Introduction

The measurements and theoretical analyses of single-nucleon removal reactions are of great value in studies on the single particle strengths of atomic nuclei, which are quantitatively represented by spectroscopic factors (SFs) [1]. It is well known that the SFs extracted from (e, e’p) and single-nucleon transfer reactions are found to be 30%-50% smaller than those predicted by the configuration-interaction shell model (CISM) [2, 3]. Such a reduction or quenching of SFs, represented by the quenching factors R_s, is supposed to originate from the limited model spaces and insufficient treatment of the nucleon-nucleon correlations in the traditional CISM [4, 5]. Unlike the results from (e, e’ p) reactions, from single-nucleon transfer reactions [6-8], and from (p, 2p) and (p, pn) reactions [2, 3, 9, 10], where the R_s values of different nuclei are nearly constant, the quenching factors from intermediate energy single-nucleon removal reactions are found to have an almost linear relationship with the proton-neutron asymmetry of the atomic nuclei, ΔS ($\text{Δ}S=S_{\text{p}}-S_{\text{n}}$ for proton removal and $\text{Δ}S=S_{\text{n}}-S_{\text{p}}$ for neutron removal with S_n and S_p being the neutron and proton separation energies in the ground states of the projectile nuclei, respectively) [11, 12]. For cases when ΔS is larger than approximately 20 MeV, which corresponds to the removal of strongly bound nucleons, the R_s values decrease to approximately 0.3; however, when ΔS is smaller than approximately -20 MeV, which corresponds to the removal of weakly bound nucleons, the R_s values are close to unity. The reasons for such a clear linear dependence observed in the results of the intermediate-energy single-nucleon removal reactions remain unknown. Because most single-nucleon removal reactions are analyzed using the Glauber model, the validity of the eikonal/Glauber model [11-13] has been questioned [14].

Owing to its simplicity, the optical limit approximation (OLA) is often used in the eikonal/Glauber model analysis of intermediate- and high-energy nuclear reactions [13, 15-19]. Only the first-order term for the expansion of the full Glauber phase shift is considered in the OLA. Higher-order interactions, such as nucleon-nucleon multiple scattering processes, are neglected [20]. In Ref. [21], B. Abu-Ibrahim and Y. Suzuki found that although the Glauber model with the OLA can reasonably reproduce the total reaction cross sections of some stable ions on ⁹Be, ¹²C, and ²⁷Al targets, it failed to reproduce the reaction cross sections and elastic scattering angular distributions of unstable nuclei.

For this, they proposed calculating the projectile-target phase shifts using nucleon-target interactions in the Glauber model calculations. This so-called nucleon-target version of the Glauber model (NTG model) has been found to considerably improve the description of the reaction cross sections and elastic scattering angular distribution data [21-23]. However, to the best of our knowledge, application of the NTG model to the analysis of single-nucleon knockout reactions and its influence on the reduction factors of single particle strengths have not yet been studied. In this study, we investigated the extent to which the R_s values of single-nucleon knockout reactions change when the NTG model is used instead of the usual OLA. Because the NTG model includes multiple scattering effects in the phase-shift functions of the colliding systems with respect to the OLA, we expect this work to provide information about the extent to which multiple scattering effects affect the description of single-nucleon removal reactions using the Glauber model.

This paper is organized as follows: The NTG model and the OLA of the Glauber model are briefly introduced in Sect. 2. The results of our calculations are given in Sect. 3, which include 1) an examination of the NTG model regarding its reproduction of the elastic scattering and total reaction cross-section data. The cases studied are the angular distributions of ¹²C elastic scattering from a carbon target at incident energies from 30 to 200 MeV/u and the ¹²C + ¹²C total reaction cross sections from 20 to 1000 MeV/u, 2) detailed study of the NTG model on single-nucleon removal at different incident energies, the case studied here is the ⁹Be ¹⁹C, ¹⁸C)X reaction, and 3) the effects of the NTG model on the reduction factors of the single particle strengths. The cases studied are single-nucleon removal cross sections of carbon isotopes ^9,10,12-20C on ⁹Be and carbon targets within 43-250 MeV/u incident energies. The range of ΔS covered in these reactions is from -26.6 to 20.1 MeV. All results are compared with those of the OLA calculations to elucidate the influence of multiple scattering effects on these reactions. Finally, the conclusions are presented in Sect. 4.

