Proton-induced knockout reactions with polarized and unpolarized beams

Accepted Manuscript Proton-induced knockout reactions with polarized and unpolarized beams T. Wakasa, K. Ogata, T. Noro

PII: DOI: Reference:

S0146-6410(17)30055-8 http://dx.doi.org/10.1016/j.ppnp.2017.06.002 JPPNP 3647

To appear in:

Progress in Particle and Nuclear Physics

Please cite this article as: T. Wakasa, K. Ogata, T. Noro, Proton-induced knockout reactions with polarized and unpolarized beams, Progress in Particle and Nuclear Physics (2017), http://dx.doi.org/10.1016/j.ppnp.2017.06.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Proton-induced knockout reactions with polarized and unpolarized beams T. Wakasaa , K. Ogatab , T. Noroa,∗ a Department

of Physics, Kyushu University, Nishi-ku, Fukuoka 819-0395, Japan Center for Nuclear Physics, Ibaraki, Osaka 567-0047, Japan

b Research

Abstract Proton-induced knockout reactions provide a direct means of studying the single particle or cluster structures of target nuclei. In addition, these knockout reactions are expected to play a unique role in investigations of the effects of the nuclear medium on nucleonnucleon interactions as well as the properties of nucleons and mesons. However, due to the nature of hadron probes, these reactions can suffer significant disturbances from the nuclear surroundings and the quantitative theoretical treatment of such processes can also be challenging. In this article, we review the experimental and theoretical progress in this field, particularly focusing on the use of these reactions as a spectroscopic tool and as a way to examine the medium modification of nucleon-nucleon interactions. With regard to the former aspect, the review presents a semi-quantitative evaluation of these reactions based on existing experimental data. In terms of the latter point, we introduce a significant body of evidence that suggests, although does not conclusively prove, the existence of medium effects. In addition, this paper also provides information and comments on other related subjects. Keywords: Spectroscopic factors, Nuclear medium effects, Nucleon knockout reaction PACS: 21.10.Jx, 21.30.Fe, 25.40.-h, 24.70.+s

∗ Corresponding

author Email address: [email protected] (T. Noro) Preprint submitted to Journal of Prog. Part. Nucl. Phys.

March 23, 2017

Contents 1 Introduction 2 Overview 2.1 Proton induced nucleon knockout 2.2 Spectroscopic factors . . . . . . . 2.3 j-dependence . . . . . . . . . . . 2.4 Medium effects on Ay . . . . . . 2.5 Isospin dependence . . . . . . . .

3 reactions . . . . . . . . . . . . . . . . . . . . . . . .

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3 Theoretical Framework 14 3.1 Nonrelativistic framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Relativistic framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.3 Theoretical activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4 Experimental Methods 4.1 Spectrometer and telescope at TRIUMF 4.2 Two-arm spectrometers at RCNP . . . . 4.3 Two-arm spectrometers at PNPI . . . . 4.4 Spectrometer and telescope at iThemba

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5 Spectroscopic Studies 5.1 Kinematics of (p, 2N) reactions . . . . . . . . . . . . . . . . . . . . . . . 5.2 Experimental studies at 505 and 392 MeV . . . . . . . . . . . . . . . . 5.3 Worldwide data and a comparison with systematic DWIA calculations . 5.4 Multipole decomposition analysis . . . . . . . . . . . . . . . . . . . . . . 5.5 Uncertainties in extracting S-factors and comparison with (e, e′ p) results

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30 30 32 35 36 36 38 38 45 50 53

6 Nuclear Medium Effects 58 6.1 Definition of effective mean density for (p, 2N) reactions . . . . . . . . . . 58 6.2 g-matrix and correlation studies and (p,2N) reactions . . . . . . . . . . . . 60 6.3 Analyzing powers for the 1s1/2 knockout . . . . . . . . . . . . . . . . . . . 61 6.4 Analyzing powers and induced polarizations for the 1s1/2 knockout under zero recoil conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 6.5 Polarization transfer coefficients and spin correlation parameters . . . . . 65 6.6 Analyzing powers for reactions other than the 1s1/2 knockout . . . . . . . 66 7 Supplementary Subjects 69 7.1 Comparison of (p, 2p) and (p, np) reactions . . . . . . . . . . . . . . . . . 69 7.2 Rescattering process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 7.3 Spectroscopic factors from the (p, p′ α) reaction . . . . . . . . . . . . . . . 76 8 Summary and Outlook

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Appendix Transition A matrix density and effective mean density

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2

1. Introduction

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In essence, a proton induced knockout reaction is a nuclear reaction in which an incident proton interacts with either a nucleon or a nuclear cluster in a target nucleus and knocks this entity out of the nucleus, generating a one-hole or a cluster-hole state. This process, in particular the one nucleon knockout reaction, is the most dominant reaction at intermediate (200 to 1000 MeV) energies. Even though the initial and final state interactions contribute significantly to this process, the above simplified definition is generally satisfactory. Therefore, proton induced knockout reactions, as well as other types of knockout reactions involving incident electrons, provide a uniquely direct means of investigating the single particle structure of a target nucleus. From the late 1950s to the early 1970s, this reaction was intensively studied in order to investigate the nuclear shell structure, using various accelerators around the world. In this type of work, the angular momenta of hole states are assigned as functions of separation energies for many kinds of target nuclei, from light to medium heavy, and the energies and widths from 1s to 2s states are systematically determined. The prior work in this field has been summarized in two review papers by Jacob and Maris in 1966 and 1973 [1, 2], although similar studies continued even into the 1990s, at PNPI, Gatchina, and this survey was extended up to 208 Pb [3–6]. Beginning in the mid 1970s, considerable progress was made in this field. Intense polarized proton beams with high duty-cycles became available, and the ability to analyze (p, 2p) reactions, in addition to differential cross sections, was greatly enhanced. In addition, (p, pn) measurements were also performed. One of the most outstanding results related to polarization measurement was the discovery of the j-dependence of (p, 2p) reactions, an effect that was first proposed by Jacob et al. [7] and later experimentally confirmed by Kitching et al. [8]. This discovery resulted in these reactions becoming very unique tools in nuclear spectroscopy, which is not replaceable with the other knockout reaction, (e, e′ p). This prior research was reviewed in a paper by Kitching et al. in 1985 [9]. The (p, 2N) reaction is a knockout reaction based on a nucleon-nucleon interaction. With regard to utilizing this process in nuclear studies, the reaction has two different applications; as a tool to study the single particle properties of a target nucleus and as a method of assessing in-medium nucleon-nucleon interactions. In the case of the former application, significant progress in experimental and theoretical studies followed the review by Kitching et al. and this has made it possible to evaluate these reactions quantitatively. Recently, unstable nuclei generated by nuclear fragmentation processes have been used as incident beams to study the structures and properties of these nuclei through nuclear reactions. The knockout reactions that occur in inverse kinematics are expected to play a crucial role in investigating the structures of those unstable nuclei, with the evaluation of these reactions serving as a spectroscopic tool. In the case of the latter application, a meaningful comparison between (p, 2N) and the nucleon-nucleon interactions in free space can be realized by polarization observables, which are ratios of cross sections and thus less sensitive to the initial and final state interactions. Even though the results are still not conclusive, there is much evidence suggesting that medium modification of the interactions has been observed. Essentially, this paper summarizes the work performed since the previous review by Kitching et al., paying special attention to the above two specific subjects because of their importance. In the incident energy 3

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region significantly higher than 1GeV, (p,2N) reactions have been used in investigating the color transparency phenomena but those works are excluded from this work. Refer a recent review on this subject by Dutta and coworkers [10]. At the same time, this work does not cover the (p,2N) studies in inverse kinematics, which started recently [11, 12]. This paper is organized as follows. In Section 2, the basic concepts and current knowledge related to proton-induced knockout reactions are provided as an overview. Section 3 presents the theoretical formalism associated with distorted wave impulse approximations in the non-relativistic and relativistic frameworks, while Section 4 describes the experimental methods, including the instrumentation employed at several laboratories. Subsequently, the review focuses on spectroscopic studies and medium effect investigations, in Section 5 and Section 6, respectively. Comments regarding several related subjects are provided in Section 7 and a summary and future outlook are given in Section 8. 2. Overview

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2.1. Proton induced nucleon knockout reactions Notation and coordinate systems. In the following section, we consider a proton-induced nucleon knockout process with the kinematics shown in Fig. 1. Here, the indices 0 and 3 denote the incoming proton and the nucleon to be knocked out from the target nucleus, respectively, while the indices 1 and 2 correspond to the outgoing nucleons in the forward and backward directions, respectively. The initial and residual nuclei are referred as A and B (or i and f ), respectively. In the nucleon-nucleus laboratory frame, we denote the scattering angles by θi and the energies and momenta by Ei and pi , respectively. The separation energy of the knocked-out nucleon for a specific final state of the nucleus is given by ES = Ei − Ef = T0 − (T1 + T2 + TB ) ,

(2.1)

where Ti is the kinetic energy of particle i. Based on the conservation of energy and momentum, we can then write E0 + MA = E1 + E2 + EB , p0 = p1 + p2 + pB ,

(2.2a) (2.2b)

with EB = MB + Ex + TB ,

70

(2.3)

where Mi is the mass of particle i in its ground state and Ex is the excitation energy of the residual nucleus. Note that Ex equals the difference between the separation energies of the ejected nucleon and the least bound nucleon. The coordinate system conventionally used is defined by the sets of orthogonal vectors ˆ= n and

p0 × p1 , |p0 × p1 |

ˆ′ = n ˆ , n

lˆ = pˆ0 ,

lˆ′ = pˆ1 ,

ˆ × lˆ , sˆ = n

ˆ ′ × lˆ′ . sˆ′ = n

(2.4)

(2.5) ′

′

′

These are shown in Fig. 2 and are denoted by n, l, s, n , l , and s . 4

Figure 1: Kinematics of the proton-induced nucleon knockout process.

sˆ′ ˆ′ l  J ] 3  Je  q   ˆ′  n 

 3  AK  A   A qlab ] A θ1   -A p0 p1

sˆ 6 lˆ e q -

ˆ n

Figure 2: Specification of directions in the laboratory frame.

Polarization observables. A complete set of polarization observables in the laboratory frame, that is, the analyzing power Ay , induced polarization P , and polarization transfer coefficients Dij (i=s′ , n′ , l′ , j=s, n, l), relates the three orthogonal components of the polarization of the outgoing nucleon in the forward direction p′ = (p′s′ , p′n′ , p′l′ ) to those of the incident nucleon polarization p = (ps , pn , pl ) through  ′       ps′ Ds ′ s 0 Ds ′ l ps 0 1  p′n′  =  0 Dn ′ n 0   pn  +  P  . (2.6) 1 + p n Ay p′l′ Dl ′ s 0 Dl ′ l pl 0

75

(p, 2N) measurements. In order to summarize the essential physics of the nucleon knockout reaction, it is important to discuss an idealized situation in which the incoming proton interacts with only one nucleon in a shell model orbit. In this case, the residual nucleus has a hole in the shell from which the nucleon is lost, and the separation energy equals the single-particle energy of this hole state. Furthermore, the nuclear recoil momentum pB equals −p3 ; that is, the negative value of the momentum that the knocked-out nucleon had in the target nucleus. Thus the separation energy spectrum and the recoil 5

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momentum distribution of the residual nucleus directly correspond to the single-particle energies and momentum distributions of the single-particle states, respectively. It should also be noted that the momentum distribution is a useful means of identifying the angular momentum of the single-particle state. The recoil momentum distribution can be obtained by determining the effects of the momenta p1 and p2 on the yields of nucleon pairs generated by quasi-elastic scattering. 2.2. Spectroscopic factors Nucleon knockout reactions. In the plane-wave impulse approximation (PWIA), the nucleon knockout (a, aN) cross section (a = e or p) can be expressed as [13] d4 σ = CK σaN S(ES , p3 ) , dΩ1 dΩ2 dE1 dE2

(2.7)

where CK is a kinematical factor, σaN is the elementally off-shell a-N (electron-nucleon or proton-nucleon) cross section, and S(ES , p3 ) is the spectral function representing the probability of finding a nucleon with separation energy ES and momentum p3 in the ˜ the spectral function can be target nucleus. For a discrete state transition with ES = E, rewritten as the momentum distribution ρ(p3 ) multiplied by a delta function for energy conservation: ˜ . S(ES , p3 ) = ρ(p3 )δ(ES − E)

90

(2.8)

This spectral function cannot actually be determined experimentally since the incoming and outgoing nucleons interact strongly with the target and residual nuclei, respectively; a phenomenon termed the distortion effect. Furthermore, the factorization of the differential cross section into the product of the elementally cross section and the spectral function is not possible because of the distortions of the nucleon waves. When such distortions are taken into account, the cross section can be evaluated theoretically using the distorted-wave impulse approximation (DWIA) for a given momentum distribution ρth (p3 ) with ∫ ρth (p3 )d3 p3 = 1 . (2.9) The experimental spectroscopic factor (S-factor) S exp can then be defined as [14] S exp =

σ exp , σ th

(2.10)

where σ exp and σ th are the experimental and theoretical cross sections, respectively. S-factors by (e, e′ p). In the independent particle shell model (IPSM), an orbit with a given total angular momentum j that is fully occupied in the target nucleus has an associated spectroscopic factor of SjIPSM = 2j + 1, since this factor denotes the number of nucleons in the single particle state. Experimentally, fragments of the single particle state have been observed as the result of residual interactions [15]. Therefore, the SjIPSM value gives the sum-rule limit for the experimental Sjexp values as ∑ exp Sj ≤ SjIPSM = 2j + 1 . (2.11) 6

Figure 3: Summed spectroscopic strengths from the (e, e′ p) reaction relative to the IPSM limits for valence orbitals as a function of the target mass number. These data were obtained at the NIKHEF facility [15].

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∑ exp Figure 3 shows the summed spectroscopic strengths Sj relative to the IPSM limit SjIPSM for various nuclei from 7 Li to 208 Pb. Here it is evident that the experimental strengths are substantially smaller than the sum-rule limits obtained from the IPSM. This discrepancy can be partly attributed to the fragmentation of some portion of the strengths to higher excitation energies. However, the primary reasons for the reduction for Sjexp are short- and long-range correlations [16]. The short-range and tensor correlations couple low-lying and high-lying states, which reduces the single-particle strength by approximately 15%, while long-range correlations provide another 20% reduction by coupling the single-particle motion to low-lying states. Consequently, the combined effects of long- and short-range correlations result in an approximately 35% reduction of the single-particle orbits, as observed in the (e, e′ p) reaction. Occupation probability and quasi-hole strength. It should be noted that the S-factor relative to the IPSM limit is not exactly equivalent to the occupation probability for a single-particle orbit [17–19]. The spectral function S(ES , p3 ) is theoretically given by ∑ S(ES , p3 ) = |⟨i|ap3 |0⟩|2 δ(ES − Ei + E0 ) , (2.12) i

where ap3 is the annihilation operator, |0⟩ is the ground state of the target nucleus A, |i⟩ are eigenstates of the residual nucleus B, and E0 and Ei are the corresponding energies. The greatest contributions to S(ES , p3 ) at relatively small p3 values are from the intermediate |i⟩ states that are close to the one-hole states of the target nucleus. In the (e, e′ p) data, several discrete peaks corresponding to one-hole states are observed at 7

a specific momentum p3 . However, it should be noted that other intermediate states corresponding to two or more nucleon emission processes also contribute to S(ES , p3 ), and these contributions are associated with a wide energy range. It follows that S(ES , p3 ) in the (e, e′ p) reaction provides the quasi-hole strength, the value of which is given by Z = |⟨h|ap3 |0⟩|2 ,

(2.13)

where |h⟩ is the quasi-hole state. In addition, the occupation probability is equal to the energy integral of the full S(ES , p3 ), with the result that ∑ n= |⟨i|ap3 |0⟩|2 = ⟨0|a†p3 ap3 |0⟩ . (2.14) i

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The correlations generated by the inter-nucleon interactions are reflected by the deviation of Z from unity, such that the value of Z provides the probability that a quasi-particle is a physical particle. Figure 4 plots the occupation probability n(ϵ) and quasi-hole strength Z(ϵ) values for nuclear matter as functions of energy ϵ relative to the Fermi energy ϵF [18]. Here, the relevant Fermi momentum is pF = 1.33 fm−1 . The correlated nucleons (the shaded region) corresponding to two or more nucleon emissions evidently account for all of the n(ϵ > ϵF ), and also contribute in part to the n(ϵ < ϵF ). Therefore, Z is significantly less than n, particularly near the Fermi energy ϵF . In Fig. 4, the 208 Pb(e, e′ p) results relative to the IPSM limits are also shown, and it can be seen that the data agree reasonably well with the predictions of ZPb for 208 Pb. The spectroscopic factors discussed in Section 5 are primarily associated with states near the Fermi energy, and thus they would be 60 to 70% of the IPSM limit. The (d, 3 He) and (p, 2N) reactions. The spectroscopic factors Sjexp have also been determined using the proton-pickup (d, 3 He) reaction. Table 1 presents comparison of the ∑ aexp results for the (e, e′ p) and (d, 3 He) reactions, and the ratio of the Sj values obtained from the (d, 3 He) and (e, e′ p) reactions are also shown in Fig. 5. It is apparent that the Sjexp values obtained from (e, e′ p) and (d, 3 He) reactions are mutually consistent. The nucleon-knockout (p, 2N) reaction is also a useful tool for the assessment of single particle properties in nuclei, including Sjexp , and this reaction can also be applied to unstable nuclei in inverse kinematics [12]. Establishing the (p, 2N) reaction as a spectroscopic tool is therefore given priority as an important subject in this paper, and this topic is reviewed in Section 5. 2.3. j-dependence

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A significant advantage of the (p, pN) reaction compared with the (e, e′ N) reaction is that spin-dependent information can be obtained by employing a polarized proton beam and/or by assessing the polarization of the knocked-out nucleon. Here, as an example, we provide a brief quantitative discussion of the j-dependence of the analyzing power Ay for the (⃗ p, pN) reaction, which is useful with regard to assigning a j value to a single-particle state.

8

Table 1: Spectroscopic factors deduced from the (e, e′ p) and (d, 3 He) reactions.

Target 12 C

16

O

31 40

P Ca

51

V

90

Zr

142

Nd

208

Pb

Ex (MeV) jπ 0.000 3/2− − 5.020 ∑ exp 3/2 S3/2− = 0.000 1/2− 6.320 3/2− 0.000 1/2+ 0.000 3/2+ 2.522 1/2+ 0.000 7/2− 1.554 7/2− 2.675 7/2− − 3.199 ∑ exp 7/2 S7/2− = 0.000 1/2− 1.507 3/2− 0.000 5/2+ 0.145 7/2+ 0.000 1/2+ 0.350 3/2+ 1.350 11/2−

Sjexp (e, e′ p) 1.72(11) 0.20(2) 1.92(11) 1.27(13) 2.25(22) 0.40 (3) 2.58(19) 1.03(7) 0.37(3) 0.16(2) 0.33(3) 0.49(4) 1.33(12) 0.72(7) 1.86(14) 1.39(26) 3.14(53) 0.98(9) 2.31(22) 6.85(68)

9

Ref. [14]

[20, 21] [22] [20, 23] [24]

[24] [25] [26]

Sjexp (d, 3 He) 1.72 0.11 1.83 1.02 1.94 0.36 2.30 1.03 0.30 0.15 0.26 0.39 1.10 0.60 1.20 1.25 3.79 1.5 2.2 5.4

Ref. [20]

[20] [20] [20] [20]

[20] [20] [20]

Figure 4: Occupation probabilities n(ϵ) and quasi-hole strengths Z(ϵ) for nuclear matter (N.M.) at pF = 1.33 fm−1 [18]. The shaded area represents the contributions from the correlated nucleons to n(ϵ). The experimental data for 208 Pb are indicated by empty squares and the relevant theoretical predictions for ZPb (ϵ) are shown by the thick solid line.

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Effective polarization. Let us consider a coplanar, quasi-free (p, 2p) reaction in an asymmetrical (θ1 < θ2 ) geometry, as shown in Fig. 6. The energy dependence of the absorption (distortion) effects can be evaluated from the energy dependence of the nucleon-nucleon (NN) total cross sections. Figure 7 plots the NN total cross section as a function of incident energy. Here, the dotted and dashed curves indicate the elastic and inelastic total cross sections, respectively, while the solid curve represents their sum. In the case of the asymmetric (p, 2p) reaction at T0 values of 200 to 400 MeV, the nuclear absorption effects at T2 are significantly greater than those at T0 and T1 . It should also be noted that, in Fig. 6, the knocked-out protons at T2 from the upper side of the nucleus would be expected to pass through more nuclear matter than those from the lower side, and thus the observed (p, 2p) events primarily originate from the lower side of the nucleus. As discussed in Section 2.1, we can select p3 by measuring p1 and p2 . If the p3 direction is selected, as indicated in Fig. 6, the protons in the target nucleus with clockwise angular momenta will make a greater contribution than those with counter-clockwise momenta. In addition, if the proton is in the j> = l + 1/2 (j< = l − 1/2) state, its spin will be predominantly down (up). It follows that, in the target nuclei, the knocked-out protons will be effectively polarized in opposite directions for the j> = l + 1/2 and j< = l − 1/2 single particle states. A more quantitative relationship between the effective polarizations p3;j> and p3;j< for the j> = l + 1/2 and j< = l − 1/2 states has been reported as [27, 28] p3;j> ≃ −

l p3;j< . l+1

(2.15) 10

Figure 5: Ratio of the summed spectroscopic strengths deduced from the (d, 3 He) reaction relative to the relevant strengths obtained from the (e, e′ p) reaction as a function of the target mass number. The band at 1.0 ± 0.2 is intended solely as a visual guide.

Analyzing power. The asymmetry value A is the product of the polarization p0 of the incoming proton beam and the analyzing power Ay , and can be obtained by measuring the asymmetry of the cross sections A = p0 Ay =

σ up − σ down , σ up + σ down

(2.16)

where σ up and σ down represent the cross sections for the up- and down-spin proton beams, respectively. The free NN cross section σ free can be written as [ ] free σ free = σ0free 1 + p0 Afree + p3 P free + p0 p3 Cnn , (2.17) y

where p3 is the polarization of the knocked-out nucleon, σ0free is the unpolarized cross free section, P free (= Afree y ) is the induced polarization, and Cnn is the spin-correlation coefup ficient. Inserting Eq. (2.17) into Eq. (2.16) with p0 = −pdown = p0 and P free = Afree y , 0 we obtain A = p0 Ay = p0

free Afree + p3 Cnn y , 1 + p3 P free

(2.18)

and the effective polarization p3 can also be obtained as p3 =

Ay − Afree y . free Cnn − Ay Afree y

(2.19) 11

Figure 6: The quantitative explanation for the effective polarization associated with the (p, 2N) reaction.

Note that Ay = Afree if p3 = 0. y As discussed above, the target nucleon in the nucleus is effectively polarized, and thus p3 might be not zero. In the case of Fig. 6, the effective polarization p3 for the proton free values are in the j> = l + 1/2 (j< = l − 1/2) state is ≃ −1 (≃ +1). Since typical Cnn ≃ 1 as shown in Fig. 8, the corresponding analyzing power Ay would be Ay;j> ≃ −1, 155

Ay;j< ≃ +1.

