Neutron density distributions of 204,206,208Pb
deduced via proton elastic scattering
at E p = 295 MeVZENIHIRO, Juzo

Doctoral Dissertation

Department of Physics
Faculty of Science
Kyoto University

2011Abstract
Cross sections and analyzing powers for polarized proton elastic scattering from 58 Ni, and204,206,208
Pb have been measured at intermediate energy E p = 295 MeV. An eﬀective relativistic
Love-Franey interaction is tuned to reproduce 58 Ni scattering data within the framework of therelativistic impulse approximation.
The neutron densities of the lead isotopes are deduced using model independent sum-ofGaussian distributions. Their error-envelopes are estimated by a new χ2 criterion including
uncertainties associated with the reaction model as well as the experimental uncertainties.
The systematic behaviors of extracted error-envelopes of the neutron density distributions in 204,206,208 Pb are presented. The extracted neutron and proton density distributions+0.048of 204,206,208 Pb give neutron skin thicknesses of ∆rnp = 0.178+0.047−0.059 fm, 0.180−0.064 fm, and+0.0540.211−0.063 fm, respectively. Even including the model uncertainties, the errors of the extracted∆rnp are at the level of 1 % of the neutron rms radius and still comparable to the previous
results which reflect the experimental errors only.
It is found that the neutron skin thicknesses are strongly correlated with the symmetryenergy at saturation S(ρsat ) and the slope coeﬃcient of the symmetry energy L according topredictions by various relativistic and non-relativistic mean-field models. By using the linear correlations our results of the neutron skins for 204,206,208 Pb lead to the weighted mean ofS(ρsat ) = 33.0 ± 1.1 MeV and L = 67.0 ± 12.1 MeV. The extracted ranges of S(ρsat ) and L
include the experimental uncertainties only, but they are consistent with and smaller than theprevious results.

Contents
1 INTRODUCTION
1.1 Equation of state of asymmetric nuclear matter . . . . . . . . . . . . . . . .
1.2 Symmetry energy and neutron skin thickness of 208 Pb . . . . . . . . . . . . .
1.3 Nucleon density distributions . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Proton density distributions . . . . . . . . . . . . . . . . . . . . . .
1.3.2 Neutron density distributions . . . . . . . . . . . . . . . . . . . . . .
1.4 Purpose of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
4
6
6
6
8

2 EXPERIMENTAL PROCEDURE
2.1 Beam line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Polarized proton beam . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Beam line polarimeter . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Faraday Cups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Magnetic spectrometer, “Grand Raiden” . . . . . . . . . . . . . . . . . . . .
2.5 Focal plane detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Trigger system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Data acquisition system . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13
13
13
15
16
17
17
20
21
22

3 DATA REDUCTION & RESULTS
3.1 BLP data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Beam polarization . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Beam intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Particle identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Multi-wire drift chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Track reconstruction of scattered particles . . . . . . . . . . . . . . .
3.3.2 Tracking eﬃciency . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Diﬀerential cross sections and analyzing powers . . . . . . . . . . . . . . . .

25
25
25
28
28
31
31
36
37

4 ANALYSIS
4.1 Relativistic impulse approximation . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Realistic point proton density distributions . . . . . . . . . . . . . .

41
41
43

i

CONTENTS

ii

4.1.2 Scalar and Vector density distributions . . . . . . . . . . . . . . . . .
Medium modification of NN interaction . . . . . . . . . . . . . . . . . . . .
Calibration of the medium eﬀect parameters . . . . . . . . . . . . . . . . . .
Extraction of neutron density distributions . . . . . . . . . . . . . . . . . . .
4.4.1 Estimation of error-envelopes . . . . . . . . . . . . . . . . . . . . .

44
46
47
50
52

5 RESULTS AND DISCUSSION
5.1 Charge, proton, and neutron rms radii . . . . . . . . . . . . . . . . . . . . .
5.2 Neutron skin thicknesses . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63
63
64

6 SUMMARY79

A Data table

83

B Relativistic impulse approximation

91

C Equation of state of nuclear matter

97

4.2
4.3
4.4

Chapter 1
INTRODUCTION
Since 1950s, various nuclear ground-state properties such as the masses of nuclei, the shapes
and sizes of nuclear matter or nuclear charge distributions, the binding energy per nucleon,
and the nuclear incompressibilities, have been measured by a variety of experimental methods
in many accelerator facilities in the world. Based on the measurements of such properties,
a number of theoretical studies have been also performed over the ages. However even forthe stable nuclei it is not easy to understand the enormous aspects of the nuclear many-bodysystem.
For the last few decades, in addition, new phenomena such as skin and halo structures,and the change of the nuclear magic number have been found from the experimental studies
of unstable nuclei, whose neutron number N is very diﬀerent from its proton number Z. In
the light unstable nuclei, for example, 11 Li was found to have a large neutron halo through themeasurements of interaction cross sections [1]. The excess two neutrons in 11 Li are remarkably
widespread by the tunneling eﬀect and very weekly bound around 9 Li core. The radius of
11
Li is comparable to that of 208 Pb. It is now expected that the complementary research forsuch exotic nuclei as well as for stable nuclei is powerful tool not only to understand these
new features of nuclei, but also to establish more precise and comprehensive framework of
nuclear many-body system including a large isospin asymmetry. For this purpose, several nextgeneration facilities in the world, which can provide high-energy and high-intensity radioactive
heavy-ion beams, has been constructed or planned in recent years. Above all, the RI Beam
Factory (RIBF) at RIKEN in Japan has already started providing heavy-ion beam such as
238
U at 345 MeV/u and some fruitful results of the productions of new rare isotopes has beenreported [2].

1.1 Equation of state of asymmetric nuclear matter
Among these new phenomena, the role of isospin asymmetry in nuclei and nuclear mattershave received a lot of attention in both the nuclear physics and astrophysics, and have been
investigated using many experimental results and theoretical models. Particularly, the isospindependent term of the equation of state (EOS), i.e., the density dependence of the nuclear

1

CHAPTER 1. INTRODUCTION

2

Figure 1.1: Rendition of the assumed structure and phases of a neutron star. Taken from
Ref. [33].

Figure 1.2: Neutron star mass versus central density obtained from various types of
the EOSs by relativistic hadronic field models,
taken from Ref. [34]symmetry energy is now considered as one of the most important question to be solved.
There have been quite a lot of microscopic and phenomenological, non-relativistic andrelativistic approaches to understand the equation of state of asymmetric nuclei and nuclear
matters, such as the Brueckner-Hartree-Fock [3, 4] and the Dirac-Brueckner-Hartree-Fock [5–8] approach with bare NN interaction, and the non-relativistic mean-field model based on the
Skyrme interactions [9–15] and the relativistic mean-field model [16–27]. In addition, the EOSof the asymmetric nuclear matter is essential to the understanding of astrophysical phenomena
such as supernova explosions, neutron stars structure and cooling system, and x-ray bursts [28–32]. Figure 1.1 shows the assumed structure ans phases of a neutron star. The compositionof a neutron star depends on the EOS of the neutron matter as shown in Fig. 1.2. Figure 1.3
shows the typical calculations of the EOS of the symmetric nuclear and pure neutron matterby microscopic many-body theories and phenomenological approaches.
For asymmetric nuclear matter, the equation of state, i.e., the energy per nucleon can begenerally approximated asE(ρ, δ) = E(ρ, 0) + S(ρ)δ2 + O(δ4 ),(1.1)

in terms of baryon density ρ = ρn + ρ p , local isospin asymmetry δ = (ρn − ρ p )/ρ, energy
per nucleon in symmetric nuclear matter E(ρ, δ = 0), and so-called nuclear symmetry energy
S (ρ). This equation (1.1) is known as the empirical parabolic law for the EOS of asymmetricnuclear matter. The odd-order terms of δ is vanishing due to the exchange symmetry between
protons and neutrons. Although it has been reported that the higher order terms O(δ4 ) become
important for determining the transition density and pressure at the inner edge separating the1.1. EQUATION OF STATE OF ASYMMETRIC NUCLEAR MATTER

3

Figure 1.3: The EOS of symmetric nuclear matter and pure neutron matter calculated by the
non-relativistic Brueckner-Hartree-Fock and variational approaches using AV18 interaction
with three body force (dotted red), the Dirac-Brueckner-Hartree-Fock using Bonn A (solidred), chiral perturbation theory (dashed black), and several phenomenological mean field models (blue) taken from Ref. [34].

liquid core form the solid crust of neutron stars [32], the higher order terms are generally
negligible around normal density [35]. Therefore, the density-dependent symmetry energy
S(ρ) can be obtained from S(ρ) ≈ E(ρ, 1) − E(ρ, 0). This means the energy diﬀerence between
the EOS of the neutron matter and the symmetric nuclear matter.
The EOS of the symmetric nuclear matter E0 (ρ) = E(ρ, 0) around the saturation density
ρsat can be expanded as
K0 2
 + O( 3 ),22dE0 (ρ)= 9ρ2sat,dρ2 ρ=ρsat

E0 (ρ) = E0 +
K0

(1.2)
(1.3)

where  = (ρ − ρsat )/3ρsat . The coeﬃcient K0 is the so-called incompressibility of symmetricnuclear matter. The binding energy per nucleon E0 and the incompressibility K0 which characterize the symmetric nuclear matter are now known to be about −16 MeV and 230 MeV,
respectively. The incompressibility, in particular, has been recently constrained by many measurements of giant resonances for various nuclei via α-inelastic scattering at 0 degree [36–40].

CHAPTER 1. INTRODUCTION

4

1.2 Symmetry energy and neutron skin thickness of 208Pb
On the other hand, the symmetry energy S(ρ) was previously studied by the systematic measurements of the masses of stable nuclei, small δ and ρ around the saturation, based on theearlier empirical mass formula [41–43]. Thus it was diﬃcult to verify the isospin-asymmetry
dependence of S(ρ). The importance of the symmetry energy of asymmetric nuclear matterhas been recognized by the study of the exotic nuclei with a large neutron excess at the radioactive ion beam facilities. However the isospin dependence of the symmetry energy S(ρ) is
still less certain and the predictions vary widely among many theoretical models [12, 44, 45].
The nuclear symmetry energy S(ρ) which characterizes the asymmetric nuclei and nuclear
matter is also expanded around the saturation density as follows:
Ksym 2
S(ρ) = S(ρsat ) + L +
 + O( 3 ),
(1.4)
2
dS(ρ)
L = 3ρsat(1.5),
dρ ρ=ρsat
Ksym = 9ρ2sat

d2 S(ρ),dρ2 ρ=ρsat(1.6)

where L and Ksym are the slope and curvature coeﬃcient of the symmetry energy at the saturation, respectively. From Eq. (1.1), (1.2), and (1.4), the saturation properties of the asymmetricnuclear matter such as the saturation density ρsat (δ), energy per nucleon E0 (δ), and the incompressibility K0 (δ) as a function of the isospin asymmetry δ, are also derived. The details of theevolution of the saturation point are described in Appendix C.
In order to describe not only the symmetric but also asymmetric nuclei or nuclear matter, the non-relativistic mean-field models based of the Skyrme interactions and the relativistic
mean-field models including various types of medium density-dependence have been developed together with the experimental results of the ground-state properties of finite nuclei over
the ages. It has been shown that the thickness of the neutron skin (∆rnp ), defined as the difference between the neutron (rn ) and proton (r p ) root-mean-square (rms) radii of a nucleus(∆rnp ≡ rn − r p ), is determined by the balance among the various nuclear matter properties.
Indeed, the neutron skin thickness of 208 Pb is strongly correlated with the nuclear symmetryenergy coeﬃcients S(ρsat ), L, and Ksym of the equation of state in asymmetric nuclear matter[25, 26, 44–50]. Figures 1.4–1.6 show that the strong correlations between neutron skin thickness for 208 Pb and some symmetry energy parameters such as S(ρsat ), L, and Ksym exist by boththe non-relativistic and relativistic mean field models.
208
Pb is the heaviest stable nucleus who has doubly-closed shell structure (Z = 82 andN = 126) having major shell gaps, and hence its nuclear properties such as the binding energy,
charge radius, have been precisely determined.
Therefore the precise extraction of the neutron skin thickness of 208 Pb is related to constrain the isospin dependence of the eﬀective nuclear interactions and EOS of the isospin
asymmetric nuclear matter, particularly, the density dependence of the nuclear symmetry energy. Furthermore, it has been suggested that a precise measurement of the skin thickness of1.2. SYMMETRY ENERGY AND NEUTRON SKIN THICKNESS OF 208 PB

5

Figure 1.4: Neutron skin thickness for 208 Pb as a function of (a) L, (b) Ksym , and (c) Esym (ρ0 ) =S(ρsat ) for 21 sets of Skyrme parameterizations. Taken from Ref. [51].

Figure 1.5: Correlation between neutron skin Figure 1.6: Correlation between neutron skinfor 208 Pb and the symmetry energy a4 = S(ρsat ) for 208 Pb and the slope of the symmetry energy
p0 = ρsat L/3. Take from Ref. [45].
at saturation. Take from Ref. [45].

208

Pb is very important for studying the radius, composition, and cooling system of neutronstars [23, 24, 26, 30–32, 52, 53].

6

CHAPTER 1. INTRODUCTION

1.3 Nucleon density distributions
The nucleon density distribution is one of the most fundamental properties of nuclei. The
precise extraction of the nucleon density distributions (ρ p , ρn ) and hence the neutron skin
thickness ∆rnp is now known to be indispensable for studying the EOS of the asymmetricnuclear matter as mentioned in the previous section.
However, compared to other ground-state properties of finite nuclei such as charge densities, charge radii, and binding energies, it is diﬃcult to measure the size and shape of theneutron densities precisely due to the large ambiguity of the hadronic process.

1.3.1 Proton density distributions
In the case of stable nuclei, the proton density ρ p and radius r p can be derived from the nuclear
charge density ρch and the intrinsic nucleon charge density. Nuclear charge distributions fora variety of stable nuclei are now known accurately from a lot of experiments [54–56]. For
example, the charge radius of 208 Pb is 5.5010(9) fm with an accuracy of 0.02 % according to
a combined analysis of electron scattering, muonic atom X-rays, and isotope shift as shown
in Ref. [56]. An electromagnetic probe, due to its simple reaction mechanism, can extract
precise information about charge density deep inside a nucleus.

1.3.2 Neutron density distributions
Neutron density and radius have been studied by many researchers using proton, α, pion scattering, and antiprotonic atoms [57–68]. However, typical experimental uncertainties of extracted neutron rms radius are more than ten times as large as those of the charge radius.
Hadronic probes exhibit uncertainties in the reaction mechanism which is mainly caused byan incomplete knowledge of the nucleon–nucleon (NN) scattering amplitude inside the nuclear medium. In order to extract precise information about the neutron density distribution an
appropriate probe and an eﬀective NN interaction must be carefully chosen.
“Proton elastic scattering” Proton–nucleus elastic scattering at intermediate energies (E p ∼300 MeV) is a relatively good probe for extracting information about both the nuclear surface
and interior. The energy of the incident protons is low enough to suppress meson production
and hence allow a shallow absorptive potential to be obtained. It is also high enough to be
described by an impulse approximation. As mentioned above, however, uncertainties in the
NN interaction in the nucleus are a problem.
In our previous papers [69, 70], we reduced the uncertainties in the NN interaction inside
the nuclear medium by introducing a phenomenological medium modification in the relativistic Love–Franey interaction in terms of the density dependent parameters. This correction iseﬀectively described as a modification of the masses and coupling constants of the exchanged
mesons according to the nucleon density. To calibrate these “medium eﬀect” parameters, we
use proton elastic scattering from 58 Ni, because the proton and neutron densities are supposed1.3. NUCLEON DENSITY DISTRIBUTIONS

7

Figure 1.7: Neutron density distributions of 116,118,120,122,124 Sn in the model-independent formof SOG function extracted from polarized proton elastic scattering at E p = 295 MeV [70].

to have almost the same shape and radius in 58 Ni [8, 13, 58, 66]. Then, using the mediumeﬀect parameters calibrated for 58 Ni, we have extracted the neutron density distributions of tin
isotopes in the form of model-independent Sum-of-Gaussians (SOG) distributions. The accuracy of the neutron radii of tin isotopes were found to be less than 0.03 fm including both thestatistical and model uncertainties [70] as shown in Fig. 1.7.
After the success of our analysis method for Ni and Sn isotopes, we have performed several experiments to measure proton elastic scattering from heavier or lighter stable nuclei suchas 204,206,208 Pb, 40,42,44,48 Ca, 36 Ar, 32 S, 24 Mg, 20 Ne, and 16,18 O. Now we have examined the applicability of our analysis method to these isotopes. In addition, since our recent analysis
has showed that the absolute values of the cross sections are very important to determine themodel-independent density distributions, a separate experiment has been performed to know
absolute eﬃciencies of some detectors [71]. Several improvements to reduce the experimental
uncertainties have been also considered in recent analyses.
Above all, we have focused on the neutron density distributions of lead isotopes since theneutron skin of 208 Pb, which is strongly correlated with the EOS parameters as mentioned
above, is expected to constrain the EOS of asymmetric nuclear matter. In this thesis, thus, wereport the most recent results from the analysis of the proton elastic scattering data of leadisotopes.

8

CHAPTER 1. INTRODUCTION

1.4 Purpose of this work
Pb isotopes The purpose of this work is to evaluate the sensitivity of proton elastic scatteringto the neutron densities in heavy nuclei such as lead isotopes 204,206,208 Pb with a new χ2 method
including the ambiguity associated with the interaction model used in the present analysis. It
is important for the understanding of isospin asymmetry to study the systematic behavior of
the extracted neutron densities. We also focus on the extracted neutron skin thicknesses of
204,206,208
Pb to constrain on the EOS parameters such as S(ρsat ) and L.
Furthermore, if our analysis method to describe the scattering observables and to extract
the precise information of the density distributions works successfully even for lead isotopes
with large isospin asymmetries, we could apply this approach to unstable nuclei on the basisof this work. Indeed, we have already proposed and performed the so-called ESPRI project(Elastic Scattering of Protons with RI beams) at NIRS-HIMAC in Chiba, Japan, GSI in Germany, and RIBF in RIKEN, Japan. This project aims to measure the cross sections of proton elastic scattering from unstable nuclei by missing mass spectroscopy. Protons, which are
inverse-kinematically scattered from solid hydrogen target by heavy ion beam of unstable nuclei at E p = 300 MeV/A, are detected up to the momentum transfer of 2.5 fm−1 with recoil
particle spectrometer (RPS) which consists of plastic scintillators, multi-wire-drift chambers,
and NaI(Tl) calorimeter arrays. We have already performed developments and experiments
for several unstable nuclei such as 9,10,11 C, 20 O, and 66,70 Ni, and achieved 1-mm-thick solidhydrogen target and the excitation energy resolution of σ ∼ 500 keV. An approved experimentis planned to be performed at RIBF in FY2011.
Previous or planned experiments for 208 Pb Since 1980s, there have been several attempts
to extract the neutron density distribution or the neutron skin thickness of 208 Pb as listed inTable 1.1. They were all deduced by using hadronic reaction and consistent with each otherwithin the errors. However, except for the 208 Pb(p, p) at 650 MeV analysis, they must assume
a specific model-dependent function or a specific theoretical calculation as the neutron density
distribution of 208 Pb because they cannot reduce the large uncertainties caused by the various
model assumptions. Thus, the uncertainties associated with the theoretical models are inherent
in all the results listed in Table 1.1. It is diﬃcult to estimate the errors due to the model uncertainties, which are usually not mentioned, but they really should do unless they use hadronic
probes. On the other hand, wide-ranged angular distributions of scattering observables are
powerful tools to evaluate how precisely the reaction and structure models can explain the realdata. In addition, we used the realistic proton density and the eﬀective NN interaction tunedby the real data. This enables us to reduce the ambiguity of the reaction model and to use themodel-independent density distributions.
There have been also several attempts to extract the neutron density distributions of 208 Pb
from proton elastic scattering data [58, 62]. The energy of the incident protons, E p = 650 and
800 MeV used in Refs. [58, 62] are rather high, where the real part of the optical potential
is completely repulsive and the imaginary part is about 50 MeV deep. Thus the information
about the nuclear interior is masked by the strong absorption. To avoid this large ambiguity

1.4. PURPOSE OF THIS WORK

9

Table 1.1: The previous results of rn , r p and ∆rnp for 208 Pb deuced from several experiments
using various hadronic probes (all in fm). Except for X-ray cascade from p̄-atom (2), the errors
are experimental only.
Experiment208
Pb(p, p) at 800 MeV [58]∗208
Pb(p, p) at 650 MeV [62]†Isospin diﬀusion data in Sn + Sn at
50 MeV/u [51]‡GDR from (α, α0 ) at 120 MeV[61]PDR from 129−132 Sn(γ, γ0 ) [68]¶
PDR in 68 Ni and 132 Sn [72]k
X-ray cascade from p̄-atom (1)[67]∗∗
X-ray cascade from p̄-atom (2)[73]††

rp
5.45
5.46
-

rn
5.59(4)5.66(4)
-

5.44
5.60
5.45 5.65(5)mdl∆rnp
0.14(4)0.20(4)0.22(4)0.19(9)0.18(4)0.194(24)0.16(2)stat (4)syst
0.20(4)exp (5)mdl∗

2nd-order KMT potential assuming 3pG neutron density.
Density-dependent t-matrix using model-independent FB neutron density.
‡
21 sets of the Skyrme interaction parameters.
¶
PDR strengths analyzed by RHB+RQRPA calculations with DD-ME parametrization.
kEnergy weighted sum rule by PDR data globally fitted with various mean field models∗∗
antiproton-nucleus optical potentials assuming 2pF proton and neutron density.
††
newly developed Skyrme parametrization Skxsxx constrained by antiproton data of Ref. [67].
†of the interior structure, Starodubsky and Hintz [62] assumed that the nuclear matter density
in the nuclear interior is almost constant. They did not use the model-independent function
itself to extract the neutron density, but instead used the Skyrme–Hartree–Fock density plusa small correction expanded in a Fourier–Bessel series to estimate the error-envelopes of theneutron densities which reflect only the statistical errors of the experimental data. They also
used a density-dependent t matrix calibrated by p-40 Ca scattering data in the framework of
the non-relativistic formalism. Their approach is very similar to ours, as seen in our previous
study for tin isotopes [70], but they did not estimate the errors due to the model ambiguity,which is an unavoidable problem in describing hadronic reactions.
There are several theoretical approaches for energy-independent global analysis. Skyrme–
Hartree–Fock models with modern parameterizations have been tested employing the g-folding
optical potential to explain the data obtained from nucleon-nucleus elastic scattering from208
Pb at 40, 65, and 200 MeV [63]. It has been suggested that 208 Pb has a neutron skin thickness of ∼ 0.17 fm, because the SkM* model gives the best agreement with the experimental
data. Another global analysis of proton-nucleus elastic scattering data in the energy range
from 500 to 1040 MeV based on the Dirac phenomenological optical model in Ref. [64] gavea range of ∆rnp from 0.083 to 0.111 fm for 208 Pb. The obtained range of ∆rnp changes depending on the momentum transfer range and the NN interaction used for the global fit. In

10

CHAPTER 1. INTRODUCTION

these theoretical approaches, statistical errors and the eﬀect of the model-dependence must be
considered in the error-estimation of ∆rnp .
As a diﬀerent experimental approach, a parity-violating electron elastic scattering measurement to extract the neutron radius in 208 Pb (PREX) has been proposed at Jeﬀerson Laboratory[74, 75]. This challenging experiment is planned to measure the neutron rms radius for 208 Pb
with an accuracy of 1 % assuming a simple model-dependent function such as a Woods-Saxon
for the neutron density of 208 Pb. The advantage of this method is said to be its “model independence” of the interaction mechanism, but the planned accuracy of 1 % is larger than the
recent measurements using hadronic probes [62, 67, 68]. Besides this electroweak probe measurement has many experimental diﬃculties because of the required accuracy to measure thevery “weak” violation (∼ 0.1 ppm = 10−7 ).
Previous analyses for the EOS parameters Several studies including listed Refs. in Table 1.1 deduced the S(ρsat ) and L from various measurements such as isospin diﬀusion [51],pigmy dipole resonance [68, 72], and world data of various nuclear reactions [76]. The extracted regions are not so diﬀerent each other, but widely spread, even though the uncertainties
associated with the dependence of a certain theoretical model were not considered.Figures 1.8
and 1.9 taken from Refs. [72, 76], show that the recent results of the slope coeﬃcient L aresummarized. Thus, the situation of the EOS parameters is similar to that of the neutron skin
thickness for 208 Pb as mentioned above. Our approach may be also eﬀective to constrain thetheoretical models and the EOS parameters. The study for other nuclei as well as lead isotopes
will give us more strict constraint on the EOS paramters.