The NTG model and the OLA

The NTG model was introduced in Refs. [21, 22]. Details of its formulae can be found in Ref. [20]. For convenience, we have summarized the necessary ones here. Let us start with the phase-shift function of a nucleon-target system ${\chi }_{\text{NT}}$, which is defined in the Glauber model framework as [20]: $e^{i{\chi }_{\text{NT}}\left(b\right)}=\langle {\text{Φ}}_0^{\text{T}}|{\displaystyle \prod _{j=1}^{A_{\text{T}}}}\left[1-{\text{Γ}}_{\text{NN}}\left(b-t_j\right)\right]{\text{Φ}}_0^{\text{T}}\rangle \mathrm{,}$ (1) where b is the impact factor vector; t_j is the projection vector of the position of the jth nucleon in the target nucleus on the x-y plane (the beam direction is the z-axis); Γ_NN is the nucleon-nucleon (NN) profile function, which is the Fourier transform of the NN scattering amplitude; and $|{\text{Φ}}_0\rangle$ is the wave function of the target nucleus, which has a mass number $A_{\text{T}}$. When an independent particle model wave function is used, which is usually assumed in the Glauber model calculations, the density of the target nucleus can be written as [20] ${\left|{\text{Φ}}_0^{\text{T}}\left(r_1\mathrm{,}{r}_2\mathrm{,}\cdots \mathrm{,}{r}_{A_{\text{T}}}\right)\right|}^2={\displaystyle \prod _{j=1}^{A_{\text{T}}}}{n}_j\left(r_j\right)\mathrm{,}$ (2) where n_j(r_j) denotes the normalized density distribution of the jth nucleon in the target nucleus. The nucleon density distribution is then ${\rho }_{\text{T}}\left(r\right)={\displaystyle \sum _{j=1}^{A_{\text{T}}}}{n}_j\left(r\right)\mathrm{.}$ (3) Using an uncorrelated wave function that satisfies Eq. (2), the nucleon-target phase shift function has the form [20]: $e^{i{\chi }_{\text{NT}}\left(b\right)}={\displaystyle \prod _{j=1}^{A_{\text{T}}}}\left[1-{\displaystyle \int }\text{d}r{n}_j\left(r\right){\text{Γ}}_{\text{NN}}\left(b-t\right)\right]\mathrm{,}$ (4) where t is the projection of r on the x–y plane. When the range of the NN interaction is smaller than the radius of the target nucleus, which is satisfied in most cases, integral ${\displaystyle \int }\text{d}r{n}_j\left(r\right){\text{Γ}}_{\text{NN}}\left(b-t\right)$ is less than unity [20]. Then, the following approximation can be obtained [20]: $1-{\displaystyle \int }\text{d}r{n}_j\left(r\right){\text{Γ}}_{\text{NN}}\left(b-t\right)\approx e^{-{\displaystyle \int }\text{d}r{n}_j\left(r\right){\text{Γ}}_{\text{NN}}\left(b-t\right)}\mathrm{.}$ (5) Then, we obtained the nucleon-target phase shift of the OLA [20]: $\begin{array}{c}{e}^{i{\chi }_{\text{NT}}^{\text{OLA}}\left(b\right)}={\displaystyle \prod _{j=1}^{A_{\text{T}}}}\exp \left[-{\displaystyle \int }\text{d}r{n}_j\left(r\right){\text{Γ}}_{\text{NN}}\left(b-t\right)\right]\\ =\exp \left[-{\displaystyle \sum _{j=1}^{A_{\text{T}}}}{\displaystyle \int }\text{d}r{n}_j\left(r\right){\text{Γ}}_{\text{NN}}\left(b-t\right)\right]\\ =\exp \left[-{\displaystyle \int }\text{d}r{\rho }_{\text{T}}\left(r\right){\text{Γ}}_{\text{NN}}\left(b-t\right)\right]\mathrm{.}\end{array}$ (6) This results in the following nucleon-nucleus phase-shift function using the OLA: ${\chi }_{\text{NT}}^{\text{OLA}}(b)=i{\displaystyle \int }dr{\rho }_{\text{T}}\left(r\right){\text{Γ}}_{\text{NN}}\left(b-t\right)\mathrm{.}$ (7) Note that in Eqs. (1) and (4), multiple scattering terms appear through the cumulant expansion of the phase-shift functions. However, after applying the approximation in Eq. (5) in Eq. (4), the resulting nucleon-nucleus phase-shift with the OLA in Eq. (7) no longer contains multiple scattering terms [24].

Similar to the nucleon-nucleus case in Eq. (1), the nucleus-nucleus phase shift function, ${\chi }_{\text{PT}}\left(b\right)$, for a composite projectile and target nucleus is [20] ${\text{e}}^{i{\chi }_{\text{PT}}\left(b\right)}=\langle {\text{Φ}}_0^{\text{P}}{\text{Φ}}_0^{\text{T}}\left|{\displaystyle \prod _{i=1}^{A_{\text{P}}}}{\displaystyle \prod _{j=1}^{A_{\text{T}}}}\left[1-{\text{Γ}}_{\text{NN}}\left(b+s_i-t_j\right)\right]\right|{\text{Φ}}_0^{\text{P}}{\text{Φ}}_0^{\text{T}}\rangle \mathrm{,}$ (8) where ${\text{Φ}}_0^{\text{P}}$ is the many-body wave function of the projectile (with mass number $A_{\text{P}}$) in its ground state. The integrals are over the coordinates of all nucleons i and j in the projectile and target nuclei, whose coordinates are $r_i$ and $r_j$, respectively. $s_i$ and $t_j$ are projections on the x-y plane. The nucleus-nucleus phase shift in this equation contains contributions from single collisions and all-order multiple scattering among the constituent nucleons in the projectile and target nuclei. Equation (8) is cumbersome to evaluate directly, although it is possible. Therefore, the optical limit approximation is typically used, and the phase-shift function with this approximation is [20] ${\chi }_{\text{PT}}^{\text{OLA}}\left(b\right)=i{\displaystyle \int }d{r}_{\text{P}}{\rho }_{\text{P}}\left(r_{\text{P}}\right){\displaystyle \int }d{r}_{\text{T}}{\rho }_{\text{T}}\left(r_{\text{T}}\right){\text{Γ}}_{\text{NN}}\left(b+s-t\right)\mathrm{,}$ (9) where ${\rho }_{\text{P}}$ and ${\rho }_{\text{T}}$ are the nucleon density distributions of the projectile and target nuclei, respectively; and $r_{\text{P}}$ and $r_{\text{T}}$ are the positions of their constituent nucleons, whose projections on the x-y plane are $s$ and $t$, respectively. As in the nucleon-nucleus case in Eq. (7), only a single NN collision contributes to the phase shift. The contributions from multiple scatterings are missing, which could be recovered to some extent by the nucleon-target version of the Glauber model (the NTG model) proposed by Abu-Ibrahim and Suzuki [20-23].