(2.20)

This relationship for this extreme case clearly demonstrates that the analyzing power data is applicable to the definitive assignment of a j value to a single particle state. A more quantitative relationship between Ay;j> and Ay;j< can be deduced by considering that in Eq. (2.15). We can write Eq. (2.19) as p3;j≷ =

Ay;j≷ − Afree y

free − A free Cnn y;j≷ Ay

.

(2.21)

free free ≲ 0.3, ≃ 0.9, Ay;j≷ ≃ 0.4, and Afree Since Cnn ≫ Ay;j≷ Afree for typical values of Cnn y y we can approximately rewrite Eq. (2.21) as

p3;j≷ ≃

Ay;j≷ − Afree y free Cnn

.

(2.22)

Based on Eqs. (2.15) and (2.22), we obtain Ay;j> − Afree l y . ≃− Ay;j< − Afree l+1 y

160

(2.23)

This relationship indicates that the Ay;j> and Ay;j< data should be at opposite sides of the corresponding Afree value with a difference ratio of l/(l + 1). y Even though the experimental data do not necessarily follow the simple relationship in Eq. (2.23), the analyzing power data are still very useful for distinguishing between j> and j< states, as discussed in Section 5. 12

Figure 7: Total (solid line), elastic (dotted), and inelastic (dashed) cross sections for pp collisions as functions of the laboratory kinetic energy.

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2.4. Medium effects on Ay Throughout the 1980s, extensive work was performed involving measurements of inclusive (p, p′ ) reactions, and it was found that the analyzing power for the quasi-elastic process was significantly suppressed compared with the relevant value for free NN scattering in free space [29–33]. The top panel of Fig. 9 shows the results for the 12 C(p, p′ ) reaction at Tp = 800 MeV [29, 34]. Relativistic effects [29, 30] can provide a reasonable explanation for the observed reduction of Ay , although this reduction might also be due to the medium effects on NN interactions in a nuclear medium. It is natural to consider that a higher nuclear density would result in a greater medium effect, and thus NN scattering with a more deeply bound nucleon would be more affected by the medium. In this regard, the (p, pN) reaction is a promising tool for the investigation of the density dependence of the Ay reduction, by allowing measurements of the Ay values for different single particle orbits. In fact, assessments of Ay for the 16 O(p, 2p) reaction at T0 = 504 MeV at TRIUMF showed that the Ay values were suppressed to a greater extent in the case of the deeply-bound 1s1/2 knockout [35, 36]. In contrast to the (p, p′ ) reaction, the Ay data for the inclusive (p, n) reaction are not suppressed compared to the free NN value [37–40]. This is demonstrated by the bottom panel of Fig. 9, which presents the results obtained from the 12 C(p, n) reaction at Tp = 494 MeV [37]. The different behavior of the Ay values for (p, p′ ) and (p, n) reactions suggests that the medium effects differ between pp and pn scattering, such that these effects on the NN interactions might be isospin dependent. For this reason, the measurement of Ay for the exclusive (p, pn) reaction is extremely helpful when studying the variations in the medium effects with changes in both the density and isospin. Based on all of the above, this review especially considers the medium effects on Ay as well as on Dij , and recent data and analyses for both (p, 2p) and (p, pn) reactions are presented in Section 6. 13

Figure 8: Spin correlation coefficients Cnn for pp scattering at θlab = 30◦ as a function of the laboratory kinetic energy.

2.5. Isospin dependence 190

195

200

205

One of the advantages of the (p, 2N) reaction compared with the (e, e′ p) reaction is that we can compare the (p, 2p) and (p, pn) data for N = Z target nuclei employing the same kinematics. In this case, the spectroscopic factors and the distortion effects are expected to be similar between these two reactions, and the R = σ(p, pn)/σ(p, 2p) cross section ratio should be approximately equal to the ratio of the corresponding free pn and pp cross sections [41]. The in-medium NN cross sections have been derived based on the Bonn NN potential and the Dirac-Brueckner approach for nuclear matter [42, 43], and it has been shown that the medium effects are significantly different between pp and pn scattering. Therefore, it is to be expected that the cross section ratio could be a useful means of assessing the medium effects on the NN interaction. The first experiments meant to determine the R value were performed at TRIUMF using a 400 MeV proton beam, employing 2 H and 12 C [41]. The R/Rfree double ratios for 2 H were found to be 1.05 ± 0.07 and 1.13 ± 0.14 depending on the kinematical conditions, results that were consistent with unity and therefore indicated the lack of a medium effect in the case of 2 H. On the contrary, the same ratios for 12 C were 1.53 ± 0.14 and 1.71 ± 0.23, suggesting possible medium effects when targeting 12 C. The subject of medium effects on the cross section will be reviewed in more detail in Section 7.1, in which recent data are also presented. 3. Theoretical Framework 3.1. Nonrelativistic framework

210

The distorted-wave impulse approximation (DWIA) has successfully been applied to the description of proton-induced knockout reactions. There are two major approaches 14

Figure 9: Analyzing powers for the 12 C(p, p′ ) reaction at Tp = 800 MeV (top) [29, 34] and for the 12 C(p, n) reaction at T = 494 MeV (bottom) [37]. The solid curves indicate the analyzing powers for p the relevant free NN scatterings.

15

to the DWIA; nonrelativistic and relativistic. In this section we briefly review the nonrelativistic DWIA (NRDWIA) formalism, essentially following Ref. [44], as applied to the spectroscopic studies in Section 5. Let us first consider a reaction process triggered by a proton (particle 0) impinging on a target nucleus (A). Following the nucleon-nucleon (NN) collision inside the nucleus, two nucleons (particles 1 and 2) are emitted, one of which is particle 0 while the other is a nucleon bound in A in the initial channel. The residual nucleus (B) will also be present in the final channel as the third particle. B will be in either the ground state or an excited state, either of which can be specified by the appropriate choice of the kinematics of particles 1 and 2. This process can be reasonably described with a three-body model consisting of particles 1, 2 and B, with the understanding that the eigenenergy (or mass) of B is specified. The excitation/de-excitation of B during the scattering process is not explicitly described, although the flux loss due to absorption is taken into account by using optical potentials for particles 0, 1 and 2. Based on these assumptions, the post form transition matrix is given by ⟨ (1) (1) (2) (2) Tµ1 µ2 µB µ0 µA = ϕK 1 (R0 ) η1/2,µ1 ζ1/2,ν1 ϕk2B (R2 ) η1/2,µ2 ζ1/2,ν2 ΦIB µB ,tB νB (ϵB , ξB ) ⟩ ˆ (+) ϕK (R0 ) η (0) ζ (0) ΦI µ ,t ν (ϵA , ξA ) , × |Vβ | Ω (3.1) 0 A A B B 1/2,µ0 1/2,ν0

where µi (νi ) with i = 0, 1, 2, A or B is the spin (isospin) projection of particle i, the ϕ (i) (i) are the plane waves, and η1/2,µ (ζ1/2,ν ) is the spin (isospin) 1/2 wave function. For the sake of simplicity, the suffixes to T for the isospin projections have been dropped. The definition of the coordinates is given in Fig. 10. The momentum (in units of ℏ) of particle i in the three-body center-of-mass (c.m.) frame, which we term the G frame, is denoted by K i and, unless otherwise stated, relativistic kinematics are adopted. ΦIA µA (ΦIB µB ) is the wave function of A (B) having total spin and isospin IA (IB ) and tA (tB ), respectively. ξC (C = A or B) is the internal coordinates set for C, ϵC represents its eigenenergy, and ˆ (+) is the Møller operator. The Coulomb interaction proportional to 1/R is treated Ω in the standard manner, employing Coulomb wave functions in the asymptotic region, although this is omitted in the formulae below for the sake of simplicity. In addition, we do not explicitly argue the antisymmetrization of nucleons. The interaction Vβ is given by Vβ = U1B + U2B + τ12 ,

215

220

(3.2)

where τ12 is the effective interaction between particles 1 and 2 as defined in multiple scattering theory [45]. It is extremely challenging to obtain an explicit form of τ12 , and so an NN effective interaction in free space, tfree , is adopted as τ12 . tfree is assumed to be a local interaction consisting of direct and exchange terms, although not shown explicitly, while U1B (U2B ) is the distorting potential for particle 1 (2) by B. In practice, an optical potential that is determined so as to describe the elastic scattering observables is adopted as UiB (i = 1 or 2); UiB is assumed to consist of a central part and a spin-orbit part. In DWIA it is further assumed that A remains in the ground state in the initial channel; that is, prior to the transition by Vβ . In this case, we have (0) ˆ (+) ϕ0,K (R0 ) η (0) ζ (0) ΦI µ (ξA ) → χ(+) Ω 0 A A 0,K 0 ,µ0 (R0 ) ζ1/2,ν0 ΦIA µA (ϵA , ξA ) , (3.3) 1/2,µ0 1/2,ν0 16

Figure 10: Definition of the coordinates.

where the distorted wave for particle 0 satisfies ( ) ℏ2 ℏ2 (+) − ∇2R0 + U0A − K02 χ0,K 0 ,µ0 (R0 ) = 0. 2M0A 2M0A

(3.4)

The distorting potential for particle 0 by A is denoted as U0A , and Mij is the reduced energy of particles i and j: Mij =

225

Ei Ej , Ei + Ej

(3.5)

where Ek is the total (relativistic) energy of particle k. In some instances, U0A in Eq. (3.4) is replaced with an interaction between particle 0 and nucleus B embedded in A, meaning that the contribution of τ12 is subtracted. This is believed to correspond to a first-order calculation with respect to τ12 . In contrast, considering the post-form representation of the transition matrix, Eq. (3.1), there is no clear reason for eliminating (+) τ12 in the calculation of χ0,K 0 ,µ0 . Rather, τ12 should be omitted in the calculation of the final-state wave function, as outlined below. Fortunately, in most cases the results of the two calculations are very close, such that the difference does not exceed the uncertainty in the optical potential itself. As the result of spin-orbit interaction, the spin direction of particle 0 may change (+) during the scattering process. χ0,K 0 ,µ0 can then be written in the following form ∑ (+) (+) (0) χ0,K 0 ,µ0 (R0 ) = χ0,K 0 ,µ′ µ0 (R0 ) η1/2,µ′ , (3.6) µ′0

0

0

(+)

and the distorted-wave matrix χ0,K 0 ,µ′ µ0 is obtained under the condition of 0

(+)

χ0,K 0 ,µ′ µ0 (R0 ) → ϕK 0 (R0 ) δµ′0 µ0 + fµ′0 µ0 (Ω0 ) 0

17

exp (iK0 R0 ) , R0

(R0 ≫ RN ) , (3.7)

230

where RN is the range of the distorting potential, fµ′0 µ0 is the scattering amplitude and Ω0 (+)

is the solid angle of particle 0. The superscript (+) on χ0,K 0 ,µ′ µ0 denotes this outgoing 0 boundary condition. The product of the plane waves in the final channel can be rewritten as [ ( )] 1 m2 ϕK 1 (R0 ) ϕk2B (R2 ) = exp iK · − R + R 1 2 1 3 mB + m2 (2π) ) ] [ ( m2 × exp i K 2 + K 1 · R2 mB + m2 1 = (3.8) 3 exp (iK 1 · R1 ) exp (iK 2 · R2 ) , (2π) where mi is the rest mass of particle i. This free wave is subsequently distorted out of Vβ by U1B and U2B . As noted, τ12 does not contribute to the distortion in the post-form representation. The resulting distorted wave is obtained as a solution of (−)

∗ ∗ (TR0 + TR2 + U1B + U2B − Ef ) ΞK 1 ,K 2 ,µ1 ,µ2 (R1 , R2 ) = 0,

(3.9)

where ℏ2 ∇2 , 2M0A R0 ℏ2 =− ∇2 , 2M2B R2

TR0 = −

(3.10a)

TR2

(3.10b)

and Ef =

ℏ2 ℏ2 K12 + k2 . 2M0A 2M2B 2B

(3.11)

Rather than applying the Faddeev theory [46] to solve this three-body scattering problem, the following approximation to the kinetic energy operators and Ef has been widely employed: TR0 + TR2 − Ef ≈ −

ℏ2 ℏ2 ℏ2 ℏ2 ∇2R1 − K12 − ∇2R2 − K 2. 2M1B 2M1B 2M2B 2M2B 2

(3.12)

One may clearly interpret this approximation at the limit of nonrelativistic kinematics as: ℏ2 ℏ2 ℏ2 ℏ2 ℏ2 2 ∇2R1 − ∇R1 · ∇R2 − ∇2R2 − K12 − k 2M1B mB 2M2B 2M0A 2M2B 2B ℏ2 ℏ2 ℏ2 ℏ2 ℏ2 2 ∇2R1 + K1 · K2 − ∇2R2 − K12 − k ≈− 2M1B mB 2M2B 2M0A 2M2B 2B ℏ2 ℏ2 ℏ2 ℏ2 ∇2R1 − K12 − ∇2R2 − K 2, =− (3.13) 2M1B 2M1B 2M2B 2M2B 2

TR0 + TR2 − Ef → −

where Mif is the reduced mass of particles i and j. This approximation replaces the coupling operator ∇R1 · ∇R2 with −K 1 · K 2 . The last expression of Eq. (3.13) is the 18

235

240

nonrelativistic form of the r.h.s. of Eq. (3.12). The uncertainty arising from the use of Eq. (3.12) is roughly on the order of 1/A, where A is the mass number of A, and so the result becomes completely accurate in the case that A is infinitely heavy. It should be noted, however, that Eq. (3.12) gives the proper asymptotics of the three-body scattering wave in the plane-wave limit (that is, Eq. (3.8)), as emphasized in Ref. [47]. This holds true irrespective of the value of A. In combination with Eq. (3.12), the Schr¨ odinger equation (3.9) becomes separable and its solution is given by (−)

(−)

(−)

ΞK 1 ,K 2 ,µ1 ,µ2 (R1 , R2 ) ≈ χ1,K 1 ,µ1 (R1 ) χ2,K 2 ,µ2 (R2 ) , where (

(3.14)

) ℏ2 ℏ2 (−) ∗ ∇2R1 + U1B − K12 χ1,K 1 ,µ1 (R1 ) = 0, 2M1B 2M1B ( ) ℏ2 ℏ2 (−) ∗ − ∇2R2 + U2B − K22 χ2,K 2 ,µ2 (R2 ) = 0. 2M2B 2M2B −

(3.15a) (3.15b)

Similar to Eq. (3.6), it is possible to write ∑ (−) (−) (i) χi,K i ,µ′ µi (Ri ) η1/2,µ′ , χi,K i ,µi (Ri ) = i

µ′i

(3.16)

i

(−)

for i = 1 and 2. The boundary condition for χi,K i ,µ′ µi (Ri ) is then i

µi −µ′i

(−)

χi,K i ,µ′ µi (Ri ) → ϕK i (Ri ) δµ′i µi +(−) i

∗ f−µ ′ ,−µ (Ωi ) i i

exp (−iKi Ri ) , Ri

(R0 ≫ RN ) , (3.17)

where Ωi is the solid angle of particle i, meaning that an incoming boundary condition is imposed. The following property is helpful with regard to practical applications: µi −µ′i

(−)

χi,K i ,µ′ µi (Ri ) = (−) i

(+)∗

χi,−K i ,−µ′ ,−µi (Ri ) . i

(3.18)

Taking the above assumptions into account, the transition matrix is given by ⟨ (1) (2) (−) (−) Tµ1 µ2 µB µ0 µA = χ1,K 1 ,µ1 (R1 ) ζ1/2,ν1 χ2,K 2 ,µ2 (R2 ) ζ1/2,ν2 tfree ⟩ (+) (0) × χ0,K 0 ,µ0 (R0 ) ζ1/2,ν0 ΨIB µB IA µA ,tB νB tA νA (R2 ) , (3.19)

where ΨIB µB IA µA ,tB νB tA νA is the overlap function between nuclei A and B:

ΨIB µB IA µA ,tB νB tA νA (R2 ) ≡ ⟨ΦIB µB ,tB νB (ϵB , ξB ) |ΦIA µA ,tA νA (ϵA , ξA )⟩ξB .

(3.20)

It should be noted that we have taken into account the lack of any effect of tfree on ΦIB µB .

19

In general, the wave function of A can be expressed as ) ∫ ∑( ∑ 1 (N) t′B νB′ νN |tA νA ζ1/2,νN ΦIA µA ,tA νA (ϵA , ξA ) = 2 ′ ′ ϵ′B tB νB

×

∑ ∑

′ µ′ ljµ IB j B

[ ( ) ] ˆ 2 ⊗ η (N) ϑnljνN IB′ t′B νB′ ;IA tA νA φnlj (R2 ) Yl R 1/2

jµj

× (jµj IB′ µ′B |IA µA ) ΦIB′ µ′B ,t′B νB′ (ϵ′B , ξB ) ,

(3.21)

where (abcd|ef ) is the Clebsch-Gordan coefficient and ) [ ( ) ] ( ) ∑( 1 (N) ˆ 2 ⊗ η (N) ˆ µ |jµ Yl R ≡ lm N j Ylm R2 η1/2,µN 1/2 2 jµj mµ

(3.22)

N

with Ylm being the spherical harmonics. The single-particle (s.p.) state of A is specified by the principal quantum number n, the orbital angular momentum l, the total s.p. spin j, and its projection µj . Inserting Eq. (3.21) into Eq. (3.20), one obtains ) ∑( 1 ΨIB µB IA µA ,tB νB tA νA (R2 ) = tB νB νN |tA νA ϑnljνN IB tB νB ;IA tA νA 2 ljµj [ ( ) ] (N) ˆ 2 ⊗ η (N) × (jµj IB µB |IA µA ) φnlj (R2 ) ζ1/2,νN Yl R . 1/2 jµj

(3.23)

With the overlap function in this form, Eq. (3.19) can be written as ⟨ (+) (−) (1) (−) (2) (0) Tµ1 µ2 µB µ0 µA = χ1,K 1 ,µ1 (R1 ) ζ1/2,ν1 χ2,K 2 ,µ2 (R2 ) ζ1/2,ν2 tfree χ0,K 0 ,µ0 (R0 ) ζ1/2,ν0 ⟩ [ ( ) ] ∑ 1/2 (N) (N) ˆ S (jµj IB µB |IA µA ) φnlj (R2 ) ζ Yl R 2 ⊗ η × , nljνN

ljµj

1/2,νN

1/2

jµj

(3.24)

where 1/2

SnljνN ≡

(

) 1 tB νB νN |tA νA ϑnljνN IB tB νB ;IA tA νA 2

(3.25)

is the spectroscopic amplitude. Note that, in order to simplify the notation, we have dropped νN , IB , tB , νB , IA , tA , and νA in the subscript. The spectroscopic factor discussed in Section 5 is given by 1/2 2 SnljνN = SnljνN . (3.26) The unpolarized triple differential cross section (TDX) in the laboratory (L) frame is given by 4 ∑ d3 σ L 1 1 (2π) 2 |Tµ1 µ2 µB µ0 µA | , (3.27) = J F LG kin ℏvα 2s0 + 1 (2IA + 1) µ µ µ µ µ dE1L dΩL1 dΩL2 1 2 B 0 A 20

where Fkin is the phase space factor: [ ]−1 K1 K2 E1 E2 E2 E2 K 1 · K 2 Fkin = 1 + + . ℏ4 c4 EB EB K22

(3.28)

Here quantities with (without) the superscript L are evaluated in the L (G) frame, and s0 is the spin of particle 0, with a value of 1/2. The relative velocity vα of particle 0 and of A, which is Lorentz invariant, is given by vα = ℏc2

K0L . E0L

(3.29)

The Jacobian JLG for the transformation from the G frame to the L frame is ( ) KL KL E2 K 0 · K 2 JLG = 1 2 γ 1 + βG , K1 K2 ℏcK2 K0 K2

(3.30)

where ℏcK0L , L E0L + EA ( ) 2 −1/2 γ = 1 − βG .

βG =

(3.31a) (3.31b)

At this point, we perform a summation of µB and µA on the r.h.s of Eq. (3.27) while expanding the absolute square of Tµ1 µ2 µB µ0 µA . Since only the Clebsch-Gordan coefficient in Eq. (3.24) depends on µB and µA , the result is ∑ ( ) 2IA + 1 j ′ µ′j IB µB |IA µA (jµj IB µB |IA µA ) = δj ′ j δµ′j µj . 2j + 1 µ µ B

(3.32)

A

As a result, the physical observables are incoherent with respect to j unless the spin direction of A or B is specified, which is consistent with the general properties of distorted-wave direct reaction theory as discussed by Satchler [48]. Furthermore, the parity conservation renders the TDX incoherent with regard to l for (p, 2N) processes, in which the intrinsic spin of the struck particle is 1/2. Equation (3.27) is then reduced to d3 σ L dE1L dΩL1 dΩL2

= JLG Fkin

with 1/2

Tµ1 µ2 µ0 µj = SnljνN

⟨

4 ∑ ∑ (2π) 1 Tµ1 µ2 µ0 µj 2 ℏvα 2 (2j + 1) µ µ µ µ

∑

lj

(−)

1

2

(1)

0

(3.33)

j

(1)

(−)

(2)

(2)

χ1,K 1 ,µ′ µ1 (R1 ) η1/2,µ′ ζ1/2,ν1 χ2,K 2 ,µ′ µ2 (R2 ) η1/2,µ′ ζ1/2,ν2 1

µ′1 µ′2

1

2

2

∑ (+) (0) (0) (N) × tfree χ0,K 0 ,µ′ µ0 (R0 ) η1/2,µ′ ζ1/2,ν0 φnlj (R2 ) ζ1/2,νN ×

∑(

mµN

µ′0

0

0

⟩ ) ( ) 1 (N) ˆ lm µN |jµj Ylm R2 η1/2,µN , 2 21

(3.34)

245

250

making use of Eqs. (3.6) and (3.16). In many studies, the kinematics of particles 1 and 2 are chosen so that the spinparity and the eigenenergy of B are identified. If IA = 0, this makes l, and j unique, and in DWIA one assumes that the TDX measured in that condition corresponds to a well-defined s.p. orbit nlj of A. Conversely, when the energy resolution is not sufficient to identify the final states of B, one has to perform the summation over l and j, and a multipole decomposition analysis (MDA) is necessary, as discussed in Section 5. When εB is above a threshold energy of B for particle decay, the spin-parity of B cannot be specified and the energy spectrum typically no longer exhibits a clear peak. Even in such cases, with the help of an MDA one can extract s.p. information. The incoherence of the TDX with respect to both l and j is crucial to the practicality of the MDA. In the usual NRDWIA framework [44, 47, 49–53], the following approximation to the distorted waves is made: (−)

(−)

(−)

χ1,K 1 ,µ′ µ1 (R1 ) = χ1,K 1 ,µ′ µ1 (R + s/2) ≈ χ1,K 1 ,µ′ µ1 (R) eiK 1 ·s/2 ,

(3.35a)

χ2,K 2 ,µ′ µ2 (R2 ) = χ2,K 2 ,µ′ µ2 (R − s/2) ≈ χ2,K 2 ,µ′ µ2 (R) e−iK 2 ·s/2 ,

(3.35b)

1

1

1

(−)

(−)

(−)

2

(+) χ0,K 0 ,µ′ µ0 0

2

2

(R0 ) =

(+) χ0,K 0 ,µ′ µ0 0

≈

(+) χ0,K 0 ,µ′ µ0 0

(R − αR R + αs s/2) (R) e−iαR K 0 ·R eiαs K 0 ·s/2

(3.35c)

with R=

R1 + R2 , 2

s = R1 − R2

(3.36)

mA + m2 . mA

(3.37)

and αR =

m2 , mA

αs =

Eqs. (3.35), termed the asymptotic momentum approximation (AMA) [54], are based on the assumption that the propagation of the distorted wave over short distances can be expressed by a plane wave having the asymptotic momentum. The accuracy of the AMA can be ascertained using Eqs. (3.35). In Ref. [55] the AMA is shown to be applicable to propagations over approximately 1.5 fm at energies above 50 MeV. It is important to recognize that the AMA is a further simplification of the local semiclassical approximation (LSCA) [56, 57]: (±)

(±)

χi,K i ,µi µi (R + t) ≈ χi,K i ,µi µi (R) eiK i (R)·t ,

(3.38)

where K i (R) is the local momentum. The direction of K i (R) is taken to be parallel (±) to the flux vector of χi,K i ,µi µi (R), whereas its magnitude is determined so as to satisfy local energy conservation. By using K i (R), one may take into account the change in the kinematics of the two colliding nucleons inside the target nucleus. Unfortunately, however, the LSCA for the spin-flipped component of the distorted wave has not yet been implemented. In the case of the spatial part of the bound state wave function ( ) ˆ2 , ψnljm (R2 ) = φnlj (R2 ) Ylm R (3.39) 22

the following Fourier decomposition is adopted: ∫ 1 ψnljm (R2 ) = dK N ψ˜nljm (K N ) eiK N ·R2 3 (2π) ∫ 1 = dK N ψ˜nljm (K N ) eiK N ·R e−iK N ·s/2 . 3 (2π)

(3.40)

Using the AMA in conjunction with Eq. (3.40), one can factorize the tfree matrix element in Tµ1 µ2 µ0 µj as ∫ ∑ 1 1/2 Tµ1 µ2 µ0 µj = SnljνN dK N t˜free κ′ µ′1 µ′2 ν1 ν2 ,κµ′0 µN ν0 νN 3 (2π) ′ ′ ′ µ1 µ2 µ0 µN ∫ (−)∗ (−)∗ (+) × dR χ1,K 1 ,µ′ µ1 (R) χ2,K 2 ,µ′ µ2 (R) χ0,K 0 ,µ′ µ0 (R) e−iαR K 0 ·R 1 2 0 ( ) ∑ 1 (3.41) × ψ˜nljm (K N ) eiK N ·R lm µN |jµj , 2 m where t˜free κ′ µ′1 µ′2 ν1 ν2 ,κµ′0 µN ν0 νN ≡ ⟩ ⟨ ′ (0) (N) (N) (2) (0) (1) (2) (1) eiκ ·s η1/2,µ′ ζ1/2,ν1 η1/2,µ′ ζ1/2,ν2 tfree eiκ·s η1/2,µ′ ζ1/2,ν0 η1/2,µN ζ1/2,νN (3.42)

with

0

2

1

κ′ = (K 1 − K 2 ) /2,

κ = (αs K 0 − K N ) /2.

