Neutron radii and neutron skin of neutron-rich nuclei deduced from proton-nucleus total reaction cross sections

TL;DR

This paper introduces a novel method to determine neutron radii of neutron-rich nuclei by comparing reaction cross sections with stable isotopes, successfully applying it to deduce the neutron radii of $^{22}$C and $^{14}$Be for the first time.

Contribution

The paper presents a new approach to extract neutron radii from reaction cross sections, enabling measurements for nuclei where direct methods are challenging.

Findings

Neutron radii of $^{22}$C and $^{14}$Be were deduced for the first time.

The method uses reaction cross sections measured at the same energy on a proton target.

Experimental data at 40A MeV was successfully applied to these nuclei.

Abstract

A new method is proposed to deduce the neutron radii of neutron-rich nuclei. This method requires measuring the reaction cross sections of both the neutron-rich nucleus and its stable isotope at the same energy on a proton target. Using this method and the available experimental data of $^{12,22}$C + $p$ and $^{9,14}$Be + $p$ at 40$A$ MeV, the neutron radii of $^{22}$C and $^{14}$Be have been deduced for the first time.

Equations

$${\sigma }_{\mathrm{A}}^{\mathrm{R}}=\int db\left(1-{\mathrm{e}}^{-2\mathrm{Im}[{\chi }_p(b)+{\chi }_n(b)]}\right),$$

$$\delta {\sigma }_{{\mathrm{A}}_2,{\mathrm{A}}_1}^{\mathrm{R}}=2\pi \int b{\mathrm{e}}^{-2\mathrm{Im}[{\chi }_{p_1}(\mathbf{b})+{\chi }_{n_1}(\mathbf{b})]}$$

$$\times \left(1-{\mathrm{e}}^{-2\mathrm{Im}[\Delta {\chi }_p(\mathbf{b})+\Delta {\chi }_n(\mathbf{b})]}\right)db,$$

$$-2\mathrm{Im}[\Delta {\chi }_N(\mathbf{b})]=-\frac{{\sigma }_{pN}^{\mathrm{tot}}}{2\pi {\beta }_{pN}}$$

$$\times \int d\mathbf{s}[{\rho }_{N_2}(\mathbf{s})-{\rho }_{N_1}(\mathbf{s})]{\mathrm{e}}^{-(\mathbf{b}+\mathbf{s}{)}^2/(2{\beta }_{pN})}.$$

$$-2\mathrm{Im}[\Delta {\chi }_N(\mathbf{b})]=-{\sigma }_{pN}^{\mathrm{tot}}({\rho }_{N2}(\mathbf{b})-{\rho }_{N1}(\mathbf{b})).$$

$$\delta {\sigma }_{{\mathrm{A}}_2,{\mathrm{A}}_1}^{\mathrm{R}}\simeq 2\pi \int b{\mathrm{e}}^{-2\mathrm{Im}[{\chi }_{p1}(\mathbf{b})+{\chi }_{n1}(\mathbf{b})]}$$

$$\times \left(1-{\mathrm{e}}^{-2\mathrm{Im}[\Delta {\chi }_n(\mathbf{b})]}\right)db.$$

$$\mathrm{FS}=\frac{\delta {\sigma }_{{\mathrm{A}}_2,{\mathrm{A}}_1}^{\mathrm{R}}[Eq.(\text{ref}dreac)]}{\delta {\sigma }_{{\mathrm{A}}_2,{\mathrm{A}}_1}^{\mathrm{R}}[Eq.(\text{ref}dreac1)]},$$

Full text

Neutron halo structure has been observed among the neutron-rich nuclei, it is manifested by the neutron density extending to extraordinary large radial distance. The neutron halo structure is indicated first by an increase in the root-mean-square point matter radii (matter radii) tanihata85 . The matter radii (${\bar{r}}_{m}=\langle r^{2}_{m}\rangle^{1/2}$) of unstable nuclei are usually calculated by measuring their reaction (interaction) cross sections on Carbon target tanihata85 or proton target tanaka10 by the Glauber theory.