The idea of the NTG model is to replace $\langle {\text{Φ}}_0^{\text{T}}\left|{\displaystyle {\prod }_{j\in \text{T}}}\left[1-{\text{Γ}}_{\text{NN}}\left(b+s_i-t_j\right)\right]\right|{\text{Φ}}_0^{\text{T}}\rangle$ for each nucleon i in the projectile in Eq. (8) by $\begin{array}{l}\langle {\text{Φ}}_0^{\text{T}}\left|{\displaystyle \prod _{j=1}^{A_{\text{T}}}}\left[1-{\text{Γ}}_{\text{NN}}\left(b+s_i-t_j\right)\right]\right|{\text{Φ}}_0^{\text{T}}\rangle \\ \equiv 1-{\text{Γ}}_{\text{NT}}\left(b+s_i\right)\mathrm{,}\end{array}$ (10) where ${\text{Γ}}_{\text{NT}}\left(b+s_i\right)$ is the profile function of its collision with the target nucleus. The nucleus-nucleus phase shift takes the following form [20]: ${\text{e}}^{i{\chi }_{\text{PT}}^{\text{NTG}}\left(b\right)}=\langle {\text{Φ}}_0^{\text{P}}\left|{\displaystyle \prod _{i=1}^{A_{\text{P}}}}\left[1-{\text{Γ}}_{\text{NT}}\left(b+s_i\right)\right]\right|{\text{Φ}}_0^{\text{P}}\rangle \mathrm{.}$ (11) This model is referred to as the NTG model. Following the same procedure used to obtain Eq. (7), the phase shift of the projectile-target system with the NTG model is ${\chi }_{\text{PT}}^{\text{NTG}}(b)=i{\displaystyle \int }\text{d}r{\rho }_{\text{P}}(r){\text{Γ}}_{\text{NT}}(b+s)\mathrm{,}$ (12) and the nucleon-target profile function Γ_NT is $\begin{array}{l}{\text{Γ}}_{\text{NT}}\left(b+s_i\right)\\ =1-\langle {\text{Φ}}_0^{\text{T}}\left|{\displaystyle \prod _{j=1}^{A_{\text{T}}}}\left[1-{\text{Γ}}_{\text{NN}}\left(b+s_i-t_j\right)\right]\right|{\text{Φ}}_0^{\text{T}}\rangle \\ =1-\exp \left[-{\displaystyle \int }d{r}_{\text{T}}{\rho }_{\text{T}}\left(r_{\text{T}}\right){\text{Γ}}_{\text{NN}}\left(b+s-t\right)\right]\mathrm{.}\end{array}$ (13) By substituting Γ_NT into Eq. (12), we obtain the nucleus-nucleus phase shift function of the NTG model: $\begin{array}{l}{\chi }_{\text{PT}}^{\text{NTG}}\left(b\right)=i{\displaystyle \int }d{r}_{\text{P}}{\rho }_{\text{P}}\left(r_{\text{P}}\right)\\ \times \left\{1-\exp \left[-{\displaystyle \int }d{r}_{\text{T}}{\rho }_{\text{T}}\left(r_{\text{T}}\right){\text{Γ}}_{\text{NN}}\left(b+s-t\right)\right]\right\}\mathrm{.}\end{array}$ (14) The nucleus-nucleus phase shift of the NTG model contains multiple scattering effects other than the OLA, which can be seen by power expansion of the nucleon-target profile function of Eq. (13): $\begin{array}{l}{\text{Γ}}_{\text{NT}}\left(b+s_i\right)\\ ={\displaystyle \int }\text{d}{r}_{\text{T}}{\rho }_{\text{T}}\left(r_{\text{T}}\right){\text{Γ}}_{\text{NN}}\left(b+s-t\right)\\ -\frac{1}{2!}{\left[{\displaystyle \int }\text{d}{r}_{\text{T}}{\rho }_{\text{T}}\left(r_{\text{T}}\right){\text{Γ}}_{\text{NN}}\left(b+s-t\right)\right]}^2+\cdots \mathrm{.}\end{array}$ (15) The first term is contributed by single scattering of the projectile nucleon from the nucleons in the target nucleus. The second term and other terms represent the contributions from multiple NN scatterings [20]. If only the first term is used in Eq. (12), the NTG phase shift is reduced to that of the OLA in Eq. (9). Considering the higher-order terms in Eq. (15), phase shifts with the NTG model recover some multiple scattering effects that are missing from the OLA. However, the contributions from the multiple scattering processes included in this method are not identical to those in the full Glauber model in Eqs. (1) and (8) [25]. Nevertheless, as we will show in the next section, the NTG model could improve the description of the elastic scattering angular distributions, especially at low incident energies, and total reaction cross sections for the ¹²C + ¹²C test case within a rather wide range of incident energies. In practice, a symmetrical version of the NTG phase shift is often calculated [21, 23] as $\begin{array}{c}{\chi }_{\text{PT}}^{\text{NTG}}\left(b\right)=\frac{i}{2}{\displaystyle \int }\text{d}{r}_{\text{P}}{\rho }_{\text{P}}\left(r_{\text{P}}\right)\\ \left\{1-\exp \left[-{\displaystyle \int }\text{d}{r}_{\text{T}}{\rho }_{\text{T}}\left(r_{\text{T}}\right){\text{Γ}}_{\text{NN}}\left(b+s-t\right)\right]\right\}\\ +\frac{i}{2}{\displaystyle \int }\text{d}{r}_{\text{T}}{\rho }_{\text{T}}\left(r_{\text{T}}\right)\\ \left\{1-\exp \left[-{\displaystyle \int }\text{d}{r}_{\text{P}}{\rho }_{\text{P}}\left(r_{\text{P}}\right){\text{Γ}}_{\text{NN}}\left(b+t-s\right)\right]\right\}\mathrm{.}\end{array}$ (16) However, the phase shifts calculated using Eqs. (14) and (16) are often very close to each other [21, 22].

The profile function Γ_NN in both the OLA and NTG model calculations is parameterized in a Gaussian form: ${\text{Γ}}_{\text{pN}}\left(b\right)=\frac{1-i{\alpha }_{\text{pN}}}{4\pi {\beta }_{\text{pN}}}{\sigma }_{\text{pN}}^{\text{tot}}\exp \left(-\frac{b^2}{2{\beta }_{\text{pN}}}\right)\mathrm{,}$ (17) where the Γ_NN parameters ${\sigma }_{\text{pN}}^{\text{tot}}$, ${\alpha }_{\text{pN}}$, and β_pN are the proton-nucleon total cross section, the ratio of the real to imaginary part of the p–N scattering amplitudes, and the corresponding slope parameter [26], respectively. Because of the lack of experimental data on neutron-neutron scattering, Γ_pp is commonly used instead of Γ_NN. In this study, ${\sigma }_{\text{pN}}^{\text{tot}}$ is obtained from Ref. [27], which is parameterized by fitting the experimental data from Ref. [28], the ${\alpha }_{\text{pN}}$ parameters were taken from those tabulated in Ref. [26] for incident energies ranging from 100 MeV/u to 2200 MeV/u. If the beam energy was lower than 100 MeV/u, we obtained the value corresponding to the lowest energy from the table. The finite-range slope parameters β_pN were taken to be 0.125 fm², in accordance with the systematic studies of single-nucleon removal reactions [13, 15, 29].