(3.43)

This prescription for tfree is termed the factorization approximation, and is also often referred to as the zero-range approximation. It is apparent that Eq. (3.41) does not contain an integral over s. It should be noted, however, that the interaction range s of tfree is taken into account in Eq. (3.42), resulting in the appearance of an NN scattering angle dependence. As well, because of Eqs. (3.42) and (3.43), the kinematics of the colliding two nucleons is specified. Here it is further assumed that the total momentum of the two nucleons is conserved, such that κ → (2αs K 0 − K 1 − K 2 ) /2 ≡ κ ¯,

(3.44)

which allows one to neglect the K N dependence of t˜free . Thus we obtain ∑ 1/2 Tµ1 µ2 µ0 µj = SnljνN t˜free κ′ µ′1 µ′2 ν1 ν2 ,¯ κµ′0 µN ν0 νN ∫

µ′1 µ′2 µ′0 µN (−)∗

(−)∗

(+)

dR χ1,K 1 ,µ′ µ1 (R) χ2,K 2 ,µ′ µ2 (R) χ0,K 0 ,µ′ µ0 (R) e−iαR K 0 ·R 1 2 0 ) ∑( 1 lm µN |jµj ψnljm (R) . × 2 m ×

23

(3.45)

The transition matrix of Eq. (3.45) is used in the calculation of the TDX, as discussed in Section 5. The analyzing power Ay is given by [ ] ∑∑ Tr Tµ2 µj σy Tµ†2 µj lj µ2 µj

Ay = ∑ ∑

lj µ2 µj

[ ] , Tr Tµ2 µj Tµ†2 µj

(3.46)

where Tµ2 µj ≡

(

T1/2,µ2 ,1/2,µj T−1/2,µ2 ,1/2,µj

T1/2,µ2 ,−1/2,µj T−1/2,µ2 ,−1/2,µj

)

(3.47)

and σy is the y-component of the Pauli spin matrix. The spin transfer coefficients Dij , defined by Eq. (2.6), are given by [ ] ∑∑ Tr Tµ2 µj σj Tµ†2 µj σi Dij =

lj µ2 µj

∑∑

lj µ2 µj

[ ] Tr Tµ2 µj Tµ†2 µj

,

(3.48)

where i (j) specifies the direction of the spin of particle 1 (0). At this point it is helpful to provide some comments on the NN transition matrix t˜free . In general, off-the-energy-shell (off-shell) t˜free is required because the NN collision takes place inside the nucleus and the energy of the NN system is not conserved, so that κ′ ̸= κ ¯. In practice, an on-shell approximation is employed for t˜free . There are two approaches to the evaluation of t˜free in Eq. (3.45): setting κ′ to κ ¯ or setting κ ¯ to κ′ , termed the initial and final energy prescriptions. In many studies, the final energy prescription is adopted, primarily because the kinematics of the two colliding nucleons in the final channel is more directly related to that in the asymptotic region; i.e., the real kinematical measurement conditions. One advantage of employing the on-shell approximation is that it allows the use of t˜free , which is well constrained by NN scattering observables. One should keep in mind, however, that choosing kinematics for which the difference between κ′ and κ ¯ is large makes the use of on-shell t˜free questionable. Furthermore, when a deeply bound nucleon is knocked out at relatively low incident energy, the in-medium modification of t˜free becomes important. As a practical means of accomplishing this modification, NN effective interactions in infinite nuclear matter (that is, g-matrix interactions) have been implemented, and some examples are presented in Section 6. Our final comment on t˜free is that it is usually defined within the c.m. frame of the two nucleons, which we term the t-frame. When this t˜free is used in Eq. (3.45), which is defined in the G frame, t˜free must be multiplied by the Møller factor defined by fMøl =

(

t E1t E2t E0t EN E1 E2 E0 EN

)1/2

,

(3.49)

where Eit is the total energy of particle i in the t-frame. 24

It is helpful to consider the form of Eq. (3.45) in the case that there is no spin-orbit term in the distorting potentials: ∑ 1/2 Tµ1 µ2 µ0 µj → SnljνN t˜free κ′ µ1 µ2 ν1 ν2 ,¯ κµ0 µN ν0 νN µN

∫

(−)∗

(−)∗

(+)

dR χ1,K 1 ,µ1 µ1 (R) χ2,K 2 ,µ2 µ2 (R) χ0,K 0 ,µ0 µ0 (R) e−iαR K 0 ·R ) ∑( 1 × lm µN |jµj ψnljm (R) . 2 m

×

(3.50)

If Eq. (3.50) is substituted into Eq. (3.33) and the following average prescription for µN is employed: t˜∗free κ′ µ1 µ2 ν1 ν2 ,¯ κµ0 µ′

N ν0 νN

2 1 ∑ ˜free t˜free tκ′ µ1 µ2 ν1 ν2 ,¯κµ0 µ¯N ν0 νN δµ′N µN , (3.51) κ′ µ1 µ2 ν1 ν2 ,¯ κµ0 µN ν0 νN ≈ 2 µ¯ N

the TDX reads

( )2 4 (2π) 2πℏ2 ∑ 1 d3 σ L dσ free (κ′ , κ ¯) → J F SnljνN LG kin 2 L L L ℏvα (mN /2) 2l + 1 dΩfree dE1 dΩ1 dΩ2 lj 2 ∑ ∫ (−)∗ (+) −iαR K 0 ·R dR χ(−)∗ , × (R) χ (R) χ (R) e ψ (R) nljm 1,K 1 ,µ1 µ1 2,K 2 ,µ2 µ2 0,K 0 ,µ0 µ0 m

(3.52)

where use has been made of )( ) ∑( 1 1 2j + 1 lm′ µN |jµj lm µN |jµj = δ m′ m 2 2 2l + 1 µ µ j

(3.53)

N

and 2 2 dσ free (κ′ , κ ¯) (mN /2) 1 ∑ ˜free tκ′ µ1 µ2 ν1 ν2 ,¯κµ0 µ¯N ν0 νN = 2 free dΩ (2πℏ2 ) 4 µ1 µ2 µ0 µ¯N

255

(3.54)

with mN being the nucleon mass and dσ free (κ′ , κ ¯ ) /dΩfree the NN differential cross section in free space. Thus, the TDX has been factorized with respect to the NN cross section. As noted, this is the result of the absence of the spin-orbit term in the distorting potentials and Eq. (3.51). In the plane wave limit (that is, the PWIA), it is possible to write an even more simplified form of the TDX: d3 σ L dE1L dΩL1 dΩL2 JLG Fkin

→

∑ 1 ℏ4 dσ free (κ′ , κ ¯) ∑ 1 2 SnljνN |φ˜nljm (Q)| , (3.55) 3 2 free ℏvα (2π) (mN /2) 2l + 1 dΩ m lj

25

where φ˜nljm (Q) =

∫

dR eiQ·R ψnljm (R)

(3.56)

and Q is the so-called missing momentum defined by Q = (1 − αR ) K 0 − K 1 − K 2 .

260

265

270

275

280

285

(3.57)

One can verify that Q is the momentum of B in the L frame, K LB , although this is only an approximation in relativistic kinematics, with a typical deviation of approximately 1% even at 2 GeV. If −Q is regarded as the momentum of the nucleon that is to be struck by particle 0 (i.e., the target nucleon), Eq. (3.55) suggests that the TDX of a (p, 2N) reaction represents a “snapshot” of the target nucleon in momentum space. This is why (p, 2N) reactions are useful for studying the s.p. properties of nuclei. In reality, however, it is not possible to eliminate distortion effects, and one may say that what is probed is the distorted momentum distribution of the target nucleon. It should be noted that the measured TDX may contain also more complicated many-body effects. In the calculations shown in Section 5, Eqs. (3.33) and (3.46) are used in conjunction with Eq. (3.45). Here we summarize the assumptions adopted in deriving these equations. Firstly, we start with a transition matrix defined by nonrelativistic reaction theory, implementing the relativistic effect with regard to the kinematics. Secondly, we do not explicitly include the excitation/de-excitation of the residual nucleus B and the target nucleus A in the initial channel. A is also assumed to remain in the ground state. The loss of flux to the channels that are not included explicitly is taken into account through employing distorting potentials, consisting of central and spin-orbit terms, for the scattering particles. Thirdly, we use a local NN transition interaction defined in free space. Fourthly, we approximate the three-body scattering wave function in the final channel by the product of two distorted waves for the two outgoing particles. This approximation may affect the result by an order of 1/A, where A is the mass number of A. It should be noted that this approximation is accurate in the plane-wave limit, meaning that the proper three-body kinematics in the asymptotic region is always guaranteed. Lastly, the NN transition matrix element t˜free is factored out from the integration of the product of the wave functions. This factorization approximation is equivalent to the asymptotic momentum approximation (AMA) to the distorted waves in Eqs. (3.35). We have further assumed that the total momentum of the NN system is conserved. The kinematics of the colliding two nucleons is thus determined by the three asymptotic momenta, K0 , K1 , and K2 , and the hypothetical momentum conservation for the NN system. If one adopts the local semiclassical approximation instead of the AMA, the change in the NN kinematics resulting from deflection may be taken into account. In actual calculations, an on-shell approximation to t˜free is made. Finally, we discuss the effect of nonlocality on the distorting potentials. It is well known that a local potential phenomenologically determined such that the solution of the Schr¨odinger equation reproduces elastic scattering observables (that is, the Schr¨ odingerbased optical potential (SOP)) does not generate a proper scattering wave inside the target nucleus because of the nonlocal properties of the true distorting potential. The distorted wave obtained with an SOP thus needs to be corrected. Following Ref. [58], in 26

most cases the distorted wave is multiplied by [ ]−1/2 Mij 2 FPR (R) = CPR 1 − β U (R) , ij NL 2ℏ2

(3.58)

which is known as the Perey factor. The range of nonlocality is denoted by βNL and the constant CPR is unity for the correction to the scattering waves. It is important to note that there is no justification for the use of the Perey factor with phenomenological SOPs in general. In the so-called Dirac phenomenology [59–62], the scalar (US ) and vector (UV ) potentials in the Dirac equation are determined in a similar manner to that for an SOP. A distinguishing feature of the Dirac phenomenology is that the optical potential used in the Schr¨odinger equation, which we term a Dirac-based OP (DOP), is unambiguously determined by US and UV . Furthermore, it can be shown that, to obtain the solution of the Dirac equation for the upper component, the distorted wave obtained by solving a Schr¨odinger equation with a DOP has to be multiplied by FDW (R) = CDW 290

295

300

305

[

Ei + US (R) − UV (R) Ei

]1/2

,

(3.59)

which we term the Darwin factor and is equivalent to CPR , CDW = 1 for scattering waves. In general, the roles of FPR and FDW are very similar; they reduce the amplitude of the distorted wave in the nuclear interior region. Although it is still an open question whether the physical origin of FPR is the same as that of FDW , we use FPR (FDW ) when an SOP (a DOP) is adopted, following convention. The above discussion can be applied to the bound state wave function ψnljm ; in the case that a local s.p. potential is adopted for ψnljm , the solution is multiplied by FPR or FDW . In contrast to the scattering waves, however, the corrected radial wave function must be normalized to unity, and CPR and CDW are determined accordingly. Here we discuss the effect of nonlocal correction semiquantitatively. In the case of scattering waves, the correction reduces the amplitude primarily in the nuclear interior region, which decreases the TDX in general. In contrast, as the result of renormalization by CPR or CDW , the correction reduces the amplitude of ψnljm in the nuclear interior and enhances it in the surface region. The net effect of the nonlocal correction thus depends on the reaction region. The nonlocal correction increases the TDX when the knockout reaction is very peripheral, and the correction to the distorted waves is very small for such peripheral reactions. Conversely, when the nuclear interior region is well probed, the correction to the distorted wave becomes dominant. 3.2. Relativistic framework There have been many theoretical investigations of (p, 2N) reactions employing the relativistic DWIA (RDWIA) [63–72], which we briefly review in this subsection. For a more detailed discussion, see, for example, Ref. [68]. The formulation of the RDWIA is in fact almost the same as that of the NRDWIA. One of the distinguishing features of the RDWIA is that the lower components (the anti-particle sectors) of the distorted waves and the bound state wave function are explicitly taken into account. As a result,

27

the transition matrix element has the following form: ∫ 1/2 (−) (1) (−) (2) Tµ1 µ2 µ0 µj = SnljνN dR1 dR2 χ ¯1,K 1B ,µ1 (R1 ) ζ1/2,ν1 χ ¯2,K 2B ,µ2 (R2 ) ζ1/2,ν2 ) ∑( 1 (N) (+) (0) (N) × tfree χ0,K 0 ,µ0 (R1 ) ζ1/2,ν0 lm µN |jµj ψnljm (R2 ) η1/2,µN ζ1/2,νN , 2 mµ N

(3.60)

(+)

where χ0,K 0 ,µ0 is the solution to the Dirac equation [

] [ ] (+) (+) cα · Pˆ + β m0 c2 + US (R1 ) + UV (R1 ) χ0,K 0 ,µ0 (R1 ) = E0 χ0,K 0 ,µ0 (R1 ) (3.61)

with US (UV ) being the scalar (vector) optical potential. As in the nonrelativistic frame(+) work, χ0,K 0 ,µ0 is decomposed into ∑

(+)

χ0,K 0 ,µ0 (R1 ) =

(+)

(0)

χ0,K 0 ,µ′ µ0 (R1 ) η1/2,µ′ . 0

µ′0

(3.62)

0

(+)

The distorted-wave matrix χ0,K 0 ,µ′ µ0 here has, however, upper and lower components 0 in the RDWIA. The explicit form of these components in the partial wave expansion is given by (+)

)1/2 E0 + m0 c2 2m0 ( )   ( ) ˆ1 ∑ LML′ 12 µ′0 |JMJ u0 (K0 R1 ) YLML′ R ) ( )   ( × ˜ ′ 1 µ′ |JMJ w0 (K0 R1 ) YLM ′ R ˆ1 i LM ′ L2 0 JMJ LML ML L ( ) ( ) 1 ∗ ˆ0 , × iL LML µ0 |JMJ YLM (3.63) K L 2

χ0,K 0 ,µ′ µ0 (R1 ) = 0

310

4π K0 R1

(

˜ = 2J − L, and u0 (w0 ) are the radial solutions to Eq. (3.61) corresponding to where L (−) the upper (lower) component. Similar expressions are obtained for χi,K iB ,µi (i = 1 or 2) and ψnljm [68]. Employing the so-called zero-range approximation, the transition matrix can be simplified to ∑ ∫ 1/2 (−) (−) Tµ1 µ2 µ0 µj = Snljν dR χ ¯ (R) χ ¯ (R) t˜free ′ ′ κ′ ,¯ κ 1,K 1B ,µ1 µ1

N

µ′1 µ′2 µ′0 µN

(+)

× χ0,K 0 ,µ′ µ0 (R) 0

∑( m

2,K 2B ,µ2 µ2

) 1 lm µN |jµj ψnljm (R) . 2

(3.64)

This zero-range approximation is, as in the NRDWIA framework, simply the AMA for the distorted waves together with an assumption of momentum conservation for the two colliding nucleons. The zero-range approximation greatly reduces the computational 28

315

320

325

330

335

340

345

350

355

360

requirements and has been widely used in prior studies [70–72]. In Ref. [68] the zerorange approximation is shown to affect the absolute value of the TDX for the 1d5/2 and 1d3/2 knockout processes from 40 Ca at 200 MeV by a factor of approximately three. In the case of the RDWIA, 16×16 matrix elements of tfree are required, only 44 components of which are independent of the symmetric properties. Although use of all 44 terms, the so-called IA2 representation [73–76], is preferable for unambiguous treatment of the NN transition matrix, this has not been implemented in the RDWIA so far because of the associated complexity. For more information concerning this implementation in the RPWIA, see Ref. [77]. Instead, only five such terms determined by the free onshell NN scattering data, equivalent to the so-called IA1 representation [78, 79], are often used. In actual calculations, the relativistic meson exchange model developed by Holowitz, Love, and Franey (HLF) [80] is adopted. The functional form of the HLF model is suitable for implementing the possible reduction of meson masses in a nuclear medium, the Brown-Rho conjecture, and to determine the effects on spin observables, as discussed in Section 6. It is evident from Eq. (3.60) that the momenta specifying the distorted waves for particles 1 and 2 are different from those in the NRDWIA framework, and that the (+) coordinate vector for χ0,K 0 ,µ0 ; K iB is the momentum of particle i in the i-B c.m. frame. Thus, in contrast to the NRDWIA, the proper asymptotics of particles 1 and 2 are not guaranteed in the plane wave limit. Furthermore, strictly speaking, the wave functions defined in different frames coexist in Eq. (3.60). These issues arise due to the recoil motion of B. Although there have been detailed discussions [65–67] regarding this subject, the proper implementation of the recoil effect in the full relativistic framework remains unclear. The RDWIA has been applied to studies of the TDX and spin observables of (p,2N) reactions. However, the absolute value of the TDX, and hence the spectroscopic factor, has not been discussed quantitatively except in a few works [68, 69], primarily because of the ambiguity in the treatment of the recoil of B in the RDWIA. Another noteworthy aspect of Eq. (3.60) is that it is based on the same approximations as in the NRDWIA: the neglect of the dynamical excitation/de-excitation of A and B, use of a local free NN interaction, and the replacement of the three-body scattering wave function with the product of two distorted waves. 3.3. Theoretical activities There have been intensive efforts aimed at realizing theoretical calculations based on the frameworks described above. With regard to the nonrelativistic framework, the studies by Chant and Roos [44, 47] are especially important. This work generated the computer code threedee, which employs a factorized approximation within an amplitude level, includes spin dependent optical potentials, and calculates a complete set of polarization observables. Continued improvements have been made to this code, and the code now includes an implementation of various global optical potential parameter sets, effective nucleon-nucleon interactions, and even an extension to treat density dependent effects. Owing to such efforts, the code has become a standard tool used by almost all experimentalists to analyze their (p,2N) data. This code was also employed in the analyses presented in the subsequent sections of the present review. In parallel with the above work, Kudo’s group at Osaka City University performed NRDWIA calculations. These calculations succeeded in reproducing the data acquired 29

365

370

375

380

385

390

395

at TRIUMF and Maryland reasonably well and also clarified various effects, such as the impact of each component of the t-matrix, the non-locality effect and the relativistic correction effect [51, 81]. Working within the relativistic framework, Maxwell and Cooper developed a formalism based on the RDWIA and carried out a fully finite range of calculations for the first time [63]. One of their most important successes was taking into account the recoil effect [65]. Their calculations reproduce the experimental data reasonably well in the case of differential cross section and analyzing power values obtained at TRIUMF, although significant disagreements were found in the case of certain geometries and for the 1s state knockout reaction. Kudo’s group also extended their work to relativistic calculations, performing the first-ever RDWIA calculations using the zero range approximation and later proceeding to finite range calculations [68]. The latter work demonstrated that finite range calculations are essential to obtaining reasonable S-factors from relativistic calculations, as mentioned in the previous subsection, and also established self-consistent calculations employing the Dirac-Hartree model [69]. Subsequently, Hillhouse’s group, in collaboration with one of Kudo’s group members, began work in this field. Neglecting the recoil effect, they performed calculations for 208 Pb target data and succeeded in obtaining excellent agreement, which had previously not been possible using a nonrelativistic approach [70]. The results of this work are presented herein, in Section 6. It is also worth noting that this prior research successfully calculated all the spin transfer coefficients contained in experimental data acquired at RCNP [82]. In addition to these works, theoretical investigations of (p,2N) reactions have been performed based on the Glauber theory [83] and the Dirac eikonal approximation [84]. In general, those frameworks are expected to be reliable at high energies of incident and outgoing nucleons. In the latter case, however, the authors have succeeded to reproduce the shape of experimental cross section data at 1 GeV and 250 MeV by introducing optical potential eikonal approximation, instead of Glauber approximation, in encountering the initial and final states interactions. Recently, the Faddeev–Alt-Grassberger-Sandhas (FAGS) theory [85, 86] was applied to the 12 C(p, 2p)11 B reaction at 400 MeV [87]. The authors selected the diagrams corresponding to the DWIA out of the full three-body diagrams, and found that the rescattering processes that are not taken into account by the DWIA formalism are negligibly small in the reaction discussed in Ref. [87]. It will be highly important to directly compare results of the FAGS with those of the DWIA, and also with experimental data. 4. Experimental Methods

400

This section reviews the facilities capable of performing proton-induced nucleon knockout reactions at intermediate energies (Tp > 100 MeV). Table 2 summarizes the (p, 2N) facilities at Tp ≳ 200 MeV. 4.1. Spectrometer and telescope at TRIUMF Both (p, 2p) and (p, pn) measurements have been performed at TRIUMF using several experimental setups, and typical setups are presented in Fig. 11. 30

Table 2: Comparison of (p, 2N) facilities.