This work aimed to extract the model-independent neutron density distributions and neutron rms radius of 204,206,208 Pb with an accuracy of 1 % (δrn /rn ) even including the model
uncertainties. This accuracy is comparable to the planned accuracy of the PREX experiment.
In this thesis we report our results of quantifying the uncertainty of both the experimental
data and the present reaction model as the errors of the neutron densities in lead isotopes. The
experimental procedures are described in Chap. 2. The experimental data of proton elastic
scattering from 58 Ni, and 204,206,208 Pb at E p = 295 MeV have been newly obtained and several
improvements have been applied at the data reduction, as explained in Chap. 3. so as toreduce the systematic uncertainties which have not been considered previously. This work
is mainly based on the approach and technique described in Refs. [69, 70]. A brief outlineof the medium-modified RIA model and the results of the extraction of the neutron density
distributions in lead isotopes are given in Chap. 4. The extracted neutron rms radii and skin
thickness are compared with several theoretical calculations in Chap. 5. Finally we provide asummary in Chap. 6.
This thesis is mainly based on the recent article Ref. [77].

1.4. PURPOSE OF THIS WORK

11

Figure 1.8: Constraint regions on the S(ρsat )-L plane by several recent analyses. Taken from
Ref. [76].

Figure 1.9: Constraint regions on the L by several recent analyses. Taken from Ref. [72].

Chapter 2
EXPERIMENTAL PROCEDURE
The experiment (E248) was performed at the ring cyclotron facility in the Research Center for
Nuclear Physics (RCNP), Osaka University. An overview of the RCNP facility is shown in
Fig. 2.1. The polarized proton ealstic scattering from 58 Ni, and 204,206,208 Pb at E p = 295 MeV
was measured by using the high-resolution magnetic spectrometer, “Grand Raiden” [78] andfocal plane detectors [79]. Diﬀerential cross sections and analyzing powers were obtained in
the angular range of 7.5◦ ≤ θlab ≤ 48.0◦ . In this chapter the details about the experimental
setup and procedure are described.

2.1 Beam line
2.1.1 Polarized proton beam
A high intensity polarized ion source (HIPIS) provided a polarized proton beam employing
the cold atomic beam technology (∼ 30 K) and an electron cyclotron resonance ionizer [80].
The polarized proton beam extracted from HIPIS was injected to an azimuthally varying field(AVF) cyclotron (K120) to keep the polarization axis in the vertical direction. The beam at
53 MeV after acceleration with the AVF cyclotron was further accelerated up to 295 MeV using a six sector ring cyclotron (K400) in a coupled mode. Since the single turn extraction fromthe ring cyclotron was achieved, the beam was very clean and free from the background causedby other bunches. The vertically polarized proton beam from the ring cyclotron was achromatically transported to the WS beam line and scattered at a target in a scattering chamber. A
typical size of the beam spot was less than 1 mm in diamter.
A schematic view of the WS beam line in the experimental hall is shown in Fig. 2.2.
The beam was tuned by using many constituent elements of the beam line such as dipole,quadratic, and steering magnets, and beam viewers so that the beam was transported along
the central axis of the beam line arranged in advance. Since the beam was achromaticallytransported on the target, if the drift of the beam energy occurred due to instabilities of thebeam conditions, we cannot recognize the drift by the beam spot on the target position only.
Therefore, during the experiment we regularly checked the two beam viewers of BV9-WS

13

14

CHAPTER 2. EXPERIMENTAL PROCEDURE

Figure 2.1: Overview of the RCNP ring cyclotron facility with the WS beam line. AVF andRing cyclotron operate in a coupled mode and provide polarized proton beam at E p = 295 MeV
to the WS beam line in west experimental hall.

and BV8-WS located at momentum-dispersive positions (∼3 m and ∼10 m upstream from thetarget position, respectively) as shown in Fig. 2.2. At BV9-WS, for example, the dispersion is2.1. BEAM LINE15

WS beam line in West Experimental Hall @ RCNP
STH12-WS QM12M-WS
QM12D-WS QM12U-WS

BV13-WS

Beam Dump

STH11-WS

QM10M-WS
QM11D-WS
Akimune Slit
QM11U-WS
QM10D-WS QM10U-WS QM9S-WS

BV12-WS BV11-WS
BV10-WS
GV10-WS

STV10-WS BV9-WS
STH10-WS

QM8D-WS QM8U-WS
BM7-WS

GV9-WS BV8-WS

Scattering
Chamber

GV8-WS BM6-WS
QM7D-WS
SX1-WS
STV9-WS
BV8a-WS
QM7U-WS
STV8-WS
GV7-WS（GR_FP_BV）

GV6-WSBLP2

QM6D-WS

Dump Q

QM6U-WS（GR_TAMIQ）

BV: Beam viewer#11

BV7-WS（GR_DUNP）

GV: Gate valve

STV6-WS

QM: Q magnet, BM: D magnet

BM5-WS
BV6-WSSLIT LR
BM4-WS
QM6D-WSS1
QM6U-WSS1

BV5-WSS1

BV5-WS
GV5-WS
GV5-WSS1OSHO STH6-WSS1

BV4b-WSS1

STV6-WSS1

BV4-WS

QM5bU-WSS1 STV5-WS
QM5bD-WSS1
BM3-WS

Figure 2.2: Schematic view of WS beam line

about ±5 mm, this value corresponds to the range of the beam incident angle −0.1◦ ≤ θinc. ≤0.1◦ (±5/3000 rad). Thus, the uncertainty of the incident beam angle was approximately,δθinc. ' ±0.02◦ after the beam spot at BV9-WS were narrowed enough to distinguish the
1 mm drift of the beam at BV9-WS by eyes. The eﬀect of this uncertainty on the scatteringobservables is referred to in Chap. 3.4.

2.1.2 Beam line polarimeter
The polarization of the proton beam was constantly monitored by using a sampling-type beamline polarimeter (BLP) placed in the WS beam line. The BLP consists of four (left, right, up,
and down) pairs of plastic scintillation counters to measure p-H scattering from the (CH2 )nfoil. In Fig. 2.3 the top view of the BLP is shown and the left and right pairs of counters in thehorizontal plane can be seen. The other (up and down) pairs are located in the vertical plane.

16

CHAPTER 2. EXPERIMENTAL PROCEDURE

Figure 2.3: Setup of the BLP (top view)

A proton in the beam was elastically or quasi-elastically scattered from another proton ofhydrogen or carbon in the (CH2 )n foil and the scattered and recoiled protons are simultaneously
detected with a pair of scintillation counters. Each pair of scintillation counters was located at
the scattering and recoil angles of 17.0◦ and 70.5◦ where p-p scattering from (CH2 )n realizes
the large eﬀective analyzing power Ay = 0.40. The y-component of the polarization wasderived from the asymmetry of the number of the events counted by the left and right pairsof scintillation counters. A typical beam polarization was about 70 %. The up and downpairs of the counters were used to determined the beam intensity during the measurements asmentioned in Sec. 2.3.
The polyethylene target was periodically inserted into the beam position during the polarization measurement, while it was removed out of the beam position during the data aquisitionof the focal-plane events. The polarization direction of the proton beam was reversed for every
one second to reduce the systematic asymmetries.

2.2 Targets
Self-supporting metal foils were used as 204,206,208 Pb and 58 Ni targets in this experiment. Twotypes of thin and thick metal foils were prepared for each target except for 204 Pb. Thin targets

2.3. FARADAY CUPS

17

were for the measurements at forward angles (7.5◦ ≤ θlab ≤ 28.5◦ ) in order to reduce the deadtime of the data acquisition less than 10 %, while thick ones were for the measurements at
backward angles (25.5◦ ≤ θlab ≤ 48.0◦ ) to increase the yields. The thicknesses and enrichmentsof the targets are listed in Table 2.1.
In order to reduce the relative systematic errors due to instabilities of the beam condition or
detectors, the four targets were automatically interchanged every ∼5 minutes at each scattering
angle. Typically the errors of the target thicknesses are about 1 %.
Table 2.1: Target thicknesses and enrichments of Ni, and lead isotopes.
NucleusThin
204
Pb
23.7 mg/cm2
206
Pb
18.4 mg/cm2
208
Pb
31.5 mg/cm2
58
Ni
39.5 mg/cm2

Thick
not prepared50.9 mg/cm2
78.8 mg/cm2
100 mg/cm2

Enrichment
99.6 %
99.3 %
99.7 %
99.39 %

2.3 Faraday Cups
The proton beam was stopped by two kinds of Faraday cups depending on the settings of themeasurement. In the measurements at forward angles (7.5◦ ≤ θlab ≤ 28.5◦ ) the Faraday cup
was mounted inside the scattering chamber (SCFC). This Faraday cup was most stable against
the fluctuations of the beam profile, and its charge collection eﬃciency has been measured bya seperate experiment [71].
In the measurements at backward angles (25.5◦ ≤ θlab ≤ 48.0◦ ) the intensity of the proton
beam was much higher than that for the forward measurements to increase the yields. The
other Faraday cup located about 25 m downstream from the scattering chamber (WallFC) wasused due to the regulation of the radiation control. The relative charge collection eﬃciency
between the SCFC and WallFC was monitored by using p-H cross section counted by up anddown pairs of BLP.

2.4 Magnetic spectrometer, “Grand Raiden”
Scattered particles from a target nucleus were analyzed using the high resolutin magnetic spectrometer “Grand Raiden” (GR) [78]. Figure 2.4 shows a schematic view of the GR spectrometer and focal plane detectors [79].
The GR spectrometer has a Q1-SX-Q2-D1(-MP)-D2(-DSR) configuration, where D, Q,
and SX denote a dipole, a quadrupole, and a sextupole magnet, respectively. Design specifications of the spectrometer are summarized in Table 2.2. The multipole magnet (MP) is designed

18

CHAPTER 2. EXPERIMENTAL PROCEDURE

Table 2.2: Design specifications of the GR spectrometer
Configuration
Q1-SX-Q2-D1(-MP)-D2(-DSR)
Mean orbit radius
3m
Total deflection angle
162◦
Measureable angle−4◦ –90◦
Focal plane length
120 cm
Focal plane tilting angle
45◦
Maximum magnetic rigidity
5.4 T·m
Momentum range
5%
Momentum resolution (p/∆p)
37000
Momentum dispersion (x|δ)
15.45 mHorizontal magnification (x|x)−0.417
Vertical magnification (y|y)
5.98

to minimize the higher-order aberrations. The third dipole magnet (DSR) is additionally used
for measurements of the spin rotation parameters or the polarization transfers. The MP and
DSR magnets were not used in the present experiment.
The GR spectrometer can rotate around the axis located at the center of the scatteringchamber from −4◦ to 90◦ . Therefore we can obtain the angular distributions of the scatteringobservables such as the cross sections, analyzing powers, and spin rotation parameters using
polarized beam. In this experiment, we measured the polarized proton scattering over the
angular range covered from 7.5◦ to 48.5◦ in 1.5◦ steps. This angular range corresponds to themomentum transfer from 0.55 fm−1 up to 3.5 fm−1 .

2.4. MAGNETIC SPECTROMETER, “GRAND RAIDEN”

Figure 2.4: Schematic view of the magnetic spectrometer, “Grand Raiden”

19CHAPTER 2. EXPERIMENTAL PROCEDURE

20

Table 2.3: Specifications of the VDCs.
Wire configuration
Active area
Number of sense wires
Anode-cathode gap
Anode wire spacing
Sense wire spacing
Sense wires
Potential wires
CathodeApplied voltage
Gas mixture
Entrance and exit window
Pre-amplifier
Digitizer

X(0◦ ), 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 gold-plated beryllium-copper wire
10 µm-thick carbon-aramid film
−5.6 kV (cathode), −0.3 kV (potential), 0 V (sense)
Argon (71.4%) + Iso-butane (28.6%) + Isopropyl-alcohol
12.5 µm aramid film
LeCroy 2735DC
LeCroy 3377 TDC

2.5 Focal plane detectors
The trajectories of the scattered particles were reconstructed from two sets (X1, U1 and X2,
U2) of vertical-type multi-wire drift chambers (VDCs) [79]. VDCs were placed at the focal
plane of the GR spectrometer. Specifications of VDCs are summarized in Table 2.3. Each
VDC consists of two sets of anode wire planes (X and U), sandwiched between three cathodeplanes. The structure of a X plane is illustrated in Fig. 2.5. Sense wire spacings are 6 mm and
4 mm for X and U planes, respectively, as shown in Fig. 2.6.
VDCs were filled with mixture gas of argon (71.4%), iso-butane (28.6%), and iso-propylalcohol. The saturated vapor of iso-propyl-alcohol at 2◦ C and 1 atm was used to avoid a gain
loss caused by aging eﬀects such as polymerizations of the quencher gas, which contaminate
the anode wires. High voltages of −5.6 kV and −0.3 kV were supplied to the cathode planes
and the potential wires, respectively, while the sense wires were grounded (0 V).
The atoms in the gas were ionized along the trajectory of the incoming charged particle.
Electrons in ionized gas moved toward a sense wire. The drift velocity was almost constant andavalanche multiplications occured near the sense wire where the electric field rapidly become
strong. Signals from sense wires were amplified and discriminated with LeCroy 2735DC cards
and timing information was digitized with LeCroy 3377 drift chamber TDC.
Two 1-cm-thick plastic scintillators (PS1 and PS2) right behind the VDCs were used togenerate the trigger signals of the data acquisition. Timing and pulse-height information of thesignals of PS1 and PS2 are digitized by using LeCroy FERA and FERET systems. A 1-cm-

2.6. TRIGGER SYSTEM

21

thick aluminum plate was placed between the two plastic scintillators to prevent δ-electrons
knocked out by scattered particles at PS1 from entering PS2.

SctteredParticle
Cathod Plane
2mm

10mm

di 1

Anode WiresdiPotential Wiredi 1

Sense Wire

Cathod Plane
Figure 2.5: Structure of a X plane of the VDCs.

2.6 Trigger system
The readout electronics and trigger systems of the focal plane scintillators for GR are shown in
Fig. 2.7. Output analog signals from plastic scintillators (PS1 and PS2) were divided into two
passages, the one was discriminated by a constant fraction discriminator (CFD; Ortec 935) andthe other was sent to a FERA (Fast Encoding and Readout ADC; LeCroy 4300B) module after
a delay. One of the CFD outputs was transmitted to TFC (Time to FERA Converter) backed
by FERA modules. A coincidence signal of the CFD outputs from both sides of the same
scintillators was generated by a Mean-Timer circuit (REPIC PRN-070), in which the times of
two signals were averaged.
The trigger system was constructed with LeCroy 2366 universal logic modules (ULM) withfield programmable gate-array (FPGA) chips [81]. The logical circuits were designed using a
CAD program on a personal computer and down-loaded to the ULMs. As shown in Fig. 2.7,
the trigger system received signals from the outputs of the Mean Timers and generated the GR
trigger by the coincidence of PS1 and PS2, internally.

22

CHAPTER 2. EXPERIMENTAL PROCEDURE

Figure 2.6: Wire configurations of the VDCs. The wire numbering orders are shown by thearrows

2.7 Data acquisition system
Schematic view of the data acquisition (DAQ) system [82, 83] is shown in Fig. 2.8. Two typesof data line, i.e., the drift-time data detected by two VDCs and the charge and timing datadetected by trigger scintillators (PS1 and PS2), existed in this experiment. They were digitizedby LeCroy 3377 system and LeCroy FERA/TFC system, respectively, event by event. For thesubsequent event reconstruction, a event header, an event counter, and input register words areattached to the each digitized data by using a specified modules, that is, the Flow Controlling
Event Tagger (FCET) [84].
The digitized data for each detector system was transfered and accumlated in pararell via
the ECL bus to high speed memory modules (HSM) in the VME crate. Each data line wasconnected to a pair of two HSMs, which worked in a double-buﬀer mode to reduce the deadtime caused by the transfer of the buﬀered data. Finally, the buﬀered data was transfered andstored in the work station server “IBM RS/6000 SP” via the Gigabit Ethernet link. The event
reconstruction and online data analysis were performed also on this computer.
This data readout flow completely starts with hardware logics, which does not include
software management nor CAMAC functions. Therefore the typical dead time for an event isless than 30 micro seconds. In the present experiment, the acquired data is about 50 GBytes.

2.7. DATA ACQUISITION SYSTEM

23

Figure 2.7: Wire configurations of the VDCs. The wire numbering orders are shown by thearrows24

CHAPTER 2. EXPERIMENTAL PROCEDURE

Figure 2.8: Schematic view of the data acquisition system

Chapter 3
DATA REDUCTION & RESULTS
The data reduction was performed by using the central computing system at RCNP, i.e., IBM
RS/6000 SP system named “kasuga”.
The acquired data consists of the BLP data and the focal plane detectors data. While the
BLP data was used to determine and calibrate the beam properties such as the beam polarizations and the beam intensities, the data of the focal plane detectors was used to identify the
incoming particles and to determine their focal plane positions, i.e., to obtain the elasticallyscattered proton events. Now the details of these procedures are described in this chapter.

3.1 BLP data
Yields on the four (L, R, U, and D) pairs of scintillators (NL , NR , NU , and ND ) at the BLP for
spin-up (↑) and spin-down (↓) modes are described asp↑a↑↑− NL(U)= σ0 N0 Nb↑ L(U) ∆ΩL(U) (1 + Ay p↑y(x) ),NL(U)= NL(U)

(3.1)

p↑
↑
a↑
NR(D)
= NR(D)− NR(D)= σ0 N0 Nb↑ R(D) ∆ΩR(D) (1 − Ay p↑y(x) ),

(3.2)

p↓
a↓
↓
− NL(U)= σ0 N0 Nb↓ L(U) ∆ΩL(U) (1 + Ay p↓y(x) ),NL(U)= NL(U)

(3.3)

↓
NR(D)

(3.4)

=

p↓
a↓
NR(D)
− NR(D)= σ0 N0 Nb↓ R(D) ∆ΩR(D) (1 − Ay p↓y(x) ),where N p and Na are the numbers of prompt and accidental coincidence events measured by
BLP, respectively. σ0 and Ay are the unpolarized cross section and analyzing power for p-H
scattering from polyethylene target. N0 and Nb are numbers of the target and beam particles.
py(x) is the vertical (horizontal) component of the beam polarization and p↓ ≤ 0 ≤ p↑ .  and∆Ω are the eﬃciency and solid angle of each detector. The scattering angles of forward andbackward protons were θlab = 17.0◦ and 70.5◦ for 295 MeV proton beam, respectively.

