . . . . . . . . . . . . .

25

2.11 Geometrical relation of anode planes of the MWPC and the center of the scatteringchamber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

2.12 Structure of each wire plane of the MWPCs.

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

26

2.13 Logic diagram for the GR focal plane detectors. . . . . . . . . . . . . . . . . . . . . .

27

2.14 Logic diagram for the LAS focal plane detectors. . . . . . . . . . . . . . . . . . . . .

28

2.15 Logic diagram for coincidence event of the GR and LAS. . . . . . . . . . . . . . . . .

28

2.16 Schematic view of the data acquisition system. . . . . . . . . . . . . . . . . . . . . .

29

3.1

3.2

3.3

Energy loss spectrum for the PS1 of the GR, which is the mean of the pulse-heightsignals of the left and right PMTs. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33(a): TDC spectrum that indicates the time difference between the mean-time signal fromthe PS1 of the GR and the RF signal from the AVF cyclotron. The RF signal was filteredat the half rate. (b): TDC spectrum corrected by the trajectory angle and the position at
the focal plane so that the influence of the momentum acceptance of the spectrometeron the TDC is eliminated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

Two-dimensional scatter plot of the corrected TDC and ∆E1 of the GR. The events
enclosed by the solid lines are recognized as the proton events. . . . . . . . . . . . . .

35

LIST OF FIGURES

3.4

139

TDC spectrum for the time difference between the GR and LAS trigger signals in ameasurement of the 40 Ca(p, 2p) reaction. . . . . . . . . . . . . . . . . . . . . . . . .

36

3.5

Position in a plane of MWDC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

3.6

Coordinate systems for the ray-tracing with two MWDCs. . . . . . . . . . . . . . . .

37

3.7

Two dimensional scatter plot of the horizontal and vertical scattering angles of protons
analyzed with the GR (a) and LAS (b). . . . . . . . . . . . . . . . . . . . . . . . . . .

41(a): Two dimensional scatter plot of the energies of coincidence two protons measured
with the GR and LAS in the 40 Ca(p, 2p) reaction at T 0 = 392 MeV. (b): Separationenergy spectrum for the 40 Ca(p, 2p) reaction. . . . . . . . . . . . . . . . . . . . . . .

43

Recoil-momentum distributions of the cross sections for the 1s1/2 -, 1p3/2 -, 1p1/2 -, and1d5/2 -hole states by DWIA calculations. They were calculated with spectroscopic factors of 1 at a separation energy 15 MeV under a certain kinematical condition. . . . .

48

5.1

Separation-energy spectra at the measured kinematical set1 in Table 2.1.

. . . . . . .

52

5.2

Separation-energy spectra at the measured kinematical set2 in Table 2.1.

. . . . . . .

53

5.3

Separation-energy spectra at the measured kinematical set3 in Table 2.1. Vertical dottedlines show boundaries where the momentum acceptance is guaranteed. Yields at theoutside of the vertical dotted lines decrease due to the finite momentum acceptance ofthe spectrometers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

Separation-energy spectra at the measured kinematical set4 in Table 2.1. Vertical dottedlines show boundaries where the momentum acceptance is guaranteed. Yields at theoutside of the vertical dotted lines decrease due to the finite momentum acceptance ofthe spectrometers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

Separation-energy spectra at the measured kinematical set5 in Table 2.1. Vertical dottedlines show boundaries where the momentum acceptance is guaranteed. Yields at theoutside of the vertical dotted lines decrease due to the finite momentum acceptance ofthe spectrometers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

3.8

4.1

5.4

5.5

5.6

Separation-energy spectrum for the 40 Ca(p, 2p) reaction up to Esep = 20 MeV at thekinematical set1 (Table 2.1). Two prominent peaks are observed at 8.3 MeV and 10.9 MeV.
The MDA was performed at separation energies above 12 MeV. . . . . . . . . . . . . 57

LIST OF FIGURES

140

Cross section (a) and analyzing power (b) of the 40 Ca(p, 2p) reaction for the peakat 8.3 MeV as functions of the recoil momentum p3 . The solid line represents the
DWIA calculations fitted by the spectroscopic factor for the 1d3/2 orbital. The discrepancy in the Ay (b) between the data and the DWIA calculation is discussed in the
text (Sec.5.2A.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

Results of the MDA for the peak at 10.9 MeV. The cross section (a) and the analyzing
power (b) are shown as functions of the recoil momentum p3 . The thick solid line shows
the fitted results. The thin solid line and the dotted line show the contributions of the2s1/2 and 1 f7/2 states, respectively. The discrepancy in the Ay (b) between the data andthe DWIA calculation is discussed in the text (Sec.5.2A.). . . . . . . . . . . . . . . .