Proton beam Forward detector Forward polarimetry Backward detector Backward polarimetry Neutron detector (Forward/Backward) Observables

Proton beam Forward detector Forward polarimetry Backward detector Backward polarimetry Neutron detector (Forward/Backward) Observables

TRIUMF Polarized Polarized NaI or Spectrometer Plastic (TOF) (MRS) N/A N/A

IUCF Polarized Si+Ge(+NaI) or NaI+MWDC N/A

NaI or Plastic (TOF) N/A

Spectrometer or NaI (LSAA) N/A

Plastic (TOF) or Spectrometer N/A

Plastic (TOF) σ Ay

σ Ay

RCNP Polarized Spectrometer (GR) Focal-plane polarimeter Spectrometer (LAS) N/A

PNPI Unpolarized Spectrometer (MAP) Focal-plane polarimeter Spectrometer (NES) Focal-plane polarimeter Plastic (TOF)

Plastic (TOF) (NPOL3) σ Ay P Dij

31

σ P

Plastic (TOF) or NaI+MWDC σ Ay

iThemba Polarized Spectrometer (K600) N/A Solid detector (Si+Ge) N/A N/A σ Ay

405

410

415

420

425

430

435

440

445

Setups with a NaI scintillator. In setup A, pairs of outgoing protons are detected simultaneously by means of four detector telescopes [88, 89]. Each detector telescope consists of a multiwire proportional chamber (MWPC), a plastic scintillator (Sci.), and a NaI(Tl) scintillation crystal (NaI). The positions measured by the MWPC give the angles of the detected protons, whereas the energies measured by the plastic and NaI scintillators provide the ∆E and E information, respectively, and the plastic scintillators also give timing information. In the case of the forward-angle telescopes (LF and RF), thick copper plates (Cu) are mounted in front of the NaI scintillators to degrade the proton energies sufficiently such that the protons stop in the NaI. In setup B, a similar detector telescope with a NaI scintillator is used to capture the outgoing protons [41]. The coincident nucleon (proton or neutron) is detected by a time-of-flight (TOF) arm consisting of an MWPC, a large plastic scintillator (Large Sci.) and a thin plastic scintillator (Sci.) used to identify the charge of the particles detected in the large plastic scintillator. TOF information is obtained by determining the time difference between the thin plastic scintillator (Timing Sci.) in another arm and the large plastic scintillator. Setups with an MRS spectrometer. A medium resolution spectrometer (MRS) [90] was used in setups C and D to detect the forward protons on the left hand side (LHS). This spectrometer consists of one quadrupole (Q) and one dipole (D) magnet, and has an effective solid angle of approximately 3 msr with a scattering-angle acceptance of about 3◦ . The positions and angles of particles analyzed by the MRS are determined by two sets of VDC’s (X1/Y 1 and X2/Y 2), employing a plastic scintillator (Sci.) as a trigger. The information from two wire planes (X0 and Y0 ) between the target and the MRS is also used to trace particle trajectories back to the target position. In setup C, the right hand side (RHS) detectors consist of 16 neutron/proton plastic scintillators (n) with a size of 114 mmϕ × 76 mmt with thin plastic scintillators (v) to identify the particles as charged or neutral [91]. The energy of the outgoing nucleon is measured by the TOF technique, with a flight path length of 5 m, and the TOF is determined with respect to a timing reference derived from a thin plastic scintillator (∆Eb ) situated at the other side of the apparatus. Scattering over the 5 m flight path is minimized by a helium-filled polyethylene bag (He Bag), as shown in Fig. 11(c). In setup D, a non-focusing magnetic spectrometer consisting of a dipole magnet with four sets of VDCs is used instead of the RHS detectors in setup C [36]. Each of the first two VDCs (VDC1 and VDC2) has two anode wire planes (x and u planes), and the wires in the x-plane are stretched vertically, while those in the u-plane are tilted by 30◦ relative to x. Horizontal and vertical angles are determined from the information generated by these VDCs. Another two sets of VDCs after the magnet have only x-planes, and their position information is used to determine the energies of particles momentum-analyzed by the dipole magnet. 4.2. Two-arm spectrometers at RCNP Spectrometers. The two outgoing protons are momentum analyzed simultaneously using a two-arm spectrometer system consisting of the Grand Raiden spectrometer (GR) [92] and the Large Acceptance Spectrometer (LAS) [93]. The GR consists of two dipole (D1 and D2) magnets, two quadrupoles (Q1 and Q2), a sextupole (SX) and a multipole 32

(a) Setup A

(b) Setup B Sci. MWPC

LB

Target Chamber

Sci.

Large Sci.

NaI LF

Proton Beam

Proton Beam Timing Sci.

MWPC Cu

MWPC Cu Sci.

To Beam Dump

To Beam Dump

NaI

RF RB

(c) Setup C

(d) Setup D

Target Chamber Proton Beam

Focal Plane Detector Sci.

MRS Q

Target Chamber Proton Beam

D

Timing Sci.

Focal Plane Detector Sci.

MRS

Q

D

VDC1 VDC2 He Bag To Beam Dump

To Beam Dump

VDC3 VDC4

RHS Detectors

Figure 11: Schematic views of typical experimental setups for (p, 2p) and (p, pn) measurements at TRIUMF.

33

MP D1 D

Q2 D2 Grand Raiden

SX Q1 Q

DSR DSR-

FPP BEAM DSR+

0

2

Large Acceptance Spectrometer 4

6m

Figure 12: Schematic view of the two-arm spectrometer system at RCNP.

450

455

460

465

470

(MP), as shown in Fig. 12. The SX magnet is used to satisfy the second-order ionoptical requirements, whereas the MP magnet compensates for ion-optical aberrations in the system. The GR and LAS are characterized by a high momentum resolving power of R = p/∆p ≃ 37000 and a large momentum bite of δp/p ≃ 35%, respectively. The full acceptances of the GR and LAS are 4.3 and 20 msr, respectively, corresponding to horizontal acceptances of ±20 and ±50 mr. In some measurements, these acceptances are limited, using collimators, to 2.4 msr (±20 mrH × ±30 mrV ) and 9 msr (±50 mrH × ±45 mrV ) for the GR and LAS, respectively. The positions and angles of particles analyzed by each spectrometer are determined by a focal-plane detector system consisting of two sets of vertical drift chambers (VDC’s) [94] with two anode wire planes (x and u planes) followed by two planes of plastic scintillators. Wires in the x-plane are stretched vertically, while those in the u-plane are tilted. Horizontal and vertical positions as well as angles can be separately determined in each VDC by using the measured drift time information obtained from the ionization charges collected on the anode wires in the x and u planes. Focal plane polarimeter. The polarization of protons at the focal plane of the GR can be determined using a focal plane polarimeter system (FPP) [95], following a dipole magnet for spin rotation (DSR). The polarization vector of the outgoing protons is precessed by the dipole field of the GR and only the two components of the vector perpendicular to the momentum vector at the FPP are actually measurable. The DSR magnet can bend the protons by either +18◦ or −17◦ , and the polarization vector is also precessed in the horizontal plane. Thus, by using the DSR magnet, the longitudinal component of the vector can be obtained as the sideways component at the FPP, such that the FPP in conjunction with the DSR magnet enables the complete measurement of all three components. The FPP consists of a 4 to 12 cm thick carbon-slab analyzer, two sets of MWPCs and 34

(a) Neutron-forward setting

(b) Neutron-backward setting

Scintillator

Scintillator

Charged Particle Veto

Figure 13: Schematic views of the neutron detector setups for (p, pn) measurement at RCNP.

475

480

485

490

495

two layers of scintillator hodoscopes. Each MWPC has three anode wire planes: the x, u and v planes. The wires in the x-plane are stretched vertically, while those in the u(u) planes are tilted by +45◦ (−45◦ ). Horizontal and vertical positions can be separately determined in each MWPC using the measured drift time information, and thus the information from these two MWPCs also provides the horizontal and vertical angles. The polarization is evaluated from the left-right or up-down asymmetry of proton scattering from the carbon-slab, employing raytracing with the VDCs and MWPCs. The effective analyzing power of the FPP at 284 MeV is approximately 0.40, which is consistent with the expected value based on the data for inclusive p-C scattering. Neutron detector. Neutrons from the (p, pn) reaction are measured over an 11 to 23 m flight path length at θlab = 25.5◦ –54.5◦ [96]. As illustrated in Fig. 13, the neutron detector system consists of 20 sets of one-dimensional position-sensitive plastic scintillators (BC408) with a size of 100 × 10 × 5 cm3 , which form part of the NPOL3 system [97, 98]. The detector system is composed of four planes of neutron detectors, each with an effective solid angle of 0.9 to 3.7 msr. 4.3. Two-arm spectrometers at PNPI Spectrometers. The two outgoing protons are simultaneously momentum analyzed using a two-arm spectrometer system consisting of a high momentum MAP and low momentum NES spectrometers [99]. Both spectrometers are composed of two quadrupole (Q1 and Q2) and one dipole (D) magnet, as can be seen from Fig. 14, The MAP and NES spectrometers are characterized by a high magnetic rigidity of Bρ = 5.7 Tm and a large acceptance of 3.1 msr, respectively. Focal plane polarimeters. The positions of particles at each spectrometer are measured by two sets of MWPCs. Each MWPC has two anode wire planes for the measurement of horizontal and vertical positions, and thus the information on these two MWPCs also 35

C-block PC2 PC1 D

Beam dump M1-M3

C-block S2 S1

D Q2 Q1 Col PC1 PC2 PC3 MAP spectrometer PC4

PC4 S2 PC3

S1

NES spectrometer Q2 Q1 Col Scattering chamber

1GeV beam

Figure 14: Schematic view of the two-arm spectrometer system at PNPI.

500

provides the horizontal and vertical angles. The typical kinetic energies measured by the MAP and NES spectrometers are approximately 800 and 200 MeV, respectively, and the thickness of the carbon block is varied between 3 to 20 cm depending on the proton energy. 4.4. Spectrometer and telescope at iThemba

505

510

The two protons from the (p, 2p) reaction are detected simultaneously with a K = 600 spectrometer and a ∆E-E detector telescope [100], as shown in Fig. 15. The K = 600 spectrometer consists of one quadrupole and two dipole magnets. This spectrometer is characterized by a high momentum resolving power of R = p/∆p ≃ 28000 and a moderate momentum bite of δp/p ≃ 10%. The positions and angles of particles analyzed by the spectrometer are measured by two sets of VDCs and one horizontal drift chamber (HDC). The two VDCs are used to find the horizontal positions and angles, whereas the HDC determines the vertical positions. The ∆E-E telescope is composed of a 1000-µm Si surface detector and a 15 mm high-purity Ge detector. 5. Spectroscopic Studies

515

520

It is possible that (p, 2N) reactions may play an important role in investigations of the single particle properties of nuclear ground states. Extensive experimental and theoretical studies have been performed in this regard, and the momentum dependence of the differential cross sections was found to be reproduced reasonably well in most cases, even at incident energies lower than 100 MeV. In the 1970s, the j-dependence of the analyzing power of the (p, 2p) reactions was found as mentioned in Introduction and Section 2. This indicates that these reactions utilizing polarized proton beams are 36

To Beamstop

Dipole 2 Dipole 1

Drift Chambers

Scintillators

Detector Telescope

Quadrupole

Target

Beam

Figure 15: Schematic view of the single-arm spectrometer and ∆E-E telescope system at iThemba.

525

530

535

540

545

550

useful for the determination of the total angular momentum j, in addition to the orbital angular momentum l that can be obtained from the cross sections of the single particle orbit. However, the evaluations of (p, 2N) reactions as a tool for extracting S-factors have not been conclusive. This is primarily because the extraction of S-factors requires estimations of the absolute values of the cross sections, and these cross sections in turn vary significantly with the optical potential parameters, which are primary inputs into the DWIA calculations. The previous review paper on quasifree nucleon-nucleon scattering by Kitching and others [9] states only that “the experimental cross sections confirm semiquantitatively the states expected from the shell model constitute a large fraction of the nuclear-wave function.” This paper also shows that the experimental cross sections are generally smaller than calculations which shell model limit for S-factors are assumed. During late 1980s and 1990s, the situation changed significantly. Firstly, a global optical potential parameterized in the Dirac approach framework, which fits experimental data precisely for wide ranges of incident energies and target masses, became available. A high quality global optical potential was also provided in the Shr¨odinger based framework. In order to perform reliable DWIA calculations for the (p, 2N) reactions, reliable optical potentials for a wide energy range of incident and outgoing nucleons are essential, since these potentials significantly reduce uncertainty in the theoretical calculations. Secondly, extensive high resolution (e, e′ p) experiments were performed beginning in the mid 1980s, and precise S-factors and geometrical parameters for proton bound states were determined. Since the absorption effects are much more moderate in these cases, compared with the (p, 2N) reactions, the resulting S-factors were useful for assessing the reliability of calculations regarding the (p, 2N) reactions. It is also worth noting that the two arm spectrometer system was constructed at RCNP in this time frame, and high resolution measurements (better than 300 keV) in the intermediate energy region became possible. This resolution allowed us to obtain typical (p, 2p) reaction data, corresponding to knockout from various orbits with almost complete separation. Based on the above, this section is organized as follows. Initially, the results of TRIUMP and RCNP (p, 2p) experiments performed at incident energies of 505 and 397 MeV, respectively, are presented, and the extent to which theoretical calculations reproduce the experimental data are discussed. In addition, the degree to 37

555

560

which the extracted S-factors are consistent with the (e, e′ p) results is examined. At these incident energies, the energies of the outgoing protons are typically higher than 100 MeV and the absorption effect is less serious than at lower energies. These studies also employed DWIA calculations using the modern global potential referenced above to extract the S-factors. Subsequently, the available low energy cross section data are surveyed and the reliability of the (p, 2N) reactions as a spectroscopic tool is examined by comparing these data with the results of systematic theoretical calculations. The majority of this section is devoted to discussions related to the cross sections, although some comments on the analyzing powers from the viewpoint of spectroscopy are provided. Finally, we discuss possible uncertainties included in the extracted S-factors. 5.1. Kinematics of (p, 2N) reactions

565

570

575

580

Prior to examining treatment of the experimental data, we briefly look at the kinematics of these reactions. A consideration of kinematics is important when performing this type of knockout reaction experiment, as well as in the extraction of spectroscopic information. As noted in Section 2.1, the final channel kinematics are completely determined by measuring the momentum vectors of two outgoing nucleons and, in simple terms, the nuclear recoil momentum is regarded as the reverse of the momentum which the knocked-out nucleon had in the target nucleus. In the classical scenario, a (p, 2N) reaction represents the knockout of a nucleon moving in the target nuclei. Accordingly, two outgoing nucleons are emitted in different directions with varying energies and it is necessary to select the kinematical conditions that best allow the extraction of the required information. Figure 16 presents contour maps of the recoil momentum in two dimensional planes of (a) the kinematic energy of one outgoing nucleon T1 and the angle of the other outgoing nucleon θ2 , and (b) the angles of both outgoing nucleons θ1 and θ2 . Other kinematical parameters and the excitation energy of the residual nucleus are fixed. Measurements are normally performed along lines in one such plane and the recoil momentum dependence of the differential cross sections or other observables is obtained. These kinematics are related to the experimental data that are provided in the following subsection, in which a more detailed explanation is provided. 5.2. Experimental studies at 505 and 392 MeV

585

590

595

The 505 MeV data were acquired at TRIUMF with a 16 O target, using the MRS spectrometer and a TOF scintillation array [91], as shown in Fig. 11. The S-factors were subsequently extracted by the same experimental group employing a renewed threedee code and optical potential parameters [36]. The missing mass spectrum is shown in Fig. 17. Here the ground 1/2− state peak of the residual 15 N nucleus is well separated from the first excited 3/2− state peak, which has an excitation energy of 6.3 MeV, although the 3/2− peak includes several other minor levels and its yields were extracted using a peak fitting procedure. The S-factor was determined from experimental data at six different angle pairs, using 22.15◦ as the forward angle and six backward angles, as indicated in Fig. 16(a) by the thick arrows. A comparison of these data with the results of DWIA calculations is provided in Fig. 18. In this figure, both experimental data and calculations are plotted as functions of T1 , the energy of the forward outgoing protons, while the recoil momentum values are indicated on the upper axes. The agreement 38

Figure 16: Contour maps of the recoil momentum pR in two dimensional planes of (a) the kinetic energy of one outgoing nucleon and the emitted angle of the other outgoing nucleon, and (b) the emitted angles of the two outgoing nucleons. The contour in (a) corresponds to the TRIUMF experiment discussed in Section 5.2, performed using the MRS spectrometer set at θ1 = 22.15◦ and TOF detectors at θ2 angles, as indicated by the thick arrows. The thin arrows are additional available data that are discussed in Section 5.3. The contour (b) corresponds to the RCNP experiment described in the same subsection, 1 ⃝ 2 and ⃝ 3 lines, keeping the energies of the forward emitted protons constant, and performed along the ⃝, 4 It is noted that the two-body energy E NN is approximately changing detection energies at the point ⃝. 2 and the two-body scattering angle θ NN is almost constant along the line ⃝. 3 constant along the line ⃝

600

605

610

615

between the experimental data and the calculation results is rather good, particularly in the peak region in the vicinity of the recoil momentum 100 MeV/c. The extracted S-factors are 1.05 for the 1/2− ground state and 2.0 for the 3/2− excited state, which are consistent with those obtained from the analysis of 16 O(e, e′ p) [21]: 1.17 and 2.24, respectively. The authors state that the differences between experimental and theoretical values, approximately 11% for either state, were significantly less than the experimental errors in the absolute cross sections, which were about 15%. It is noteworthy that these DWIA calculations employed the same rms radii for the bound states as in the (e, e′ p) analysis. An RCNP group studied 12 C and 40 Ca(p, 2p) reactions at 392 MeV, leading to lowlying states of the residual nuclei [101, 102]. This study was intended to examine the reliability of (p, 2p) reactions as a spectroscopic tool, and the effects of recoil-momentum on the cross sections and analyzing powers were determined while employing four different sets of kinematical parameters, as shown in Fig. 16(b). In this experiment, the majority of the low-lying states were observed to be well isolated from one another, although the three states in the Ex = 2.5–3.0 MeV region were not separated and so were jointly analyzed. Figures 20 and 21 present some of the resulting data and a comparison with theoretical calculations. This figure demonstrates that the DWIA calculations reproduced the experimental data quite well, except for some limited regions, as seen in the rightmost regions of two plots for the 12 C target, and the leftmost region of one plot for the 40 Ca target. As in the case of the above 505 MeV analysis, these DWIA calculations employed the Dirac-based global optical potentials and bound state wave functions with 39

Figure 17: Missing mass spectrum for the

16 O(p, 2p)

reaction at 505 MeV [91].

Figure 18: The differential cross sections for the 16 O(p, 2p)15 N reaction at 505 MeV and comparison with the results of DWIA calculations. The six plots on the left hand side correspond to the 1p1/2 knockout while the rightmost six correspond to the 1p1/2 knockout.

40

620

the same geometrical parameters as those used in (e, e′ p) analyses [20]. A comparison of the S-factors with the results of (e, e′ p) analyses is given in Table 5.3. From these studies, it is concluded that the S-factors extracted from (p, 2p) data in the 400 to 500 MeV region are consistent with (e, e′ p) results within a variation of 15%.

41

Figure 19: Typical spectra obtained from RCNP experiment at 392 MeV. Each residual nuclei low-lying state is well isolated from others, with the exception of the 1/2+ state of 39 K.

42

Figure 20: The differential cross sections and analyzing powers for 12 C(p, 2p)11 B reactions leading to (upper plot) the ground 3/2− state and (lower plot) the excited 1/2− state of the residual nuclei. The circled numbers correspond to the four different kinematical conditions shown in Fig. 16(b). The data 1 to ⃝ 3 and calculations are plotted as functions of the angle of the rightward outgoing protons for ⃝ 4 The upper abscissas provide scales for the recoil and the energy of the leftward outgoing protons for ⃝. momentum p3 . The solid lines (dashed lines) indicate the results of DWIA (PWIA) calculations.

43

Figure 21: These plots are the same as those in Fig. 20 but with data for 40 Ca(p, 2p)39 K reactions leading to (upper plot) the ground 3/2+ state and (lower plot) the excited 1/2+ state of residual nuclei. In the lower plot, the DWIA calculations employed 2s1/2 and 1f7/2 components that are not separated in the experimental data, and those components are indicated by thin solid and dotted lines, respectively.

44

625

630

635

640

645

5.3. Worldwide data and a comparison with systematic DWIA calculations Presently available (p, 2N) data intended to study discrete final states are summarized in Table 5.3. We exclude very old data for which the energy resolution is insufficient compared to more recent data, as well as lower energy data for which the reaction mechanism appears not to be a simple knockout reaction. From the remaining data, we chose reasonably separated 16 O and 40 Ca target values and extracted S-factors for comparison with DWIA calculations using modern optical potential parameters. The purpose of this exercise was to examine the incident energy effects when extracting S-factors. We also selected experimental data for incident energies in the vicinity of 200 MeV, and similar analyses were applied in order to examine the target dependence. Finally, an analysis of the highly excited states of the residual nucleus in the case of a 48 Ca target is shown as a means of examining the binding energy dependence. Theoretical calculations were carried out using the latest version of the code threedee [44], which employs non-relativistic DWIA. The spin degree of freedom was fully included in these calculations and a detailed formalism is given in Section 3. Even though this code allows the use of the density dependent nucleon-nucleon t-matrix, we instead employed a factorized approximation of the amplitude level and utilized the SP07 phase-shift solution [103] to extract the experimental p-p scattering amplitudes. Distorted waves were calculated using the Schr¨odinger equivalent reduction of the global Dirac-type parameter, EDAD1, developed by Cooper et al. [61]. Each of the scattering wave functions were multiplied by the Darwin factors, Eq. (3.59), but no additional nonlocality corrections were applied. The bound state wave functions were generated by the conventional well-depth method using Woods-Saxon potentials, and the geometrical parameters were taken from DWIA analyses of (e, e′ p) data [20]. In addition, a nonlocality correction using the Perey factor, Eq. (3.58), with a range parameter βNL value of 0.85 fm [58] was applied to the bound state wave functions. The S-factors were obtained using ordinary least squares fitting, although in some cases experimental data in the small cross section region or a very few irregularly deviated data points had a tendency to force the S-factor values up or down. To avoid this, we employed cross section data with limited ranges of recoil momentum, p3 ≤ 50 MeV/c for s-states and 35 MeV/c ≤ p3 ≤ 150 MeV/c for p-, d-, and f -states, which correspond to the regions in which the differential cross sections are greater than a half or one third of the nearby peak values. Even in these regions, data points that deviated more than three error bars from the fitted lines were excluded. We also attempted to estimate the uncertainty in the S-factors to reflect the fitting quality. When a data includes many data sets of recoil momentum dependences, as in the case of the TRIUMF or the RCNP values shown above, simple statistical treatment gives an estimate for the S-factor with unrealistically small uncertainty. In such case, the reduced chi-square of the fitting is much more than unity, since the theoretical calculations does not necessarily reproduce relative peak heights very well. Accordingly, we introduced the parameter α to represent the level of disagreement, and re-defined the reduced chi-square as n

χ2 /d.o.f. =

1 ∑ (σi − SσiDWIA )2 , n − 2 i=1 δi2 + (αSσiDWIA )2

(5.1)

as well as searched for the optimum values of S-factor S and α that generate a minimum of this reduced chi-square value equal to unity. In this expression, σi and δi (i = 1 ∼ n) are 45

650

the i-th experimental cross section data and the associated error, σiDWIA is the theoretical cross section for which the kinematical conditions correspond to the i-th data, and n is the number √ of data points. Through this procedure, we determined the S-factor value to be S ± ∆S 2 + α2 S 2 , where ∆S is the uncertainty of the derived S-factor resulting from a typical statistical treatment when α = 0 and d.o.f. = n − 1.