3.1.1 Beam polarization
The beam polarization was determined by using the left-right asymmetry at BLP as describedin Sec. 2.1.1. The analyzing powers of Ay = 0.40±0.01 for the inclusive (p, 2p) scattering from25

CHAPTER 3. DATA REDUCTION & RESULTS

Asymmetry βLR

Beam polarization |py|

26

0.9 (a)
0.8
0.7
0.6
1000

1020

1040

1060

1080

1100

0.9 (b)
0.8
0.7
0.6
1000

1020

1040

1060

1080

1100

1020

1040

1060

1080

1100

1020

1040

1060

1080

1100

0.9
0.8
0.7
0.6

(c)

1000
1.2

(d)

1.1
1
1000

Run number

Figure 3.1: Three types of beam polarization (a, b, c) and instrumental asymmetry (d) for
each run. Red and blue circles mean |p↑(↓) | (spin-up and -down mode). Beam polarizations (a)assuming asymmetry βLR = 1, (b) using each βLR , and (c) using the averaged βLR fitted by aGaussian as shown in Fig. 3.2 are compared with each other. In this data reduction the type of(c) was used.

the polyethylene target, which was previously calibrated by another BLP installed between the
AVF and ring cyclotrons [85, 86], was used to determine the beam polarization.
The beam polarization can be obtained from Eq. (3.1), (3.2), (3.3), (3.4) as follows:
p↑(↓)
y

↑(↓)
↑(↓)
1 NL − βLR NR=,
Ay NL↑(↓) + βLR NR↑(↓)

(3.5)

βLR

↓ ↑
↑ ↓
NL↓ Q↑ − NL↑ Q↓L ∆ΩL NL Nb − NL Nb=≡=,R ∆ΩR NR↑ Nb↓ − NR↓ Nb↑ NR↑ Q↓ − NR↓ Q↑(3.6)

where Q is the charge of the proton beam collected by SCFC or WallFC. Since βLR in Eq. (3.6)
is the form of the fraction, the collection eﬃciency of each Faraday cup is canceled out. The
instrumental asymmetry βLR should be constant and independent from the beam polarization
during the measurement. However, due to instabilities of the beam condition or detectors the3.1. BLP DATA

27

35
Constant
Mean
Sigma

30

3.987 / 4
31.65
1.061
0.2373E-01

Counts/bin

25

20

15

10

5

0
0.9

0.95

1

1.05

1.1

1.15

Instrumental asymmetry βLR

1.2

1.25

Figure 3.2: The projection of the instrumental asymmetries βLR in Fig. 3.1(d) was fitted witha Gaussian.

asymmetry βLR calculated from Eq. (3.6) fluctuated for each run as shown in Fig. 3.1(d). Since
the trigger rate dependence of βLR was not seen, the main source of this fluctuation is thought
to be the beam instabilities.
Figure 3.1(a), (b), and (c) show three types of the beam polarization |p↑(↓) | for each run.
Red and blue circles represent spin-up and spin-down mode, respectively. The three types (a),(b), and (c) in Fig. 3.1 means the calculations of beam polarizations using three diﬀerent βLR ;(a) assuming no instrumental asymmetry, that is, βLR = 1, (b) using βLR calculated for each run,and (c) using the averaged value of βLR . The averaged βLR was determined from the projectionof the instrumental asymmetries βLR of Fig. 3.1(d). The histogram in Fig. 3.2 was fitted witha Gaussian. The half width at half maximum (HWHM) of the fitted Gaussian is used as an
uncertainty of the averaged βLR . The solid lines in (a), (b), and (c) are the arithmetic meansof spin-up and -down beam polarizations for each type of polarization. It is found that thesolid lines are almost constant and the same with each other independently from run number
and the types of the βLR . This means that the polarization of the protons produced by the ion
source kept stable during the experiment and independent from the instability of the beam
conditions such as the energy and the transmission which mainly depend on the beam line andthe cyclotrons.
While the polarizations calculated by type (b) are unstable due to the fluctuated βLR , thetype (c) calculation realized stable |py | since the constant value of βLR = 1.06(3) was used. The
beam polarization the type (c) was used in this data reduction. The typical beam polarization
was about 75 % during the measurement.

CHAPTER 3. DATA REDUCTION & RESULTS

28

3.1.2 Beam intensity
In the measurements at backward angles the thick target and WallFC were used for highintensity proton beam (∼ 400 nA) as mentioned in Sec. 2.3. Because of the multiple scattering
at a target, the transmission eﬃciency of the proton beam from the target to the WallFC is lessthan 100 %. Then we used the up and down pairs of the BLP detectors (U and D) to determine
the relative charge collection eﬃciency of the WallFC to the SCFC including the loss of thebeam transmission behind the target.
The absolute charge collection eﬃciency of the SCFC has been already calibrated by an
separate experiment [71]. The SCFC, which is located right behind the target, can catch almost
all the incident protons which enter the target area. Thus the SCFC is suitable for the reliablereference.
From Eqs. (3.1)-(3.4), the intensity of the incident proton beam (Nb↑(↓) ) at the BLP is givenasNb↑(↓)=βUD ≡ND↑(↓)↑(↓))/(2σ0 N0 D ∆ΩD ),
(NU +
βUD
N ↓ N ↑ − NU↑ Nb↓
U ∆ΩU= U↑ b↓.
D ∆ΩDND Nb − ND↓ Nb↑

(3.7)
(3.8)

Assuming the transmission from the BLP to the Faraday Cup and the ∆Ω are constant during
the measurement, the beam charge QSC(Wall) collected by the SCFC (WallFC) is proportional
to the number of the beam and the yield (NU + ND /βUD ) from Eq. (3.7) as follows:()
RSC(Wall)ND
QSC(Wall) = RSC(Wall) · Nb =
NU +
.
(3.9)
2σ0 N0 D ∆ΩDβUDThus the ratio RSC(Wall) is described as
RSC(Wall) = 2σ0 N0 D ∆ΩDQSC(Wall)
NU + βNUDD.

(3.10)

From Eq. (3.10) the relative ratio R = RWall /RSC was calculated for each run and target. The
averaged ratio R for each target was determined as listed in Table 3.1. The ratio of 204 Pb isrelatively large compared to the other three targets. This is because the thin target was usedfor 204 Pb only in the backward measurements. Compared to the case of the beam polarizationin the previous subsection, the fluctuation of the up-down instrumental asymmetry βUD wererelatively small and hence, the errors of the ratios are about 1 % as shown in Table 3.1.

3.2 Particle identification
Timing and pulse-height information of trigger scintillators (PS1 and PS2) were used to identify the charged particle which triggered the event.

3.2. PARTICLE IDENTIFICATION

29

Table 3.1: The relative ratio R = RWall /RSC of the beam collection eﬃciencies of SCFC and
WallFC
206
208
58
Target 204 Pb
Pb
Pb
Ni
R
0.97(1) 0.95(1) 0.92(1) 0.95(1)x 10 2
1600

208

protonPb(p,p)1400

Ep=295 MeV

1200

°

θlab.=36.0

Counts/ch

1000
800

deuteron

600
400
200
0
200

400

600

800

1000

Pmean of PS1 (ch)Figure 3.3: The spectrum of the geometrical mean P̄ of PS1. proton and deuteron peaks are
identified.

The energy loss of a charged particle passing through a plastic scintillator (∆E) depends onthe charge and velocity as described by the well-known Bethe-Bloch formula. The scintillation
photons produced at a position x where the charged particle passed through were detected by
PMTs attached on both sides of a scintillator. The number of the scintillation photons initially
produced at x was proportional to ∆E, but it was attenuated due to the absorption in the materialof the scintillator during the transmission. The number of photons entering a PMT (I) is simply
described as a function of position x,( x)
I(x) = I0 exp − ,(3.11)lwhere I0 is the initial number of photons at x and l is the attenuation length of the material.
When the length of the scintillator is L, the numbers of photons at the both sides of the scintillator are I(x) and I(L − x). The output charge of the left and right side of PMTs (PL and PR )

CHAPTER 3. DATA REDUCTION & RESULTS

30

1000

10 4

208

Pb(p,p)800

θlab.=36.0°

Pmean of PS1 (ch)10 3deuteron

600
10 2

400

10

proton

200

0

1
-400

-200

0

200

400

X position (mm)Figure 3.4: Colors plot of P̄ of PS1 versus the position spectrum at the focal plane. It is foundthat P̄ is independent of the focal plane position.

are proportional to I(x) and I(L − x) respectively. The geometrical mean P̄ of PL and PR is( L)√√
P̄ = PL · PR ∝ I(x) · I(L − x) = I0 exp −.
(3.12)
2lP̄ is independent of the position x and proportional to the initial number I0 and ∆E. The P̄
spectra are useful for the particle identification (PID) as shown in Fig. 3.3. Figure 3.4 shows
the P̄ of PS1 versus the X position at the focal plane. The X position at the focal plane wasdetermined by VDCs data as written in the next section. The geometrical mean P̄ of PS1 isfound to be independent of the position at the focal plane.
However the proton peak in the ∆E spectrum has a tail structure over the deuteron peakregion as clearly seen in Fig. 3.4. If only the ∆E information was used for the particle identification, the proton events in the tail was cut oﬀ together with the deuteron and triton events.
This causes the reduction of the yields, which accounts for about one percent of the total and is
not negligible for the precise measurement of the cross sections, especially at the angles where
the background events are relatively large.
Then the information about time of flight (TOF) was also used for PID. Figure 3.5 shows
the time diﬀerence between the trigger produced by PS1 and PS2 and the radio-frequency(RF) signal of the AVF cyclotron. There exist two peaks in the Fig. 3.5 because the RF signals
were downscaled to a half of its frequency. Two-dimensional colors plot of TOF versus the X
position at the focal plane is also shown in Fig. 3.6. One can clearly distinguish the elastically3.3. MULTI-WIRE DRIFT CHAMBERS

31

Figure 3.5: TOF spectrum between the trigger and the downscaled RF.

scattered protons from deuterons. It is found that the TOF spectrum is useful to identify the
proton events. Figure 3.7 shows the two-dimensional plot of TOF versus ∆E which was finallyused for PID. The area surrounded by the dashed lines was used as the gate of the proton events.

3.3 Multi-wire drift chambers
3.3.1 Track reconstruction of scattered particles
As mentioned in Sec. 2.5, the scattered particles analyzed by the GR spectrometer were detected with VDCs at the focal plane. Since a charged particle enter VDCs in the 45◦ direction,
the number of hit wires in one plane is usually more than three as shown in Fig. 2.5 while thebackground events such as the X- and γ-rays mostly hit only one wire.
The following procedures were applied for the reconstruction of each wire-plane position
where the charged particle passed through in VDCs as reported in Ref. [87].
• A cluster had at least two hit wires. A single hit wire is not considered as a cluster andignored.
• If there existed more than two hit wires in a single cluster, the hit wire of the shortest
drift time was ignored. This is because the uniformity of the electric field was distorted
near the sense wire.

32

CHAPTER 3. DATA REDUCTION & RESULTS

Figure 3.6: Colors plot of the X position at the focal plane versus TOF. Elastically scatteredprotons can be clearly identified as shown in the area surrounded by the solid circle.

Figure 3.7: Colors plot of P̄ versus TOF of PS1 for PID.

Counts/bin

3.3. MULTI-WIRE DRIFT CHAMBERS

70000
60000
50000
40000
30000
20000
10000
0

(a)

T0
0

50

100

Counts/bin

150

200

250

300

350

400

1

1.2 1.4

TDC X1 (ch)

x 10
10000

33

(b)

8000
6000
4000
2000
0
-0.4 -0.2 0

0.2 0.4 0.6 0.8

Drift Length X1 (cm)Figure 3.8: (a) TDC spectrum on X1 plane obtained from LeCroy 3377. (b) Drift length (cm)converted from TDC spectrum (a) by using so-called x-t relation.

• The intercept position and angle for each cluster were calculated by a least-square
method.
The position and angle of a trajectory at each wire plane is calculated by a least-square fittingof the drift lengths of the hit wires in one cluster The trajectory is uniquely determined by
combining the deduced positions at four wire planes. The residual position resolution for eachplane was about 300 µm in full width at half maximum (FWHM). It corresponds to the energy
resolution of about 10 keV at E p = 295 MeV, which is much smaller than the energy spreadof the beam (≥ 100 keV in FWHM). Figure 3.8 shows the typical TDC spectrum (drift time)
and the converted drift length.
If more than one clusters existed in the X1(U1) and/or X2(U2) plane, the consistency
among the angles was checked for all the possible combinations by χ square method. In
addition, If there were more than one good track (multi-track event), this event is ignored
because we could not judge whether each track is true or not. However, the number of multitrack event is less than 1 % and neglected here. This multi-cluster treatment is eﬀective forCHAPTER 3. DATA REDUCTION & RESULTS

34

6
208

Theta at VDC (deg.)

4

Pb(p,p)θlab.=36.0

°

10 2
2

0
10

-2
1st
g.s.

-4

-6
-200

-100

0

100

200

300

400

500

1

X position (mm)Figure 3.9: θ versus X plot at the focal plane of the spectrometer reconstructed from the VDCs
data in the case of 208 Pb at θlab. = 36.0◦ .

avoiding severe reduction of the yields and eﬃciencies of VDCs especially at the scattering
angles where the background events are relatively large. The yield ratios of single-cluster overmulti-cluster events are about 80 ∼ 95 %.
After the reconstruction of the trajectories of the scattered particles, the position and angle
spectra at the focal plane were obtained as shown in Fig. 3.9. Figure 3.10 shows the typical
position spectra of the four target nulcei at θlab. = 36◦ . Elastically scattered proton events
can be distinguished from other inelastic events which represent excited states of each target
nucleus. Finally their yields for 204,206,208 Pb and 58 Ni targets were counted within the 3σ regionof the elastic peaks in Fig. 3.10.

0
-100

-

2.648MeV (3 )+0
200 -100+++

2.460MeV (4 )500

1.454MeV (2 )-2.615MeV (3 )++g.s. (0 )1000

g.s. (0 )+2.460MeV (4 )

-

0
100
Spin UP

0.899MeV (2+)
1.274MeV (4 )0.803MeV (2 )+1.684MeV (4 )+200

2.615MeV (3 )+g.s. (0 )

+

400
0
800
600
400
200
0

1.454MeV (2 )+g.s. (0 )-2.648MeV (3 )+++g.s. (0 )Ni
500g.s. (0 )

Pb

0
800
600
400
200
0

35

800
600
400
200
0
600

0.899MeV (2+)
1.274MeV (4 )0.803MeV (2 )+1.684MeV (4 )+g.s. (0 )

+

400

1000
58

g.s. (0 )

206

800
600
400
200
0
600
200

208

Counts/bin

Pb

204

Pb

3.3. MULTI-WIRE DRIFT CHAMBERS

0100
Spin DOWN

200

X position (mm)Figure 3.10: Typical position spectra for proton elastic scattering from 204,206.208 Pb and 58 Ni atθlab. = 36.0◦ . Elastic events (g. s. (0+ )) can be distinguished from other excited states in each
spectrum.

CHAPTER 3. DATA REDUCTION & RESULTS

36

3.3.2 Tracking eﬃciency
The tracking eﬃciencies of each plane of VDCs was estimated as follows:
NX1∩U1∩X2∩U2,
NU1∩X2∩U2
NX1∩U1∩X2∩U2=,
NX1∩X2∩U2
NX1∩U1∩X2∩U2=,
NX1∩U1∩U2
NX1∩U1∩X2∩U2=,
NX1∩U1∩X2

X1 =(3.13)

U1(3.14)

X2
U2(3.15)(3.16)

where NX1∩U1∩X2∩U2 denotes the number of events in which the cluster positions can be determined for all planes, while NU1∩X2∩U2 , for example, denotes the number of three planes except
for X1 plane. The total tracking eﬃciency for the elastic events is estimated by multiplying
the eﬃciencies of all the four planes:total = X1 U1 X2 U2 .

(3.17)The obtained eﬃciencies were about 95 ∼ 98% per plane. Thus the total eﬃciencies total
were about 86 ∼ 92%. The eﬃciencies gradually decreased as the scattering angle becomelarge. This is because at the backward angles the rate and yield of the background eventsis getting large compared to the elastic events, and thereby the ineﬃcient moment, when theelectric field is eﬀectively lowered, becomes large. If the mulcti-cluster treatments were not
performed, the eﬃciency of each plane would get down to ∼ 90% because of the high amountof the background events.
It should be noted that even if one analyzes only single cluster events for the track reconstruction, in order to calculate the eﬃciency for each plane by using Eqs. (3.13)–(3.16)
one must use all the events including multi-cluster events for the denominators of Eqs. (3.13)–(3.16). Otherwise, the eﬃciency is misestimated and seems to be much higher than it should be
especially where the background events relatively increase and make the multi-cluster events.
This is mainly caused by the strong correlations of the background events among four
planes. Now we think about the case of the X1 plane eﬃciency, for example. In case of nocorrelation between four planes, from Eq. (3.13), the tracking eﬃciency X1 for the singlecluster events is written as∏4
i (1 − mi )YX1 = ∏i=1
= 1 (1 − m1 ),(3.18)
4

(1
−
m
)Y
i
i
i=2
where i , mi , and Y are the intrinsic eﬃciency against elastically scattered protons for eachplane, the probability of making multi-cluster for each plane, and the real yield, respectively.
However, if there existed correlations between four planes, the X1 is(∏4 )i=1 i (1 − m)YX1 = (∏4 )= 1 .
(3.19)
i=2 i (1 − m)Y

3.4. DIFFERENTIAL CROSS SECTIONS AND ANALYZING POWERS

37

Here, for simplicity, it is assumed that all four planes are simultaneously correlated with one
another and thus the probability of detecting multi-cluster events is m. The two equations
above shows that the existence of the correlations causes the overestimation of the eﬃcienciesof single-cluster tracks.
Indeed, it is found that there exits the correlations among four planes and the eﬃciencies
calculated by Eqs. (3.13)–(3.16) are little diﬀerent between the single- and multi-cluster analyses. Besides, the yield reconstructed by single-cluster analysis is, of course, smaller than bymulti-cluster analysis. Therefore, when one uses such a overestimated eﬃciency, the crosssections finally obtained are doubly underestimated. It is known that this estimation causes
about 5 ∼ 20 % decrease of the cross sections depending on the scattering angles. The previously measured data for 58 Ni in Ref. [69] used only single cluster events to estimate the VDCs
eﬃciencies. This is one of the reasons that we newly measured the data for 58 Ni as well as leadisotopes.

3.4 Diﬀerential cross sections and analyzing powers
Elastically scattered proton events for each target nucleus can be chosen from the position
spectrum of the focal plane as shown in Fig. 3.10. Yields of the elastic events for each spin
direction of the beam (↑ and ↓) are expressed asY ↑(↓) =)dσ (↑(↓) ↑(↓) ↑(↓)
1 + p↑(↓)
A ∆ΩGR ,y Nt Nb lydΩ(3.20)

where py , Nt , Nb , l, , and ∆ΩGR are the y component of the beam polarization, number oftarget nuclei in unit area, number of protons in the beam, live-time ratio of the DAQ system,
total detector eﬃciency, and solid angle of the Grand Raiden spectrometer, respectively. The
number of proton in the beam Nb was derived from the collected charge by each Faraday Cup
and the collection eﬃciency of the SCFC as mentioned in Sec. 3.1.2. The total eﬃciency 
was determined by the product of the total eﬃciency of the VDCs and the trigger eﬃciency
of PS1 and PS2 for protons at 295 MeV. The absolute trigger eﬃciency of PS1 and PS2 hasbeen also measured in a separate experiment recently as written in Ref. [71]. The solid angle∆ΩGR was defined by the slit of ±20 mrad wide and ±30 mrad high which was located at theentrance of the Grand Raiden spectrometer.
Therefore, diﬀerential cross sections dσ/dΩ and analyzing powers Ay at each scattering
angle are derived from Eq. (3.20) as follows: ↓ ↑ Y pyY ↑ p↓y 
1
1
dσ


=
− ↑
,
(3.21)↑↓↓dΩpy − py Nb  ↓ l↓ Nb  ↑ l↑ Nt ∆ΩGR
1−αAy =(3.22),
αp↑y − p↓y
/
Y↑
Y↓
.
α =
(3.23)Nb↓  ↓ l↓ Nb↑  ↑ l↑

38

CHAPTER 3. DATA REDUCTION & RESULTS

Note that p↓y is a negative value, thus dσ/dΩ becomes positive definite. Statistical errors ofcross sections and analyzing powers (∆(dσ/dΩ) and ∆Ay ) are estimated by the simple way ofthe propagation of uncertainty and described as
( )
dσ
1
dσ
1
·
∆
=
· ↑
dΩ
dΩ py − p↓y αp↑y − p↓y{[]× (1 − α)2 (p↓y ∆p↑y )2 + (p↑y ∆p↓y )2
 ↓2
1/2
2 ↑2 

p
α
p


y
y

+(p↑y − p↓y )2  ↑ +,(3.24)


↓

Y
Y
{
1
↓2
∆Ay =
(1 − α)2 (α2 ∆p↑2
y + ∆py )↑↓ 2(αpy − py )
(
)}1/2
1
2 ↑
↓ 2 1
+α (py − py )+.
(3.25)Y↑ Y↓√
The uncertainties of the counting statistics (1/ Y) are so small that the statistical errors aremainly determined by the uncertainty of the polarization (py ∼ 0.75(4) determined from BLPdata) especially for the analyzing powers. The statistical errors are typically about 1 ∼ 2 %.
The systematic errors of this measurement mainly consist of the uncertainties of the beam
instability and the target thicknesses, and these are about 1 % respectively. The ambiguity of
the incident angle is the main component of the beam instability which aﬀects the scatteringobservables for heavy nuclei such as lead isotopes. This uncertainty was estimated from thechange of the cross sections when the the scattering angles change with ±0.02◦ , which were
calculated by the RIA calculation. a
Therefore, the systematic errors account for a main part of the uncertainties of this experimental data. The error of Nb was due to the error of the charge collection eﬃciency as written
in Sec. 3.1.2. In this data reduction this error was added to the systematic errors, not to thestatistical errors described above. We totally estimated the the systematic errors as ±3 % of
the obtained data. Therefore the experimental uncertainties are finally obtained by adding the±3 % systematic errors to the statistical errors calculated from Eq. (3.24) and (3.25)
Compared to previous measurements at RCNP as performed in Ref. [69] and [70], there
are several improvements in both the measurement and data reduction which are already mentioned above. Thus newly obtained data is systematically improved and especially the data setof cross sections for 58 Ni is around 10 ∼ 15 % larger than the previous set used in Ref. [69] and[70]. Although the previous data cannot be compared directly with the new data since the cona
As mentioned in Chap. 2.1, we monitored the beam positions to keep the beam conditions stable after eachmeasurement. That means that we did not recognize the short-term and real-time stability of the beam just as themeasurement is running. The BLP data in Figs. 3.1 and 3.2 partly shows the instabilities of the beam conditions
during the measurement. Thus, we estimated the uncertainty from the beam angle a little bit larger than the simply
calculated from the angular range of θlab. ± 0.02◦ . For more sophisticated measurement with small uncertainty ofthe beam conditions, we need new instruments for nondestructive monitoring of the real-time beam position withan accuracy of less than 1 mm.

3.4. DIFFERENTIAL CROSS SECTIONS AND ANALYZING POWERS

39

ditions of the measurements are diﬀerent from each other, we summarize these improvements
and the eﬀects on the cross section data:• Monitoring the beam condition by beam line viewers after each run. (keep the variationof the cross section due to the angular fluctuation about ±1 %.)• Particle identification using dE-TOF information. (increase of ≥ 1 %: see in Ref. [71])• Multi-cluster treatments for the VDCs data reduction. (increase of 5 ∼ 20 % if one used
Eqs. (3.13)–(3.16) to estimate the tracking eﬃciencies.)• Calibrations of absolute eﬃciencies of the trigger scintillators and absolute collection
eﬃciencies of the SCFC by a separate experiment in Ref. [71]. (increase of ∼ 2.0 % and1.5 %, respectively )Data sets of angular distributions of diﬀerential cross sections and analyzing powers forpolarized proton elastic scattering from 58 Ni and 204,206,208 Pb were obtained over an angularrange of 7◦ < θ < 50◦ , corresponding to a range of momentum transfer from 0.55 fm−1 to
3.5 fm−1 . The digital data are listed in Appendix A. In the next chapter the analysis method
to extract the neutron density distributions in lead isotopes via the experimental data of proton
elastic scattering are given.