Comparisons between the NTG model and OLA in Glauber model calculations

In Ref. [30], T. Nagashisa and W. Horiuchi demonstrated the effectiveness of the NTG by comparing the description of the total reaction cross sections using the full Glauber model calculation, the NTG model, and the OLA for cases of ^12,20,22C on a ¹²C target at various incident energies. In this work, our main purpose was to study how much the single-nucleon removal cross sections (σ_-1N) change when the NTG model is used instead of the OLA. Before calculating σ_-1N, we first compared our calculations for the elastic scattering angular distributions and total reaction cross sections with the experimental data and with the predictions of the OLA. The calculations were made for the ¹²C + ¹²C system. Thus, we verified the effectiveness of the Γ_NN parameters used in our calculations, which were further used in the calculations of σ_-1N. Single-nucleon removal reactions were calculated using a modified version of the computer code MOMDIS [31].

3.1

Elastic scattering angular distributions and total reaction cross sections

The angular distributions of ¹²C elastic scattering from a ¹²C target at 30, 85, 120, and 200 MeV/u were calculated with both the OLA and NTG model. The results are presented in Fig. 1 together with the experimental data. The dots are experimental data from Refs. [32, 33]. Clearly, the NTG improved the description of the ¹²C + ¹²C elastic scattering considerably with respect to the OLA, especially when the incident energy was below approximately 100 MeV/u. This is expected because the multiple scattering effect, which is included in the NTG model but not in the OLA, is more important at low incident energies than at higher incident energies. Note that other corrections owing to, for instance, the antisymmetrization of the projectile and target wavefunctions [34], Fermi motion of the nucleons in the colliding nuclei [35], and distortion of the trajectories [18] can also affect the low-energy cross sections. More complete calculations that consider these aspects together may be an interesting subject for the future.

Fig. 1Full-screen View

(Color online) Elastic scattering angular distributions of ¹²C on a carbon target at incident energies of 30, 85, 120, and 200 MeV/u. The red solid and blue dashed curves are results of Glauber model calculations with the NTG model and the OLA, respectively

A comparison of the NTG and OLA predictions and the total reaction cross sections of the ¹²C + ¹²C system is shown in Fig. 2. The symbols represent experimental data from Ref. [36-43]. Again, we see that the results of the NTG model are in better agreement with the experimental data than those of the OLA, especially for incident energies of several tens of MeV/u and above, where most of the one-nucleon removal cross-sectional data were measured [12]. In both elastic scattering and total reaction cross section calculations, the proton and neutron density distributions of the ¹²C nucleus are taken to be a Gaussian form with a root mean square radius of 2.32 fm [12], which is very close to the $2.33\pm 0.01\text{ fm}$ from elastic electron scattering data [44].

Fig. 2Full-screen View

(Color online) Reaction cross sections of ¹²C on a carbon target. The red solid and blue dash-dotted curves are results of Glauber model calculations with the NTG model and the OLA, respectively

Note that the Γ_NN parameters are the same in both the NTG and OLA calculations. The only difference between these two methods is that the former introduces multiple scattering effects in the calculation of the eikonal phase functions. The improvement provided by the NTG model in the description of elastic scattering angular distributions and total reaction cross sections suggests that nuclear medium effects, such as the multiple scattering effect studied here, should be considered in the Glauber model description of nuclear reactions induced by heavy ions. In the following section, we study how the NTG model could affect the theoretical predictions of the single-neutron removal cross sections and single particle strengths obtained from the experimental data.

3.2

Single-nucleon removal cross sections at different incident energies

In an inclusive single-nucleon removal reaction $A(a\mathrm{,}b)X$, where only the core nucleus b ($A_b=A_a-1$) is detected, two processes may occur: diffraction dissociation and stripping, which correspond to the escape of valence neutrons or their capture by the target nucleus, respectively. Within the Glauber model framework, their cross sections, ${\sigma }_{\text{sp}}^{\text{dd}}$ and ${\sigma }_{\text{sp}}^{\text{str}}$, respectively, are calculated by [45] $\begin{array}{c}{\sigma }_{\text{sp}}^{\text{dd}}=\frac{1}{2j+1}{\displaystyle \sum _m}{\displaystyle \int }db[\langle {\psi }_{nljm}\left|{\left|1-S_{\text{v}}{S}_{\text{c}}\right|}^2\right|{\psi }_{nljm}\rangle \\ -{\displaystyle \sum _{m^{\prime }}}{\left|\langle {\psi }_{nlj{m}^{\prime }}\left|\left(1-S_{\text{v}}{S}_{\text{c}}\right)\right|{\psi }_{nljm}\rangle \right|}^2]\mathrm{,}\end{array}$ (18) and $\begin{array}{c}{\sigma }_{\text{sp}}^{\text{str}}=\frac{1}{2j+1}{\displaystyle \sum _m}{\displaystyle \int }db{\left|S_{\text{c}}\right|}^2\times \\ \langle {\psi }_{nljm}\left|\left(1-{\left|S_{\text{v}}\right|}^2\right)\right|{\psi }_{nljm}\rangle \mathrm{.}\end{array}$ (19) Here, $S_{\text{c}}=e^{i{\chi }_{\text{cT}}}$ and $S_{\text{v}}=e^{i{\chi }_{\text{vT}}}$ are the core-target and valence nucleon-target Smatrices, respectively. The valence nucleon-target phase shift function ${\chi }_{\text{vT}}$ is calculated using Eq. (7), and the core-target phase shift function ${\chi }_{\text{cT}}$ is calculated using Eq. (9) for the OLA and Eq. (16) for the NTG model; $b$ is the impact factor vector of the projectile in the plane perpendicular to the beam direction; ψnljm is the single-particle wave function (SPWF) and n, l, and j are the principal, angular momentum, and total angular momentum numbers, respectively; and m is the projection of j. Equations (7, (9), (14), and (16) concern only nuclear phase shifts. The Coulomb phase shift must also be considered for charged particles [31]: ${\chi }_{\text{C}}=2\eta \ln (kb)\mathrm{,}$ (20) where $\eta =Z_1{Z}_2{e}^2\mu /{\hslash }^{2}k$ is the Sommerfeld parameter and Z₁ and Z₂ are the charge numbers of the two colliding particles, whose reduced mass is μ, and k is the wave number in the center-of-mass system. Single-particle wave functions are associated with the specific states of the core with spin Ib and the composite nuclei with spin Ia by spectroscopic factors ${\left(C^{2}S\right)}_{I_a{I}_b\mathrm{,}nlj}$. Therefore, the single-particle cross section of removal of a nucleon from the ground state of a projectile leaving the core nucleus in a specific state with the SPWF having quantum numbers nlj is ${\sigma }_{\text{sp}}\left(I_a{I}_b\mathrm{,}nlj\right)={\left(\frac{A}{A-1}\right)}^N{\left(C^{2}S\right)}_{I_a\mathrm{,}{I}_b\mathrm{,}nlj}\times \left({\sigma }_{\text{sp}}^{\text{dd}}+{\sigma }_{\text{sp}}^{\text{str}}\right)\mathrm{,}$ (21) where the ${[A/(A-1)]}^N$ factor represents the center-of-mass corrections to the spectroscopic factor $C^{2}S$ [46], and $N=2n+l$ is the number of oscillator quanta associated with the major shell of the removed particle (the minimum value of n is taken to be zero).