46

47

6)

5)

4)

3)

2)

1)

0.0[1p1/2 ], 6.32[1p3/2 ] 0.0[1s1/2 ] 0.0[1p3/2 ], 2.13[1p1/2 ] 0.0[1d3/2 ], 2.52[2s1/2 ] 0.0[3s1/2 ], 0.35[2d3/2 ], 1.35[1h11 ], 1.67[2d5/2 ] 0.0[1s1/2 ] 0.0[1d3/2 ], 2.52[2s1/2 ] 0.0[1s1/2 ] 0.0–1.9[3s1/2 +2d3/2 +1h11 +2d5/2 ] 0.0[3s1/2 ], 0.35[2d3/2 ], 1.35+1.67[1h11 +2d5/2 ] 0.0+2.13[1p3/2 +1p1/2 ] 0.0[1p3/2 ],2.13[1p1/2 ] 0.0[1p1/2 ], 6.32[1p3/2 ] 0.0[1d3/2 ], 2.52[2s1/2 ], 6.74[1d5/2 ]6) 5.8[1h11/2 +3s1/2 +2d3/2 +2d5/2 ] 0.0[1p3/2 ], 2.13[1p1/2 ] 0.0[1d3/2 ], 2.52[2s1/2 ] 0.0[1d3/2 ], 2.52[2s1/2 ] 0.0[2p1/2 ], 1.51[2p3/2 ]+1.75[1f5/2 ] 0.0[1p1/2 ], 6.32[1p3/2 ] 0.0[1d3/2 ], 2.52[2s1/2 ] 0.0[1p1/2 ], 6.32[1p3/2 ] 0.0[1d3/2 ], 2.52[2s1/2 ] 0.0[1p3/2 ], 22[1s1/2 ] 0.0[1p3/2 ] 0.0[1d3/2 ], 2.52[2s1/2 ] 0.0[1p], 10.6[1s] 0.0[1p], 10.6[1s]

many O 4 He 12 C 40 Ca 208 Pb 4 He 40 Ca 4 He 208 Pb 208 Pb 12 C 12 C 16 O 40 Ca 197 Au 12 C 40 Ca 48 Ca 90 Zr 16 O 40 Ca 16 O 40 Ca 6 Li 7 Li 40 Ca 7 Li 9 Be 16

Ex (MeV) [orbit]

Target 1)

PNPI TRIUMF TRIUMF RCNP RCNP RCNP TRIUMF TRIUMF TRIUMF iThemba iThemba iThemba4) IUCF5) TRIUMF TRIUMF iThemba4) RCNP RCNP RCNP RCNP iThemba4) IUCF U. Maryland U. Maryland U. Maryland U. Maryland U. Maryland Ukraine Ukraine

Lab. [104]2) [36, 91] [105] [101, 102] [101, 102] [106, 107] [105] [63, 108]3) [105] [109] [100] [110] [111, 112] [88] [89] [113] [114] [115] [114] [114] [116] [117] [118] [118] [119] [120] [118] [121] [122]

Refs.

Many levels are included.

For differential cross sections, the summed values of 1p3/2 and 1p1/2 contributions are given.

Named NAC when the data were published.

It is written in Ref. [64] that the cross section data is arbitrary as a private communication with a member of the experimental group.

and references therein.

Only absolute cross sections are provided in this paper. Relative values of differential cross sections and induced polarizations are given in [6, 123],

Named LINP when a part of data were published.

76 70

100

151 148 101

197

350 300 250 204 202 200

Energy (MeV) 1000 505 500 392

Table 3: Worldwide (p, 2p) data intended for the study of discrete final states. The + sign in the Ex and orbit columns indicates plural peaks are convoluted.

655

660

665

670

675

680

685

690

695

700

Since the 505 MeV data and the fitting quality are shown above in Fig. 18, we present here a lower energy example. Figure 22 summarizes 40 Ca target data obtained by Samanta et al. [118] and the results of the present reanalysis. These data were acquired using two sets of counter telescope settings at five angle pairs. In the original paper, the cross section data were plotted as functions of the kinetic energies of forward outgoing protons, but the recoil momenta are employed in this figure in order to demonstrate that the peak position is stable in this type of plot. The finite thick lines in the figure are the results of a best-fit DWIA calculation using a single S-factor for all five distributions. The thickness of the lines indicate the statistically calculated standard deviations, while the shaded areas are the uncertainties estimated by introducing the parameter α discussed above. As shown in the figure, α was 0.29 and the S-factor including this uncertainty was 2.60±0.76, which means the resulting S-factor is reliable within approximately 30%. In the original work, an S-factor value of 5.0 at this energy and 4.0 at 101 MeV are reported. The improvements in the optical potentials and the calculation code, performed by two authors of the work, are therefore significant [44]. At this point, we present the results of the systematic re-analysis. Firstly, we focus on the S-factors determined using the 16 O and 40 Ca target data acquired at various incident energies, plotted in Fig. 23. Even though the quantity of data is relatively small, this figure implies that the S-factors were determined with uncertainties ranging from 10 to 15% at energies higher than 200 MeV, and approximately 30% or more at 100 MeV and below. Within these uncertainties, the S-factors are consistent with the (e, e′ p) results, and these values and the associated uncertainties are indicated by thick lines and shaded areas, respectively. It is evident that the extremely small error bar for the 40 Ca target S-factor in the vicinity of 200 MeV results from the inclusion of just one data set that contains a single cross section peak. In such case, the process used to estimate uncertainties as described above does not work properly. Next we consider the results obtained using various target data at approximately 200 MeV. Figure 24 is similar to the previous plot, except that the S-factors are plotted relative to those reported from (e, e′ p) studies. Here, the horizontal axis scale represents the target mass number, although different orbits having the same target nucleus have been slightly shifted to avoid overlapping. The error bars for the majority of the Sfactors in this plot are significantly shorter than those in Fig. 23, although this results from using a similar quantity of data points as in the previous 40 Ca data set. That is, only one data set was included. Therefore, we would expect that similar uncertainties as in Fig. 23 would also be included in each of the S-factors in Fig. 24. In this case, these S-factors are approximately consistent with the (e, e′ p) results, although some systematic deviations are apparent, which are smaller for light targets and larger for heavy targets. As noted in Section 2, Kramers et al. compared the spectroscopic factors obtained from (e, e′ p) and (d,3 He) experiments and showed that consistent results were possible if consistent analyses were applied [20]. They examined the S-factors for both proton knockout and pick up from 48 Ca up to an excitation energy 8.1 MeV of the residual 39 K. Here we present a similar comparison using recent RCNP data acquired with a 48 Ca target [114]. Figure 25 presents a sample of the spectrum. In this figure, each of the excited levels of the residual 37 K studied in Ref. [20] are shown. In this analysis, however, two peaks at approximately 5.3 MeV, and those at approximately 8.0 MeV as well, are jointly analyzed and peaks at 6.51 MeV and 6.87 MeV are excluded from the analysis because of low statistics. An example of the differential cross section and the analyzing 48

Figure 22: Experimental data obtained for

40 Ca(p, 2p)39 K(g.s.)

and results from DWIA calculations.

Figure 23: Spectroscopic factors obtained using 16 O and 40 Ca(p, 2p) data at various energies. The thick lines and surrounding shaded areas indicate S-factor values and their error ranges from (e, e′ p) analyses [20]. The ±20% deviations from the (e, e′ p) results are also indicated. See the text for a discussion of the error bars. The references for these experimental data are summarized in Table 5.3.

49

Figure 24: Spectroscopic factors obtained using various target (p, 2p) data at incident energies close to 200 MeV plotted relative to the (e, e′ p) results. The heights of the blocks on the (e, e′ p) line, the unity lines, indicate uncertainties in the (e, e′ p) results. Also shown are ±20% deviations from the (e, e′ p) results. The data in parentheses are results of multipole decomposition analysis. The references for these experimental data are summarized in Table 5.3.

705

710

715

720

725

power are provided in Fig. 26, along with the results of DWIA calculations. The data are seen to be well reproduced by the calculations and the j value of the hole state is confirmed to be 5/2. A comparison of the S-factors obtained from the (p, 2p) and (e, e′ p) reactions is given in Fig. 27. Even though the S-factors of the 2s1/2 knockout associated with the (p, 2p) reaction are uniformly less than the values obtained from the (e, e′ p) reaction, the same consistency between the values as seen in Figs. 23 and 24 is evident. These comparisons demonstrate that the (p, 2p) and (e, e′ ) results are consistent, typically within 15% at energies above 200 MeV, and within approximately 30% in the vicinity of 100 MeV. uncertainties in the determination of the S-factors and possible differences between S-factors deduced from (p, 2p) and (e, e′ p) data are discussed in Section 5.5. Table 5.3 provides the S-factor values derived from the above consistent analysis. 5.4. Multipole decomposition analysis As shown in Figs. 20, 21 and 26, (p, 2p) data obtained at incident energies higher than 200 MeV are fairly well reproduced by DWIA calculations. This enables multipole decomposition analysis (MDA), in which multiple S-factors associated with different orbits are extracted using a single recoil momentum dependence of differential cross sections, or differential cross sections and analyzing powers. One example is already shown in the lower plots of Fig. 21. In the case of the 40 Ca(p, 2p)39 K measurement performed at RCNP, the 1/2+ state of the residual nucleus is not separated from nearby states and the contribution of the adjacent 7/2− state is not negligible. Even though the differential cross sections at around the zero recoil momentum point are dominated by 2s1/2 knockout, the corresponding theoretical calculation, the thin solid lines, significantly underestimate the experimental cross sections in the pB ≈ 200 MeV/c region. This discrepancy is resolved by adding the cross sections of the 1f7/2 knockout, shown by the dotted lines, and from the optimized strengths, the S-factors are determined. The resultant S-factor values are 0.89 and 0.34 for 2s1/2 and 50

Table 4: Spectroscopic factors extracted from the consistent analysis of (p, 2p) data. The errors given in the parentheses do not include those caused by uncertainties in the absolute values of the experimental cross sections, which are typically 10% to 15%. Spectroscopic factors from (e, e′ p) data and r0 values are from [20, 23].

Target nucleus 12 C

16

40

48

Ca

Ca

90

Zr Pb

208

∗

O

Ex (MeV) 0.00

3/2−

2.13

1/2−

5.02

3/2−

0.00

1/2−

6.32

3/2−

0.00

3/2+

2.52

1/2+

2.81

7/2−

0.00 0.36 3.42 3.85 3.95 5.24 5.49 7.81 8.13 0.00 0.00 0.35

1/2+ 3/2+ 5/2+ 1/2+ 3/2+ 5/2+ 5/2+ 5/2+ 5/2+ 1/2− 1/2+ 3/2+

Jπ

T0 (MeV) 392 197 392 197 392 197 505 200 150 101 505 200 150 101 392 200 197 101 76 392 197 101 76 392 197 197

197 202 202

Results of multi-pole decomposition analyses.

51

S(p, 2p) reanalysis 1.82(3) 1.30(7) 0.30(2) 0.23(3) 0.23(2) 0.15(1) 1.00(26) 1.04(8) 1.07(10) 1.23(20) 2.06(50) 2.46(12) 1.95(16) 2.01(54) 2.11(12) 3.47(7) 2.72(41) 2.48(46) 2.57(56) 0.89(5)∗ 0.92(9)∗ 0.91(18) 1.14(19) 0.34(4)∗ 0.58(31)∗ 0.75(5) 2.15(24) 0.71(2) 0.11(1)∗ ∗ } 0.39(2) 1.08(7) } 0.75(6) 0.70(2) 1.04(15) 2.56(46)

1.72(11)

r0 (fm) 1.35

0.26(2)

1.65

0.20(2)

1.51

1.27(13)

1.37

2.25(22)

1.28

2.58(19)

1.30

1.03(7)

1.28

0.38(4)

1.35

1.07(7) 2.26(16) 0.68(5) 0.17(1) 0.32(3) 0.29(2) 0.75(5) 0.43(3) 0.23(2) 0.72(7) 0.98(9) 2.31(22)

1.23 1.25 1.13 1.29 1.29 1.19 1.18 1.24 1.30 1.32 1.25 1.23

S(e, e′ p)

(p, 2p) data [101, 102] [114] [101, 102] [114] [101, 102] [114] [36, 91] [88] [113] [118] [36, 91] [88] [113] [118] [101, 102] [89] [115] [118] [118] [101, 102] [115] [118] [118] [101, 102] [114] [114]

[114] [100] [100]

Figure 25: A typical spectrum of a 48 Ca(p, 2p) reaction at RCNP. The arrows indicate peak positions possibly resulting from 16 O contamination of the target. These contribution were subtracted using measurements with a 48 CaO target.

Figure 26: Differential cross section and analyzing power for the 48 Ca(p, 2p) reaction leading to a residual 47 K nucleus with a 8.0 MeV excitation energy. The full lines indicate the results of DWIA calculations assuming a 1d5/2 knockout. In the Ay plot, calculation results corresponding to a 1d3/2 knockout are shown by the dashed line.

52

Figure 27: Spectroscopic factors determined from the 48 Ca(p, 2p) reaction are plotted relative to those obtained using an (e,e’p) reaction. The heights of the blocks indicate the errors in the (e, e′ p) S-factors. The dashed lines show ±20% deviations.

730

735

740

745

750

755

1f7/2 knockouts, respectively, which are consistent with the values deduced using (e, e′ p) reactions, 1.03 and 0.38. At the same time, the analyzing power data is also better reproduced by this MDA. Figure 28 presents an MDA using a 48 Ca(p, 2p)47 K data at 197 MeV performed for demonstration purposes. As described in the caption, the experimental data were generated by summing two adjacent peaks in the separation energy spectra, corresponding to the ground state and first excited state peaks in Fig. 25, and the two components were separated using MDA. In this case, the S-factor for the 1d3/2 orbit deduced using MDA was 2.70, which is meaningfully larger than the result by state-by-state analysis, 2.15. This is caused by the fact that the DWIA calculation for 2s1/2 orbit slightly underestimate the experimental data in the large T1 region. Even though the difference may not be important in determination of 2s1/2 S-factor, it is clear from the figure that the discrepancy causes meaningful increase of the S-factor for the 1d3/2 component. The similar situation is appearing in the analysis of the peak at approximately 3.9 MeV in Fig. 25. The same combination of 1/2+ and 3/2+ peaks are not separated and so a similar MDA was applied. As a result, obtained S-factor for the 1d3/2 component is 0.39, which is considerably larger than the (e, e′ p) result, 0.32. Careful examinations of fitting quality are required in order to evaluate the reliability of MDA results. Here we have only concerned ourselves with the application of MDA to the differential cross sections. However, similar analyses are expected to be possible even for analyzing powers because these are also reproduced by DWIA calculations, with the exception of those cases discussed in Section 6, and a remarkable j-dependence, caused by the effective polarizations, is observed, as shown in Fig. 29. 5.5. Uncertainties in extracting S-factors and comparison with (e, e′ p) results Non-locality correction and optical potential. As discussed in the last part of Section 3.1, DWIA calculations include uncertainties in the nonlocal corrections. In the case of the Shur¨odinger-based optical potential, the use of the Perey factor for the phenomenological potential is not necessarily justified theoretically, and in the case of the Dirac-based 53

Figure 28: A demonstration of MDA using 48 Ca(p, 2p) data acquired at an incident energy of 197 MeV, the sample spectrum for which is shown in Fig. 25. The full circles indicate experimental data obtained from sums of the ground state and first excited state peaks. The solid line shows the fit obtained by MDA, which consists of the 2s1/2 and the 1d3/2 components indicated by the short dashed and long dashed lines, respectively. As shown in Fig. 25, these two peaks are practically separated and data for each state are obtained, as indicated by the squares and triangles. The dashed line is a result of state-by-state analysis for the 1d3/2 component, while the result for the 2s1/2 component is essentially the same as the short dashed line.

Figure 29: The differential cross sections and analyzing powers for the 48 Ca(p, 2p) reaction leading to the ground 3/2+ state and Ex = 5.5 MeV 5/2+ state of the residual 47 Ca. A clear j-dependence, which is quantitatively reproduced by DWIA calculations, is seen in the case of the analyzing power.

54

760

765

770

775

780

785

790

795

800

potential and its non-relativistic reduction, it is unknown whether or not sources of nonlocality other than the Darwin term exist. In addition, even when employing the modern global optical potential parameter, the calculation results may depend on the chosen parameter set, and a numerical examination of these effects is presented herein. Figure 30 shows the S-factor values extracted from comparisons of experimental data and the results of DWIA calculations with various corrections or different potential parameters. The experimental data represent the differential cross section of the 1p1/2 proton knockout from a 16 O target at an incident energy of 200 MeV, as acquired by Kitching et al. [88]. In the figure, the upper full circle shows the standard calculation result, employing the EDAD1 potential with nonlocality correction using the Darwin factors for all three scattering wave functions and the Perey factor for the bound state wave function. This S-factor value and error bar are the same as those of the corresponding points in each of Figs. 23 and 24. The next three points are the results obtained when the nonlocality corrections are modified. In the first case, the open circle, the Perey factors for three scattering waves are multiplied in addition to the Darwin factor. In the second case, the full triangle, the Darwin factor is excluded, and in the third case, the open triangle, the Perey factor for the bound state wave function is excluded, retaining the Darwin factor for scattering waves. From these points, it is apparent that the Darwin correction pulls the S-factor up by almost 20%, while using the Perey factors for scattering wave functions results in a shift of about 12%. In addition, the Perey correction to the bound state wave function lowers the S-factor and this effect is somewhat stronger than applying the same correction to all three scattering wave functions. The next three points represent results obtained using different potential sets parametrized by the same group that developed EDAD1. The parameter sets EDAD2 and EDAD3 are global potentials similar to EDAD1 but with different parameterizations, while EDAIO is a parameter set optimized for p + 16 O scattering only. As can be seen from the figure, the former two potentials give essentially the same results as EDAD1, based on a comparison of the error bars, while the EDAIO potential generates a somewhat smaller value. This is likely the result of also using the EDAIO parameters for the scattering waves in the final channel, where the potentials are produced by residual 15 N, not 16 O. In fact, our analysis confirmed that the similar EDAICA potential gives almost the same results as using EDAD1 to EDAD3 in the case of a 40 Ca target, where the mass difference between the target nucleus and the residual nucleus is only 2.5%. Lastly, the two diamonds are the results obtained by Koning and Delaroche using the Shr¨odinger-based potential set [124]. One remarkable observation is that the open diamond, representing the analysis for which the Perey corrections were excluded, is quite similar to the EDAD1 result generated without using the Darwin correction, shown by the full triangle. That is, the Dirac-based potential and the Shr¨odinger-based potential give almost the same DWIA results, and their different nonlocal corrections cause some deviation between the two, represented by the difference between the top full circle and the full diamond. Therefore, if we wish to study the S-factors at a 10% level of accuracy, it is essential to justify the nonlocality correction. In the present case, we employed Dirac-based potentials for the scattering states and Shr¨odinger-type potentials for bound states. However, if the same potentials had been employed for both states, the resulting values would have been somewhat different.

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Figure 30: Effects of non-local corrections and optical potentials on the extracted S-factors. See the text for details.

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Absorption effect. In (p, 2N) reactions, nuclear distortion effects play a crucial role and, among these effects, absorption is the most important. Since the nuclear force is highly energy dependent and the size of the nucleus varies with the mass number, the absorption effect is also energy dependent and target dependent. As such, the absorption effect changes significantly in Fig. 23 and in Fig. 24. In order to estimate the extent of such changes, we compared the DWIA and PWIA cross sections at peak points, and the resulting ratios of DWIA and PWIA calculation results are plotted in Fig. 31. Here the left panel presents the energy dependence of the ratio for 2s1/2 and 1d3/2 knockouts from a 40 Ca target. The ratio is seen to be roughly constant at energies higher than 250 MeV, and the ratio values are close to 0.3 and 0.2 for the two kinds of knockout. However, these similarities decrease significantly in the lower energy region, and drop to only several percent around 100 MeV. This decrease leads to the increases in the uncertainties in determining S-factors at lower energies discussed above. The right panel summarizes the target dependence of the ratios corresponding to the targets and knocked out orbits in Fig. 24. These ratios are scattered as the result of orbit-dependent fluctuations, but still exhibit a general decrease as the mass number increases. For a 208 Pb target, the ratio is only 3%, such that 97% of the flux is escaping to different final channels. It is important to stress again that intensive efforts in optical potential parametrization and in improvement of theoretical calculations have made it possible to extract S-factors with the uncertainty level shown in Figs. 23 and 24 in spite of such strong absorption effects. Differences in nuclear regions accessed by (e, e′ p) and (p, 2N) reactions. The (p, 2N) reactions are known to suffer significantly from absorption. For this reason, there have been attempts to visualize the manner in which each part of the nuclear region contributes to the cross section and to examine the reliability of factorized calculations. One means of doing so is to calculate differential cross sections excluding the inner regions defined by various cut off radii from the integration, and to plot the differences between calculations for adjacent cut off radii [100, 118]. A similar procedure was employed in a recent work that compared (e, e′ p) and (d,3 He) results [20]. In a study of the medium effect in 56

Figure 31: The ratios of calculated DWIA and PWIA cross sections. The left panel presents the energy dependence of the ratio for the given orbit knockout from a 40 Ca target. The right panel summarizes the target dependence at an incident energy of 200 MeV, corresponding to Fig. 24. The ratios were taken at a specific cross section peak in each knockout cross section.