Chapter 4
ANALYSIS
In this chapter, we report our analysis method and results of the neutron density distributions
extracted from the experimental data of proton elastic scattering from 204.206,208 Pb.
The analysis method is based on the framework of the relativistic impulse approximation(RIA) using the relativistic Love-Franey (RLF) NN interaction [88] proposed by Murdock andHorowitz (MH model) [89, 90].

4.1 Relativistic impulse approximation
In the MH model the nucleon–nucleus optical potential is calculated by folding the RLF NN
interaction with the nucleon vector and scalar density of the target nucleus. The RLF NN
interaction is based on the Lorentz invariant NN scattering amplitude ofFµ55γ(1)µ + F PS γ(0)γ(1)= F S + F V γ(0)µ 5A 5+F T σµν(0) σ(1)µν + F γ(0) γ(0) γ(1) γ(1)µ ,(4.1)

where F S , F V , F PS , F T , and F A denote the scalar, vector, tensor, pseudoscalar, and axial
vector amplitudes, and the subscripts (0) and (1) represent the incident and recoil nucleon,
respectively. For a spin-zero spherical nucleus, only the scalar and the zeroth-component of
the vector amplitudes are dominant contribution to the optical potential. The details of the RIA
calculations by MH model are mentioned in Appendix B.
Figure 4.1 shows the obtained experimental data of cross sections and analyzing powers
for elastic scattering from 204,206,208 Pb, and 58 Ni compared with two kinds of model calculations. The solid and dashed lines are the RIA calculations [89, 90] with Dirac–Hartree (DH)[17] densities of the target nuclei, and the calculations using the recent global Dirac opticalmodel by Cooper, Hama and Clark [91], respectively. Although both calculations well reproduce the analyzing powers, only the global Dirac optical model is in good agreement with thecross sections. The MH model poorly reproduces the angular distributions of the cross section
especially at backward angles. This is because neither the RLF interaction nor the nucleon
densities used in the MH model are realistic. However, even though a realistic nucleon density41

CHAPTER 4. ANALYSIS

42

10 8

1

RIA(MH model)
Global Potential
RIA(MH model)

105+ ρp : unfolded ρchρn=N/Z×ρp

Pb 4(×10 )

10
10
10
10

Ay

208

Pb

58

Pb

0

-0.5
-1
1

-2
-3

208

0.5

-2

Ni(×10 )0.5

Aydσ/dΩ (mb/sr)

Pb 2(×10 )

1

10

Pb

-1
1

206

-1

206

0

-0.5

2

10

Pb

0.5

10 3
10

204

-1
1

204

10 4

0

-0.5

Ay

10

6

0.5

Ay

10 7

-4

0

-0.5
-5

10

20

30

θc.m. (deg)

40

50

-1

58

Ni

10

20

30

θc.m. (deg)40

50

Figure 4.1: Obtained data of diﬀerential cross sections and analyzing powers for elastic scattering from 58 Ni and 204,206,208 Pb at E p =295 MeV, whereas the lines are due to Murdock andHorowitz (solid) [89, 90] and the global Dirac optical potential (dashed) [91]. The dash-dottedlines show the MH model calculations for 58 Ni with the realistic nucleon density by an unfolding charge density.

of 58 Ni, as mentioned in the next subsection, was used in place of the DH density for the MH
calculation (dash-dotted lines in Fig. 4.1), the disagreement with the experimental data of 58 Ni
still remains. In order to explain the experimental data we need the eﬀective NN interaction
inside the nuclear medium in place of the RLF interaction.

4.1. RELATIVISTIC IMPULSE APPROXIMATION

43

4.1.1 Realistic point proton density distributions
A realistic point proton density distribution was determined by unfolding the nuclear charge
distribution extracted from electron elastic scattering [54, 55, 92–94] with the intrinsic charge
distributions of the proton and neutron. Using the Fourier transform of the radial densityρ̃(q) = F {ρ(r)} where q is the momentum transfer, the relationship between the charge, proton,pneutron, intrinsic proton-charge, and intrinsic neutron-charge densities ρch , ρ p , ρn , ρch, ρnch isapproximately described aspρ̃ch (q) ' ρ̃ p (q)ρ̃ch(q) + ρ̃n (q)ρ̃nch (q)' ρ̃ p (q)G Ep (q2 ) + ρ̃n (q)GnE (q2 ),

(4.2)

p(n)
where in the non-relativistic limit, ρ̃ch
' G Ep(n) (q2 ) which is the experimentally determined
Sachs electric form factor in proton (neutron). The correction term by the spin–orbit andrelativistic eﬀects on the Sachs form factors in the Breit frame, the so–called Darwin–Foldy
correction, are negligible because their eﬀects on the nuclear size are one order smaller than
the neutron-charge radius and comparable with the error of charge radius as shown in Refs.
[58, 95].
For the nuclear charge distributions of 204,206,208 Pb and 56 Ni, we used the parameter sets ofthe model-independent analyses of electron scattering and/or muonic X-ray data as listed inRef. [54]. The Sum-of-Gaussians (SOG) expansion were used for 206,208 Pb, 56 Ni, whereas theFourier-Bessel (FB) series expansion for 204 Pb. The model-independent analyses in the formof SOG and FB have been originally developed by Sick [92], and Friar and Negele [94, 96],
respectively.
p(n)For the intrinsic nucleon charge form factor ρ̃ch, we adopted a new set of nucleon electromagnetic form factors which have been recently extracted from the reanalysis of the world e-p
and e-d scattering data at low-momentum transfers from 0.3 to 1 GeV2 /c2 where the nucleon
charge form factor is very sensitive to its model-independent charge radius [97–100]. Thepproton-charge radius of 0.895(18) fm obtained from ρ̃chin Refs. [97–99] is much larger thanthe value of 0.863(4) fm used in the previous work [101], but is consistent with the value of0.883(14) fm obtained from the recent measurement of the hydrogen 1S Lamb shift [102]. a b
In order to know the neutron density distribution, not only the nucleon-charge radius but
also its density is required because the realistic point proton distribution is derived from
nucleus- and nucleon-charge density. In this work we adopted the parameterization of nucleon charge distribution reported in Ref. [99], which can reproduce the model-independentaCompared to the new analysis of elastic e-p scattering data, the previous analyses [101, 103] did not include various higher-order corrections at both low and high momentum transfers such as the Darwin-Foldy term,
Coulomb distortion, etc. [104–106], while the analyses of the atomic Lamb shift measurements have developed including various QED corrections. Thus the previously extracted nucleon form factors are rather modeldependent and can not give the intrinsic charge radii which should be compared with the Lamb shift results.
b
It should be noted, however, that the very recent measurement of the muonic-hydrogen Lamb shift haspresulted in rch= 0.84184(67) fm with a remarkable accuracy of 0.08% [107]. This new value is very inconsistent
with, and 5σ smaller than the ones mentioned above. The origin of this large discrepancy is not understood atthis time and requires further validation of this measurement and the QED eﬀects on muonic-hydrogen atoms.

44

CHAPTER 4. ANALYSIS

nucleon charge radius. The procedure of unfolding method follows the same way as mentioned in Ref. [70]. Figure 4.2 shows the extracted proton density distributions of 58 Ni and
204,206,208
Pb (solid lines) by unfolding method. Compared to the theoretical DH calculations(blue dotted lines), the extracted distributions show the gradual change of density as a functionof radius and particularly the slopes of the surface diﬀusenesses are relatively small. On thecontrary, the Hartree–Fock–Bogolyubov calculations with several Skyrme interactions (green
lines) have more gradual slopes than the extracted proton densities.
In the case of 58 Ni, the neutron rms radius is expected to be almost the same as the proton
rms radius as mentioned later. Therefore we assumed that the neutron density of 58 Ni has thesame shape as the proton density, that is, ρn = (N/Z)ρ p .

4.1.2 Scalar and Vector density distributions
For the RIA calculations, scalar density distributions are necessary as well as baryon density
distributions. While the vector density is normalized to the mass number by baryon number
conservation, there is no analogous normalization condition for the scalar density. Scalardensity distributions can not be extracted directly, unless the wave functions of the ground state
are known in advance by solving the field equations for the scalar meson. In the relativistic
mean field theory [108], however, the scalar density ρS is approximated in the form of thevector density ρV as follows:
{
}
3 kF2
S
ρ (r) ≈ 1 −ρV (r),
(4.3)
∗2
10 M
where kF and M ∗ are the local Fermi momentum and the eﬀective mass, respectively. The
density ratio of ρS /ρV derived from Eq. (4.3) is usually about 0.93 at the saturation (kF '1.30 fm−1 and M ∗ ' 0.6M), but around the surface of nuclei the ratio gradually approaches
one. According to the DH calculations for heavy nuclei the volume integral of the ratios arealmost constant at 0.96, as reported in Refs. [69, 70]. Therefore, we simply assumed a scalardensity ρS of ρS = 0.96ρV for a vector density ρV using the realistic nucleon density.
The eﬀect of the negative-energy states in the filled ”Dirac sea”, namely, the vacuum polarization on the scalar density was reported in Ref. [109] and causes 10% reduction of the scalardensity in the nuclear center. In this work we did not deal with the eﬀect explicitly, but it may
be included somewhat by the density-dependent modification of the eﬀective NN interaction
as written in the next section.

4.1. RELATIVISTIC IMPULSE APPROXIMATION

58

Ni

0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0

2

4

6

-3

8

r (fm)
206Pb

ρp(r) (fm )-3ρp(r) (fm )

0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0

2

4

6

r (fm)

8

0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
10 0
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
10 0

45

204

Pb

2

4

6

8

10

r (fm)
208

Pb

2

4

6

8

10

r (fm)

Figure 4.2: Proton density distributions of 58 Ni and 204,206,208 Pb by unfolding charge distributions with the new parameter sets of nucleon charge form factor (solid lines). The blue dottedlines are due to the DH calculations for comparison, while the green long-dotted, short-dotted,
dashed, and dash-dotted lines are due to Skyrme–Hartree–Fock–Bogolyubov calculations with
SIII, SKM*, SKP, and SLY4 parametrizations, respectively.

CHAPTER 4. ANALYSIS

46

4.2 Medium modification of NN interaction
The RLF interaction in the MH model is described by a set of five Lorentz covariant functions
and was determined from the free NN phase shift analysis [88]. It was found that the Dirac
optical potential folded by the free NN interaction can not fully reproduce the experimental
data at high momentum transfers even using the realistic nucleon density. In order to explain
the experimental data, we introduced a medium modification into the RLF NN scattering amplitudes by varying the coupling constants and masses of the σ and ω mesons depending on
the local density as follows:
g2j , ḡ2jm j , m̄ j

g2j

ḡ2j

,
1 + a j ρ(r)/ρ0 1 + ā j ρ(r)/ρ0
[
]
[
]
ρ(r)
ρ(r)
→ mj 1 + bj
, m̄ j 1 + b̄ jρ0
ρ0j = σ, ω,

→

(4.4)
(4.5)

where m j , m̄ j , g j , and ḡ j are the masses and coupling constants of σ and ω mesons for realand imaginary amplitudes, respectively. The normal density ρ0 is 0.1934 fm−3 . In free space,
where the density of the target nucleus is zero, the masses and coupling constants of the exchanged mesons are set to be the same as those of the free NN interaction, but inside thenucleus the modification is assumed to be proportional to the nucleon density ρ(r) with thephenomenological parameters a j , ā j , b j , and b̄ j . This density-dependence describes the first–
order approximation in terms of the nuclear density and partially explains various many-bodyeﬀects such as Pauli blocking c , vacuum polarization [109, 110], multi-step processes [111],
and partial restoration of chiral symmetry [112]. This density-dependent correction has theanalogous expression of the nuclear eﬀective interaction in a nucleus reported by Nagata andco-workers in Ref. [113].
Since the modification has a universal form of local-density dependent terms, we can apply
it to any other nuclei once the parameters are calibrated with a real nucleus. At the present
stage we have used four parameters assuming the same modification for the real and imaginaryparts of the NN scattering amplitude (a j = ā j , b j = b̄ j ) because with eight parameters there areso many degrees of freedom that these parameters have strong correlations with each other.
The extraction of the neutron densities of lead isotopes is based on the medium-modified
RIA calculation with realistic proton densities. For this purpose we first need to determine
the medium-modification of the eﬀective RLF interaction. In the next subsection, we report
the calibration of the medium-eﬀect parameters using the experimental data of proton elastic
scattering from 58 Ni.
cThough in the MH model the correction of the Pauli blocking eﬀect is introduced by modifying the opticalpotentials with a local-density approximation based on the Dirac-Brueckner calculation [89, 90], when using the
medium-modified RIA we omit the Pauli blocking correction in the MH model. For the Pauli blocking correction
we phenomenologically modified the NN interaction, not the optical potential.

4.3. CALIBRATION OF THE MEDIUM EFFECT PARAMETERS47

4.3 Calibration of the medium eﬀect parameters
For the calibration of the four medium-eﬀect parameters, a j and b j ( j = σ, ω), we havechosen p-58 Ni elastic scattering because 58 Ni is the heaviest spherical nucleus with N ≈ Z
and the neutron density of 58 Ni is expected to have a similar distribution to the proton density(ρn = (N/Z)ρ p ). Both the experimental and theoretical results [8, 13, 27, 58, 66, 114] support
a neutron rms radius for 58 Ni that is almost the same as the proton rms radius, while for
N = Z nuclei, such as 40 Ca or 56 Ni, the proton rms radii are larger than the neutron rms radii
due to Coulomb repulsion. Figure 4.3 and 4.4is neutron-number dependence of neutron skin
thicknesses for Ni isotopes by several relativistic and non-relativistic calculations, respectively,
and shows that all the listed theoretical calculations also agree that the neutron skin thicknessof 58 Ni is almost zero. The realistic proton density of 58 Ni is extracted by unfolding the charge
density with the new intrinsic nucleon-charge density.
A fit to the 58 Ni data of cross sections and analyzing powers obtained in this experimentand spin rotation parameters previously measured at the same energy [87] was carried out by
χ2 test [115] with the four medium-eﬀect parameters. The value of χ2 is given by∑calc 2
2
(yexp
χ2 =
(4.6)
θi − yθi ) /∆yθi ,θicalcwhere yexp
θi , ∆yθi , and yθi are the ith experimental data, error, and medium-modified RIA calculation for 58 Ni at each θi . The best-fit parameters providing a minimum of χ2 (χ2min ), arelisted in Table 4.1. The solid line in Fig. 4.5 is the medium-modified RIA calculation withthese best-fit parameters and the realistic nucleon densities deduced from the charge distribution. Compared to the previous work [70], the modifications by the best-fit parameters arevery small. The best-fit calculations are in better agreement with the experimental data thanthe previous data even though the number and angular range of the data points is much larger
than for the previous data [69, 70]. This means that the improvements to reduce the experimental systematic uncertainties work very well. The standard error of each best-fit parameterin Table 4.1 was determined from a contour corresponding to an increase of 1 in χ2 from χ2minby allowing all the other parameters to vary freely to minimize χ2 for each chosen value of theparameter.

Table 4.1: Best-fit medium-eﬀect parameters a j and b j ( j = σ, ω) in Eq. (4.4), (4.5).
j
aj
bj

σ
−0.044 ± 0.026
0.097 ± 0.013

ω
0.037 ± 0.040
0.075 ± 0.021

CHAPTER 4. ANALYSIS

48

Figure 4.3: The neutron skin thicknesses for even-Ni isotopes calculated by DBHF with theGroningen (triangles) and Bonn A (circles) interactions, and RMF with the NL3 [19] (crosses)
parametrization. All of them shows that the neutron skin thickness of 58 Ni is almost zero.
Taken from Ref. [8].
1rn-rp (fm)

0.8

Ni (N=20-62)

0.6

SIII

Sly4

0.4

SkM*

Sly5

0.2

SkP

0
-0.2
-0.4

60

80

Mass Number (Z=28)
Figure 4.4: The neutron skin thicknesses for even-Ni isotopes calculated by Hartree–Fock–
Bogolyubov (HFB) plus pairing corrections using five types of Skyrme forces, that is, SIII[116] (filled circles), SkM* [9] (filled squares), SkP [10] (asterisks), Sly4 [11] (open circles),and SLy5 [11] (open squares) parameterizations. As is the case with relativistic models in
Fig. 4.3 nonrelativistic models show that the neutron skin thickness of 58 Ni is almost zero.

4.3. CALIBRATION OF THE MEDIUM EFFECT PARAMETERS

10 4

49

1
RIA(MH model)+ DH density

0.5

RIA(MH model)

10 3

RIA(medium effect)+ realistic density

dσ/dΩ (mb/sr)

10 2

Ay

+ realistic density

0

-0.5

10

-1

-2

58

40

θc.m. (deg)

50

10

20

30

40

50

-0.5

-3

10

0

58

Ni(p,p) Ni

Ep=295 MeV
10

30

0.5

-1

Q
10

20

1

1

10

10

20

30

40

θc.m. (deg)50-1

θc.m. (deg)Figure 4.5: Calibration of medium-eﬀect parameters by fitting to the experimental data for
Ni. The solid line is the medium-modified RIA calculation with best-fit parameters. The
dashed and dash-dotted lines are from the original MH model with DH and realistic nucleon
densities.
58

CHAPTER 4. ANALYSIS

50

4.4 Extraction of neutron density distributions
Using the eﬀective NN interaction calibrated by 58 Ni data we extracted the neutron density
distributions of 204,206,208 Pb. The realistic proton density distributions of lead isotopes werealso used by unfolding charge distributions.
In order to find the best-fit neutron density distribution, we used a sum-of-Gaussian (SOG)
function, which has been originally investigated in the model-independent analysis of charge
distributions [54, 55]. The SOG neutron density is described as
ρn (r) =

N

12
∑

Qi
2π3/2 γ3 i=1 1 + 2R2i /γ2×(e−(r−Ri ) /γ + e−(r+Ri ) /γ ),
2

2

2

2

(4.7)

where N and Qi are the number of neutrons, and the fraction of N in the ith Gaussian with∑
the normalization condition i Qi = 1, respectively. For simplicity we fixed the width γ andposition Ri of the ith Gaussian to the same values used for the charge distributions of 208 Pb in
Ref. [54] since the e-208 Pb elastic scattering data covers a wider range of momentum transferfrom 0.44 to 3.7 fm−1 than our data from 0.55 to 3.5 fm−1 .
We searched for the best-fit values of Qi by the χ2 method. Figure 4.6 shows the resultsof the medium-modified RIA calculations with the best-fit neutron density distributions of
204,206,208
Pb (solid lines). The medium-modified RIA calculations with the DH nucleon density (dash-dotted lines) well reproduce the analyzing powers and the angular positions of thediﬀraction peaks and minima compared with the original MH calculations (dashed lines), but
still overestimate the absolute values of the cross sections. The best-fit calculations (solidlines) are in good agreement with the experimental data of both the cross sections and analyzing powers even in the high momentum transfer region up to 3.5 fm−1 . It was found thatthe absolute values of the cross sections aﬀect the nucleon density distributions quite directly.
Thus when using the model-independent densities it is very important to determine the crosssections precisely. The typical reduced χ2min , namely χ2min /ν, where ν is the number of degrees
of freedom, is about 4.
Coulomb potential eﬀect in lead isotopes In the search for the best-fit parameters of theneutron densities, it was also found that the the Coulomb potentials, i.e., the charge radii oflead isotopes are sensitive to the scattering observables especially at high momentum transfers.
For the calculation of the Coulomb potential the charge density of the target nucleus is assumed
to be that of a uniformly charged sphere of radius r0 A1/3 where r0 = 1.25 fm is usually known
to be the reduced Coulomb radius. Where the radius is larger than R0 = r0 A1/3 , non-relativistic
Coulomb wave functions are used. In the case of 208 Pb, for example, r0 is about 1.20 fm, which
reproduces experimentally determined value of the rms charge radius of 208 Pb. If the reduced
Coulomb radius r0 is changed by ±5% (r0 = 1.14, 1.26 fm), the reduced χ2 of the fittingwith the experimental data increases by about δχ2 /ν = ±50/47 ∼ ±1. This χ2 increase issurprisingly larger than the value corresponding to 1 standard deviation (details are mentioned

4.4. EXTRACTION OF NEUTRON DENSITY DISTRIBUTIONS

10 8

1

RIA(MH model)+ DH density+ DH density

RIA(medium effect)+ ρp : realistic
ρn : best fit

10 6
5

10

-1
1

204

Pb
4
(×10 )

10 4

0

-0.5

204

10

Pb
20

30

40

50

30

40

50

30

40

50

0.5

3

Ay

dσ/dΩ (mb/sr)

10

Ay

10

0.5

RIA(medium effect)

7

51

10 2

0

-0.5

206

206

Pb
2
(×10 )

10

-1
1

10

Pb
20

1

10
10

Pb

Ay

10

0.5

208

-1
-2

0

-0.5
-3

10

20

30

θc.m. (deg)

40

50

-1

208

10

Pb
20

θc.m. (deg)Figure 4.6: Best fit results for neutron density distributions in 204,206,208 Pb, are shown as solidlines. The original MH and medium-modified RIA calculations with the DH nucleon density
are also shown by dashed and dash-dotted lines.

in the next subsection) and shows that the precise measurement of the experimental data withsmall systematic uncertainties has been achieved to such an extent that the small change of
Coulomb radius can be distinguished. The diﬀerence of the calculated observable between
r0 = 1.20 fm and r0 = 1.14, 1.26 fm occurs almost at the backward angles (θlab. > 30◦ ), andhence arises from the change of the depth of the Coulomb potential rather than its radius dueto the nuclear charge conservation. Therefore in the RIA calculations in this thesis we used theCHAPTER 4. ANALYSIS

52

adjusted values of the reduced Coulomb radii of target nuclei which reproduce the rms charge
radii reported in Ref. [54, 56].

4.4.1 Estimation of error-envelopes“Experimental uncertainties” The standard error-envelopes of the neutron density distributions due to the experimental errors can be estimated by the increase ∆χ2 corresponding to
1 standard deviation from χ2min , expressed as the following inequality:χ2 ≤ χ2min + ∆χ2 .