Experimentally, single-nucleon removal cross sections are usually measured inclusively, that is, only the core nucleus b is measured without discriminating its energy states. Correspondingly, theoretical calculations for these measurements should also include the contributions from all the bound excited states of the core nucleus b [13], which corresponds to a summation of all the single-particle cross sections associated with all possible single particle wave functions: ${\sigma }_{-1\text{N}}^{\text{th}}={\displaystyle \sum _{nlj\mathrm{,}{I}_b}}{\sigma }_{\text{sp}}\left(I_a\mathrm{,}{I}_b\mathrm{,}nlj\right)\mathrm{.}$ (22)

To see how much difference the NTG model predicts in the single-nucleon removal cross sections with respect to the OLA, we study the (C¹⁹C, ¹⁸C) reaction on a ⁹Be target at 64, 100, 200, and 400 MeV/u incident energies. The excited states of the ¹⁸C nucleus, the associated single-particle wave functions, and their corresponding shell model predicted spectroscopic factors are taken to be the same as those in Ref. [15]. The single-particle wave functions are calculated with single particle potentials of the Woods–Saxon forms with the depths adjusted to provide the experimental separation energies of the valence nucleon. The radius and diffuseness parameters were taken to be r₀=1.25 fm and $a=0.7\text{ fm}$, respectively, the same as those used in Ref. [15]. The results are shown in Table 1. Single-nucleon removal cross sections with the NTG model and OLA are denoted as ${\sigma }_{-1\text{N}}^{\text{NTG}}$ and ${\sigma }_{-1\text{N}}^{\text{OLA}}$, respectively. Note that the ${\sigma }_{-1\text{N}}^{\text{OLA}}$ values at 64 MeV/u agree well with those reported in Ref. [15]. The ratios of ${\sigma }_{-1\text{N}}^{\text{NTG}}$ and ${\sigma }_{-1\text{N}}^{\text{OLA}}$ are shown in Fig. 3.

Table 1Full-screen View

Single-neutron removal cross sections of ¹⁹C on a beryllium target at incident energies of 64, 100, 200, and 400 MeV/u calculated using the NTG model, ${\sigma }_{-1\text{N}}^{\text{NTG}}$, and the OLA, ${\sigma }_{-1\text{N}}^{\text{OLA}}$. The state of the core nucleus and their corresponding single-nucleon spectroscopic factors are taken from Ref. [15]

$E_{\text{inc}}$ (MeV/u)	Ex (MeV)	$J^{\pi }$	nlj	$C^{2}S$	${\sigma }_{-1\text{N}}^{\text{OLA}}$ (mb)	${\sigma }_{-1\text{N}}^{\text{NTG}}$ (mb)	${\sigma }_{-1\text{N}}^{\text{NTG}}/{\sigma }_{-1\text{N}}^{\text{OLA}}$
64	0.000	0+	1s1/2	0.580	104.31	109.3	1.050
	2.144	2+	0d5/2	0.470	18.93	21.16	1.118
	3.639	2+	0d5/2	0.104	3.53	3.98	1.127
	3.988	0+	1s1/2	0.319	17.82	19.72	1.107
	4.915	3+	0d5/2	1.523	46.18	52.21	1.131
	4.975	2+	0d5/2	0.922	27.83	31.46	1.130
	Inclusive				218.42	237.83	1.089
100	0.000	0+	1s1/2	0.580	87.58	90.14	1.029
	2.144	2+	0d5/2	0.470	17.95	19.13	1.066
	3.639	2+	0d5/2	0.104	3.41	3.64	1.067
	3.988	0+	1s1/2	0.319	16.43	17.44	1.061
	4.915	3+	0d5/2	1.523	45.05	48.24	1.071
	4.975	2+	0d5/2	0.922	27.15	29.08	1.071
	Inclusive				197.57	207.67	1.051
200	0.000	0+	1s1/2	0.580	61.66	63.55	1.031
	2.144	2+	0d5/2	0.470	15.46	16.52	1.069
	3.639	2+	0d5/2	0.104	3.01	3.23	1.073
	3.988	0+	1s1/2	0.319	13.47	14.30	1.062
	4.915	3+	0d5/2	1.523	40.59	43.61	1.071
	4.975	2+	0d5/2	0.922	24.48	26.31	1.075
	Inclusive				158.67	167.52	1.056
400	0.000	0+	1s1/2	0.580	54.76	57.04	1.042
	2.144	2+	0d5/2	0.470	14.61	16.00	1.095
	3.639	2+	0d5/2	0.104	2.87	3.16	1.101
	3.988	0+	1s1/2	0.319	12.54	13.57	1.082
	4.915	3+	0d5/2	1.523	38.80	42.91	1.106
	4.975	2+	0d5/2	0.922	23.41	25.89	1.106
	Inclusive				146.99	158.57	1.079