59

Cross section for each 2-MeV bin in the separation energy at the set2 as a function of therecoil momentum p3 . The thick solid line shows the MDA results including 2s1/2 -holestate as a s-hole state. The dotted lines, the short-dashed lines, and the dash-dotted lines
show the contributions of 2s1/2 -, 1p-, and 1d5/2 -hole states, respectively. . . . . . . .

65

5.10 Analyzing power for each 2-MeV bin in the separation energy at the set2 as a functionof the recoil momentum p3 . The thick solid line shows the MDA results including 2s1/2 hole state as a s-hole state. The analyzing power data was not included in the MDA. The
dotted lines, the short-dashed lines, and the dash-dotted lines show the contributions of
2s1/2 -, 1p-, and 1d5/2 -hole states, respectively. . . . . . . . . . . . . . . . . . . . . .

66

5.11 Cross section for each 2-MeV bin in the separation energy at the set2 as a function of therecoil momentum p3 . The thick solid line shows the MDA results including 1s1/2 -holestate as a s-hole state. The dotted lines, the short-dashed lines, and the dash-dotted lines
show the contributions of 1s1/2 -, 1p-, and 1d5/2 -hole states, respectively. . . . . . . .

67

5.12 Analyzing power for each 2-MeV bin in the separation energy at the set2 as a functionof the recoil momentum p3 . The thick solid line shows the MDA results including 1s1/2 hole state as a s-hole state. The analyzing power data was not included in the MDA. The
dotted lines, the short-dashed lines, and the dash-dotted lines show the contributions of
1s1/2 -, 1p-, and 1d5/2 -hole states, respectively. . . . . . . . . . . . . . . . . . . . . .

68

5.13 Cross section for each 2-MeV bin in the separation energy at the set3 as a function of
the recoil momentum p3 . The thick solid line shows the MDA results. The dotted lines,
the short-dashed lines, and the dash-dotted lines show the contributions of 1s1/2 -, 1p-,
and 1d5/2 -hole states, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

5.14 Analyzing power for each 2-MeV bin in the separation energy at the set3 as a function of
the recoil momentum p3 . The analyzing power data was not included in the MDA. The
dotted lines, the short-dashed lines, and the dash-dotted lines show the contributions of
1s1/2 -, 1p-, and 1d5/2 -hole states, respectively. . . . . . . . . . . . . . . . . . . . . .

70

5.7

5.8

5.9

LIST OF FIGURES

141

5.15 Cross section for each 2-MeV bin in the separation energy at the set4 as a function of
the recoil momentum p3 . The thick solid line shows the MDA results. The dotted lines,
the short-dashed lines, and the dash-dotted lines show the contributions of 1s1/2 -, 1p-,
and 1d5/2 -hole states, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

5.16 Analyzing power for each 2-MeV bin in the separation energy at the set4 as a function of
the recoil momentum p3 . The analyzing power data was not included in the MDA. The
dotted lines, the short-dashed lines, and the dash-dotted lines show the contributions of
1s1/2 -, 1p-, and 1d5/2 -hole states, respectively. . . . . . . . . . . . . . . . . . . . . .

72

5.17 Cross section for each 2-MeV bin in the separation energy at the set5 as a function of
the recoil momentum p3 . The thick solid line shows the MDA results. The dotted lines,
the short-dashed lines, and the dash-dotted lines show the contributions of 1s1/2 -, 1p-,
and 1d5/2 -hole states, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

5.18 Analyzing power for each 2-MeV bin in the separation energy at the set5 as a function of
the recoil momentum p3 . The analyzing power data was not included in the MDA. The
dotted lines, the short-dashed lines, and the dash-dotted lines show the contributions of
1s1/2 -, 1p-, and 1d5/2 -hole states, respectively. . . . . . . . . . . . . . . . . . . . . .

74

5.19 Cross section and analyzing power as a function of the recoil momentum p3 for theselected separation-energy bins at 17 [(a) and (d)], 33 [(b) and (e)] and 53 MeV [(c)and (f)]. The thick solid lines show the MDA results. The thin solid lines, the dottedlines, the short-dashed lines, and the dash-dotted lines show 1p-, and 1d5/2 -hole states,
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

5.20 Strength distributions obtained in the separation-energy region of 12–84 MeV. The solidlines are shown to guide the eyes. The vertical dotted line at 24 MeV in (a) shows
the border where the s-hole state contribution was divided into the contribution of the2s1/2 -hole state and that of the 1s1/2 -hole state. . . . . . . . . . . . . . . . . . . . . .

76

5.21 Strength distributions for the hole states of the (a) 1s1/2 , (b) 1p, (c) 2s1/2 , and (d) 1d5/2
orbitals in the separation energy region of 12–84 MeV. The solid lines and the dashdotted lines in the figures for the 1s1/2 -, 1p- and 1d5/2 -hole states show the fitted curves
and the fitted asymmetric Lorentzian functions, respectively. The short-dashed lines
show the estimated contribution of the four-body background. . . . . . . . . . . . . .