(p, 2p) reactions, the group concerned defined an effective mean density and introduced Tr a weighting function as a tool to estimate the density. This function, written as δreal (R), has properties that represent the real part of the integrand of the transition matrix and is hereafter termed the transition matrix density. However, as shown in Appendix A, the integration of this function by the radius parameter R gives the differential cross section as ∫ ∞ Tr σ= (R) dR. δreal (5.2) 0

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Here, σ is the calculated differential cross section from Eq. (3.33) in the case of DWIA Tr calculations for (p, 2N) reactions. Therefore, δreal (R) shows the contribution of each part of the radius to the differential cross section. The transition matrix densities are calculated and plotted in Fig. 32. Here, the (e, e′ p) reactions are simulated by switching off the distortion for incoming and forward (32.5◦ ) outgoing protons. As seen in the figure, the (e, e′ p) results are very close to the PWIA results, but the (p, 2p) data are slightly shifted to the outside, meaning that the inside is significantly suppressed compared to the other two types of calculations. This effect can be seen more clearly in the negative contribution region in (b). Therefore, even though the difference between S-factors derived using (e, e′ p) data and those obtained using (p, 2p) data will not be large, meaningful differences may arise. It may also be possible to study bound state wave functions more precisely by comparing the S-factors determined from (p, 2p) and (e, e′ p) reactions, provided that the reliabilities of nonlocal corrections and of other corrections are improved. In this figure, the transition matrix density for a (d,3 He) reaction at an incident energy of 56 MeV is also plotted on the right hand side. The shape of this plot is quite different from that of knockout reactions and the internal portion exhibits significant fluctuations, as has been previously discussed in Ref. [20]. It should be noted that the scattering angle of this calculation corresponds to the first peak of the 1d3/2 orbit transfer, and that the transferred momentum q is 53 MeV/c, intermediate between the recoil momenta of the knock out reactions in (a) and (b). These oscillation-like fluctuations most likely result from the difference in the potential depths of deuterons and 3 He particles. 57

Tr (R) for a 40 Ca(p, 2p)39 K reaction leading to the 2s Figure 32: The transition matrix densities δreal 1/2 state (a) and the 1d3/2 state (b) at an incident energy of 392 MeV. The recoil momenta are 0 MeV/c and Tr (R) 100 MeV/c, respectively, corresponding to the cross section peak. The thick full lines indicate δreal for (p, 2p) reactions, while the thin full lines show simulated (e, e′ p) reactions for which distortions are switched off except for one outgoing proton, and the dashed lines indicate PWIA results for which all distortions have been switched off. The density distribution of the 40 Ca target is also plotted with dotted Tr (R) for (d,3 He) reaction leading to the 1d lines. Plot (c) presents data for the δreal 3/2 state. All lines have been normalized to unity.

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The modification of the properties of mesons and nucleons in the nuclear medium is one of the most interesting topics in current nuclear physics. The reduction of hadron masses in this medium has been predicted as an effect of the partial restoration of chiral symmetry in nuclear matter [125], and the modification of the nucleon spinor in nuclear matter has also been discussed in the framework of a Dirac approach [126]. These modifications may result in variations of the nucleon-nucleon (NN) interaction with density. Within the nucleon degree of freedom, the effect of Pauli blocking has been investigated and density dependent g-matrices have been constructed. It is anticipated that it will be possible to examine these in-medium modifications of the NN interaction through (p, 2N) reactions, which are regarded as nucleon-nucleon scatterings in a nuclear field. In this section, we review those studies that have investigated possible modifications of nuclear interactions in the nuclear medium. 6.1. Definition of effective mean density for (p, 2N) reactions In order to probe the nuclear medium effect, it is necessary to employ nuclear reactions that take place in the interior of the nucleus, and it is preferable to know to what extent the nuclear interior contributes to each such reaction. The effective mean density defined in Ref. [127] or Ref. [84] provide a scale for this purpose. Here we introduce it following the former work based on DWIA, which is consistent with the description in Section 3. Some results of the latter work are presented in Appendix A. Initially, we can suppose that the matrix for nucleon-nucleon scattering is dependent on density in a linear manner, and can be written as t = t0 + t1 ρ(R),

(6.1)

where ρ(R) denotes the nuclear density at radius R. In this case, neglecting the spin degree of freedom and assuming a spherical target nucleus, the differential cross section 58

of a (p, pN) reaction in a non-relativistic factorized DWIA can be expressed as ∫ dσ 3 =Fk t0 χ1 (R)∗ χ2 (R)∗ ϕ(R)χ0 (R)dR dΩ1 dΩ2 dE1 2 ∫ +t1 χ1 (R)∗ χ2 (R)∗ ρ(R)ϕ(R)χ0 (R)dR 2 ∫ 2 =Fk |t0 + t1 ρ¯| χ1 (R)∗ χ2 (R)∗ ϕ(R)χ0 (R)dR ,

(6.2)

where Fk is a kinematical factor, and χi (R) and ϕ(R) are the distorted waves of the incident (i=0) and outgoing (i=1,2) particles and the bound state wave function, respectively. The effective mean density ρ¯ is defined as ∫ ∞ ∫ ∞ ρ¯ D(R)dr ≡ ρ(R)D(R)dr (6.3a) 0 ∫0 D(r) = χ1 (R)∗ χ2 (R)∗ ϕ(R)χ0 (R)R2 dΩ. (6.3b) In general, ρ¯ is a complex number because D(R) is a complex function. As shown in Appendix A, the real and imaginary parts of ρ¯ can be calculated as ∫ ∫ 1 ∞ 1 ∞ Tr Tr Re ρ¯ = ρ(R) δreal (R)dR, and Im ρ¯ = ρ(R) δimag (R)dR, σ 0 σ 0 (6.4) 3

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dσ Tr Tr where σ = dΩ1 dΩ , δreal (R) and δimag (R) are the transition matrix density and its 2 dE1 imaginary counterpart as defined in Section 5 and Appendix A. Equation (6.2) indicates that ρ¯ represents the sensitivity of the (p, 2N) cross section to the density dependent term t1 of the NN amplitude. The above definition and calculation of the effective mean density are robust, at least semi-quantitatively. Therefore, if some medium effect exists that depends on the local nuclear density, and if we select a nuclear reaction for which the value of ρ¯ is sufficiently large, it should be possible to observed the medium effect in the reaction experimentally. If no ρ¯ dependence is observed, we can conclude that this medium effect is unimportant. Because of the strong effect of absorption on the incident and outgoing nucleons, (p, 2N) reactions occur primarily in the peripheral regions of the target nuclei. Therefore, the effective mean density introduced above typically has a low value, such as 15% or less of the nuclear saturation density, in the vicinity of the cross section peak regions where the reaction mechanism is expected to be simple. This is consistent with the treatment in the previous section, in which the NN amplitudes in free space have been employed in order to extract spectroscopic information from these reactions. However, the knockout of deeply bound 1s1/2 nucleons represents an exception. In the case of the 1s1/2 state, the bulk of the wave function is distributed over the entire nuclear region, including the small radius zone, as a result of zero angular-momentum, while its exterior exponential tail is suppressed by the effect of a large binding energy value. As a result, the effective mean density can equal more than 30% of the saturation density, which gives us a possibility to study the medium effect through the (p,2N) reactions. 59

6.2. g-matrix and correlation studies and (p,2N) reactions As mentioned in Section 3, the transition interaction for the (p,2N) process in the nonrelativistic framework is the effective interaction τ12 between particles 1 and 2 defined in multiple scattering theory [45]: τ12 ≡ v12 + v12

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1 τ12 , E − K − HA − m0 − mA + iϵ

(6.5)

where E, K, and HA are the total energy, the total kinetic energy operator, and the Hamiltonian of A, respectively. ϵ is an infinitesimal constant that is taken to be 0 after all the calculation. In the usual NRDWIA, τ12 is approximated to an on-shell NN effective interaction in free space, tfree . Although the RDWIA takes into account the contribution of anti-particle sectors, the interaction is still evaluated in free-space, meaning that no external field exists. In some cases, especially when the NN scattering takes place in a region where the nuclear density is high, many-body effects on tfree will become important. As a practical means of accomplishing this, the so-called g-matrix interactions obtained in infinite nuclear matter have been implemented. Up to about 200 MeV of the incident nucleon energy, the Pauli blocking plays a significant role in NN scattering in nuclear medium. Accordingly, many g-matrices have been constructed by solving Eq. (6.5) with approximating HA by an infinite nuclear-matter description; the mean-field potential of the nuclear matter and the Pauli blocking effect are taken into account in non-relativistic [128–134] and relativistic [135, 136, and references therein] frameworks. However, most of these theoretically constructed g-matrices have been examined by using experimental data of nucleon elastic and inelastic scattering. Contrarily, comparison with experimental (p,2N) data is quite limited. We will mention those limited comparisons in the following subsections. At higher energies, the Pauli blocking effect is expected to be small and other correlation effects become important relatively. Ryckebusch and others theoretically examined the short range correlation effect using the relativistic multiple-scattering Glauber approximation [137]. Their result shows that the distorted momentum distribution for the 12 C(p, 2p) reaction with knockout from the 1s1/2 orbit increases about 10% by the effect and about 8% in the case of the 1p3/2 orbit. They examined the effect at an incident energy of 1.4 GeV, but they are suggesting that the energy dependence of this effect is moderate and similar amount may appear even in the several hundred MeV region. As mentioned in the previous section, the theoretical estimation of the absolute cross section includes significant uncertainties. In the case of the 1s1/2 knockout, the absorption effect is more serious than the knockout from the peripheral orbits and it is not an easy task to extract conclusive results from above 10% or 8% effect, or even the effect is stronger than those. One of the practical way to study these nuclear effects is to examine the ratio of the cross sections, spin observables in particular. As such, the s-state knockout has another advantage over other knockout types. That is, the simple relationship between the spin observables of the (p, 2N) reactions and those of the elementary p-N processes that results from the absence of effective polarization noted in Section 2 does hold. Accordingly, the following subsections focus primarily on spin observables for the 1s1/2 knockout. In addition, since the nuclear medium effects on spin observables are not necessarily studied systematically, the discussions are forced to be in a phenomenological manner based on the existing experimental data. 60

Figure 33: Analyzing powers for the s1/2 knockout (p, 2p) reactions of three target nuclei at an incident energy of 392 MeV [127]. The data are plotted as functions of effective mean density. The forward emitted proton detection angle was 25.5◦ while the forward emitted proton energy value and the backward emitted proton angle corresponded to free proton-proton scattering.

6.3. Analyzing powers for the 1s1/2 knockout

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As discussed in Section 2, a pioneering study concerning the 1s1/2 knockout using a polarized beam was performed by a TRIUMF group at 500 MeV with a 16 O target [35]. This research determined that the analyzing power is significantly less than the values predicted by DWIA calculations using the free NN t-matrix. These results were subsequently confirmed by an RCNP experiment at 392 MeV [127] and this effect more clearly demonstrated, as shown in Fig. 33, which plots the Ay data for three target nuclei as functions of the above-discussed effective mean density. The experimental data show a clear monotonic decrease of the density, while the results of PWIA and DWIA calculations do not reproduce this trend, strongly suggesting the existence of some nuclear effect on the NN interaction. It is worth noting that the nucleon knockout reaction is the most dominant process at intermediate energies, and the kinematical condition employed when obtaining these data generated close to zero recoil, such that the differential cross section of the 1s1/2 knockout reaction was maximized. As a result, this phenomenon may be observed at the cross section maximum of the primary process, as opposed to attempting to obtain evidence from a small fraction of a process. It is also worth mentioning that the admixture of multistep processes estimated from the separation energy spectra is only several percent of the cross section and does not cause significant shift of the analyzing power in the case of 12 C target [101]. This estimation is also consistent with the analysis of the rescattering processes in Section 7. The TRIUMF group subsequently published new analyzing power data for a 16 O target at 504 MeV. In this work, the kinematical conditions were chosen to emphasize interactions in the nuclear interior, including zero-recoil points for the 1s1/2 knockout and an asymmetrical angle setting at which the analyzing power of the elementary p-p process was significant. They compared their data with the results obtained from various DWIA calculations, non-relativistic calculations using the free NN interaction, an empirical 61

Figure 34: Analyzing power for the 16 O(p, 2p) reaction at 500 MeV. The data are plotted as functions of the energy of forward emitted protons and the zero-recoil points are indicated by arrows. The dashed, dotted and solid lines in the left plot are non-relativistic DWIA calculations using a free p-p t-matrix, a t-matrix including the m∗ effect and an empirical density dependent t-matrix, respectively. The long dashed line shows the result of fully relativistic DWIA calculations. In the right plot, the solid line is the same calculation as the solid line in the left plot, while the dotted-dashed (long dotted-dashed) line is a calculation with the spin orbit potential of the incident proton (both the incident and emitted protons) set to zero. Theese data and calculations are taken from Ref. [36].

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density dependent interaction parameterized by Kelly [138], a density dependent t-matrix with a so-called m∗ correction according to Horowitz and Iqbal [80], and a fully relativistic calculation [66]. The results demonstrated that none of these theoretical models produced values that were in good agreement with the experimental data and confirmed that the empirical analyzing power values obtained for the 1s1/2 knockout are typically much less than those derived from calculations, as shown in the left plot of Fig. 34. This same study also found that the analyzing power values obtained from fully relativistic calculations exhibit a steeper slope when plotted as a function of outgoing energy, compared to the plot resulting from the non-relativistic values, and that the results of these two methods are similar to one another only in the vicinity of the zero-recoil point. In addition, it was demonstrated that spin orbit distortions greatly affect the DWIA predictions of the 1s1/2 knockout analyzing power over the majority of the data range, with the exception of the region close to the zero-recoil momentum point. From these findings, they concluded that the spin-orbit part of the optical potential requires improvement, and that the analyzing power data at the points of negligible recoil momentum constrain the two-body effective interaction independent of the uncertainty in the optical potential. Taking these results into account, the following two subsections examine the experimental studies regarding spin observables at the zero recoil points. 6.4. Analyzing powers and induced polarizations for the 1s1/2 knockout under zero recoil conditions. A similar experiment to the RCNP work described above was performed at PNPI, Gatchina, at an incident energy of 1 GeV [99]. In this case, an unpolarized proton beam was employed and the induced polarizations of the emitted protons were measured using 62

Figure 35: Induced polarizations for s1/2 knockout (p, 2p) reactions at 1 GeV under zero-recoil conditions [99]. Free p-p scattering data are also plotted in the right hand panel. The data are plotted as functions of the effective mean density in the left hand panel, and the momentum transfer, which corresponds to the detection angles, in the right hand panel. The solid lines (dashed lines) provide the results of DWIA (PWIA) calculations using the free t-matrix, while the dotted lines show data from non-relativistic DWIA calculations including the m∗ effect. The dotted-dashed line in the right panel presents the results of a phase-shift analysis for free p-p scattering [103].

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a double scattering technique. The results are summarized in the left panel of Fig. 35. The experimental data, representing the polarizations of both the forward and backward emitted protons, are only slightly decreased from the free p-p values as functions of density in this case, whereas the results of PWIA and DWIA calculations increase according to the shift in the kinematics. As a result, the variations from the calculated values exhibit a similar trend to the 392 MeV results. This suppression effect in comparison to the simple DWIA and PWIA calculations is much greater even than the m∗ effect, which is consistent with the TRIUMF results obtained at 500 MeV. The angular dependence of the polarizations in the case of a 12 C target are plotted in the right panel of the figure along with data for p-p scattering and the results of theoretical IA calculations. These data show almost constant reductions from the p-p results and IA calculations. The observation that essentially the same reduction rates are observed over wide ranges of incident and outgoing proton energies, over which the contributions of multistep processes are expected to change significantly, essentially excludes the possibility that this phenomenon is caused by some reaction mechanism other than a single step process. An additional experimental study was performed at 392 MeV, [139]. In this experiment, Ay values were determined for various light target nuclei and the results were plotted as functions of the effective mean density and of the proton separation energy. The original purpose of this work was to assess the Ay suppression phenomenon while employing numerous targets and to examine whether or not the suppression phenomenon is actually an effect of density. In general, the effective mean density is highly correlated with the proton separation energy because deeper binding significantly reduces the tail portion of the bound state wave function. However, employing either 3 He or 4 He as targets gives exceptionally high mean densities if a simple local density concept is applied to these small, compact nuclei. The results obtained from this prior study are shown in 63

Figure 36: Analyzing powers for 1s1/2 knockout (p, 2p) reactions of various target nuclei at 392 MeV for a forward emitted angle of 25.5◦ , with zero recoil conditions [139]. The dashed line and the solid line in the right side panel are the results of PWIA and DWIA calculations using the free NN t-matrix, while the dotted-dashed and dotted lines are the results of DWIA calculations using an empirical densitydependent g-matrix [138] and the m∗ correction in the Dirac approach [29], respectively.

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Fig. 36. As can be seen, the Ay data exhibit a better correlation with the separation energy, implying that the Q value of the reaction (the difference between the incident and final two-body energies) could be the key parameter characterizing the suppression of Ay . Nevertheless, the situation is not conclusive. Based on the correlation between the Ay data and the Q values, it is natural to speculate that the inclusion of the energy off-shell effect of the NN interaction could possibly explain the suppression effect. However, it is noted in Ref. [139] that a trial calculation incorporating the energy off-shell effect based on the AV18 NN potential within the non-relativistic PWIA model results in only a negligible change in the value of Ay under the kinematical conditions of the experimental trials. Since the definition of the effective mean density is robust, at least semi-quantitatively, and since it is not unreasonable to treat nuclear effects, such as the lower component of the Pauli blocking effect, as density effects, these plots likely demonstrate that the 3 He and 4 He nuclei are too small to allow the application of the concept of nuclear density. In addition, as shown in the right hand panel of the figure, the distortion effect (the difference between the solid and dashed lines), is remarkably large in the case of a 4 He target, while the density dependent calculation, the dotted line, does not show significant deviations at the 4 He target point. However, the density dependent effect, the distance between the solid and dotted lines, is more pronounced than in the case of the other target. Further theoretical considerations will be required regarding this topic. In addition to the above, it has also been confirmed that the Ay values associated with a 4 He target exhibit even less suppression than those for a 6 Li target case in a PNPI experiment at 1 GeV [140]. Lastly, we present 1s1/2 knockout data that shows the isospin dependence of the suppression [96]. The measurement conditions used to obtain these data were similar to 64

Figure 37: Analyzing powers for (p, 2p) and (p, pn) reactions at 392 MeV. The dashed and solid lines indicate the results of non-relativistic PWIA and DWIA calculations, respectively. The data are from Ref. [96].

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those employed for the data in Fig. 36, except that a TOF scintillator was used instead of one spectrometer and the (p, pn) and (p, 2p) reactions were observed simultaneously. The results are shown in Fig. 37. The Ay values for the (p, 2p) reactions exhibit density dependent suppression, which is consistent with data from previous measurements. The (p, pn) reaction with the forward emission of protons also shows similar Ay suppression relative to the results of IA calculations. However, when neutrons are emitted forward, the reaction gives almost the same Ay values as the calculations, with enhanced values in some cases. To date, no theoretical consideration has been given to these experimental results. This difference between the (p, 2p) or proton-forwared (p, pn) analyzing power and the neutron-forward (p, np) analyzing power is similar as the analyzing powers of inclusive (p, p′ ) and (p, n′ ) scattering and with the values of angular integrated (p, 2p) and (p, np) analyzing powers [112]. 6.5. Polarization transfer coefficients and spin correlation parameters In a meson-exchange force model, nucleon-nucleon interactions are mediated by various kinds of mesons. Therefore, if properties of mesons are modified in nuclear field, the nucleon-nucleon interactions are also expected to be modified. Considering (p, 2p) reactions, Krein and others theoretically examined the effects of coupling constant and meson mass modifications based on the Bonn one-boson exchange model for the NN interaction [141]. This work attempted to reduce the meson masses using a scaling conjecture suggested by Brown and Rho [142] and also to reduce the coupling constants, gσNN and gωNN , by the same amount. It was determined that either or both of these modifications lead to pronounced suppression of the p-p analyzing power in the medium and it was claimed that the apparent reduction in the p-p analyzing power observed in the case of 1p3/2 and 1p1/2 knockout reactions in a TRIUMF experiment (discussed in the next subsection) can be explained by this kind of modification. These groups also pointed out the importance of polarization transfer measurements, especially Dnn measurements, in order to distinguish between meson mass modification and coupling constant modification. The first measurement of polarization transfer coefficients, solely Dss and Dsl , in a 65

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(p, 2p) reaction were performed at TRIUMF [36], while more complete measurements took place at RCNP [82]. These studies measured all five polarization transfer coefficients, Dij , associated with forward emitted protons in addition to the analyzing power and the induced polarization for 1s1/2 knockout reactions with the same three targets used for previous analyzing power measurements. The results are shown in Fig. 38. Roughly speaking, the majority of the Dij values, with the exception of Dl′ s , are not greatly shifted from the values predicted by both relativistic and non-relativistic IA calculations. They also found that the meson-mass modification using the Brown-Rho scaling failed to reproduce some of Dij values but demonstrated that all the data could be reproduced by proper choice of each meson mass. In fact, these polarization transfer coefficients are quite sensitive to small changes in the meson masses and varying meson mass modifications result in different changes in the Dij values. Therefore, this type of data potentially provides a critical test of theoretical models regarding medium modifications of hadrons and interactions. Recently, the spin correlation parameters Cnn and Cs′ s′′ were determined at PNPI using two focal line polarimeter systems [123, 143] (see Fig. 39). In addition, the PNPI researchers improved their induced polarization measurements on the 1s1/2 knockout to obtain better statistics and extended the measurement to a 28 Si target, thouth the shape of the 1s1/2 bump was less pronounced than in previous measurements [3]. The results obtained for the spin correlation parameter were similar to the RCNP results for polarization transfer coefficients. That is, no meaningful deviations from the PWIA predictions were observed. However, the improved polarization measurements clarified that the induced polarizations of the two outgoing protons, having asymmetric geometries, deviate from one another, and this deviation increases as a function of the nuclear density or the proton separation energy. 6.6. Analyzing powers for reactions other than the 1s1/2 knockout In a simplified scenario, the analyzing power Ay of a (p, 2N) reaction consists of the Ay of the in-medium NN scattering as well as the contribution of the effective polarization, as shown in Eq. (2.18). In the review paper by Kitching et al., it is pointed out that the analyzing powers of in-medium NN scattering deduced from experimental p-state and d-state knockout data (the weighted means of j-upper and j-lower analyzing powers as noted in Section 2) in asymmetrical angular settings evidently vanish, as shown in Fig. 40. The possibility of medium modification of NN amplitudes has been discussed, along with speculation that the nuclear density is significantly higher for p- and d-state knockouts because the typical recoil momenta of these reactions are on the order of 100 MeV/c. This problem, apparent disappearance of NN-amplitude originated analyzing power in pand d-state knockout, remains unsolved. However, the effective mean densities calculated for these knockout reactions are approximately 10% of the saturation density, as shown in the figure, at least in the cross section peak region where the apparent disappearance is still observed. In addition, in the case of Figs. 20 and 21, the Ay values are reasonably well reproduced by DWIA calculations even though the data were obtained with an asymmetrical angular setting. Therefore, this phenomenon does not likely result from the appearance of a nuclear density effect. It is also worth considering experimental data acquired at iThemba LABS, South Africa, for the 208 Pb(p, 2p) reaction at 202 MeV, and the corresponding theoretical analysis based on the NRDWIA and RDWIA [70, 71, 100]. The Ay data for the 3s1/2 knockout 66

Figure 38: Polarization transfer coefficients, Dnn , Ds′ s , Ds′ l , Dl′ s , Dl′ l , analyzing powers, Ay , and induced polarizations, P , for the 1s1/2 knockout 6 Li and 12 C(p, 2p) reactions and the 2s1/2 knockout 40 Ca(p, 2p) reaction under zero recoil conditions. The definition of D ij is given in Section 2, Eq. (2.6). Here the data, shown by closed circles, are plotted as functions of the effective mean density (in units of the nuclear saturation density, ρ0 ), while the closed squares denote the data for p-p scattering. The thin solid and dashed lines respectively indicate the non-relativistic PWIA and DWIA predictions, whereas the thick solid and dashed lines respectively represent the relativistic PWIA and DWIA results. The dotted lines indicate the results of non-relativistic DWIA calculations including m∗ corrections. In the case of the 1s1/2 knockout from 12 C, the open circles correspond to Rho-Brown corrections to the relativistic distorted wave model, whereas the open squares denote the relativistic distorted wave results with optimized meson-nucleon coupling constants. The open triangles show results when only the gσNN value is reduced by 4% from parameters used in the calculation denoted by the open squares. The original paper is Ref. [82].