(4.8)

For the error-estimation of the neutron density we need to know the 1–standard deviation
region encompassed by the joint variation of multi parameters. ∆χ2 for multi parameters obeys
the χ2 probability distribution function for m degrees of freedom, where m is the number offitted parameters [115], and is roughly equal to m for 1 standard deviation (∆χ2 ≈ m = 11
in this case). By using the Monte Carlo technique, we determined both the minimum andmaximum envelopes of all the possible neutron density distributions which satisfy Eq. (4.8) as
the standard error-envelope. The errors of the neutron rms radii are also determined from the
maximum and minimum value of the rms radii of the allowed neutron density distributions.
Figure 4.7 shows the case of 208 Pb. The hatched area surrounded by the solid lines in Fig.
4.7 (c) shows the standard error-envelope of the neutron density in 208 Pb estimated by Eq. (4.8),
together with the DH neutron density distribution (dashed line). The three-parameter-Gaussian(3pG) neutron density extracted from the p-208 Pb elastic scattering at 800 MeV using thesecond-order Kerman-McMaus-Thaler (KMT) model [58] (dash-dotted line) is also shownfor comparison. The inset in Fig. 4.7 (c) is a magnification of the density distributions in theradial range from 4.5 fm to 6.5 fm.
The lines presented with the experimental observables for p-208 Pb in Figs. 4.7 (a) and (b),are the medium-modified RIA calculations with the best-fit neutron density (solid), the 3pG
neutron density (dash-dotted), and the DH nucleon density (dashed). Although there seems to
be little diﬀerence between the medium-modified RIA calculations with the best-fit and 3pGneutron density since the solid line overlaps with the dash-dotted line extensively, χ2 for the
3pG neutron density (= 255.3) is about 5σ (= 5∆χ2 ' 55) larger than χ2min (= 192.5) especiallyat data points of high momentum transfer. The diﬀerence also appears in the rms radii. Our
analysis gave a neutron rms radius for 208 Pb of rn = 5.653+0.026−0.029 fm, which is 0.06 fm larger
than the value of rn =5.593 fm of the 3pG neutron density. This means that the data even athigh momentum transfers can aﬀect the rms radius. Because of the conservation of neutron
number N, the information of the scattering observables at high and low momentum transfers,
which reflects the bulk and surface structure, respectively, can doubly constrain the neutron
radius. This allows us to extract the precise information about the neutron density from proton
elastic scattering.
Figure 4.8 shows the percent deviation d between the experimental data and the medium-4.4. EXTRACTION OF NEUTRON DENSITY DISTRIBUTIONS

53

Figure 4.7: Results of fitting to the experimental data and extracted neutron density of 208 Pb
with its standard error-envelope (solid lines). The dashed and dash-dotted lines are mediummodified RIA calculations but using the DH nucleon densities and the 3pG neutron density by
Ray [58] respectively. The dotted line in (c) is the realistic proton density.

modified RIA calculations for cross sections for 208 Pb, which is written as
d = 200 ×

σexp − σcal,
σexp + σcal

(4.9)

where σexp(cal) is the experimental (theoretical) cross section at each scattering angle. The red,green, and blue marks stand for the same definitions as in Fig. 4.7. The agreement with theexperimental data is almost at the level of ±10 %. Compared to the calculation of the 3pG
neutron density (green marks), our results (red marks) are found to be improved particularly

CHAPTER 4. ANALYSIS

54

60

σ −σ
d = 200 × σexp + σcal (%)
exp
cal

d (%)

40
20
0

-20
-40
10

15

20

25

30

35

θc.m. (deg)

40

45

50

Figure 4.8: Percent deviation between the experimental data and the medium-modified RIA
calculations of diﬀerential cross section at each angle defined as Eq. (4.9). The marks are thesame definitions as Fig. 4.7. The lines are just for the guide.

at the backward angles. The percent deviation fluctuate largely at forward angles, where theangular distribution of the cross section rapidly changes.
The results for 204,206 Pb are qualitatively similar to that for 208 Pb and they are shown in
Figs. 4.9 and 4.10.
It was found that our method to extract the neutron densities works successfully even forheavy nuclei like lead isotopes. The results of the fitting and the error estimation show that thesystematic uncertainties of the experimental data are small enough to deduce the neutron radiiwith a good accuracy of 0.5 % and thereby the medium-modified RIA works well. However,the diﬀerence between the experimental data and the medium-modified RIA predictions still
exists as the minimum reduced χ2 is about 4.

4.4. EXTRACTION OF NEUTRON DENSITY DISTRIBUTIONS(a)+ DH density

RIA(medium effect)+ ρp(r) : realistic
ρn(r) : best fit

dσ/dΩ (mb/sr)

10 2

0.5

Ay

10

1 (b)

RIA(medium effect)

3

0
-0.5

10

10
0.2

-1

0.15

10

ρp(r)

Pb(p,p)204Pb

-3

50

DH

0.05

204

40

(c)

-3

-2

30

θc.m. (deg)

ρn(r)

0.1

10

20

ρ(r) (fm )

1

10

55

this work

Ep=295 MeV
10

20

30

40

θc.m. (deg)

50

0
0

2

4

6

8

10

radius (fm)Figure 4.9: Results of fitting to the experimental data and extracted neutron density of 204 Pb
with its standard error-envelope (solid lines). The dashed lines are medium-modified RIA
calculations but using the DH nucleon densities.

CHAPTER 4. ANALYSIS

56

(a)

+ DH density

3

RIA(medium effect)+ ρp(r) : realistic
ρn(r) : best fit

dσ/dΩ (mb/sr)

10 2

0.5

Ay

10

1 (b)

RIA(medium effect)

0
-0.5

10

10
0.2

-1

0.15

10

10

-2

0.05

ρp(r)

206

Pb(p,p)206Pb

-3

50

DH
ρn(r)

0.1

40

(c)

-3

10

30

θc.m. (deg)ρ(r) (fm )120

this work

Ep=295 MeV
10

20

30

40

θc.m. (deg)

50

0
0

2

4

6

8

10

radius (fm)Figure 4.10: Results of fitting to the experimental data and extracted neutron density of 206 Pb
with its standard error-envelope (solid lines). The dashed lines are medium-modified RIA
calculations but using the DH nucleon densities.

4.4. EXTRACTION OF NEUTRON DENSITY DISTRIBUTIONS

57“Model uncertainties” In addition to the experimental uncertainty, we evaluated the errorenvelopes including the model uncertainties in the medium-modified RIA. If the theoreticalmodel was ideal, the reduced χ2min should be nearly equal to 1. However, the value of χ2min /ν inthis case is about 4, which is far from 1 and shows the incompleteness of this medium-modified
RIA model as well as the unknown systematic errors of the experiment. To incorporate thisincompleteness into Eq. (4.8) as the model uncertainties, we simply multiplied the experimental errors by a constant factor S which realizes χ2min /ν = 1. Therefore, we defined a new
chi-square as χ̃2 ≡ χ2 /S 2 assuming that χ̃2 follows the same χ2 distribution and Eq. (4.8):
χ̃2 ≤ χ̃2min + ∆χ̃2 ,(4.10)

where χ̃2min /ν ≡ (χ2min /S 2 )/ν = 1 and ∆χ̃2 = ∆χ2 . Thus Eq. (4.10) is equivalent toχ2 ≤ χ2min + ∆χ2 × S 2= χ2min + ∆χ2 × (χ2min /ν).

(4.11)The error-envelopes including the model uncertainties were estimated by Eq. (4.11) for 204,206,208 Pb
data d , where ∆χ2 for the 204,206,208 Pb data is defined as the number of fitted parameters m,
namely 11. The cross-hatched areas in Fig. 4.11 show the error-envelopes of the neutron density distributions for 204,206,208 Pb obtained from Eq. (4.11) and the hatched areas are the standard error-envelopes due to Eq. (4.8). The radial distributions of the neutron surface densities
4πr2 ρ(r) are also shown in Fig. 4.12 whose definitions are the same as Fig. 4.11. The neutron
surface density distributions show that the errors of the neutron density distributions are large
at both the inner (0–3 fm) and surface (8–10 fm) regions. Particularly, the error-envelopes ofthe surface regions causes the errors of the rms redii.
In this analysis for lead isotopes we neglected the density dependence of ρ and π mesonsin the medium modification because the diﬀerence between the proton and neutron density is
about 1/5 less than the normal density and the modification for ρ and π mesons is expected to
be much smaller than for σ and ω mesons. The error from this assumption is also includedin the model uncertainties described by Eq. (4.11). When using model-dependent hadronic
process, it is important to show these error-envelopes including the model uncertainties, inparallel with the conventional method of showing the experimental uncertainties only.
Table 4.2 lists the best-fit values of the SOG parameters Qi and χ2min /ν for each lead isotope.
In addition, a fit to the upper and lower error-envelopes including the model uncertainties foreach isotope was made by using the SOG parameters Qi of Eq. (4.7) with the same γ, N, andRi as used in the investigation of the neutron densities. In this case the normalization condition∑
is not satisfied ( Qi , 1) because the envelope is not the density distribution itself. The fitted
parameters Qi are listed in Table 4.3.
d
In Ref. [70] it was reported that the error-envelopes of tin isotopes were estimated by Eq. (4.11) for theneutron-density parameters together with Eq. (4.8) for the medium-eﬀect parameters, simultaneously. In thatcase, however, the envelope is doubly-estimated because the fluctuation of the medium-eﬀect parameters indicated by Eq. (4.8) is a part of the uncertainties associated with the various model assumptions described by
Eq. (4.11). Thus, Eq. (4.11) is enough to evaluate the error-envelopes including the model uncertainties.

CHAPTER 4. ANALYSIS

58

0.18

204

-3

ρ(r) (fm )

0.16
0.14

0.18

Pb

0.1

0.08

0.08

0.06

0.06

ρp(r)

0.02
2

4

6

8

10

r (fm)
0.18

208

0.16

-3

ρ(r) (fm )

ρp(r)

0.04

0.02
0
0

ρn(r)

0.12

0.1

0.04

Pb

0.14

ρn(r)

0.12

206

0.16

0.14

6

8

10

ρn(r) : DHρp(r) : unfolded ρch

0.08

ρp(r)

0.04

4

standard
error-envelope
error-envelope
w/ model uncertainty

Pb

0.1
0.06

2

r (fm)

ρn(r)

0.12

0
0

0.02
0
0

2

4

6

8

10

r (fm)
Figure 4.11: Extracted neutron densities for 204,206,208 Pb with two types of error-envelopes
shown together with DH neutron densities (dotted lines) and point proton densities by unfolding charge densities (dash-dotted lines). The cross-hatched blue and hatched red errorenvelopes were estimated by Eq. (4.11) and Eq. (4.8), respectively.

4.4. EXTRACTION OF NEUTRON DENSITY DISTRIBUTIONS

40

40

30

ρn(r)

2

2

ρp(r)

10

2.5

5

-1

7.5

10

r (fm)

Pb

30

ρn(r)

20

2.5

5

7.5

10

standard
error-envelope
error-envelope
w/ model uncertainty
ρn(r) : DHρp(r) : unfolded ρch

15

ρp(r)

10

0
0

r (fm)

208

35

2

ρp(r)

5

40

4πr ρ(r) (fm )

15
10

5

25

ρn(r)

20

15

0
0

Pb

30
25

20

206

35

-1

-1

4πr ρ(r) (fm )Pb

4πr ρ(r) (fm )

204

35

25

59

5
0
0

2.5

5

7.5

10

r (fm)
Figure 4.12:
The radial distributions of neutron surface densities for 204,206,208 Pb(4πr2 ρ(r) fm−1 ) with two types of error-envelopes shown together with DH neutron surface
densities (dotted lines) and proton surface densities by unfolding charge densities (dash-dottedlines). The cross-hatched blue and hatched red error-envelopes were the same as Fig. 4.11.

60

CHAPTER 4. ANALYSIS

204,206,208
Table 4.2: Best-fit
Pb. The√ SOG parameters of the neutron density distributions forwidth γ (= 1.70/ 3/2) and position Ri of the ith Gaussian in Eq. (4.7) are fixed to the valuesof the charge distribution [54]. The number in the parentheses are the χ2min over the degrees of
freedom ν for each isotope.
∑iRi (fm)Qi (the fraction of N in ith Gaussian, i Qi = 1)
204
Pb (180.1/47) 206 Pb (185.3/47) 208 Pb (192.5/47)
1
0.1
0.0065961
0.0068899
0.0053549
2
0.7
0.0126023
0.0123201
0.0106038
3
1.6
0.0063489
0.0000021
0.0208715
42.1
0.0323212
0.0231717
0.0096117
52.7
0.0703161
0.0998543
0.0767189
6
3.5
0.1150041
0.0677595
0.0639530
74.2
0.0413854
0.0961086
0.1324368
85.1
0.3317518
0.2860324
0.2484495
96.0
0.2994768
0.3249540
0.3467533
106.6
0.0315217
0.0333252
0.0271199
11
7.6
0.0526614
0.0495821
0.0581130
128.70.0000142
0.0000001
0.0000137

4.4. EXTRACTION OF NEUTRON DENSITY DISTRIBUTIONS

61

Table 4.3: Fitted SOG parameters Qi of the upper and lower error-envelopes of the neutron
density distribution for each lead isotope determined by Eq. (4.11). γ, N, and Ri are the sameas in Table 4.2, but the normalization condition is not satisfied in the case of the envelopes∑( Qi , 1).
i

Ri (fm)
204

1
2
3
4
5
6
7
8
9
10
11
12

0.1
0.7
1.6
2.1
2.7
3.5
4.2
5.1
6.0
6.6
7.6
8.7

upper
0.0094688
0.0127647
0.0000598
0.0311879
0.1455760
0.0080875
0.1394140
0.2843110
0.3396610
0.0181303
0.0593428
0.0000224

Pb
lower
0.0000016
0.0189556
0.0140777
0.0063748
0.0595549
0.1396962
0.0019071
0.3534691
0.2677440
0.0464501
0.0452618
0.0000093

Qi
206
Pb
upper
0.0140275
0.0007678
0.0000035
0.0704769
0.0864652
0.0398421
0.1516393
0.2477640
0.3580471
0.0231355
0.0566526
0.0000018

lower
0.0001227
0.0155131
0.0003936
0.0046326
0.0624612
0.0909420
0.0225707
0.2273172
0.2415563
0.0216326
0.0330576
0.0000028

208

upper
0.0084768
0.0096719
0.0132542
0.0280440
0.0850354
0.0764422
0.1243973
0.2607873
0.3540381
0.0245054
0.0663033
0.0000095

Pb
lower
0.0001292
0.0172562
0.0157425
0.0000877
0.0723946
0.0516049
0.1416711
0.2296302
0.3434331
0.0257194
0.0502610
0.0000095

Chapter 5
RESULTS AND DISCUSSION
In the previous chapter, we succeeded in extracting the neutron density distributions of 204,206,208 Pb
and estimated their error-envelopes even including the uncertainties associated with our reaction model. The result of the density distributions and their parameters are already shown in
Fig. 4.11, Tables 4.2 and 4.3. Now we show and discuss more details of the extracted quantities
such as the neutron radii and the neutron skin thicknesses.

5.1 Charge, proton, and neutron rms radii
The diﬀerences between the rms radius of the best-fit neutron density and the maximum (minimum) rms radii in all the neutron densities constituting each error-envelope determined by
Eq. (4.11) are listed in Table 5.1 as the upper (lower) errors of the rms radius of neutron δrnmdl ,while the errors of the rms radii δrnstd are the experimental ones determined by Eq. (4.8).
It is found that the eﬀect of the model uncertainty on the errors of rn for lead isotopes is aslarge as that of the experimental uncertainty. The neutron rms radii rn were determined with agood accuracy of about 1 % ('0.06 fm) even including the model uncertainty.

, and neutron rn used inTable 5.1: Root-mean-square radii of the charge rch , proton runfoldpmdl
this work. The two types of errors for rn , namely, δrn determined by Eq. (4.11) and δrnstd by
Eq. (4.8), are listed (all in fm).
Nucleusrch
204
Pb
5.479(2)206
Pb
5.490(2)208
Pb
5.503(2)runfoldp5.420(2)5.433(2)5.442(2)

63

rn
5.598
5.613
5.653

δrnstdδrnmdl+0.029−0.020+0.026−0.026+0.026−0.029+0.047−0.059+0.048−0.064+0.054−0.063

CHAPTER 5. RESULTS AND DISCUSSION

64

Table 5.2: Obtained values of rn , r p and ∆rnp for 208 Pb compared with the previous results
deuced from several experiments using various hadronic probes (all in fm). Except for thiswork and X-ray cascade from p̄-atom (2), the errors are experimental only. The errors of thiswork stand for the total errors including the model uncertainties δrnmdl determined by Eq. (4.11).
ExperimentThis work from Table. 5.1
208
Pb(p, p) at 800 MeV [58]∗208
Pb(p, p) at 650 MeV [62]†Isospin diﬀusion data in Sn + Sn at
50 MeV/u [51]‡GDR from (α, α0 ) at 120 MeV[61]PDR from 129−132 Sn(γ, γ0 ) [68]¶
PDR in 68 Ni and 132 Sn [72]k
X-ray cascade from p̄-atom (1)[67]∗∗
X-ray cascade from p̄-atom (2)[73]††rp5.442(2)
5.45
5.46
-

rn
5.653+0.054−0.0635.59(4)5.66(4)-∆rnp
0.211+0.054−0.0630.14(4)0.20(4)0.22(4)

5.44
5.45

5.60
5.65(5)mdl

0.19(9)0.18(4)0.194(24)0.16(2)stat (4)syst
0.20(4)exp (5)mdl∗

2nd-order KMT potential assuming 3pG neutron density.
Density-dependent t-matrix using model-independent FB neutron density.
‡
21 sets of the Skyrme interaction parameters.
¶
PDR strengths analyzed by RHB+RQRPA calculations with DD-ME parameterization.
kEnergy weighted sum rule by PDR data globally fitted with various mean field models∗∗
antiproton-nucleus optical potentials assuming 2pF proton and neutron density.
††
newly developed Skyrme parameterization Skxsxx constrained by antiproton data of Ref. [67].
†5.2 Neutron skin thicknesses“Systematic change of ∆rnp ” The systematic behavior of the extracted neutron skin

thicknesses ∆rnp for lead isotopes with the two types of error bars due to δrnstd (red) and δrnmdl(blue) is shown with previous experimental and theoretical results [9–11, 19, 22, 23, 62, 67,
68, 73, 117] in Fig. 5.1. The errors due to δrnstd (red) are the experimental only and thus, should
be compared with other experimental results. The present result shows a slight increase of theneutron skin thickness at 208 Pb.
As seen in Fig. 5.1, the resulting values of the neutron skin thicknesses have similar values
and tendency to other results except for the case of NL3 parameterization [19] within the error
bars. The recently developed parameterizations of DD-ME2, DD-PC1, FSUGold, and Skxs20
are particularly consistent with our result. The neutron skin thickness and density of 204 Pb was
firstly extracted by our measurements and will be useful to constrain the various theoreticalmodels.

5.2. NEUTRON SKIN THICKNESSES

65“∆rnp for 208 Pb” The obtained neutron skin thickness ∆rnp for 208 Pb is compared withthose obtained from previous experiments in Table 5.2. The listed experimental results including this work are not very diﬀerent within the error bars. However, none of the previous results
considers the model ambiguities used in their analyses. In addition, except for the proton elastic scattering approaches, other analysis methods assumed the model-dependent shape of theneutron density, and thus it is inherently diﬃcult to extract the precise values of the neutron
skin thicknesses and the density distributions as well as to estimate the model uncertainties
due to many theoretical assumptions.
In Table 5.3 we list the typical relativistic and non-relativistic mean field models with various parameter sets. Non-relativistic mean field models with D1S [118], D1N [118], SIII [116],
SkM* [9], SkI3 [119], SkI4 [119], SkP [10], SkX [12], Sly4 [11], SSk [120], GSkI [120],
GSkII [120], BSk14 [121], BSk17 [122], and BSk20 [123] parameterization are shown. Since
the relativistic mean field (RMF) model was originally proposed by Walecka in Ref. [124], theeﬀective Lagrangian has been adjusted to the well-known many nuclear properties and developed from many theoretical researches. They are classified into several types of the RMF parameterizations such as NL1 [125], NL3 [19], NL-SH [126], TM1 [18], G2 [127], PK1 [114],
FSUGold [23], FSUGZ03 [24], BKA20 [25], BKA22 [25], BKA24 [25], and IU-FSU [128]for the nonlinear meson-meson coupling model, PKDD [114], DD-ME1 [129], DD-ME2 [22],and DD-PC1 [117] for the density-dependent meson-nucleon couping model, PC-F1 [21], PCLA [130], PC-PK1 [131], and FKVW [132] for the point-coupling model, SL1 [26], SLC[26] for the Brown-Rho scaling model, QMC-I and QMC-II [133] for the quark-meson coupling model. These paramterizations are tuned to reproduce the properties of various finite
nuclei. Previously, predictions of the neutron skin thicknesses widely diﬀered between the
non-relativistic and relativistic mean field models, but recent studies using newly developed
relativistic parameterizations such as DD-ME and FSUGold have reported ∆rnp results closer
to those of non-relativistic models than the relativistic model with NL1, NL3, NL-SH, andTM1 parameterization.
In the case of FSUGold parameterization, for example, two additional coupling constants
which represent non-linear vector and isoscalar–isovector couplings, were introduced basedon relativistic eﬀective field theory [16]. The new relativistic parameterization has been calibrated to explain the measurements of the giant monopole resonance in 90 Zr and 208 Pb andthe isovector giant dipole resonance in 208 Pb, without compromising the quality of the otherground state properties such as the binding energy per nucleon and the charge radii. The predicted value of ∆rnp = 0.21 fm for 208 Pb by FSUGold parameterization is in close agreementwith our result.
The most recent theoretical studies [51, 72, 73] as listed in Table. 5.2 which are constrained by the most recent experimental data, namely, the X-ray cascade of antiprotonic atoms[67], isospin diﬀusion in heavy-ion collisions [134], and pigmy dipole resonance (PDR) data[68], have reported neutron skin thicknesses of 208 Pb of 0.20(4) fm [73], 0.22(4) fm [51], and
0.194(24) fm [72], respectively. These obtained values are in remarkable agreement with oursof ∆rnp = 0.211+0.054−0.063 fm.