77

6.1

Centroids (symbols) and FWHM widths (solid lines) of the strength distributions forthe 1p- and 1s1/2 -hole states with those obtained from the previous experiments, from
Table 5.2. Vertical lines indicate uncertainties of the centroids as standard deviation.
Mougey et al. did not give the uncertainties. . . . . . . . . . . . . . . . . . . . . . .

80

LIST OF FIGURES142

DWIA calculations of the 40 Ca(p, 2p) reaction for the 1d3/2 -hole state at 8.3 MeV withdifferent optical potentials. The parameter sets of Cooper et al. (EDAD2, EDAD3,EDAICA) [72] and of Hama et al. (DP1, DP2) [90] were used for the distorting potential. The strengths of the 1d3/2 -hole state were obtained by fitting to the experimental
data. The thick solid line indicates the PWIA calculation. . . . . . . . . . . . . . . .

81

DWIA calculations of the 40 Ca(p, 2p) reaction for the 2s1/2 -hole state at 10.9 MeV withdifferent optical potentials. The used optical potentials were same as those for the 1s3/2 hole state in Fig. 6.2. The strength of the 2s1/2 -hole state was determined by the MDA
for the peak. The only contribution of 2s1/2 -hole state are displayed for simplicity. The
thick solid line indicates the PWIA calculation. . . . . . . . . . . . . . . . . . . . . .

82

Spectroscopic factors obtained with the different distorting potentials for the 1d3/2 -,2s1/2 -, and 1 f7/2 -hole states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

DWIA calculations of the 40 Ca(p, 2p) reaction for the 1d3/2 -hole state with the different
radius parameters of the Woods-Saxon potential. The strengths of the 1d3/2 -hole state
were obtained by fitting to the experimental data. . . . . . . . . . . . . . . . . . . . .

84

DWIA calculations of the 40 Ca(p, 2p) reaction for the 2s1/2 -hole state with the different
radius parameters of the Woods-Saxon potential. The strength of the 2s1/2 -hole state
was determined by the MDA for the peak. The only contribution of 2s1/2 -hole state aredisplayed for simplicity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

Spectroscopic factors obtained with the different radius parameters of the Woods-Saxon
potential for the 1d3/2 -, 2s1/2 - and 1 f7/2 -hole states. . . . . . . . . . . . . . . . . . .

86

Spectroscopic factors obtained with the different diffuseness parameters of the WoodsSaxon potential for the 1d3/2 -, 2s1/2 -, and 1 f7/2 -hole states. . . . . . . . . . . . . . .

86

Correlation between the spectroscopic factors for the 1d3/2 -hole state and the strengthof the spin–orbit term for the bound state. . . . . . . . . . . . . . . . . . . . . . . . .

87

6.10 Spectroscopic factors relative to the IPSM limits for the orbitals in 40 Ca as a function of
the separation energy. The solid line and dashed line are shown to guide the eyes. The
strengths shown by the squares are obtained by the MDA with the DWIA calculation
including the nonlocality correction in the bound state wave function and in the reaction.
The nonlocality corrections almost uniformly decrease the spectroscopic factors for allof the orbitals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

6.11 Spectroscopic factors relative to the IPSM limits for the orbitals in 40 Ca as a function of
the separation energy. The dashed lines are shown to guide the eyes. . . . . . . . . . .

90

6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

LIST OF FIGURES

6.12 Spectroscopic factors relative to the IPSM limits for the orbitals in 40 Ca as a functionof the separation energy with the predictions by Fabrocini et al. [13] and Bisconti etal. [14]. For the separation energies of the predictions, the values of the present result areused since no predicted values are given in their papers. The separation energies for the
1p1/2 and 1p3/2 orbitals are plotted deviating ±2 MeV from 29.6 MeV respectively. The
solid line (present result), dashed line (Fabrocini et al.), and dash-dotted line (Bisconti etal.) are shown to guide the eyes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.1 (A) Two dimensional plot of the horizontal scattering angle (θGR ) and the position at
the focal plane (XGR ), which corresponds to the kinetic energy, for the GR. (B) Two
dimensional plot of the horizontal scattering angle corrected so that the kinetic-energy′ ) and X . (C) Cordependence of the horizontal scattering angle was canceled (θGRGR′
rected angle (θGR ) projected on the horizontal angle axis in −50 ≤ XGR ≤ 50. . . . . .
B.2 Momentum |p3 | estimated in pp scattering event. The scattering angles from the MWPCs were randomly deviated in the angular resolution by Monte Carlo technique forsmoothing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143

91

100

101