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Fig. 3Full-screen View

(Color online) Ratios of the NTG and OLA predicted single-particle cross sections associated with different core states of the ⁹Be (¹⁹C, ¹⁸C)X reaction at incident energies 64, 100, 200, and 400 MeV/u. The black dots represent the results calculated with Eq. (21). The excitation energies of the core nucleus ¹⁸C and the properties of their corresponding single-particle wave functions—their nlj values and root mean square radii—are also shown. The lines are to guide the eyes

It is interesting to note the following:

1. The one-nucleon removal cross sections calculated with the NTG model are larger than those calculated with the OLA within the whole energy range from 50 to 400 MeV/u.

2. Such differences are larger at incident energies smaller than approximately 100 MeV/u, almost constant around 100-200 MeV/u, and increase slightly when the incident energy is larger than approximately 200 MeV/u,

3. The differences are also more significant when the root mean square radius of the single-particle wave function is smaller, which means that the NTG model is especially important for one-neutron removal cross sections of a given reaction when the single nucleon is tightly bound.

The same was observed for the other nuclei examined in this study. The difference between the NTG model and the OLA is in the core-target S-matrix, S_c, only. However, as expressed in Eqs. (18) and (19), we cannot separate S_c from S_v and the single-particle wave functions when calculating the single-nucleon removal cross sections. Thus, we cannot show how the NTG model affects the σ_-1N values with respect to the OLA. In the following subsection, we discuss how the spectroscopic factors extracted from the experimental data and their reduction factors change when the NTG model is used instead of the OLA.

3.3

Reduction factors of single particle strengths

The spectroscopic factors in Eq. (22) are often obtained from configuration interaction shell model (CISM) calculations to determine one-nucleon removal cross sections. Owing to limited model spaces and insufficient treatment of nucleon-nucleon correlations, it is well known that CISM-predicted SFs are usually larger than the experimental ones. The reduction factors of the SFs, R_s, which are ratios of the experimental and theoretical SFs, are defined to quantify the differences. In the case of inclusive single-nucleon knockout reactions, the reduction factors are defined as the ratios between the experimental and theoretical cross sections [11, 12]: $R_{\text{s}}={\sigma }_{-1\text{N}}^{\exp }/{\sigma }_{-1\text{N}}^{\text{th}},$ For nuclei with more than one set of available experimental data, the weighted mean of the R_s values for each measurement is used [47]: $\Re =\frac{{\displaystyle \sum {}_i}{R}_{\text{s}}{}_i{w}_i}{{\displaystyle \sum {}_i}{w}_i}\mathrm{,}$ (23) where the weights are defined by the errors in the individual R_s values ${\left(\text{Δ}{R}_{\text{s}}\right)}_i$: $w_i={\left[\frac{1}{\text{Δ}{R}_{\text{s}}{}_i}\right]}^2\mathrm{,}$ and the errors on average $\overline{R_{\text{s}}}$ are $\text{Δ}\Re =\frac{1}{\sqrt{{\displaystyle \sum {}_i}{w}_i}}\mathrm{,}$ The effective neutron-proton asymmetry ΔS_eff is given by [15] $\begin{array}{l}\text{Δ}{S}_{\text{eff}}=S_{\text{n}}+{\overline{E}}_{\text{f}}-S_{\text{p}}\mathrm{,}\text{ for neutron removal}\mathrm{,}\\ \text{Δ}{S}_{\text{eff}}=S_{\text{p}}+{\overline{E}}_{\text{f}}-S_{\text{n}}\mathrm{,}\text{ for proton removal}\mathrm{,}\end{array}$ where ${\overline{E}}_{\text{f}}$ is obtained by weighting the excitation energy E^* of each final state using the single-nucleon removal cross section of that state.

We analyzed a series of single-nucleon removal reaction data by using the method described in the previous subsection. The details of these reactions, such as the target nuclei used and incident energies, are listed in Table 2. The theoretical predicted single-nucleon removal cross sections using the NTG and the OLA, ${\sigma }_{-1\text{N}}^{\text{NTG}}$ and ${\sigma }_{-1\text{N}}^{\text{OLA}}$, respectively, are also listed together with the experimental single-nucleon removal cross sections, ${\sigma }_{-1\text{N}}^{\text{exp}}$, and the reduction factors, ${\Re }^{\text{NTG}}$ and ${\Re }^{\text{OLA}}$, respectively. The single-particle spectroscopic factors ($C^{2}S$) used in these calculations were obtained from references corresponding to the experimental data and Ref. [47]. The reduction factors are presented in Fig. 4 as functions of neutron-proton asymmetry. Because many σ_-1N were measured inclusively, that is, they include all bound states of the core nuclei, which correspond to different separation energies of the removed nucleon, an effective neutron-proton asymmetry is used here: $\text{Δ}{S}_{\text{eff}}=S_{\text{n}}+{\overline{E}}_{\text{f}}-S_{\text{p}}$ for neutron removal and $\text{Δ}{S}_{\text{eff}}=S_{\text{p}}+{\overline{E}}_{\text{f}}-S_{\text{n}}$ for proton removal, where ${\overline{E}}_{\text{f}}$ is the weighted mean excitation energy of the core nucleus, ${\overline{E}}_{\text{f}}=\left({\displaystyle {\sum }_i}{E}_{\text{ex}\mathrm{,}i}{\sigma }_{\text{sp}\mathrm{,}i}\right)/{\displaystyle {\sum }_i}{\sigma }_{\text{sp}\mathrm{,}i}$, with $E_{\text{ex}\mathrm{,}i}$ and ${\sigma }_{\text{sp}\mathrm{,}i}$ being the excitation energy of the core nucleus in its i-th state and the corresponding single-particle cross section with Eq. (21) [11]. In all these calculations, the single-particle wave functions are calculated with Woods–Saxon potentials whose radius parameters, r₀, are determined with the HF calculations [48] and the diffuseness parameters being fixed as $a=0.65\text{ fm}$ except for the ^15,17,18C projectiles, for which the r₀=1.15 fm and $a=0.50\text{ fm}$ are used following Ref. [49]. And for proton removal of ¹⁶C, r₀=1.40 fm and $a=0.70\text{ fm}$ are used following Ref. [50]. The proton and neutron density distributions of the nucleus ⁹Be are taken to be a Gaussian form with a root mean square radius of 2.36 fm [12].