CHAPTER 5. RESULTS AND DISCUSSION

66

0.4
0.35

∆rnp (fm)

0.3

Model

Experiment

NL3

SkM*

This work

DD-ME2SkP(p,p) at 650 MeV

DD-PC1

Sly4

antiproton

FSUGold

Skxs20pigmy DR

0.25
0.2
0.15
0.1
204

206

208

Mass Number
Figure 5.1: Systematic behavior of the neutron skin thicknesses for 204,206,208 Pb. The filled
circles are the results of this work with the two types of error bars. The red and blue lines
represent the errors due to the experimental only δrnstd and including the model uncertainties
δrnmdl , respectively. The filled squares, triangle, and inverted triangle are from the analysis ofproton elastic scattering at 650 MeV [62], X-ray cascade from antiprotonic atoms [67], andPDR strengths from 129−132 Sn(γ, γ0 ) [68], respectively, with their statistical errors only. The
open triangles, crosses, squares and diamonds show the calculations of relativistic mean fieldmodels (pale blue marks) with NL3 [19], DD-ME2 [22], DD-PC1 [117], and FSUGold [23]parameterization while the open circles, squares, stars and asterisks are from non-relativistic
mean field models (pink marks) with SkM* [9], SkP [10], Sly4 [11], and Skxs20 [73] parameterization. DD-ME2, FSUGold, and Skxs20 parameterizations have been recently developed
and conform closely to all the extracted neutron skin thicknesses.

5.2. NEUTRON SKIN THICKNESSES

67“Symmetry energy coeﬃcients” Correlations between the neutron skin for 208 Pb andthe symmetry energy coeﬃcients of nuclear matter have also been reported by various recent
theoretical studies [44–48, 51] as mentioned in Chap. 1 (the correlation factor is almost 1 for
208
Pb). In Table 5.3, the nuclear matter incompressibility K0 , the symmetry energy S(ρsat )and its slope L and curvature Ksym at saturation density ρsat as defined by Eqs. (1.2)–(1.6),
are listed compared with the neutron skin thicknesses for 208 Pb. The incompressibility K0
is now expected to be about 210–250 MeV [22–25, 38, 39, 135, 136] constrained by manymeasurements of isoscalar giant resonances for various nuclei [36, 37, 40]. The symmetryenergy S(ρsat ) at saturation is also known to be ∼32 MeV, but L and Ksym are still less certain
and their predicted values vary widely among many theoretical models. Particularly the slope
coeﬃcient L of the symmetry energy at saturation density is strongly correlated with ∆rnp for208
Pb [35, 51].
Then we tried to extract L by examining the global correlation between ∆rnp for 208 Pb and Lof typical theoretical predictions as listed in Table 5.3. First, the correlations between neutron
skin thickness for 208 Pb and several EOS coeﬃcients such as K0 , S(ρsat ), L, and Ksym calculatedwith the listed parameterizations, are shown in Figs. 5.2, 5.3, 5.4, and 5.5, respectively. The
incompressibility K0 is not correlated with the neutron skin. The hatched areas in Fig. 5.2
represent the two range of K0 , namely, 205–240 MeV and 230–250 MeV constrained by Itohet al. [36] and Li et al. [40], respectively.
On the other hand, the symmetry energy S(ρsat ) and slope L coeﬃcients predicted by thoseparameterizations, whether relativistic or non-relativistic, show strong linear correlations with∆rnp . Assuming that these linearity were valid as reported in many articles, linear fittings
were performed. The solid straight lines in Figs. 5.4 and 5.5 show the linear fittings. Since
fitting errors are almost negligible with respect to the errors of the neutron skin thickness, we
deduced the range of acceptable values for S(ρsat ) and L corresponding to the errors of ∆rnp .
+37.9
Our results are S(ρsat ) = 33.7+4.0−4.6 MeV and L = 74.3−20.3 MeV by using the extracted ∆rnp for208
Pb. The red and blue thick lines on the y-axis in all the Figures represent the two types of theerrors of ∆rnp for 208 Pb due to the experimental errors only (red) and the total errors including
the model uncertainties (blue), respectively. Those on the x-axis in Figs. 5.4 and 5.5 are theresulted range of S(ρsat ) and L. The red and blue boxes in Fig. 5.6 are the constraint regionson S(ρsat ) - L plane by the extracted ∆rnp for 208 Pb.
For 204,206 Pb, similar correlations can be seen whereas there are small number of theoreticalmodels we can use. Thus, the same fittings as in the case of 208 Pb were carried out. Figures 5.7,
5.8, 5.9, and 5.10 show the obtained ranges of S(ρsat ) and L for 204,206 Pb, respectively.
Figure 5.11 and 5.12 show all the obtained regions S(ρsat ) - L plane for 204,206,208 Pb due tothe experimental errors and the total errors, respectively. Since the model uncertainties are notindependently determined for each isotopes, the weighted average of three ranges due to thetotal errors in Fig. 5.12 must not be performed. On the other hand, the experimental errors
are independent among three isotopes. Therefore, we deduced the weighted average of threeranges due to the experimental errors as shown in Fig. 5.11. The combined ranges of S(ρsat )and L resulted in S(ρsat ) = 33.0 ± 1.1 MeV and L = 67.0 ± 12.1 MeV, respectively, which is

CHAPTER 5. RESULTS AND DISCUSSION

68

consistent with L = 88 ± 25 MeV obtained from the recent analysis of isospin diﬀusion data[51] and L = 65 ± 16 MeV determined by the energy-weighted sum rule of PDR data [72].
However, the errors of ∆rnp including model uncertainties is still so large that the constraint
on the slope coeﬃcient L is wide with the range of 30–90 MeV by such a rough estimation as
the global correlation approach mentioned above. Therefore we need to develop our analysis
and experimental data for a more precise measurement of the neutron skin thickness. Extending our analysis to other nuclei is also important to improve the constraint on the value of theslope L.

Table 5.3: Neutron skin thickness for 208 Pb and the nuclear matter properties at saturation
density, such as the incompressibility coeﬃcient K0 , the symmetry energy S(ρsat ), its slope L,
and the curvature Ksym in various relativistic and non-relativistic theoretical models.
Model∆rnp (fm)Relativistic
nonlinear coupling
NL1 [125]
0.321
NL3 [19]
0.280
NL-Z [137]0.307
NL-SH [126]0.266
TM1[18]0.271
G2 [127]
0.257
PK1[114]
0.260
FSUGold [23]0.207
FSUGZ03 [24]0.190
BKA20 [25]
0.200
0.220
BKA22 [25]
BKA24 [25]0.240
IU-FSU [128]0.160
density dependent

E0 (ρ)
K0 (MeV)S(ρsat ) (MeV)

S(ρ)
L (MeV)

212
271.8
173
355
281
215
282.6
230
233
240
227
228
231.2

43.5
37.4
41.7
36.1
36.9
36.4
37.6
32.6
31.6
32.3
33.3
34.3
31.3

140.1
118.6
133.3
113.6
110.8
100.7
116
60.6
64
76
79
85
47.2

Ksym (MeV)

143
100
80
34
−7
55
−52
−11
−15
−9
−15

5.2. NEUTRON SKIN THICKNESSES

69

Continued.
Model∆rnp (fm)
PKDD [114]0.240
DD-ME1 [129]0.200
DD-ME2 [22]0.193
DD-PC1 [117]0.200
point coupling
PC-F1 [21]0.270
PC-LA [130]
0.250
0.260
PC-PK1 [131]
FKVW [132]0.200
Brown-Rho scaling
SLC [26]
0.210
quark-meson coupling
QMC-I [133]0.26
QMC-II [133]0.27
Non-relativistic
Gogny
D1S[118]
0.135
D1N[118]
0.142
SkyrmeSIII [116]0.13
SkM*[9]0.17
SkI3 [119]0.22
SkI4 [119]0.18
SkP [10]0.145
SkX [12]0.16
Skxs20 [73]0.200
Sly4 [11]0.16Sk255 [136]0.250
Sk272 [136]
0.243
0.17
SSk [120]
GSkI [120]0.19

K0 (MeV)
262.2
245
251
230

S(ρsat ) (MeV)
36.8
33.1
32.3
33

L (MeV)
90
55
51
70

Ksym (MeV)
−80
−101
−87
−108

255
264
238
250

37.8
37.2
35.6
33.1

117
108
113
80

75
−61
95
11

230

31.6

280
382

35
35

210
230

32.0
29.3

22.4
31.9

−252

355
216
258
248
201
271
200
229.9
255.0
271.6
229.2
230.2

28.2
30.0
34.8
29.5
30.0
31.1

9.9
45.8
100.5
60.4
19.7
33.2

−394
−156
73
−41
−267
−252

32.0
37.4
37.4
33.5
32.0

46.0
95.0
91.7
52.8
63.5

−120
−58.3
−67.8

CHAPTER 5. RESULTS AND DISCUSSION

70

Continued.
∆rnp (fm)
0.20
0.16
0.15
0.14
0.211+0.054−0.063

Model
GSkII [120]
BSk14 [121]
BSk17 [122]
BSk20 [123]This work

K0 (MeV)
233.6
239.3
241.7
241.4
-

S(ρsat ) (MeV)
34.2
30.0
30.0
30.0
-

Ksym (MeV)

−137
-

0.35

0.35

0.3

L (MeV)
66.8
43.9
36.3
37.4
-

Relativistic
Non-relativistic

0.3

NL-Z

Relativistic
Non-relativistic

NL-SH

0.25

∆rnp (fm)∆rnp (fm)QMC-II

0.2

0.25

0.2

0.15

0.15
SIII

M. Itoh, et. al. (2004)

0.1
100

0.1

T. Li, et. al. (2010)
150

200

250

300

K0 (MeV)350

400-500 -400 -300 -200 -100

0

100

200

300

Ksym (MeV)

Figure 5.2: ∆rnp for 208 Pb (fm) vs K0 (MeV). Figure 5.3: ∆rnp for 208 Pb (fm) vs Ksym (MeV).

5.2. NEUTRON SKIN THICKNESSES

71

0.35

0.25

0.2

(33.7, 0.211)

0.15

29.1 31.6

0.2(74.3, 0.211)

25

30

0.1

35.7 37.7

35

S(ρsat) (MeV)
Figure 5.4:
(MeV).

0.25

0.15

0.1
20

Relativistic
Non-relativistic

0.3∆rnp (fm)∆rnp (fm)

0.3

0.35

Relativistic
Non-relativistic

40

45

0

30.2

20

54.0

40

60

92.6 112.2

80

100

120

140

L (MeV)∆rnp for 208 Pb (fm) vs S(ρsat ) Figure 5.5: ∆rnp for 208 Pb (fm) vs the slope
coeﬃcient L (MeV).

CHAPTER 5. RESULTS AND DISCUSSION

72

140

Relativistic
Non-relativistic120

L (MeV)

100
80
60
40
20
0
20

25

30

35

40

45

S(ρsat) (MeV)
Figure 5.6: L (MeV) vs S(ρsat ) (MeV). Surrounded areas are the extracted ranges of S(ρsat )and L from ∆rnp for 208 Pb as shown in Figs. 5.4 and 5.5

5.2. NEUTRON SKIN THICKNESSES

73

0.35

Relativistic
Non-relativistic∆rnp for 204Pb (fm)0.25

∆rnp for0.2(33.1, 0.178)

0.15

0.1
20

29.3 32.0

25

30

Relativistic
Non-relativistic

0.3

204

Pb (fm)

0.3

0.35

0.2

0.15

0.1

35.1 36.3

35

0.25

40

45

0

(67.1, 0.178)

20.9

20

51.4

40

S(ρsat) (MeV)
Figure 5.7:
(MeV).

∆rnp for 206Pb (fm)Pb (fm)120

140(32.6, 0.180)

0.15

0.1

30

35

S(ρsat) (MeV)

0.25

0.2

0.15

0.1

28.6 31.0 34.2 35.6

25

Relativistic
Non-relativistic

0.3

206

∆rnp for

100

0.35

Relativistic
Non-relativistic

0.2

Figure 5.9:
(MeV).

80∆rnp for 204 Pb (fm) vs S(ρsat ) Figure 5.8: ∆rnp for 204 Pb (fm) vs the slope
coeﬃcient L (MeV).

0.25

20

60

L (MeV)

0.35

0.3

89.8 103.9

40

45

0

(59.7, 0.180)

12.4

20

40.5

40

89.9 95.2

60

80

100

120

140

L (MeV)∆rnp for 206 Pb (fm) vs S(ρsat ) Figure 5.10: ∆rnp for 206 Pb (fm) vs the slope
coeﬃcient L (MeV).

CHAPTER 5. RESULTS AND DISCUSSION

74

140

Relativistic
Non-relativistic

120

208

Pb

L (MeV)

100
80
60

206

Pb

40

204

20
0
20

Pb

25

30

35

40

45

S(ρsat) (MeV)
Figure 5.11: All the extracted regions of L vs S(ρsat ) for 204,206,208 Pb correspond to the red lines
in Figs. 5.4, 5.5, 5.7, 5.8 5.9, and 5.10 due to the experimental errors of the extracted ∆rnp only.
Black filled circle and solid error bars are the weighted average of these three regions.

5.2. NEUTRON SKIN THICKNESSES

140

75

Relativistic
Non-relativistic

120

208

Pb

L (MeV)

100
80
60

206

Pb

40

204

20
0
20

Pb

25

30

35

40

45

S(ρsat) (MeV)
Figure 5.12: Similar to Fig. 5.11, but the extracted regions correspond to the blue lines in
Figs. 5.4, 5.5, 5.7, 5.8 5.9, and 5.10 due to the total errors of the extracted ∆rnp including themodel uncertainties.

CHAPTER 5. RESULTS AND DISCUSSION

76“Isobaric incompressibility” We also examined a more advanced topic of the EOS
briefly. It has been found that the contributions of the higher order terms in Eqs. (1.2) and (1.4)
are not necessarily negligible especially for the incompressibility of the asymmetric nuclear
matter as discussed in Ref. [138, 139]. The EOS of the symmetric nuclear matter E0 (ρ) is
expanded to one-order higher than Eq. (1.2). The coeﬃcient of third-order derivative J0 in the
expansion of E0 (ρ) is expressed as
K0 2 J0 3
 +  + O( 4 ),2!
3!
3
d E0 (ρ)= 27ρ3sat.
dρ3 ρ=ρsat

E0 (ρ) = E0 +
J0

(5.1)
(5.2)

Recently the isospin dependence of incompressibility has been studied by the high-quality
data of isoscalar giant monopole resonances (ISGMR) in Sn isotopes measured from (α, α0 )reaction at RCNP [39, 40, 139, 140]. The extended, so-called, isobaric incompressibility K(δ)
with isospin asymmetry δ can be expressed as
K(δ) = K0 + Kτ δ2 + O(δ4 )J0Kτ ≈ Ksym − 6L −
L,
K0

(5.3)
(5.4)

where the third term of the right hand in Eq. (5.4) is derived from the higher-order coeﬃcient J0 in Eq. (5.1). The asymmetry term Kτ characterizes the isospin dependence of theincompressibility K(δ) and also very important to understand the properties of the asymmetricnuclear or neutron matter such as neutron stars. Chen et al. in Ref. [138] obtained a value ofKτ = −370 ± 120 MeV from the analysis of isospin diﬀusion data considering the higher-orderterms of the EOS and the incompresibility at saturation. However, Li et al. in Ref. [40] resulted in Kτ = −550 ± 100 MeV much diﬀerent from that of Chen et al. The result of Li etal. were obtained from the ISGMR strength distributions in Sn isotopes by using the empiricalexpression of the finite nucleus incompressibility KA with the lowest order terms only [141]:
KA ≈ K0 + Ksurf A−1/3 + Kτ α2 + KCoul

Z2
,
Z 4/3

(5.5)

where A, Z, Ksurf , and KCoul are the mass and atomic number, the surface, and Coulomb termswith α = (N − Z)/A. The higher-order terms of this expansion, which was neglected in theiranalysis, are not negligible as pointed out in Ref. [142]. That is the reason of the discrepancy
between the results of Chen and Li. Thus, since the analysis method of Ref. [40] are still
ambiguous to extract the Kτ only, we used the result of Chen et al. [138] as shown in thehatched area in Fig. 5.13. While the eﬀect of L on Kτ accounts for about 75%, no correlation
is found between Kτ and ∆rnp (or L) in Fig. 5.13. However, combined with the range ofKτ = −370 ± 120 MeV, our result of ∆rnp for 208 Pb put some constraints on the theoreticalmodels.

5.2. NEUTRON SKIN THICKNESSES

77

0.35

Relativistic
Non-relativistic∆rnp for208

Pb (fm)

0.3

NL3

TM1
G2
BKA24

0.25
SkI3
DD-ME1
DD-ME2

0.2

FSUGold

BKA20
SkM*
SkX
SkP
SIII

0.15

0.1

BKA22
FSUGZ03
SkI4
Sly4
BSK21

Chen et al.(2009)

-800

-700

-600

-500

-400

-300

-200

-100

Kτ (MeV)Figure 5.13: Hatched area is the constraint region on the Kτ and ∆rnp for 208 Pb plane withseveral non-relativistic (triangles) and relativistic (circles) mean-field models. The constraint
on ∆rnp is our extracted value (blue and red lines on the y-axis), whereas on Kτ is taken from
Ref. [138].

78

CHAPTER 5. RESULTS AND DISCUSSION“Future perspectives” Since unknown systematic errors are also included in the model
uncertainties, further progress from both experiment and theory are necessary. To achieve
more precise extraction of the neutron density distributions with an accuracy of less than 1%of neutron radius rn , we need to reveal such a systematic ambiguity. Then, we need to develop
both the experimental and theoretical improvements. Some clues for the improvements werealso obtained from this work for lead isotopes.
• To reduce the systematic uncertainties due to the instabilities of the beam conditions
just while the data is been taking, the check of the beam-line viewers is not enough.
Therefore we plan to install beam position monitors (BPM), especially, cavity-BPM foron-line non-destructive monitoring. The high-quality beam is also needed.
• In the relativistic framework, Scalar density is also necessary. Since the Scalar-to-Vector
ratio is changing depending on the change of the eﬀective mass around surface regionof a nucleus, we need more careful treatment of Scalar density than that of the present
as the use of the constant value of ρS /ρV = 0.96.
• Coulomb potentials are now assumed to be the rigid sphere, but its eﬀect on the scattering observables is found to be not so small. we need to modify the calculation of
Coulomb potentials with the folding form of proton density distributions.
• For unstable nuclei, charge and proton density distributions are also unknown parameters, thus we need to investigate the simultaneous extraction method of proton andneutron density distributions, otherwise, to wait for other experiments such as electron
scattering, isotope shift measurements. Since the sensitivity of Coulomb potentials tothe scattering observables was recognized, we are examining the simultaneous extraction by using proton elastic scatterings of diﬀerent incident energies, that is, diﬀerent
NN amplitudes.
• We will proceed the reanalyses of the obtained experimental data for lighter nuclei suchas 40,42,44,48 Ca, 16,18 O with several improvements to reduce systematic uncertainties aslisted above. We will verify how applicable our analysis method is to extract the neutron
densities of light nuclei. Particularly, 48 Ca is expected to have a neutron skin structure.
The neutron skin thickness of 48 Ca is also important to determine the EOS parameters.
These improvements listed above are the next steps towards the precise extraction of the nucleon density distributions of various nuclei, and are now in progress.

Chapter 6
SUMMARY
We measured the angular distributions of cross sections and analyzing powers for polarized
proton elastic scattering from 58 Ni and 204,206,208 Pb at E p = 295 MeV. Using the experimental
data, we have extracted the neutron density distributions of 204,206,208 Pb. In order to explain
the proton elastic scattering at intermediate energies, phenomenological medium modifications were introduced into the free NN interaction with density-dependent parameters. The
medium-eﬀect parameters were determined from the experimental observables for 58 Ni, whose
nucleon density is well known. After the calibration of the eﬀective NN interaction, we deduced the neutron density distributions of 204,206,208 Pb in the form of a model-independent SOG
distribution.
Furthermore, we have evaluated the error-envelopes of the neutron densities due to bothexperimental uncertainties and uncertainties associated with the various model assumptionsin the medium-modified RIA by means of a new χ2 criterion. The model ambiguity due tothe theoretical assumptions, especially in use of hadronic probes, is very important to extract
physical quantities. However it is very diﬃcult to estimate such an ambiguity and hence it isusually not mentioned in many cases.
The extracted rms radius of the neutron density for 208 Pb is consistent with past results andrecent theoretical predictions such as FSUGold or BKA20 parameterization. The experimental
standard errors of the neutron rms radii (δrnstd ' 0.03 fm) are slightly smaller than the valueof Ref. [62]. Even including the model ambiguity, the estimated errors of the neutron rms
radii (δrnmdl ' 0.06 fm) were found to be relatively small with an accuracy of about 1 %, whichis comparable or better than the estimated accuracy of the PREX experiment using parityviolating electron scattering at its present status.
We also deduced the constraint range of the symmetry energy at saturation S(ρsat ) andthe slope coeﬃcient L of the symmetry energy corresponding to the results of the extracted
neutron skin thicknesses ∆rnp for 204,206,2208 Pb by using the global correlations between S(ρsat ),L, and ∆rnp . Since the the errors including the model uncertainties are not independent among
204,206,2208
Pb, the method of the weighted average was not be able to be applied to combining
the three constraint ranges due to the total errors including the model uncertainties. Therefore,
we obtained the combined range of S(ρsat ) = 33.0 ± 1.1 MeV and L = 67.0 ± 12.1 MeV, which

79

80

CHAPTER 6. SUMMARY

reflect the experimental errors of ∆rnp only. The obtained ranges are very consistent with andmuch smaller than the previous results. However, the errors of S(ρsat ) and L due to the model
uncertainties (∼ 3 MeV and 40 MeV) are much larger than the ones due to the experimental
uncertainties (1.1 MeV and 12.1 MeV). Further progress from both the experiment and theory
are necessary.

ACKNOWLEDGEMENTSFirst of all, I would like to express my sincere gratitude to Prof. Harutaka Sakaguchi. He is mysupervisor and has guided me from the beginning of my research career. He has continuously
advised, supported, and encouraged me at every stage of this work. Discussions and talks withhim have always taught me the pleasure and attitude toward physics as well as very skillfulexperimental technique.
I would like to thank Prof. K. Hatanaka, Prof. A. Tamii, Prof. T. Kawabata, Prof. M. Yosoi,Prof. T. Murakami, Prof. I. Tanihata, Prof. H. J. Ong, Prof. Y. Maeda, Dr. S. Terashima, Dr.
Y. Matsuda, and the late Prof. H. Okamura for their support and helpful discussion regarding
the present work during my research days at RCNP.
I would address my thanks to all the members of RCNP-E248 collaboration: Prof. H.
Sakaguchi, Prof. T. Murakami, Prof. M. Yosoi, Dr. Y. Yasuda, Dr. S. Terashima, Mr. Y. Iwao,
Dr. H. Takeda, Dr. M. Itoh, Dr. H. P. Yoshida, and Dr. M. Uchida for their grate contributions
to perform this experiment successfully.
I gratefully acknowledge the continuing support of the RCNP cyclotron staﬀ for providing
the clean and stable beam.
I would like to thank Prof. K. Imai and Prof. T. Nagae for their thoughtful suggestions andencouragements.
I have spent five years as a graduate student of the experimental nuclear and hadron physics
group at Kyoto University and the last three years as a researcher at the Research Center
for Nuclear Physics, Osaka University. I enjoyed research activities, discussions, drinkings,
recreations, and other things in the pleasant and comfortable atmosphere. I am grateful to all
my colleagues and friends.
Finally, I wold like to express my great thanks to my parents, brother, and grand motherfor their continuous encouragement, patience, and generous aﬀection and support.