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Figure 39: Spin correlation functions (left) and induced polarizations (right) acquired at PNPI using a 1 GeV unpolarized beam. The data and calculations are taken from Ref. [123]

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are significantly suppressed relative to the results of the NRDWIA calculations, while fully relativistic DWIA calculations successfully reproduce the data, as can be seen from Fig. 41. This figure also provides an estimation of the effective mean density, which is only several percent of the saturation density. Therefore, this improvement is not reasonably attributed to a simple lower component effect of the Dirac spinor, which should be a density effect. A similar result involving an improvement using relativistic calculations for the 208 Pb(p, 2p) reaction was also obtained at 392 MeV, although the differences between the non-relativistic and relativistic calculation results were less significant than in the case of the 202 MeV data [72, 107]. It should be noted here that the experimental data are also compared with NRDWIA calculations using density dependent g-matrices in Ref. [100]. In the calculations, sizable differences from those using the free nucleonnucleon interaction are observed in some kinematical conditions, though the fitting to the experimental data is not necessarily improved quite meaningfully. Even though above analyzing power data and a 200 MeV 40 Ca target data by TRIUMF as well are significantly deviated from conventional calculations, the cross sections are reasonably well reproduced by those calculations. Actually, the S-factors deduced using these data are already included in Fig. 24 and show consistent feature with other target results, though the deviation is somewhat large in the the 40 Ca case. In this section, we examined those studies aimed at understanding nuclear medium effects, although at present such work is still in the initial stages. The suppression of the analyzing powers and induced polarizations for the 1s1/2 knockout are observed over wide ranges of incident energies, target nuclei and detection kinematics, except in the case of neutron forward (p, np) measurements, and cannot be attributed to distortion effects or nuclear reaction mechanisms. Therefore, this evidence strongly suggests that a medium effect is at work, although at present such evidence is not sufficiently conclusive, and it is important to identify other observables in the 1s1/2 knockout that are modified. 68

Figure 40: The analyzing powers and differential cross sections for the 16 O(p, 2p) reaction acquired at TRIUMF with an asymmetrical angle pair. The cross points of the experimental 1p1/2 and 1p3/2 analyzing powers remain on the Ay = 0 line while the calculated points deviate significantly toward the positive region. The bottom panel shows the effective mean densities corresponding to the data.

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One direction such efforts could take would be to extend the measurements in isospin space, namely, to compare the p-p scattering and the p-n scattering in nuclear field. We have already discussed a comparison of the (p, 2p) and (p, pn) analyzing powers for the 1s1/2 knockout, but a comparison of the differential cross sections would be of interest and an extension of the nuclear field in isospin space using unstable nuclei would also be valuable. 7. Supplementary Subjects

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7.1. Comparison of (p, 2p) and (p, np) reactions Section 5 did not discuss the (p, np) reaction, the reason being that a relatively minimal amount of experimental data was accumulated after the review by Kitching et al., and thus it is difficult to provide a systematic assessment. In this subsection, we introduce individual works performed following the review, with the aim of examining the consistency between the (p, 2p) and (p, np) reactions. Both 16 O(p, 2p) and (p, np) data for the 1p1/2 and 1p3/2 knockout reactions were acquired simultaneously at TRIUMF using the MRS magnetic spectrometer and TOF scintillators at an incident energy of 505 MeV [91]. The results demonstrated that the experimental (p, np)/(p, 2p) cross section ratios were generally higher than those obtained from DWIA calculations. We applied a similar procedure as employed in Section 5 to analyze these data. That is, we compared the data with DWIA calculations using currently available parameters and extracted S-factors. The fitting uncertainties were estimated using the parameter α introduced in Section 5, where the value of αS indicates the uncertainty in the S-factor S. The data and the fitting results are summarized in Fig. 42. This 69

Figure 41: Differential cross sections (left) and analyzing powers (right, upper) for the 3s1/2 knockout (p, 2p) reaction of a 208 Pb target at 202 MeV. In the cross section plot, the three lines present the results of NRDWIA calculations using three kinds of distorting optical potential parameter sets. In the Ay plot, the dashed line indicates the results of the NRDWIA calculations that corresponds to the dashed line in the cross section plot, the solid and dotted-dashed lines present zero-range and finite-range RDWIA calculation data, respectively, and the dotted line indicates relativistic PWIA calculation results. In the right, lower panel, estimations of the effective mean densities for these measurement conditions are plotted. The cross section plot is taken from Ref. [100] and the Ay plot is from Refs. [70] and [71].

70

Figure 42: 16 O(p, 2p) and (p, pn) data obtained at TRIUMF. The lines indicate the results of DWIA calculations used to extract S-factors. The thickness of each line represents uncertainties resulting from statistical errors, while the shaded areas indicate uncertainties caused by fitting quality, representing α as explained in Section 5.3. uncertainties caused by experimental errors in the absolute cross sections are not taken into account. These data are from Ref. [91].

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figure demonstrates that the S-factors for the (p, np) reaction are consistently larger than those for the (p, 2p) reaction. The S-factor ratios derived are 1.24 ± 0.13 and 1.20 ± 0.12 for the 1p1/2 and 1p3/2 knockouts, respectively. These values would be expected to be close to unity given simple isospin symmetry, and the derived deviations are about twice of the uncertainties. Further precise measurements are required to make the result conclusive. Simultaneous measurements of the (p, 2p) and (p, np) cross sections using a magnetic spectrometer and TOF scintillators were also performed at PNPI in a series of studies to determine the energies and widths of the deep hole states, using a 1 GeV proton beam, as noted in Section 1. This work obtained the (p, np)/(p, 2p) cross section ratios for various nuclear targets and compared these values with the same ratios for a D target. The relative values of these double ratios are well explained by DWIA predictions employing the Hartree-Fock-Skyrme wave function [104]. A portion of the results obtained in the case of a 40 Ca target is shown in Fig. 43, from which it is evident that the agreement is excellent. Here we consider an experiment preformed at IUCF by Carman et al. [111]. The original intent of the work was to determine whether or not the inclusive (p, p′ ) yield arises primarily from simple (p, 2p) and (p, np) hard scattering processes in connection 71

Figure 43: Ratios of (p, np)/(p, 2p) cross sections for a DWIA predictions. The data are from Ref. [104].

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1185

40 Ca

target and a D target. The lines indicate

with nuclear correlation or via a medium effect as discussed in Section 6. In the analysis process, they compared the angle-integrated (p, 2p) and (p, np) cross sections with their DWIA estimations at an incident energy of 200 MeV, and Figure 44 presents the comparison. The lines in this figure represent the results of DWIA calculations using the same S-factors for the (p, 2p) and (p, np) reactions. The S-factor values, 3.7 for the 3/2− state and 0.85 for the 1/2− state, are more than twice the values determined from the (e, e′ p) reactions, but the ratios between the (p, 2p) and (p, np) values are consistent with the calculations. Finally, we present more recent experimental results obtained at RCNP. Figure 45 shows differential (p, 2p) and (p, np) cross sections acquired with the LAS and TOF scintillators at 392 MeV [96]. This (p, 2p) reaction is the same reaction presented in Fig. 20 except for the angular setting, although there is a significant difference in the energy resolution caused by the use of different detector combinations. In the case of the data in Fig. 20, the separation energy resolution, about 300 keV, was sufficient to separate the low lying states of the residual 11 B nucleus, as shown in Fig. 19. However, the data in Fig. 45 have an associated FWHM resolution of 3.5 MeV, and up to 8 MeV of the excitation energy region was integrated to obtain the cross section. The solid lines in Fig. 45 indicate the sum of the DWIA cross sections leading to the ground 3/2− state and the excited 1/2− (Ex = 2.1 MeV) and 3/2− (Ex = 5.0 MeV), using the S-factor values extracted from the data in Fig. 20, but multiplied by a factor 1.25 to reproduce the experimental data. In the case of the (p, np) reaction, the experimental data shown in the right hand panel of Fig. 45 are 15% greater than the values denoted by the solid line. Although a portion of these discrepancies may be attributed to experimental 72

Figure 44: Angle integrated 16 O(p, 2p) and (p, np) cross sections at an incident energy of 200 MeV. The lines indicate the results of DWIA calculations using the same S-factors for both the (p, 2p) and (p, np) reactions. The data are from Ref. [111].

73

Figure 45: Differential 12 C(p, 2p) and (p, pn) cross sections at an incident energy of 392 MeV. The solid lines are the results of DWIA calculations using the same S-factor values extracted from the data in Fig. 20 but multiplied by a factor of 1.25. These data are from Ref. [96].

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uncertainties, it is believed that the contributions of minor peaks also significant. In cases such as these, it is evident that high resolution measurements are essential to performing a reliable comparison. 7.2. Rescattering process In a series of experiments [113, 144–150] by Cowley and collaborators, the reaction mechanism of the rescattering process during two-proton coincidence measurements was investigated. As indicated in Section 3, by specifying the kinematics of the outgoing two protons, the single particle (s.p.) state that the struck proton occupied in the target nucleus, A, can be identified. This holds, however, only for the assumed reaction: a one-step proton knockout process. In reality, the knocked-out proton can hit another nucleon in the residual nucleus, B, before its emission, which may be followed by a further sequence of nucleon-nucleon collisions. Such rescattering is well known to be the dominant process in nucleon knockout reactions at high incident energies. The emitted protons in a rescattering process such as this may satisfy, albeit accidentally, the kinematical conditions of the measurements for the (p, 2p) process of interest. If this is the case, the observed cross section will not reflect in general the s.p. structure of A, and should therefore be regarded as a background. The rescattering process is denoted by (p, p′ p′′ ), and should be differentiated from the usual (p, 2p) process. In work reported in Ref. [150], two-proton coincidence measurements were carried out at RCNP with a 40 Ca target at a proton beam energy of 392 MeV. The kinetic energy T1 and the scattering angle θ1 of particle 1 were fixed at 220 MeV and 25.5◦ , respectively. Particle 2 was detected on the opposite side on the reaction plane to that of particle 1, while varying the energy T2 and the scattering angle θ2 . The measured cross section was interpreted as the sum of the (p, 2p) and (p, p′ p′′ ) cross sections, which were calculated by the DWIA model and a model based on the multistep direct process, respectively. The latter, first described in Ref. [151], is given by [ ] ∑ d3 σppN ∑ ∫ d4 σresc 1 d2 σNp′ x dΩN = (7.1) tot dT d (Ω − Ω ) . dT1 dΩ1 dT2 dΩ2 dT1 dΩ1 dΩN σpN 2 2 N N=p,n

λ

74

Figure 46: Binding energy spectrum of the results obtained using Eq. (7.1).

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1230

40 Ca(p, 2p)39 K

at 392 MeV for θ2 = 60◦ . The line represents

This model assumes that, in the initial stage, a typical (p, pN) process takes place, where N represents the type of nucleon that is struck (p or n), and λ is the s.p. orbit of N inside the 40 Ca target. In this case, N acts as an intranuclear projectile and triggers all possible reactions capable of emitting a proton with T2 , and Ω2 ; d2 σNp′ x /dT2 d (Ω2 − ΩN ) is the N-induced inclusive cross section for which experimental data are employed. The (p, pN) cross section d3 σppN /dT1 dΩ1 dΩN in Eq. (7.1) is calculated using the DWIA model, tot neglecting the absorption of N. Here σpN is the pN total cross section at incident energy TN , as determined by the kinematics of the (p, pN) process with λ specified. In Fig. 46, the experimental data for θ2 = 60◦ are compared with the results generated using Eq. (7.1). The horizontal axis in this plot indicates the sum of the kinetic energies of the two emitted protons, Etotal , which specifies the binding energy of the struck proton inside A in the DWIA limit. One can see that Eq. (7.1) properly describes the background portion of the data. The remainder of the experimentally derived spectrum is attributed to the (p, 2p) cross section, as shown in Fig. 5 of Ref. [150]. This success in the description of the background is crucial to the eventual quantitative analysis of experimental (p, 2N) data, especially with regard to knockout processes of deeply bound nucleons. These results also offer significant insight into the overall two-proton emission process at intermediate energies, and suggest the importance of the knockout process as a trigger for all the processes that follow. In Ref. [152], Eq. (7.1) was applied directly to the inclusive (p, p′ x) process and was found to successfully reproduce the experimental high-energy emission data. A semiclassical distorted wave model (SCDW) [56, 153] was also adopted in the analysis of the (p, p′ x) data. Several aspects of the SCDW were subsequently improved [154–156] and this model was shown to reproduce the (p, p′ x) data very well, including up to three-step processes [156]. It is worth mentioning here that the Fig. 5 of Ref. [150] indicates that the admixture of multistep step processes is about 25% of the cross section in the peak region of the 1s1/2 knockout from 40 Ca target, but at the same time, it shows the admixture is almost negligible in the discrete states region. Explicit estimation of the rescattering process 75

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introduced in this subsection supports the reliability of the spectroscopic studies treated in Section 5. 7.3. Spectroscopic factors from the (p, p′ α) reaction The proton-induced knockout reaction is also a direct and convenient means of investigating alpha clustering in nuclei. The existence of alpha clusters in nuclei can be determined by comparing the momentum distributions of (p, p′ α) reactions with the results of theoretical calculations based on preformed alpha clusters bound in a target nucleus. In addition, the experimental alpha spectroscopic factors (Sα ) extracted from the (p, p′ α) reaction should be in agreement with the theoretical predicted values. Theoretically, the (p, p′ α) cross section at a recoil momentum of −⃗ pB in the factorized approximation can be expressed as d3 σ = CK σpα Sα |ϕ(−⃗ pB )|2 , dΩp dΩα dEp

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(7.2)

where CK is a kinematical factor, σpα is the elemental off-shell p-α cross section, Sα is the alpha spectroscopic factor, and ϕ is the distorted momentum distribution. Note that, in PWIA, ϕ is the Fourier transform of the α-cluster wave function at an α-cluster momentum of −⃗ pB , and thus |ϕ(−⃗ pB )|2 reduces to the α-cluster momentum distribution. In practice, distortion effects significantly reduce the contribution of the nuclear interior to ϕ, and so the (p, p′ α) reaction is highly localized in the vicinity of the nuclear surface. Figure 47(a) compares the experimental Sα values for p-shell nuclei deduced from (p, p′ α) data [157–161] with those predicted using the shell model [162] and the no-core shell model [163]. The experimental distributions of the cross sections are in reasonable agreement with the result of DWIA calculations, and the relevant Sα values have been determined as normalization factors. It should be noted that analyzing power data [158, 160, 161] are also reproduced reasonably well in the same manner, meaning that a quasifree knockout mechanism dominates the reaction at Tp ≳ 100 MeV. The experimental Sα values for 9 Be at Tp = 100 to 200 MeV also agree very well with the predicted values, although the values at Tp = 296 MeV (open squares) [160] are significantly greater than those at Tp = 100 MeV (filled circles) [157], with the exception of the 7 Li data. The significant uncertainty in Sα for 12 C at Tp = 100 MeV (filled diamond) [161] results from the strong dependence of the experimental data on the kinematical conditions, as shown in Fig. 48. This large fluctuation is primarily attributed to the uncertainty in the results of DWIA calculations, since the theoretical cross sections vary greatly with the optical potential parameters. Thus, in order to resolve the discrepancy between the predicted and experimental results, reliable DWIA calculations based on precise global optical potentials for a wide range of energies and target masses are required, as discussed in Section 5 in the case of the (p, 2N) data analysis. In these studies, the Sα values for pshell nuclei extracted from (p, p′ α) data were also found to be consistent with theoretical predictions, within a deviation of approximately 30%. Carey et al. [164] has reported Sα values for several heavier target nuclei, from 16 O to 66 Zn based on experimental (p, p′ α) data at Tp = 101.5 MeV. Figure 47(b) presents their results for sd-shell nuclei together with the recent large-scale shell model predictions [165]. Here the experimental and theoretical data agree reasonably well not only with regard to the oscillatory mass number dependence but also in the case of the absolute 76

Figure 47: Comparison between experimental and theoretical Sα values. (a) For p-shell nuclei at Tp = 100 to 296 MeV [157–161]. The solid and dashed lines indicate the shell model [162] and no-core shell model [163] predictions, respectively. The shaded areas represent the ±30% values obtained from theoretical calculations. (b) For sd-shell nuclei at Tp = 100 MeV [164]. The solid lines indicate the largescale shell model predictions [165]. The shaded areas denote the ±30% values obtained from theoretical calculations.

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values, within approximately 30%. It should, however, be noted that the analyzing power data for the 40 Ca(p, p′ α) reaction at Tp = 100 MeV could not be reproduced by the DWIA calculations [166]. In light of the above, further theoretical and experimental studies concerning the derivation of Sα values using (p, p′ α) data are needed. 8. Summary and Outlook

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As noted in the Introduction, this review first outlined the theoretical basis for the discussions herein and discussed the associated experimental apparatus, before focusing on two subjects: spectroscopic studies and examinations of the nuclear medium effect. In the case of the former subject, substantial progress has been made during the last two decades, both in terms of theory and experimental work. Accordingly, the data that have been reported worldwide were compared with the results of DWIA calculations using presently available optical potentials. This comparison confirmed that the spectroscopic factors extracted using (p, 2N) data at intermediate energies are consistent with those determined using (e, e′ p) data, typically within a deviation of 15%, when the geometrical parameters used to generate the bound state wave functions are the same as those employed in the (e, e′ p) analyses. However, when the goal is to use the (p,2N) reactions as a spectroscopic tool, it is preferable to determine the geometrical parameters in a self-consistent manner. At present, (p, 2N) reactions are expected to be used in unstable nuclear studies involving inverse kinematics. In this scenario, both the radii of the bound state orbits and the shape of the optical potential are not necessarily known and may in fact deviate significantly from those for stable, known nuclei. Therefore, in order to estimate the sensitivities of the differential (p,2N) cross sections to the geometrical parameters of both of the bound state wave functions and the optical potentials, we attempted a simulation. This simulation involved an examination of the manner in which the peak value and 77

Figure 48: The Sα values extracted from the 12 C(p, p′ α) data at Tp = 101.5 MeV under different kinematical conditions for (θp , θα ). The solid line and shaded area represent the average value and uncertainty, respectively.

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the width of the recoil momentum dependence of the cross section vary as the radii of the potentials used to generate the bound state wave function and the scattering wave function are changed when employing DWIA calculations. As shown in Fig. 49, the peak values and widths are relatively sensitive to both potential parameters, although these sensitivities involve changes in opposite directions. These results demonstrate that careful interpretations of comparisons between experimental data and theoretical calculations are required. In present-day studies regarding unstable nuclear physics, the strong dependence of S-factors on nucleon binding energy is attracting interest [167] and (p,2N) reactions involving inverse kinematics are expected to play a crucial role in clarifying this phenomenon. In order to obtain convincing data, however, further theoretical improvements in the construction of the optical potential would be greatly beneficial, and close cooperation between experimentalists and those studying nuclear reactions and nuclear structure theory will be indispensable. With regard to medium effect studies, there has been much evidence to strongly suggest modification of the in-medium nucleon-nucleon interactions that is not explained by the distortion effect nor by a mixture of different reaction mechanisms. However, a significant problem associated with such studies is that the relevant data involve solely analyzing power and induced polarization. As noted in the previous review paper by Kitching et al., an extension of investigations to isospin space is desirable. In addition, theoretical efforts are required to improve the reproducibility of analyzing power data in the region most sensitive to spin-dependent distortion that is not reproducible in present-day analyses.

78

Figure 49: Sensitivity of the peak value (left) and the width of the recoil momentum distribution (right) of the differential cross section to the size of the potential radius. The radius parameter of the bound state potential (b.s. potential) and those for all components of the optical potential (opt. potential) have been modified by multiplying the abscissa (R/R0 ) values. The associated nuclear reaction is that of 40 Ca(p, 2p) leading to the 1/2+ state of residual 39 K, and the optical potential parameterization of Schwandt et al. [168] is employed.

Appendix A. Transition matrix density and effective mean density

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The effective mean density was first defined in Ref. [127] for the real part and this definition was later extended to the imaginary part [169]. The transition matrix density that is employed in the present paper was introduced in those prior publications as a weighting function for the calculation of the density. Similar definitions are also provided in Ref. [137] for the distorted momentum distributions used in (e, e′ p) calculations. The triple differential cross section, TDX, and the transition matrix of the (p, pN) reactions are given by Eqs. (3.33) and (3.45) in Section 3, but here we write them simply as σ= T =

d3 σ L dE L dΩL dΩL ∫ ∞1 1 2

= |T |2 ,

(A.1a) (A.1b)

D(R)dR.