Table 2Full-screen View

Experimental (${\sigma }_{-1\text{N}}^{\text{exp}}$) and theoretical inclusive single-nucleon removal cross sections calculated with the OLA (${\sigma }_{-1\text{N}}^{\text{OLA}}$) and the NTG model (${\sigma }_{-1\text{N}}^{\text{NTG}}$), and the corresponding reduction factors ${\Re }^{\text{OLA}}$ and ${\Re }^{\text{NTG}}$

Reaction	ΔSeff (MeV)	Target	Einc (MeV/u)	${\sigma }_{-1\text{N}}^{\text{exp}}$ (mb)	${\sigma }_{-1\text{N}}^{\text{OLA}}$ (mb)	${\sigma }_{-1\text{N}}^{\text{NTG}}$ (mb)	${\Re }^{\text{OLA}}$	${\Re }^{\text{NTG}}$
(20C, 19C)	-26.574	C	240	58(5) [51]	47.55	51.88	1.22(11)	1.12(10)
(19C, 18C)	-24.142	Be	57	264(80) [52]	179.06	201.62	1.47(45)	1.31(40)
	-24.104	Be	64	226(65) [53]	176.69	195.48	1.28(37)	1.16(33)
	-23.754	C	243	163(12) [51]	134.75	146.63	1.21(9)	1.11(8)
Average	-24.022						1.22(8)	1.12(8)
(18C, 17C)	-21.793	C	43	115(18) [49]	103.20	128.70	1.11(17)	0.89(14)
(17C, 16C)	-20.130	C	49	84(8) [49]	92.80	109.70	0.91(9)	0.77(7)
	-20.121	Be	62	115(14) [52]	87.80	100.77	1.31(16)	1.14(14)
	-20.121	Be	79	116(18) [54]	90.37	100.48	1.28(20)	1.15(18)
Average	-20.124						1.03(7)	0.88(6)
(15C, 14C)	-18.275	C	54	137(16) [49]	180.56	196.44	0.76(9)	0.70(8)
	-18.242	C	62	159(15) [49]	176.11	189.78	0.90(8)	0.84(8)
	-18.169	C	83	146(23) [36]	166.44	176.08	0.88(14)	0.83(13)
	-17.879	Be	103	146(23) [53]	142.52	149.89	0.98(3)	0.94(3)
Average	-18.155						0.95(3)	0.90(0)
16C, 15C)	-18.055	C	55	65(6) [49]	90.90	103.73	0.72(7)	0.63(6)
	-18.053	C	62	77(9) [49]	89.78	101.10	0.86(10)	0.76(9)
	-18.045	Be	75	81(7) [50]	81.99	90.94	0.99(9)	0.89(8)
	-18.094	C	83	65(5) [52]	86.75	94.87	0.75(6)	0.69(5)
Average	-18.051						0.80(4)	0.71(3)
(14C, 13C)	-10.807	C	67	65(4) [49]	133.284	148.61	0.49(3)	0.44(3)
	-10.800	C	83	67(14) [36]	130.74	142.66	0.51(13)	0.47(12)
	-10.767	C	235	80(7) [55]	110.92	121.39	0.72(6)	0.66(6)
Average	-10.793						0.53(3)	0.48(2)
(12C, 11C)	3.259	C	95	53(22) [56]	102.21	111.06	0.52(22)	0.48(20)
	3.266	C	240	60.51(11.08) [57]	94.12	104.37	0.64(12)	0.58(11)
	3.265	C	250	56.0(41) [58]	93.73	104.31	0.60(4)	0.54(4)
Average	3.263						0.60(4)	0.54(4)
(10C, 9C)	17.277	Be	120	23.4(11) [59]	47.40	51.65	0.49(2)	0.45(2)
	17.277	C	120	27.4(13) [59]	49.72	54.36	0.55(3)	0.50(2)
Average	17.277						0.52(2)	0.48(2)
(9C, 8B)	-12.925	Be	67	48.6(73) [60]	62.77	66.67	0.77(12)	0.73(11)
	-12.925	Be	100	56(3) [61]	58.77	59.72	0.95(5)	0.94(5)
Average	-12.925						0.92(5)	0.90(5)
(12C, 11B)	-2.237	C	230	63.9(66) [62]	103.75	105.33	0.62(6)	0.61(6)
	-2.237	C	250	65.6(26) [58]	102.93	105.36	0.64(3)	0.62(2)
Average	-2.237						0.63(2)	0.62(2)
(13C, 12B)	13.523	C	234	39.5(60) [62]	79.69	81.55	0.43(5)	0.40(4)
(14C, 13B)	12.830	C	235	41.3(27) [62]	78.65	81.43	0.53(3)	0.51(3)
(16C, 15B)	18.303	Be	75	18(2) [50]	60.23	62.50	0.30(3)	0.29(3)
	18.303	Be	239	16(2) [63]	56.86	58.45	0.28(4)	0.27(3)
	18.303	C	239	18(2) [62]	54.57	55.87	0.33(4)	0.32(4)
Average	18.303						0.30(2)	0.28(2)
(15C, 14B)	20.134	C	237	28.4(28) [62]	55.36	57.58	0.51(5)	0.49(5)