81

Appendix A
Data table

Table A.1: Diﬀerential cross sections and analyzing powers for 58 Ni(p, p) at E p = 295 MeV.
θc.m. (degree)
7.67
9.20
10.74
12.27
13.80
14.83
15.34
16.87
18.40
19.93
21.47
23.00
24.53
26.06
27.59
29.12

dσ/dΩ (mb/sr)
1.867×103
6.474×102
1.952×102
2.194×101
1.562×101
3.263×101
3.658×101
4.174×101
3.159×101
1.652×101
7.064
2.042
1.319
1.797
2.006
1.703

∆(dσ/dΩ) (mb/sr)
0.062×103
0.224×102
0.067×102
0.067×101
0.051×101
0.124×101
0.140×101
0.163×101
0.121×101
0.057×101
0.216
0.070
0.040
0.066
0.077
0.061

83

Ay
∆Ay
0.508 0.045
0.630 0.048
0.590 0.046
0.174 0.041
0.467 0.044
0.862 0.054
0.886 0.055
0.922 0.056
0.869 0.055
0.644 0.048
0.234 0.041−0.607 0.047−0.047 0.041
0.789 0.052
0.898 0.056
0.777 0.051

APPENDIX A. DATA TABLE

84

Continued.
θc.m. (degree)
30.65
32.18
33.71
35.24
36.77
38.29
39.82
41.35
42.87
44.40
45.92
47.44
48.97

dσ/dΩ (mb/sr)
1.031
4.881×10−1
1.781×10−1
8.786×10−2
1.004×10−1
1.158×10−1
1.007×10−1
6.607×10−2
3.169×10−2
1.113×10−2
3.274×10−3
3.275×10−3
4.929×10−3∆(dσ/dΩ) (mb/sr)
0.034
0.149×10−1
0.059×10−1
0.289×10−2
0.033×10−1
0.042×10−1
0.036×10−1
0.229×10−2
0.102×10−2
0.034×10−2
0.118×10−3
0.137×10−3
0.226×10−3

Ay
0.528
0.116
−0.540−0.492
0.470
0.763
0.749
0.652
0.428
0.012
−0.649
0.365
0.786

∆Ay
0.045
0.041
0.045
0.045
0.044
0.051
0.051
0.048
0.043
0.041
0.050
0.044
0.052

85

Table A.2: Diﬀerential cross sections and analyzing powers for 204 Pb(p, p) at E p = 295 MeV.
θc.m. (degree)
7.55
9.06
10.57
12.08
13.59
14.59
15.10
16.61
18.11
19.62
21.13
22.64
24.15
25.66
27.17
28.68
30.19
31.69
33.20
34.71
36.22
37.73
39.23
40.74
42.25
Continued.

dσ/dΩ (mb/sr)
1.540×103
5.709×102
6.785×102
3.526×102
9.619×101
1.991×101
1.919×101
4.882×101
4.339×101
1.575×101
4.166
5.049
6.383
4.131
1.463
6.671×10−1
9.571×10−18.870×10−1
4.427×10−1
1.355×10−1
1.187×10−1
1.551×10−1
1.089×10−1
3.929×10−2
1.234×10−2∆(dσ/dΩ) (mb/sr)
0.063×103
0.255×102
0.240×102
0.130×102
0.349×101
0.071×101
0.069×101
0.192×101
0.171×101
0.055×101
0.128
0.178
0.244
0.150
0.044
0.210×10−1
0.338×10−1
0.351×10−1
0.139×10−1
0.044×10−1
0.037×10−1
0.060×10−1
0.037×10−1
0.121×10−1
0.040×10−2

Ay
0.325
0.666
0.680
0.793
0.757
0.672
0.705
0.937
0.936
0.646
−0.159
0.671
0.874
0.645
−0.035−0.227
0.698
0.690
0.322
−0.455
0.284
0.660
0.583
0.109
−0.337

∆Ay
0.042
0.049
0.048
0.052
0.050
0.048
0.049
0.057
0.057
0.048
0.041
0.048
0.055
0.048
0.040
0.042
0.049
0.056
0.042
0.044
0.042
0.053
0.046
0.041
0.043

APPENDIX A. DATA TABLE

86

Continued.
θc.m. (degree)
43.76
45.26
46.77
48.28

dσ/dΩ (mb/sr)
1.767×10−2
1.993×10−2
1.104×10−2
2.959×10−3∆(dσ/dΩ) (mb/sr)
0.061×10−2
0.072×10−2
0.038×10−2
0.104×10−3

Ay
0.600
0.741
0.573
0.034

∆Ay
0.047
0.051
0.047
0.047

87

Table A.3: Diﬀerential cross sections and analyzing powers for 206 Pb(p, p) at E p = 295 MeV.
θc.m. (degree)
7.55
9.06
10.57
12.08
13.59
14.59
15.10
16.60
18.11
19.62
21.13
22.64
24.15
25.66
27.17
28.68
30.18
31.69
33.20
34.71
36.22
37.72
39.23
40.74
42.25
Continued.

dσ/dΩ (mb/sr)
1.537×103
5.931×102
6.977×102
3.507×102
9.188×101
1.987×101
2.002×101
5.120×101
4.439×101
1.523×101
4.045
5.461
6.504
4.052
1.398
6.926×10−1
9.901×10−1
8.747×10−1
4.154×10−1
1.279×10−1
1.267×10−1
1.577×10−1
1.047×10−1
3.580×10−2
1.277×10−2∆(dσ/dΩ) (mb/sr)
0.063×103
0.265×102
0.247×102
0.129×102
0.334×101
0.071×101
0.073×101
0.201×101
0.175×101
0.053×101
0.125
0.195
0.250
0.145
0.042
0.214×10−1
0.353×10−1
0.343×10−1
0.129×10−1
0.043×10−1
0.040×10−1
0.059×10−1
0.035×10−1
0.111×10−2
0.041×10−2

Ay
0.326
0.640
0.675
0.796
0.754
0.668
0.726
0.936
0.935
0.635
−0.194
0.696
0.877
0.624
−0.093−0.143
0.721
0.671
0.271
−0.494
0.365
0.702
0.544
0.026
−0.263

∆Ay
0.042
0.049
0.048
0.052
0.050
0.048
0.050
0.057
0.057
0.048
0.041
0.049
0.055
0.048
0.041
0.050
0.050
0.055
0.041
0.045
0.043
0.052
0.046
0.041
0.043

APPENDIX A. DATA TABLE

88

Continued.
θc.m. (degree)
43.75
45.26
46.77
48.27

dσ/dΩ (mb/sr)
1.947×10−2
2.037×10−2
1.041×10−2
2.449×10−3∆(dσ/dΩ) (mb/sr)
0.069×10−2
0.074×10−2
0.035×10−2
0.095×10−3

Ay
0.634
0.727
0.520
−0.070

∆Ay
0.048
0.051
0.046
0.052

89

Table A.4: Diﬀerential cross sections and analyzing powers for 208 Pb(p, p) at E p = 295 MeV.
θc.m. (degree)
7.55
9.06
10.57
12.08
13.59
14.59
15.09
16.60
18.11
19.62
21.13
22.64
24.15
25.66
27.17
28.67
30.18
31.69
33.20
34.71
36.21
37.72
39.23
40.74
42.24
Continued.

dσ/dΩ (mb/sr)
1.580×103
5.958×102
7.352×102
3.521×102
9.086×101
2.002×101
2.148×101
5.260×101
4.444×101
1.467×101
4.045
5.610
6.541
3.861
1.329
7.050×10−1
9.840×10−1
8.396×10−1
3.825×10−1
1.196×10−1
1.279×10−1
1.512×10−1
9.577×10−2
3.064×10−2
1.287×10−2∆(dσ/dΩ) (mb/sr)
0.065×103
0.262×102
0.260×102
0.130×102
0.329×101
0.071×101
0.078×101
0.206×101
0.175×101
0.050×101
0.125
0.202
0.250
0.135
0.040
0.215×10−1
0.352×10−1
0.326×10−1
0.118×10−1
0.039×10−1
0.041×10−1
0.058×10−1
0.319×10−2
0.094×10−2
0.040×10−2

Ay
0.326
0.626
0.683
0.797
0.752
0.692
0.741
0.935
0.933
0.606
−0.181
0.730
0.868
0.593
−0.138−0.056
0.736
0.653
0.220
−0.481
0.432
0.695
0.511
−0.013−0.097

∆Ay
0.042
0.048
0.048
0.052
0.050
0.049
0.050
0.057
0.057
0.047
0.041
0.050
0.055
0.047
0.041
0.047
0.050
0.054
0.041
0.045
0.044
0.052
0.045
0.041
0.042

APPENDIX A. DATA TABLE

90

Continued.
θc.m. (degree)
43.75
45.26
46.76
48.27

dσ/dΩ (mb/sr)
2.019×10−2
1.974×10−2
9.180×10−3
2.053×10−3∆(dσ/dΩ) (mb/sr)
0.071×10−2
0.071×10−2
0.306×10−3
0.078×10−3

Ay
0.651
0.711
0.470
−0.141

∆Ay
0.048
0.050
0.045
0.051

Appendix B
Relativistic impulse approximation
The analysis method to describe the reaction is essentially based on the framework of therelativistic impulse approximation (RIA) with the relativistic Love-Franey (RLF) interaction
developed by Murdock and Horowitz (MH model) [88–90]. In this appendix, we review theframework and formalism used in the MH model.
Since 1980s, after the success of Dirac phenomenology to explain the scattering observables, especially, polarization observables, the original RIA approach has been developed withthe microscopic description by McNeil, Ray, and Wallace [143] (MRW model). In the original RIA, the experimental NN scattering amplitude is represented by a particular set of five
Lorentz covariant functions (Eq. (4.1)), as mentioned in Chap. 4.1.1. The functions are thenfolded with the target nucleus densities to produce a first-order tρ optical potentials. This expression F of five Lorentz covariant functions is an approximated form by using an analogy
from the empirical amplitude, which is conventionally expressed in terms of the nonrelativistic
Wolfenstein amplitudes with the Pauli matrices [143].
However, the general expression of F in the two-particle Dirac space involves 44 components of Lorentz covariants under the isospin and CPT invariance. Thus the only five functions
described in Eq. (4.1) is not suﬃcient to specify F , i.e., to determine the negative energy matrix elements uniquely. Thus there are nontrivial assumptions involved in using Eq. (4.1) to
calculate the optical potential. In addition, the original RIA procedure does not separate thedirect and exchange contributions to the amplitudes. These contributions have diﬀerent characteristic dependence on the momentum transfer and energy, especially at low energies. The
original RIA model assumes that the NN interaction is unmodified by the surrounding nuclear
medium such as inside a nucleus. While this assumption is valid at high energies, the modifications from nuclear medium such as Pauli blocking are not negligible at lower intermediate
energy regions.

91APPENDIX B. RELATIVISTIC IMPULSE APPROXIMATION

92

Ū0′

Ū0′

Ū1′
k′c

Ū1′
k′c

−k′c

2

2

2

g Λ2Λ+q2 λL τ1I

g Λ2Λ+q2 λL τ0I2

g Λ2Λ+Q2 λL τ0I

1
q2 +m2ikc−k′cg Λ2Λ+Q2 λL τ1I
1
Q2 +m2i

kc

−kc

U0

−kc

U0

U1

U1

Exchange

DirectFigure B.1: Meson exchange diagrams for the relativistic Love-Franey model.

“Relativistic Love-Franey NN interaction” On the basis of this original form of the NN
amplitude, the RLF NN interaction was further developed by Horowitz [88] to improve theproblems listed above.
The NN amplitude is modeled as arising from the first Born approximation for the exchange of a set of mesons. The parameters of the mesons were fitted to reproduce NN scattering data at several energies. From direct and exchange Feynman diagrams in Fig. B.1, the NN
amplitude for each meson is calculated straightforward. The NN-meson vertex factor from the
Feynman rules is assumed to be)Λ2i
λL(i) (τ )Ii ,gi 2
q + Λ2i((B.1)

where L(i) denotes the Lorentz type of the ith meson listed in Table. B.1 and Ii is its isospin (0or 1). Then, each Lorentz invariant F L (L = S , V, PS , T , or A) can be written as]
M2 [ LF D (q) + F XL (Q) ,F L (q, Ec ) = i
2Ec kc∑LδL,L(i) {τ0 · τ1 }Ii f i (q),F D (q) ≡i

F XL (Q)≡ (−1)T∑if (q) ≡ig2i
q 2 + m2i(

BL(i),L {τ0 · τ1 }Ii f i (Q),Λ2i
Λ2i + q 2

)2

)2
(
ḡ2i
Λ̄2i
,
−i 2
q + m̄2i Λ̄2i + q 2

(B.2)
(B.3)
(B.4)
(B.5)

where D and X denote the direct and exchange terms as shown in Fig. B.1, q and Q are
direct and exchange three-momentum transfers. Ec and kc are the total energy and relative
momentum in the nucleon-nucleon cm system. BL(i),L is the (L(i), L) component of the Fierz

93

Table B.1: Dirac matrix types
L
S (scalar)
V (vector)
PS (pseudoscalar)
A (axial-vector)
T (tensor)

λL
1
γµ
γ5
γ5 γµ
σµν

transformation matrix:
0

BL,L0

0

Tr(λL λL λL λL )=,[Tr(λL0 λL )]2
 2 2 1 −2 2   S 
 8 −4 0 −4 −8   V 
1 


=
 24 0 −4 0 24   T  .
8 


 −8 −4 0 −4 8   A 

 

2 −2 1 2 2
PS

(B.6)

(B.7)

The coupling constant, mass and cut-oﬀ parameter for i-th meson denoted by gi , mi , and Λiare fitted to the free NN amplitudes at several diﬀerent laboratory energies [88]. The full setof parameters can be found in Ref. [88, 89]. T is the total isospin of the two-nucleon system,thus
F L (pp) = F L (T = 1),]
1[ L
F L (pn) =F (T = 0) + F L (T = 1) ,
2

(B.8)
(B.9)

are used to get pp and pn scattering amplitudes, respectively.
The form of NN amplitudes in Eq. (4.1) is completely local. No factors of a momentum dotted int a gamma matrix appear explicitly. However, as the energy is decreased, theNN amplitudes cannot neglect the exchange nonlocalities implied by a pion with pseudovector coupling. It is known that the pseudoscalar term of Eq. (4.1) should be replaced by thepseudovector invariant
F

PS

5
5
γ(0)
γ(1)
→ −F PV556 qγ(0)6 qγ(1)2M 2M.

(B.10)

to give meaningful results at lower energies. Note that this is not a exact medium modification
diﬀerent from such a Pauli blocking eﬀect.

94

APPENDIX B. RELATIVISTIC IMPULSE APPROXIMATION“tρ optical potential in the MH model” The first-order Dirac optical potentials for the
spherical nuclei are produced by folding NN amplitude with the target densities:
U L (r; E) = U DL (r; E) + U XL (r; E),∫
4πip
LU D (r; E) ≡ −dr 0 ρL (r 0 )tDL (|r 0 − r|; E),M ∫4πip
U XL (r; E) ≡ −dr 0 ρL (r 0 , r)tXL (|r 0 − r|; E) j0 (p|r 0 − r|),M

(B.11)(B.12)(B.13)

where tDL (|r|; E) are Fourier transforms of tDL (q; E) ≡ (iM 2 /2Ec kc )F DL (q) and similarly for theexchange pieces tXL (Q; E). The nuclear densities ρL (r) are expressed as sums over the occupied
nuclear levels:ρL (r 0 , r) ≡occ∑

Ūα (r 0 )λL Uα (r),(B.14)

αwhere Uα is a single particle four-component wave function in state α, and the relativistic
Hartree approximation is applied to obtain Uα . For the exchange term, non-local densitiesρL (r 0 , r) are approximated as
(
)(
)
3
L 0
L 1 0
ρ (r , r) ≈ ρ(B.15)(r + r)j1 (|r 0 − r|kF ),2|r 0 − r|kF
where ρV ((r 0 − r/2)) = 2k3F /3π2 .
For a spin-zero nucleus, the only nonzero densities are scalar, vector, and tensor densities
with λT = σ0i in Eq. (B.14). The optical potential takes the form
Uopt = U S + γ0 U V − 2iα · r̂U T .

(B.16)The tensor contribution is found to be small and will be negligible. Thus the optical potential
has only scalar and vector contributions.
For laboratory energy around 200 MeV, it is necessary to correct the optical potentials
for medium modifications from Pauli blocking. These can be approximated by performing a
Dirac-Brueckner calculation with a local-density approximation as follows:[( V)]ρ (r) 2/3
L
U (r; E) → 1 − a(E))(B.17)U L (r; E).
ρ0where ρ0 = 0.1934 fm−3 . In the present work, we have further modified the RLF NN interaction with the density-dependent terms described as Eqs. (4.4), (4.5) to explain the various
medium eﬀects. Since the Pauli blocking eﬀect is also included to our modification, we havenot used the correction of Eq. B.17 introduced by Horowitz.
The Dirac equation for the projectile is finally expressed as{[]}−iα·∇ + U V (r; E) + β M + U S (r; E) U0 (r) = EU0 (r),(B.18)

95

where E is the total nucleon-nucleus center-of-mass projectile energy, M is its rest mass, andU0 is the incident projectile wave function. Note that the Coulomb potential is included inU V . The Coulomb potential used in this model is assumed to be a uniform spherical charge
distribution of radius R = r0 A1/3 , where r0 ≈ 1.25 fm.
To solve the Dirac equation of Eq. (B.18), in the calculations of the MH model the wave
function is first separated into its upper and lower two-component wave functions and the
Dirac equations rewritten as two coupled, first-order diﬀerential equations: (σ·p)v0 (r) + (U V + M + U S )u0 (r) = Eu0 (r),(B.19) (σ·p)u0 (r) + (U V − M − U S )v0 (r) = Ev0 (r)






 u0 (r)  i 0 σi  1 0 
0i0 , α = γ γ  , β = γ  .
U0 (r) = (B.20)
v0 (r) 
σi 0 
0 −1 
The lower component v0 is then eliminated, leading to a single second-order diﬀerential equation for the upper components. At this moment this equation contains local, spin-orbit, andnonlocal Darwin potentials, but the nonlocality can be removed by rewriting the upper wave
function as
u0 (r) ≡ A1/2 (r; E)χ0 (r),U S (r; E) − U V (r; E)A(r; E) ≡ 1 +.
E+M(B.21)(B.22)

Then the equation for the new two-component function χ0 (r), that is, Schrödinger-like equation becomes(−∇2 + Vc + Vso σ·l + VDarwin )χ0 (r) = (E 2 − M 2 )χ0 (r),(B.23)

where the nonrelativistic energy-dependent optical potentials are
Vc (r; E) ≡ 2MU S + 2EU V + (U S )2 − (U V )2 ,1 ∂A,
Vso (r; E) ≡ −
rA ∂r
(
)
(
)2
3 ∂A
1 ∂ 2 ∂Ar+.
VDarwin (r; E) ≡ − 2
2r A ∂r∂r
4A2 ∂r

(B.24)(B.25)(B.26)

Appendix C
Equation of state of nuclear matter
Liquid drop model The study to understand the many-body properties of nuclei or nuclear
matter has started since 1930s by Bethe and von-Weizäcker [144, 145] who developed themacroscopic semi-empirical mass formula of nuclei using the liquid-drop model (LDM):
E(N, Z) = −aV A + aS A2/3 Bsurf + aC

Z2
BCoul ,
A1/3

(C.1)

aV = a1 (1 − κV I 2 ),(C.2)

aS = a2 (1 − κS I ),

(C.3)

2

where A = N + Z, the assumptions has been made that the nuclear matter in the interior is
uniform and incompressible so that the radius of a spherical nucleus is proportional to A1/3 . a1
is the binding energy of symmetric nuclear matter and I = (N −Z)/A. The term κV proportional
to I 2 is included in order to describe the dependence of the bulk binding energy on the neutron
excess, but Eq. (C.2) in the simple form has no shape dependence. Eq. (C.3) is the surface
energy term, which is analogy to Eq. (C.2). Bsurf and BCoul are the factors related to the nuclear
deformation.

Droplet model One specific approach to extending the LDM is called the droplet model(DM). Myers and Swiatecki [146–148] developed the LDM model, which is extended to one
higher order in the expansion parameters A−1/3 and I 2 . Diﬀerent from LDM, the extensions
includes the general expression of nuclear incompresiibility and density distributions. Theenergy by DM is expressed as[][]
1 2 1 49 J 2 2 2/3E(N, Z; shape) = −a1 + J δ̄ − K ¯ + M δ̄ A + a2 +
δ̄ A Bsurf
2
2
4Q+a3 A1/3 Bcurv + c1 Z 2 A−1/3 BCoul − c2 Z 2 A1/3 Bred
−c5 Z 2 BW − c3 Z 2 A−1 − c4 Z 4/3 A−1/3 ,
2

97

(C.4)

98

APPENDIX C. EQUATION OF STATE OF NUCLEAR MATTER

where
δ̄ =

I + 163 cQ1 ZA−2/3 BV

,

(C.5)

¯ = [−2a2 A−1/3 Bsurf + Lδ̄2 + c1 Z 2 A−4/3 BCoul ]/K,

(C.6)

1 + 94 QJ A−1/3 Bsurf

and the constants ci (i = 1, 2, 3, 4, 5) are()
c21 1 18
3 e2
, c2 =
c1 =
+
,
5 r0
336 J K( )2( )2/35b
53c3 = c1, c4 = c1
,
2
r0
4
2π
1 c21
c5 =
.
64 Q

(C.7)

The quantities Bcurv , Bred , BV and BW are the curvature correction to the surface energy, theinterior and surface corrections due to the Coulomb forces, respectively. The coeﬃcients J, Q,
and r0 in Eq. (C.4) are analogous to the LDM coeﬃcients κV , κS , and aC of Eq. (C.1). Three
new coeﬃcients K, L, and M are the incompressibility, the dependence of the incompressibility
on neutron excess, and the higher order coeﬃcient in asymmetry I, respectively.
Mean field approach In the late 1960s, microscopic theoretical studies of the nuclear manybody system were pioneered by Brueckner et al. [149]. Since then, many theoretical models
such as the microscopic many-body approach, the eﬀective-field theory, and the phenomenological mean-field approach, have been developed by using various bare or eﬀective, two-bodyor three-body interactions.
The phenomenological approach, in particular, the Skyrme-Hartree-Fock and the relativistic mean-field models are the most successful approaches to describe the macroscopic properties of finite nuclei and/or nuclear matter. In these theoretical approaches, the energy per
nucleon of infinite nuclear matter is also described as a function of the density ρ and isospin
asymmetry δ in an analogous fashion to LDM or DM expression. It can be expanded in aseries of δ asE
(ρ, δ) ≡ E(ρ, δ) = E0 (ρ) + S2 (ρ)δ2 + S4 (ρ)δ4 + O(δ6 )A
E0 (ρ) = E(ρ, 0)
1 ∂2 E(ρ, δ)
S2 (ρ) =2!
∂δ2 δ=0S4 (ρ) =1 ∂4 E(ρ, δ).
4!
∂δ4 δ=0

(C.8)
(C.9)
(C.10)(C.11)

where E0 (ρ) is the energy per nucleon of symmetric nuclear matter. The second-order term
S2 (ρ) ≡ S(ρ) is known as the so-called nuclear symmetry energy. The higher-order O(δ4 )99

terms are very small and often neglected. Therefore, the energy per nucleon of pure neutron
matter (δ = 1) is approximately written asE(ρ, 1) ≈ E(ρ, 0) + S(ρ).