0

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All of the factors not included in the integral in these equations, R2 from the volume element, and integrations of the angular coordinates are included in D(R). As well, the angular momentum degree of freedom is neglected for the time being. The purpose of the first part of this Appendix is to determine a function that directly represents the contribution to the cross section as a function of R, as well as to demonstrate that the Tr function δreal (R) introduced by Eq. (5.2) satisfies this requirement. Tr (R) and its imaginary counterpart as We begin by defining the function δreal 1 (σ − σR ) , 2∆R 1 Tr ′ δimag (R) = (σ − σR ), 2∆R Tr δreal (R) =

(A.2a) (A.2b)

79

where ∫ 2 ∫ R+∆R ∞ σR = D(R)dR − D(R)dR , 0 R ∫ 2 ∫ R+∆R ∞ ′ σR = D(R)dR + i D(R)dR . 0 R

(A.3a)

(A.3b)

Substituting these definitions into Eqs. (A.2) gives 2 ∫ ∞ 2 ∫ ∞ ∫ R+∆R D(R)dR D(R)dR − 0 D(R)dR − R 0 Tr δreal (R) = 2∆R ∫ ∞ ∫ ∞ 1 1 ∗ = D(R) D (R)dR + D∗ (R) D(R)dR 2 2 0 0 ∫ ∞ ∫ ∞ 2 2 ∗ D(R) + ∫ ∞D (R) = ∫∞ D(R)dR D(R)dR ∗ 2 0 D(R)dR 0 2 0 D (R)dR 0 [ ] D(R) σ, (A.4a) = Re ∫ ∞ D(R)dR 0 2 ∫ ∞ 2 ∫ ∞ ∫ R+∆R − D(R)dR D(R)dR + i D(R)dR 0 0 R Tr (R) = δimag 2∆R ∫ ∞ ∫ ∞ i i ∗ = − D(R) D (R)dR + D∗ (R) D(R)dR 2 2 0 0 [ ] D(R) = Im ∫ ∞ σ. (A.4b) D(R)dR 0 Here, ∫the second equalities of both of Eqs. (A.4) satisfy the infinitesimal limit of ∆R. ∞ Since 0 D(R)dR and σ are constants, we can write Tr δreal (R) ∝ Re D(R)

and

∫∞

Tr δimag (R) ∝ Im D(R),

(A.5)

provided that the phase of 0 D(R)dR is redefined to be real. In addition, the integration of these functions with respect to R gives ∫ ∞ ∫ ∞ Tr Tr δreal (R) dR = Re[1] σ = σ and δimag (R) dR = Im[1] σ = 0. (A.6) 0

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0

Therefore, the differential cross section can be obtained by the simple integration of Tr Tr δreal (R) by R, and the integration of δimag (R) does not make any contribution to the cross Tr section. That is, δreal (R) is the function that we are attempting to obtain. Considering Tr both Eqs. (A.1b) and (A.5) , we refer to δreal (R) as the transition matrix density. It should be noted that a similar procedure is applicable to coordinates other than R, such as polar or azimuthal angle coordinates, ϑ and φ, respectively, or even two or three dimensions. In fact, Overmeire and coworkers have presented two dimensional plots of similar kinds of functions in the R-ϑ plane [84]. They employed the relativistic eikonal 80

Tr approximation with fully antisymmetrized wave functions and derived δreal (using present symbol) and ρ¯ (defined below), following the similar procedure described above. In their derivation, the function D(R) in Fig. (A.1b) is defined as ∫ D(R) ∝ dΩ R2 e−ipm ·R ϕ(R) SIFSI (R), (A.7)

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where pm is the missing momentum, ϕ(R) is the four component bound state wave function and SIFSI (R) is the phase factor which reflects the initial and final state interactions. Since they employ eikonal approximation, the SIFSI (R) is practically the product of three terms that represent the absorption and phase rotation of the incident and two outgoing nucleons. An example of SIFSI (R), corresponding to a (p, 2p) reaction at 1 GeV, is plotted in Fig. A.501 . The absorption effects by the target and residual nuclei are visibly indicated. Up to this point, we have neglected the angular momentum degree of freedom. In general, the differential cross section is written as a summation of the coherent and incoherent partial amplitudes, as given in Eqs. (3.33) and (3.45). Here we write them simply as 2 ∑ ∑ ∫ ∞ . σ= (A.8) D (R)dR ij 0 i j However, if we accept that Di (R) ≡ σ=

∑ ∫ i

0

∞

2 Di (R)dR .

∑

j

Dij (R), the cross section can be written as

(A.9)

Therefore, the coherent sum does not affect the above derivation. In order to take the incoherent sum into account, we introduce the partial cross sections and partial transition matrix densities as ∫ ∞ 2 ∑ σi = σ= (A.10a) Di (R)dR , σi , o

δiTrreal (R)

i

1 = (σi − σiR ) , 2∆R

Tr δreal (R)

=

∑

δiTrreal (R),

(A.10b)

i

and also define σiR in a similar manner. A similar derivation to that above provides ∫ ∞ σi = δiTrreal (R)dR, (A.11) 0

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Tr δreal (R)

and as redefined in Eq. (A.10b) satisfies the present requirement. It is also possible to examine partial cross sections, for example, the spin-up and spin-down cross sections, separately. 1 The plot is based on scanned data from Figs. 3 and 7 in [84]. Therefore, it is not numerically quite accurate.

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Figure A.50: Radial (R) and polar-angle (ϑ) dependence of the real part of SIFSI , introduced in Eq. (A.7), in the scattering plane (φ = 0◦ ) for proton knockout from the 1s orbit in 12 C. The polar axis is taken to the direction of the incident beam. The panels (a), (b) and (c) show contributions from impinging, scattered and ejected protons, respectively, and the panel (d) is the product of those three contributions. The panel (e) shows the two-dimensional transition matrix density in the plane wave approximation, which is practically proportional to the product of the bound state wave function ϕ(R) and R2 sin ϑ from the volume element. The panel (f) is the product of (d) and (e), which represents the contribution to the differential cross section. The energies of the impinging and scattered (θ1 = 13.4◦ ) protons are 1 GeV and 870 MeV, respectively, and the ejected proton angle θ2 is 67◦ . The original plots of upper Tr,PWIA four panels and the one-dimensional transition matrix density, δreal (R) (using present symbol), are in [84]. 82

Figure A.51: The partial transition matrix densities for 1p1/2 and 1p3/2 knockout reactions as functions of (left) the polar angle coordinate ϑ and (right) the azimuthal angle coordinate φ of the integration variables. The emitted angles of two protons are coplanar symmetric, while those energies are asymmetric, being 105 MeV for φ = 0◦ side protons and close to 83 MeV (1p1/2 knockout) and 77 MeV (1p3/2 knockout) for φ = 180◦ side protons. The incident beam axis corresponds to the ϑ = 0 derection. The spin-up and spin-down present the spin directions, which is normal to the reaction plane, of the incident protons.

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As a demonstration, we show the transition matrix densities for 1p1/2 and 1p3/2 knockout reactions as functions of ϑ and φ in Fig. A.51. In the case of 1p1/2 knockout, upper panels, it is displayed that the large ϑ and small φ region mainly contributes to the spin-up cross section and small ϑ and large φ region contributes to the spin-down cross section. Moreover, it is also presented in this figure that the spin-up and spindown contributions are almost inverted between 1p1/2 knockout and 1p3/2 knockout. These features are corresponding to the j-dependence of the (p,2N) reactions discussed in Section 2, although a bit complicated quantum mechanical effects are seen in two dimensional plots which are not given here. Tr Tr (R) are defined, it is simple to introduce the effective mean Once δreal (R) and δimag density. In the case of a spherically symmetric nuclear density ρ(R), we can write ∫∞ ∫∞ ∫∞ Tr ρ(R)δreal (R)dR ρ(R)D(R)dR ρ(R)D∗ (R)dR 0∫ 0 ∫ 0 ∫ = + ∞ Tr ∞ ∞ δreal (R)dR 2 0 D(R)dR 2 0 D∗ (R)dR 0 ∫∞

ρ(R)δ Tr (R)dR 0 ∫ ∞ Trimag δreal (R)dR 0

because ∫ ∞ 0

Tr δreal (R)dR

= Re ρ¯, ∫∞ ∫∞ −i 0 ρ(R)D(R)dR i 0 ρ(R)D∗ (R)dR ∫∞ ∫∞ = + 2 0 D(R)dR 2 0 D∗ (R)dR

= Im ρ¯,

∫ = σ =

0

∞

2 D(R)dR ,

83

(A.12a)

(A.12b)

(A.13)

and, from the definition in Eq. (6.3a), ∫∞ ρ(R)D(R)dR 0∫ = ρ¯. (A.14) ∞ D(R)dR 0 ∫∞ In this case, the phase of 0 D(R)dR is not necessarily redefined to be real, unlike the scenario for Eq. (A.5), while Im ρ¯ will not always become negligible. At this point, it is helpful to comment on the step function employing cut off radii that is referenced in Section 5 and used in Refs. [20, 100, 118]. This function is defined as ( ∫ 2 ∫ ∞ 2 ) ∞ 1 P (r) = D(R)dR − D(R)dR . (A.15) ∆R R R+∆R At the infinitesimal limit of ∆R, this function is transformed to  2  ∫ 2 ∫ ∞ ∫ R+∆R 1  ∞  − P (r) = D(R)dR D(R)dR − D(R)dR ∆R R

∫

R

∞

= D(R) D∗ (R)dR + D∗ (R) R [ ] ∫ ∞ ∗ = Re D(R) D (R)dR .

R

∫

∞

D(R)dR

R

(A.16)

R

1355

∫∞ Therefore, P (r) includes a R D∗ (R)dR factor that goes to zero at large values of R, meaning that P (R) overestimates the contribution of the nuclear interior to a greater Tr extent than δreal (R). References

1360

1365

1370

1375

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

G. Jacob, T. A. J. Maris, Rev. Mod. Phys. 38 (1966) 121. G. Jacob, T. A. J. Maris, Rev. Mod. Phys. 43 (1973) 6. S. L. Belostotski, et al., Sov. J. Nucl. Phys. 41 (1985) 903, ; Yad. Fiz. 41 (1985) 1425. S. S. Volkov, et al., Sov. J. Nucl. Phys. 52 (1990) 848, ; Yad. Fiz. 52 (1990) 1339. A. A. Vorob’ev, et al., Phys. Atom. Nucl. 57 (1994) 1, ; Yad. Fiz. 57 (1994) 3. A. A. Vorob’ev, et al., Phys. Atom. Nucl. 58 (1995) 1817, ; Yad. Fiz. 58 (1995) 1923. G. Jacob, et al., Phys. Lett. B 45 (1973) 181. P. Kitching, et al., Phys. Rev. Lett. 37 (1976) 1600. P. Kitching, et al., Adv. Nucl. Phys. 15 (1985) 43. D. Dutta, K. Hafidi, M. Strikman, Prog. Part. Nucl. Phys. 69 (2013) 1. T. Kobayashi, et al., Nucl. Phys. A 805 (2008) 431c. T. Aumann, C. A. Bertulani, J. Ryckebusch, Phys. Rev. C 88 (2013) 064610. S. Frullani, J. Mougey, Adv. Nucl. Phys. 14 (1984) 1. G. van der Steenhoven, et al., Nucl. Phys. A 480 (1988) 547. L. Lapik´ as, Nucl. Phys. A 553 (1993) 297c. W. H. Dickhoff, C. Barbeieri, Prog. Part. Nucl. Phys. 52 (2004) 377. O. Benhar, A. Fabrocini, S. Fantoni, Nucl. Phys. A 505 (1989) 267. O. Benhar, A. Fabrocini, S. Fantoni, Phys. Rev. C 41 (1990) R24. V. R. Pandharipande, I. Sick, P. K. A. de Witt Huberts, Rev. Mod. Phys. 69 (1997) 981. G. J. Kramer, H. P. Blok, L. Lapik´ as, Nucl. Phys. A 679 (2001) 267. M. Leuschner, et al., Phys. Rev. C 49 (1994) 955.

84

1380

1385

1390

1395

1400

1405

1410

1415

1420

1425

1430

1435

[22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79]

J. Wesseling, et al., Nucl. Phys. A 547 (1992) 519. G. J. Kramer, et al., Phys. Lett. B 227 (1989) 199. J. W. A. den Herder, et al., Nucl. Phys. A 490 (1988) 507. J. B. J. M. Lanen, et al., Nucl. Phys. A 560 (1993) 811. E. Quint, Ph.D. thesis, University of Amsterdam (1988). T. A. J. Maris, M. R. Teodoro, C. A. Z. Vascondellos, Nucl. Phys. A 322 (1979) 461. T. A. J. Maris, M. R. Teodoro, E. A. Veit, Phys. Lett. B 94 (1980) 6. C. J. Horowitz, M. J. Iqbal, Phys. Rev. C 33 (1986) 2059. C. J. Horowitz, D. P. Murdock, Phys. Rev. C 37 (1988) 2032. X. Y. Chen, et al., Phys. Lett. B 205 (1988) 436. O. H¨ ausser, et al., Phys. Rev. Lett. 61 (1988) 822. R. Fergerson, et al., Phys. Rev. C 38 (1988) 2193. J. A. McGill, et al., Phys. Lett. B 134 (1983) 157. C. A. Miller, et al., J. Phys. Colloques 51 (1990) C6–595. C. A. Miller, et al., Phys. Rev. C 57 (1998) 1756. T. N. Taddeucci, et al., Nucl. Phys. A 527 (1991) 393c. X. Y. Chen, et al., Phys. Rev. C 47 (1993) 2159. T. Wakasa, et al., Phys. Rev. C 59 (1999) 3177. C. Hautala, et al., Phys. Rev. C 65 (2002) 034612. A. N. James, et al., Nucl. Phys. A 324 (1979) 253. G. Q. Li, R. Machleidt, Phys. Rev. C 48 (1993) 1702. G. Q. Li, R. Machleidt, Phys. Rev. C 49 (1994) 566. N. S. Chant, P. G. Roos, Phys. Rev. C 27 (1983) 1060. A. K. Kerman, H. McManus, A. M. Thaler, Ann. Phys. (N.Y.) 8 (1959) 551. L. D. Faddeev, Zh. Eksp. Teor. Fiz. 39 (1960) 1459, ; Sov. Phys. JETP 12, (1961) 1014. N. S. Chant, P. G. Roos, Phys. Rev. C 15 (1977) 57. G. R. Satchler, Nucl. Phys. A 55 (1963) 1. Y. Kudo, K. Miyazaki, Phys. Rev. C 34 (1988) 1192. Y. Kudo, N. Kanayama, T. Wakasugi, Phys. Rev. C 38 (1988) 1126. N. Kanayama, et al., Prog. Theor. Phys. 83 (1990) 540. Y. Kudo, H. Tsunoda, Prog. Theor. Phys. 89 (1993) 89. K. Ogata, K. Yoshida, K. Minomo, Prog. Theor. Phys. 92 (2015) 034616. K. Yoshida, K. Minomo, K. Ogata, arXiv:1603.00638 [nucl-th]. K. Minomo, et al., J. Phys. G 37 (2010) 085011. Y. L. Luo, M. Kawai, Phys. Rev. C 43 (1991) 2367. K. Ogata, G. C. Hillhouse, B. I. S. van der Ventel, Phys. Rev. C 76 (2007) 021602(R). G. Perey, B. Buck, Nucl. Phys. 32 (1962) 353. E. D. Cooper, et al., Phys. Lett. B 206 (1988) 588. S. Hama, et al., Phys. Rev. C 41 (1990) 2737. E. D. Cooper, et al., Phys. Rev. C 47 (1993) 297. E. D. Cooper, S. Hama, B. C. Clark, Phys. Rev. C 80 (2009) 034605. E. D. Cooper, O. V. Maxwell, Nucl. Phys. A 493 (1989) 468. O. V. Maxwell, E. D. Cooper, Nucl. Phys. A 513 (1990) 584. O. V. Maxwell, E. D. Cooper, Nucl. Phys. A 565 (1993) 740. O. V. Maxwell, E. D. Cooper, Nucl. Phys. A 574 (1994) 819. O. V. Maxwell, E. D. Cooper, Nucl. Phys. A 603 (1996) 441. Y. Ikebata, Phys. Rev. C 52 (1995) 890. J. Mano, Y. Kudo, Prog. Theor. Phys. 100 (1998) 91. G. C. Hillhouse, et al., Phys. Rev. C 68 (2003) 034608. G. C. Hillhouse, et al., Phys. Rev. C 67 (2003) 064604. G. C. Hillhouse, T. Noro, Phys. Rev. C 74 (2006) 064608. J. A. Tjon, S. J. Wallace, Phys. Rev. C 32 (1985) 267. J. A. Tjon, S. J. Wallace, Phys. Rev. Lett. 54 (1985) 1357. J. A. Tjon, S. J. Wallace, Phys. Rev. C 35 (1987) 280. J. A. Tjon, S. J. Wallace, Phys. Rev. C 36 (1987) 1085. B. I. S. van der Ventel, G. C. Hillhouse, Phys. Rev. C 69 (2004) 024618. M. L. Goldberger, Y. Nambu, R. Oehme, Ann. Phys. (N.Y.) 2 (1957) 226. G. E. Brown and A. D. Jackson, The Nucleon-Nucleon Interaction, North-Holland Publishing Company, Amsterdam, 1976.

85

1440

1445

1450

1455

[80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101]

1460

[102] [103] [104] 1465

1470

1475

1480

1485

1490

1495

[105] [106] [107] [108] [109] [110] [111] [112] [113] [114] [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] [125] [126] [127] [128] [129] [130] [131] [132] [133] [134] [135]

C. J. Horowitz, Phys. Rev. C 31 (1985) 1340. Y. Kudo, K. Miyazaki, Phys. Rev. C 34 (1986) 1192. T. Noro, et al., Phys. Rev. C 77 (2008) 044604. O. Benhar, et al., Z. Phys. A 355 (1996) 267. B. V. Overmeire, et al., Phys. Rev. C 73 (2006) 064603. L. D. Faddeev, Sov. Phys. JETP 12 (1961) 1014, ; Zh. Eksp. Theor. Fiz. 39 (1960) 1459. E. O. Alt, P. Grassberger, W. Sandhas, Nucl. Phys. B 2 (1977) 167. R. Crespo, A. Deltuva, E. Cravo, Phys. Rev. C 90 (2014) 044606. P. Kitching, et al., Nucl. Phys. A 340 (1980) 423. L. Antonuk, et al., Nucl. Phys. A 370 (1981) 389. C. A. Miller, TRIUMF Report TRI-83-3, p.339, 1983 (unpublished). W. J. McDonald, et al., Nucl. Phys. A 456 (1986) 577. M. Fujiwara, et al., Nucl. Instrum. Methods Phys. Res. A 422 (1999) 484. N. Matsuoka, et al., RCNP Annual Report, 1991, p. 186. T. Noro, et al., RCNP Annual Report, 1991, p. 217. M. Yosoi, et al., AIP Conf. Proc. 343 (1995) 157. Y. Yamada, Ph.D. thesis, Kyushu University (2010). T. Wakasa, et al., Nucl. Instrum. Methods Phys. Res. A 547 (2005) 569. E. Ihara, et al., Phys. Rev. C 78 (2008) 024607. V. A. Andreev, et al., Phys. Rev. C 69 (2004) 024604. R. Neveling, et al., Phys. Rev. C 66 (2002) 034602. T. Noro, et al., in: Proceedings of the Kyudai-RCNP International Symposium on Nuclear ManyBody and Medium Effects in Nuclear Interactions and Reactions, Fukuoka, 2002, World Scientific, Singapore, 2003, p. 223. H. Yoshida, Ph.D. thesis, Kyushu University (2010). R. A. Arndt, I. I. Strakovsky, R. L. Workman, Phys. Rev. C 62 (2000) 034005. S. L. Belostotsky, et al., in: Proceedings of the International Symposium on Modern Developments in Nuclear Physics, Novosibirsk, 1987, p. 191. W. T. H. van Oers, et al., Phys. Rev. C 25 (1982) 390. G. C. Hillhouse, et al., arXiv:nucl-th/0608029. T. Ishida, Ph.D. thesis, Kyushu University (2007). P. Kithcing, et al., AIP Conf. Proc. 97 (1983) 232. A. A. Cowley, et al., Rev. Lett. B 359 (1995) 300. A. A. Cowley, et al., Phys. Rev. C 40 (1989) 1950. D. S. Carman, et al., Phys. Lett. B 452 (1999) 8. D. S. Carman, et al., Phys. Rev. C 59 (1999) 1869. S. V. F¨ ortsch, et al., Phys. Rev. C 48 (1993) 743. T. Noro, et al., RCNP Annual Report, 2015. Y. Nagasue, et al., RCNP Annual Report, 2005, p. 2. A. A. Cowley, et al., Phys. Rev. C 44 (1991) 329. P. G. Roos, et al., Phys. Rev. Lett. 40 (1978) 1439. C. Samanta, et al., Phys. Rev. C 34 (1986) 1610. K. Ranjan, et al., Nucl. Phys. A 226 (1974) 365. R. K. Bhowmik, et al., Phys. Rev. C 13 (1976) 2105. V. B. Shostak, et al., Nucl. Phys. A 643 (1998) 3. V. B. Shostak, et al., Phys. Rev. C 61 (1999) 024601. O. V. Miklukho, et al., Phys. Atom. Nucl. 76 (2013) 781. A. Koning, J. Delaroche, Nucl. Phys. A 713 (2003) 713. T. Hatsuda, Nucl. Phys. A 544 (1992) 27c. B. D. Serot, J. D. Walecka, Adv. Nucl. Phys. 16 (1986) 116. K. Hatanaka, et al., Phys. Rev. Lett. 78 (1997) 1014. G. Bertsch, et al., Nucl. Phys. A 284 (1977) 399. J. P. Jeukenne, A. Lejeune, C. Mahoux, Phys. Rev. C 16 (1977) 80. H. V. von Geramb, AIP Conf. Proc. 97 (1982) 44. Y. N, S. Nagata, J. Michiyama, Prog. Theor. Phys. 70 (1983) 459. K. Nakayama, W. G. Love, Phys. Rev. C 38 (1988) 51. K. Amos, et al., Adv. Nucl. Phys. 25 (2000) 275. T. Furumoto, Y. Sakuragi, Y. Yamamoto, Phys. Rev. C 78 (2008) 044610. C. Fuchs, A. Faessler, M. El-Shabshiry, Phys. Rev. C 64 (2001) 024003.

86

1500

1505

1510

1515

1520

1525

1530

[136] [137] [138] [139] [140] [141] [142] [143] [144] [145] [146] [147] [148] [149] [150] [151] [152] [153] [154] [155] [156]

[157] [158] [159] [160] [161] [162] [163] [164] [165] [166] [167] [168] [169]

L. White, F. Sammarruca, Phys. Rev. C 88 (2013) 054619. J. Ryckebusch, W. Cosyn, M. Vanhalst, Phys. Rev. C 83 (2011) 054601. H. Seifert, et al., Phys. Rev. C 47 (1993) 1615, and references therein. T. Noro, et al., Phys. Rev. C 72 (2005) 041602(R). O. V. Miklukho, et al., Phys. Atom. Nucl. 69 (2006) 452, ; Yad. Fiz. 69 (2006) 474. G. Krein, et al., Phys. Rev. C 51 (1995) 2646. G. E. Brown, M. Rho, Phys. Rev. Lett. 66 (1991) 2720. O. V. Miklukho, et al., arXiv:1402.0308 [nucl-ex]. A. Cowley, et al., Phys. Rev. Lett. 24 (1980) 1930. G. Ciangaru, et al., Phys. Rev. C 27 (1983) 1360. A. A. Cowley, et al., Phys. Lett. B 201 (1988) 196. J. V. Pilcher, et al., Phys. Rev. C 40 (1989) 1937. D. M. Whittal, et al., Phys. Rev. C 42 (1990) 309. A. A. Cowley, et al., Europhys. Lett. 13 (1990) 37. A. A. Cowley, et al., Phys. Rev. C 57 (1998) 3185. G. Ciangaru, et al., Phys. Rev. C 29 (1984) 1289. A. A. Cowley, et al., Phys. Rev. C 62 (2000) 064604. Y. Watanabe, et al., Phys. Rev. C 59 (1999) 2136. S. Weili, et al., Phys. Rev. C 60 (1999) 064605. K. Ogata, et al., Nucl. Phys. A 703 (2002) 152. K. Ogata, et al., in: Proceedings of the Kyudai-RCNP International Symposium on Nuclear ManyBody and Medium Effects in Nuclear Interactions and Reactions, Fukuoka, 2002, World Scientific, Singapore, 2003, p. 231. P. G. Roos, et al., Phys. Rev. C 15 (1977) 69. C. W. Wang, et al., Phys. Rev. C 31 (1985) 1662. A. Nadasen, et al., Phys. Rev. C 40 (1989) 1130. T. Yoshimura, et al., Nucl. Phys. A 641 (1998) 3. J. Mabiala, et al., Phys. Rev. C 79 (2009) 054612. D. Kurath, et al., Phys. Rev. C 7 (1973) 1390. P. Navr´ atil, Phys. Rev. C 70 (2004) 054324. T. A. Carey, et al., Phys. Rev. C 29 (1984) 1273. A. Volya, Y. M. Tchuvil’sky, Phys. Rev. C 91 (2015) 044319. R. Neveling, et al., Phys. Rev. C 77 (2008) 037601. J. A. Tostevin, A. Gade, Phys. Rev. C 90 (2014) 057602. P. Schwandt, et al., Phys. Rev. C 26 (1982) 55. T. Noro, et al., in: Proceedings of the RCNP International Symposium on Nuclear Responses and Medium Effects, Osaka, 1998, Universal Academy Press, Tokyo, 1998, p. 167.

87