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Fig. 4Full-screen View

(Color online) Averaged reduction factors $\Re$ listed in Table 2 as functions of the effective neutron-proton asymmetry ΔS_eff. The red squares and blue dots are results of the neutron removal of the NTG model and the OLA, respectively. The red triangles and blue inverted triangles are the same but for proton removal. The light red and blue bands represent the widths of their distributions

As shown in Table 2, σ_-1N values predicted with the NTG model are generally larger than those predicted with the OLA. Thus, the $\Re$ values with the NTG are smaller than those with the OLA. On average, the changes in the $\Re$ values induced by the NTG model with respect to the OLA are approximately 7.8%. However, as shown in Fig. 4, the $\Re$ values with the NTG model and the OLA, ${\Re }^{\text{NTG}}$ and ${\Re }^{\text{OLA}}$, respectively, still depend linearly on the effective neutron-proton asymmetry ΔS_eff, although the slope with the NTG model is 18% smaller than that with the OLA. The parameters of this linear dependence are $\begin{array}{c}{\Re }^{\text{OLA}}=0.687-0.0154\text{Δ}{S}_{\text{eff}}\mathrm{,}\\ {\Re }^{\text{NTG}}=0.633-0.0131\text{Δ}{S}_{\text{eff}}\mathrm{.}\end{array}$ (24) Thus, the systematics of the $\Re$ values with respect to ΔS_eff observed in Refs. [11, 12] persist even when the multiple scattering effects inherited in the NTG model are included in the Glauber model calculations.

A closer examination of Fig. (4) shows that the differences between ${\Re }^{\text{NTG}}$ and ${\Re }^{\text{OLA}}$ in the most negative ΔS_eff region are larger than those in the most positive ΔS_eff region. Specifically, the average differences between ${\Re }^{\text{NTG}}$ and ${\Re }^{\text{OLA}}$ are 9.9% for ΔS_eff<-10 MeV and 5.3% for ΔS_eff>10 MeV. This suggests that the multiple scattering effect introduced by the NTG model is more important for the removal of weakly bound nucleons than for deeply bound nucleons. This is misleading. Most cases in the ΔS_eff<-10 MeV region are single-neutron removal reactions and those in the ΔS_eff>10 MeV region are single-proton removal reactions. From the ${\Re }^{\text{NTG}}$ and ${\Re }^{\text{OLA}}$ values listed in Table 2, it is clear that the average differences between ${\Re }^{\text{NTG}}$ and ${\Re }^{\text{OLA}}$ are 10.6% and 4.2% for neutron and proton removal reactions, respectively. Currently, it is unclear why the NTG model exhibits such systematic differences in these two types of reactions. As discussed at the end of the previous section, the only difference between the NTG model and the OLA is in the core-target S-matrices, S_c. However, as expressed in Eqs. (18) and (19), S_c cannot be separated from S_v and the single-particle wave functions when calculating the single-nucleon removal cross-sections. This implies that the multiple scattering effects induced in the NTG model on σ_-1N through S_c are moderated by single-particle wave functions, which are different for different cases. Therefore, we cannot explicitly show how the NTG model alone affects the σ_-1N values or why it behaves differently for proton and neutron removal reactions.

Summary

The reduction in single particle strengths, represented by the reduction factors of single-nucleon spectroscopic factors extracted from experimental data with respect to the configuration interaction shell model predictions, is supposed to be related to the nucleon-nucleon correlations in atomic nuclei. Much theoretical and experimental effort has been devoted to this field of research. One of the open questions is why the reduction factors obtained from intermediate- and high-energy single-nucleon removal cross sections such as those compiled in Refs. [11, 12] show strong linear dependence on the neutron-proton asymmetry, whereas those of other types of reactions, such as (p, pN) and single-nucleon transfer reactions, do not [2, 3, 6, 9, 64, 65]. Because single-nucleon removal reactions were analyzed using the Glauber model, the validity of the Glauber model for such reactions is being questioned. In this respect, corrections to the Glauber model and an examination of their effects on single-nucleon removal cross sections are important.

In this study, we examined how the nucleon-target version of the Glauber model (the NTG model), which introduces multiple scattering of the constituent nucleons in the projectile and the target nuclei, can change the theoretically predicted single-nucleon removal cross sections with respect to the usual optical limit approximation, which does not contain multiple scattering effects. For this purpose, we first examined the NTG model in its reproduction of the elastic scattering angular distributions and the total reaction cross sections of the ¹²C + ¹²C system, and compared their results with the experimental data and those calculated with the OLA. The NTG model was found to improve the description of the elastic scattering angular distributions, particularly at lower incident energies. Both the elastic scattering and total reaction cross sections calculated in this work agree well with those reported in previous publications, for example, Refs. [21, 23, 30].

We then compared the predictions of the inclusive single-nucleon removal cross sections using the NTG model and OLA. The case studied is the ⁹Be (¹⁹C, ¹⁸C)X reaction within the incident energy range from 64 MeV/u to 400 MeV/u. The ${\sigma }_{-1n}$ values predicted by the NTG model were larger than those predicted by the OLA within the entire energy range. This difference is larger at lower incident energies. It is also larger when the separation energy of the nucleon is larger, which corresponds to a smaller root mean square radius of the single-particle wave function.

Finally, we studied the extent to which the reduction factors of the single particle strengths obtained from single-nucleon removal reactions changed when the NTG model was used instead of the OLA. The cases studied are one-nucleon removal reactions induced by ^9,10,12-20C isotopes on carbon and ⁹Be targets. On average, the reduction factors obtained with the NTG model were found to be less than those obtained with the OLA by 7.8%. We also found that the average differences in ${\sigma }_{-1\text{n}}$ are larger than those in ${\sigma }_{-1\text{p}}$ by 10.6% and 4.2%. However, the linear dependence of the reduction factor on the neutron–proton asymmetry persisted. Thus, the question of why the reduction factors of the single particle strengths from single-nucleon removal reaction measurements depend differently on ΔS with respect to other types of reactions remains open, even when the multiple scattering effect is included in the Glauber model analysis with the NTG model.