(C.12)In order to understand the properties of the pure neutron matter such as a neutron star, we firstneed to know the EOS of symmetric nuclear matter and the symmetry energy.
The symmetric EOS E0 (ρ) around the saturation can be characterized by an equilibrium
density of ρsat ' 0.15 fm−3 , an energy per nucleon of E0 ≡ E0 (ρsat ) ' −16 MeV, and a pressureof P = ρ2 dE0 /dρ = 0. Thus, the symmetric EOS E0 (ρ) around the saturation density is
expanded up to the fourth-order in density as
E0 (ρ) = E0 +K0 2 J0 3 I0 4
 +  +  + O( 5 ),
2!
3!
4!

(C.13)

where  = (ρ − ρsat /3ρsat ) and the first-order coeﬃcient of  vanishes at saturation defined by
P = 0. The coeﬃcient K0 is the so-called incompressibility of symmetric nuclear matter.
The symmetry energy S(ρ) and the fourth-order coeﬃcient S4 (ρ) in Eq. (C.8) are similarly
expanded around the saturation density as
Ksym 2 Jsym 3 Isym 4 + + + O( 5 )2!
3!
4!
Ksym,4 2 Jsym,4 3 Isym,4 4S4 (ρ) = S4 (ρsat ) + Lsym,4  + + + + O( 5 ),2!
3!
4!
S(ρ) = S(ρsat ) + Lsym  +(C.14)(C.15)

where Lsym(,4) , Ksym(,4) , Jsym(,4) , and Isym(,4) are the slope, curvature, third-order, and fourth-order
coeﬃcient of symmetry energy S(ρ) (S4 (ρ)). The pressure of symmetry energy Lsym , whichis strongly correlated with the neutron skin thickness of heavy asymmetric nuclei, does notvanish unlike the symmetric EOS.
The coeﬃcients in Eqs. (C.13), (C.14), and (C.15), namely, E0 , K0 , J0 , I0 , S(ρsat ), Lsym ,Ksym , Jsym , Isym , S4 (ρsat ), Lsym,4 , Ksym,4 , Jsym,4 , and Isym,4 , characterize the EOS of asymmetricnuclear matter, especially, its density dependence around the saturation density. Among these
coeﬃcients, E0 , K0 , and S(ρsat ) have been well understood from recent theoretical and experimental studies. However, others are still less certain and their predictions widely diﬀer among
many mean-field models.
Isospin dependence of saturation properties By using these coeﬃcients, the evolutionof the saturation properties such as the saturation density, the binding energy per nucleon, andthe incompressibility can be described as a function of the isospin asymmetry δ. The saturation
point is defined as∂E(ρ, δ)= 0,
∂ρ ρ=ρsat (δ)(C.16)

100

APPENDIX C. EQUATION OF STATE OF NUCLEAR MATTER

where the saturation density ρsat (δ) is derived from Eqs. (C.8), (C.13), (C.14), (C.15):



2 


 4
3Lsym 2  3Ksym Lsym 3Lsym,4 3J0 Lsym
6 ρsat,0 ,

δ
+
O(δ
)
ρsat (δ) = 1 −δ + 
−−(C.17)

K0
K0
K02
2K03
where ρsat,0 is redefined as the saturation density of symmetric nuclear matter. The binding
energy per nucleon at saturation density Esat (δ) ≡ E(ρsat (δ), δ) is also expressed up to fourthorder in δ as

2 
Lsym

 4
2
 δ + O(δ6 ).
Esat (δ) = E0 + S(ρsat,0 )δ + S4 (ρsat,0 ) −(C.18)
2K0
The incompressibility of asymmetric nuclear matter is generally defined as∂P(ρ, δ)∂ρ∂E(ρ, δ)∂2 E(ρ, δ)
= 18ρ
+ 9ρ2
∂ρ
∂δ2
P(ρ, δ)∂2 E(ρ, δ)
= 18
+ 9ρ2
,
ρ
∂δ2

K(ρ, δ) = 9(C.19)

where P(ρ, δ) is the pressure of asymmetric nuclear matter and can be expressed asP(ρ, δ) = ρ2

∂E(ρ, δ).
∂ρ

(C.20)

Therefore, the incompressibility of asymmetric nuclear matter at saturation Ksat (δ) ≡ K(ρsat (δ), δ)
where P(ρsat (δ), δ) = 0, is given by
Ksat (δ) = 9ρsat (δ)(∂2 E(ρ, δ)∂ρ2 ρ=ρsat (δ))
J0= K0 + Ksym − 6Lsym −Lsym δ2K0(
J0 Lsym,4+ Ksym,4 − 6Lsym,4 −K0
2
2
9Lsym Jsym Lsym I0 Lsym
−
+
+
K0
K0
2K02
2
2 
J02 Lsym 4
J0 Ksym Lsym 3J0 Lsym δ + O(δ6 ).
+
+
−
2
2
3 
K0
K0
2K0

(C.21)(C.22)

Figures C.1 and C.2 show several typical theoretical calculations for the EOS of asymmetric nuclear matter, namely, the non-relativistic SHF model using SIII, the RMF model using
TM1, the non-relativistic BHF calculation using AV18 interaction with or without three-bodyforce, and the relativistic DBHF calculation. It can be seen that the saturation point changes
depending on the isospin asymmetry.

101

Figure C.1: The EOS of asymmetric nuclear matter from the Skyrme-Hartree-Fock (left panel)and relativistic mean-field (right panel) model calculations. The solid curves correspond to
proton-to-neutron ratios of 0, 0.2, 0.4, 0.6, 0.8, and 1 (from top to bottom). Taken from [150].

Figure C.2: Same as Fig. C.1 from non-relativistic Brueckner-Hartree-Fock calculations
with or without three-body force (left and middle windows) and from the relativistic DiracBrueckner-Hartree-Fock calculations (right window). Taken from [150].

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List of Tables
The previous results of rn , r p and ∆rnp for 208 Pb deuced from several experiments using various hadronic probes (all in fm). Except for X-ray cascadefrom p̄-atom (2), the errors are experimental only. . . . . . . . . . . . . . . .

9

2.1
2.2
2.3

Target thicknesses and enrichments of Ni, and lead isotopes. . . . . . . . . .
Design specifications of the GR spectrometer . . . . . . . . . . . . . . . . .
Specifications of the VDCs. . . . . . . . . . . . . . . . . . . . . . . . . . . .

17
18
20

3.1

The relative ratio R = RWall /RSC of the beam collection eﬃciencies of SCFC
and WallFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

1.1

4.1
4.2

4.3

5.1

5.2

5.3

Best-fit medium-eﬀect parameters a j and b j ( j = σ, ω) in Eq. (4.4), (4.5). . .
204,206,208
Best-fit SOG parameters
Pb.
√ of the neutron density distributions for
The width γ (= 1.70/ 3/2) and position Ri of the ith Gaussian in Eq. (4.7)
are fixed to the values of the charge distribution [54]. The number in theparentheses are the χ2min over the degrees of freedom ν for each isotope. . . .
Fitted SOG parameters Qi of the upper and lower error-envelopes of the neutron density distribution for each lead isotope determined by Eq. (4.11). γ,N, and Ri are the same as in Table 4.2, but the normalization condition is not∑satisfied in the case of the envelopes ( Qi , 1). . . . . . . . . . . . . . . . .
Root-mean-square radii of the charge rch , proton runfold, and neutron rn usedpin this work. The two types of errors for rn , namely, δrnmdl determined by
Eq. (4.11) and δrnstd by Eq. (4.8), are listed (all in fm). . . . . . . . . . . . . .
Obtained values of rn , r p and ∆rnp for 208 Pb compared with the previous results deuced from several experiments using various hadronic probes (all infm). Except for this work and X-ray cascade from p̄-atom (2), the errors areexperimental only. The errors of this work stand for the total errors including
the model uncertainties δrnmdl determined by Eq. (4.11). . . . . . . . . . . . .
Neutron skin thickness for 208 Pb and the nuclear matter properties at saturation density, such as the incompressibility coeﬃcient K0 , the symmetry energy S(ρsat ), its slope L, and the curvature Ksym in various relativistic and nonrelativistic theoretical models. . . . . . . . . . . . . . . . . . . . . . . . . .

113

47

60

61

63

64

68

114

A.1
A.2
A.3
A.4

LIST OF TABLES

Diﬀerential cross sections and analyzing powers for 58 Ni(p, p) at E p = 295 MeV. 83
Diﬀerential cross sections and analyzing powers for 204 Pb(p, p) at E p = 295 MeV. 85
Diﬀerential cross sections and analyzing powers for 206 Pb(p, p) at E p = 295 MeV. 87
Diﬀerential cross sections and analyzing powers for 208 Pb(p, p) at E p = 295 MeV. 89

B.1 Dirac matrix types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

List of Figures
1.1
1.2
1.3

1.4
1.5
1.6
1.7

1.8
1.9
2.1

2.2
2.3
2.4
2.5
2.6

Rendition of the assumed structure and phases of a neutron star. Taken from
Ref. [33]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Neutron star mass versus central density obtained from various types of theEOSs by relativistic hadronic field models, taken from Ref. [34] . . . . . . .
2
The EOS of symmetric nuclear matter and pure neutron matter calculated by
the non-relativistic Brueckner-Hartree-Fock and variational approaches using
AV18 interaction with three body force (dotted red), the Dirac-BruecknerHartree-Fock using Bonn A (solid red), chiral perturbation theory (dashedblack), and several phenomenological mean field models (blue) taken from
Ref. [34]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
208
Neutron skin thickness for Pb as a function of (a) L, (b) Ksym , and (c)Esym (ρ0 ) = S(ρsat ) for 21 sets of Skyrme parameterizations. Taken from Ref. [51]. 5
Correlation between neutron skin for 208 Pb and the symmetry energy a4 =S(ρsat ) at saturation. Take from Ref. [45]. . . . . . . . . . . . . . . . . . . .
5
208Correlation between neutron skin for Pb and the slope of the symmetry energy p0 = ρsat L/3. Take from Ref. [45]. . . . . . . . . . . . . . . . . . . . .
5
Neutron density distributions of 116,118,120,122,124 Sn in the model-independent
form of SOG function extracted from polarized proton elastic scattering at
E p = 295 MeV [70]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7Constraint regions on the S(ρsat )-L plane by several recent analyses. Taken
from Ref. [76]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Constraint regions on the L by several recent analyses. Taken from Ref. [72].
11
Overview of the RCNP ring cyclotron facility with the WS beam line. AVF
and Ring cyclotron operate in a coupled mode and provide polarized proton
beam at E p = 295 MeV to the WS beam line in west experimental hall. . . .
Schematic view of WS beam line . . . . . . . . . . . . . . . . . . . . . . . .
Setup of the BLP (top view) . . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic view of the magnetic spectrometer, “Grand Raiden” . . . . . . . .
Structure of a X plane of the VDCs. . . . . . . . . . . . . . . . . . . . . . .
Wire configurations of the VDCs. The wire numbering orders are shown by
the arrows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

115

14
15
16
19
21
22

LIST OF FIGURES

116

2.7

Wire configurations of the VDCs. The wire numbering orders are shown by
the arrows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.8

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

24

3.1

Three types of beam polarization (a, b, c) and instrumental asymmetry (d) for
each run. Red and blue circles mean |p↑(↓) | (spin-up and -down mode). Beampolarizations (a) assuming asymmetry βLR = 1, (b) using each βLR , and (c)using the averaged βLR fitted by a Gaussian as shown in Fig. 3.2 are comparedwith each other. In this data reduction the type of (c) was used. . . . . . . . .

26

The projection of the instrumental asymmetries βLR in Fig. 3.1(d) was fittedwith a Gaussian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

The spectrum of the geometrical mean P̄ of PS1. proton and deuteron peaks
are identified. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

Colors plot of P̄ of PS1 versus the position spectrum at the focal plane. It isfound that P̄ is independent of the focal plane position. . . . . . . . . . . . .

30

3.5

TOF spectrum between the trigger and the downscaled RF. . . . . . . . . . .

31

3.6

Colors plot of the X position at the focal plane versus TOF. Elastically scattered protons can be clearly identified as shown in the area surrounded by thesolid circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

3.7

Colors plot of P̄ versus TOF of PS1 for PID.

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

32

3.8

(a) TDC spectrum on X1 plane obtained from LeCroy 3377. (b) Drift length(cm) converted from TDC spectrum (a) by using so-called x-t relation. . . . .

33

θ versus X plot at the focal plane of the spectrometer reconstructed from the
VDCs data in the case of 208 Pb at θlab. = 36.0◦ . . . . . . . . . . . . . . . . . .

34

3.10 Typical position spectra for proton elastic scattering from 204,206.208 Pb and 58 Ni
at θlab. = 36.0◦ . Elastic events (g. s. (0+ )) can be distinguished from otherexcited states in each spectrum. . . . . . . . . . . . . . . . . . . . . . . . . .

35

3.2
3.3
3.4

3.9

4.1

4.2

Obtained data of diﬀerential cross sections and analyzing powers for elastic
scattering from 58 Ni and 204,206,208 Pb at E p =295 MeV, whereas the lines are
due to Murdock and Horowitz (solid) [89, 90] and the global Dirac optical potential (dashed) [91]. The dash-dotted lines show the MH model calculationsfor 58 Ni with the realistic nucleon density by an unfolding charge density. . .

42

Proton density distributions of 58 Ni and 204,206,208 Pb by unfolding charge distributions with the new parameter sets of nucleon charge form factor (solid lines).
The blue dotted lines are due to the DH calculations for comparison, while
the green long-dotted, short-dotted, dashed, and dash-dotted lines are due toSkyrme–Hartree–Fock–Bogolyubov calculations with SIII, SKM*, SKP, andSLY4 parametrizations, respectively. . . . . . . . . . . . . . . . . . . . . . .

45

LIST OF FIGURES

4.3

117

The neutron skin thicknesses for even-Ni isotopes calculated by DBHF with
the Groningen (triangles) and Bonn A (circles) interactions, and RMF with the
NL3 [19] (crosses) parametrization. All of them shows that the neutron skin
thickness of 58 Ni is almost zero. Taken from Ref. [8]. . . . . . . . . . . . . . 48
4.4 The neutron skin thicknesses for even-Ni isotopes calculated by Hartree–Fock–
Bogolyubov (HFB) plus pairing corrections using five types of Skyrme forces,that is, SIII [116] (filled circles), SkM* [9] (filled squares), SkP [10] (asterisks), Sly4 [11] (open circles), and SLy5 [11] (open squares) parameterizations. As is the case with relativistic models in Fig. 4.3 nonrelativistic models
show that the neutron skin thickness of 58 Ni is almost zero. . . . . . . . . . . 48
4.5 Calibration of medium-eﬀect parameters by fitting to the experimental datafor 58 Ni. The solid line is the medium-modified RIA calculation with best-fit
parameters. The dashed and dash-dotted lines are from the original MH modelwith DH and realistic nucleon densities. . . . . . . . . . . . . . . . . . . . . 49
4.6 Best fit results for neutron density distributions in 204,206,208 Pb, are shown as
solid lines. The original MH and medium-modified RIA calculations with the
DH nucleon density are also shown by dashed and dash-dotted lines. . . . . . 51
4.7 Results of fitting to the experimental data and extracted neutron density of208
Pb with its standard error-envelope (solid lines). The dashed and dashdotted lines are medium-modified RIA calculations but using the DH nucleon
densities and the 3pG neutron density by Ray [58] respectively. The dottedline in (c) is the realistic proton density. . . . . . . . . . . . . . . . . . . . . 53
4.8 Percent deviation between the experimental data and the medium-modified
RIA calculations of diﬀerential cross section at each angle defined as Eq. (4.9).
The marks are the same definitions as Fig. 4.7. The lines are just for the guide. 54
4.9 Results of fitting to the experimental data and extracted neutron density of204
Pb with its standard error-envelope (solid lines). The dashed lines are mediummodified RIA calculations but using the DH nucleon densities. . . . . . . . . 55
4.10 Results of fitting to the experimental data and extracted neutron density of206
Pb with its standard error-envelope (solid lines). The dashed lines are mediummodified RIA calculations but using the DH nucleon densities. . . . . . . . . 56
4.11 Extracted neutron densities for 204,206,208 Pb with two types of error-envelopes
shown together with DH neutron densities (dotted lines) and point proton densities by unfolding charge densities (dash-dotted lines). The cross-hatchedblue and hatched red error-envelopes were estimated by Eq. (4.11) and Eq. (4.8),
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.12 The radial distributions of neutron surface densities for 204,206,208 Pb (4πr2 ρ(r) fm−1 )with two types of error-envelopes shown together with DH neutron surface
densities (dotted lines) and proton surface densities by unfolding charge densities (dash-dotted lines). The cross-hatched blue and hatched red error-envelopes
were the same as Fig. 4.11. . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

118

LIST OF FIGURES

Systematic behavior of the neutron skin thicknesses for 204,206,208 Pb. The filled
circles are the results of this work with the two types of error bars. The red andblue lines represent the errors due to the experimental only δrnstd and including
the model uncertainties δrnmdl , respectively. The filled squares, triangle, andinverted triangle are from the analysis of proton elastic scattering at 650 MeV[62], X-ray cascade from antiprotonic atoms [67], and PDR strengths from129−132Sn(γ, γ0 ) [68], respectively, with their statistical errors only. The open
triangles, crosses, squares and diamonds show the calculations of relativistic
mean field models (pale blue marks) with NL3 [19], DD-ME2 [22], DD-PC1[117], and FSUGold [23] parameterization while the open circles, squares,
stars and asterisks are from non-relativistic mean field models (pink marks)
with SkM* [9], SkP [10], Sly4 [11], and Skxs20 [73] parameterization. DDME2, FSUGold, and Skxs20 parameterizations have been recently developed
and conform closely to all the extracted neutron skin thicknesses. . . . . . . .
5.2 ∆rnp for 208 Pb (fm) vs K0 (MeV). . . . . . . . . . . . . . . . . . . . . . . . .
5.3 ∆rnp for 208 Pb (fm) vs Ksym (MeV). . . . . . . . . . . . . . . . . . . . . . . .
5.4 ∆rnp for 208 Pb (fm) vs S(ρsat ) (MeV). . . . . . . . . . . . . . . . . . . . . . .
5.5 ∆rnp for 208 Pb (fm) vs the slope coeﬃcient L (MeV). . . . . . . . . . . . . .
5.6 L (MeV) vs S(ρsat ) (MeV). Surrounded areas are the extracted ranges of S(ρsat )and L from ∆rnp for 208 Pb as shown in Figs. 5.4 and 5.5 . . . . . . . . . . . .
5.7 ∆rnp for 204 Pb (fm) vs S(ρsat ) (MeV). . . . . . . . . . . . . . . . . . . . . . .
5.8 ∆rnp for 204 Pb (fm) vs the slope coeﬃcient L (MeV). . . . . . . . . . . . . .
5.9 ∆rnp for 206 Pb (fm) vs S(ρsat ) (MeV). . . . . . . . . . . . . . . . . . . . . . .
5.10 ∆rnp for 206 Pb (fm) vs the slope coeﬃcient L (MeV). . . . . . . . . . . . . .
5.11 All the extracted regions of L vs S(ρsat ) for 204,206,208 Pb correspond to the redlines in Figs. 5.4, 5.5, 5.7, 5.8 5.9, and 5.10 due to the experimental errors of
the extracted ∆rnp only. Black filled circle and solid error bars are the weighted
average of these three regions. . . . . . . . . . . . . . . . . . . . . . . . . .
5.12 Similar to Fig. 5.11, but the extracted regions correspond to the blue lines in
Figs. 5.4, 5.5, 5.7, 5.8 5.9, and 5.10 due to the total errors of the extracted ∆rnp
including the model uncertainties. . . . . . . . . . . . . . . . . . . . . . . .
5.13 Hatched area is the constraint region on the Kτ and ∆rnp for 208 Pb plane withseveral non-relativistic (triangles) and relativistic (circles) mean-field models.
The constraint on ∆rnp is our extracted value (blue and red lines on the y-axis),
whereas on Kτ is taken from Ref. [138]. . . . . . . . . . . . . . . . . . . . .
5.1

B.1 Meson exchange diagrams for the relativistic Love-Franey model. . . . . . .

66
70
70
71
71
72
73
73
73
73

74

75

77
92

C.1 The EOS of asymmetric nuclear matter from the Skyrme-Hartree-Fock (leftpanel) and relativistic mean-field (right panel) model calculations. The solid
curves correspond to proton-to-neutron ratios of 0, 0.2, 0.4, 0.6, 0.8, and 1(from top to bottom). Taken from [150]. . . . . . . . . . . . . . . . . . . . . 101

LIST OF FIGURES

119

C.2 Same as Fig. C.1 from non-relativistic Brueckner-Hartree-Fock calculations
with or without three-body force (left and middle windows) and from therelativistic Dirac-Brueckner-Hartree-Fock calculations (right window). Taken
from [150]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101