The root-mean-square point neutron radii (neutron radii) play an important role in nuclear physics and astrophysics. Neutron radii (${\bar{r}}_{n}=\langle r^{2}_{n}\rangle^{1/2}$) are essential to extract the neutron skin thickness ($S_{n}={\bar{r}}_{n}-{\bar{r}}_{p}$ ). The neutron skin thickness is strongly correlated to the radius of the low-mass neutron stars and to the density dependence of the symmetry energy fattoyev18 ; sagawa18 , (often encoded in a quantity denoted by $L$) which is related to the pressure of the pure neutron matter at the saturation density iida03 . The origin of the size of both the thickness of the neutron skin and the neutron star is the pressure of the neutron-rich matter, pushing either against surface tension in an atomic nucleus or against gravity in a neutron star. The stellar radius and the neutron skin are sensitive to the same equation of state fattoyev18 . Recently, neutron star physics has received renewed interest since the LIGO-Virgo collaboration made the first direct detection of gravitational waves from the coalescence of a neutron star binary system abbott17 . The analysis of the data placed constraints on the tidal effects of the coalescing bodies, which were then translated to constraints on neutron star radii abbott18 . Also, the knowledge of the nuclear symmetry energy is relevant for the Standard Model tests via atomic parity violation sil .

Electron-nucleus scattering and measurement of muonic x rays are used to determine the charge radii of stable nuclei. These techniques cannot be used for neutron-rich isotopes due to the weak beam intensities. High precision measurements of the charge radii (${\bar{r}}_{c}=\langle r^{2}_{c}\rangle^{1/2}$) of He mueller07 ; brodeur12 , Li sanchez06 , Be nortershauser09 ; krieger12 , and Mg Yordanov12 isotopes have been achieved using the isotope-shift technique. Combining the isotope-shift data with the data deduced from the reaction (interaction) cross section, one can obtain the neutron radii of the neutron-rich isotopes from the relation ${\bar{r}}^{2}_{n}=(A/N){\bar{r}}^{2}_{m}-(Z/N){\bar{r}}^{2}_{p}$ and hence their neutron skin thickness Suzuki95 .

Due to the limitation of the isotope-shift technique, mainly due to the low luminosity of rare isotopes close to the neutron drip line, the measurement of the charge-changing cross section ($\sigma_{cc}$) has been used to determine the proton radii of neutron-rich Carbon-isotopes yamaguchi11 . The authors of Ref. yamaguchi11 report that they scaled the experimental cross section to reproduce the known charge radius of 12C. They also combined the charge-changing cross section data with the reaction cross section measurements to obtain the neutron radii of 15,16C, and hence their neutron skins. Refs. estrade ; kanungo used charge-changing cross section to deduce the proton radii of 12-17B and 12-19C without any scaling factor.

The knowledge of the neutron radii of stable nuclei is scarce compared to that of proton radii, its main sources are parity-violation in electron scattering abrahamyan and hadron scattering experiments, see for example clark . Currently, these methods cannot be used for neutron-rich nuclei. In this letter, a new method is proposed in order to deduce the neutron radii of neutron-rich nuclei. The method requires only measuring the difference between the reaction cross sections of the studied neutron-rich nucleus and its stable isotope on proton targets at the same energy. Also, the efficiency of this method is demonstrated and it is applied to determine the neutron radii of 22C and 14Be.

The total reaction cross section of a nucleus ($A=p+n$) incident on a proton is expressed in the Optical Limit Approximation (OLA) of Glauber theory glauber as b

[TABLE]

where ${\bm{\bf}b}$ is the impact parameter vector perpendicular to the beam (z) direction, and $\chi({\bm{\bf}b})$ is the phase-shift function defined as $i\chi_{N}({\bm{\bf}b})=-{\int{d{\bm{\bf}s}}}\rho_{N}({\bm{\bf}s})\Gamma_{pN}({\bm{\bf}s}+{\bm{\bf}b})$, where $\chi_{N}$ implies the phase shift due to the protons ($N=p$) or neutrons ($N=n$) inside the nucleus. The function $\rho_{N}({\bm{s}})$ is the z-integrated density distribution of the protons (neutrons). The finite-range profile function, $\Gamma_{pN}$, for $pp$ and $pn$ scatterings, is usually parameterized in the form $\Gamma_{pN}({\bm{\bf}b})=\frac{1-i\alpha_{pN}}{4\pi\beta_{pN}}\,\,\sigma_{pN}^{\rm tot}\,{\rm e}^{-{\bm{\bf}b}^{2}/(2\beta_{pN})}$, where $\alpha_{pN}$ is the ratio of the real to the imaginary part of the $pp$ ($pn$) scattering amplitude in the forward direction, $\sigma_{pN}^{\rm tot}$ is the $pp$ ($pn$) total cross section, and $\beta_{pN}$ is the slope parameter of the $pp$ ($pn$) elastic scattering differential cross section. For the zero-range approximation, it is parameterized in the form $\Gamma_{pN}({\bm{\bf}b})=\frac{1-i\alpha_{pN}}{2}\,\,\sigma_{pN}^{\rm tot}\,\delta({\bm{\bf}b})$. The validity of the OLA has already been tested with stable and unstable nuclei incident on a proton target badawy08 ; nagahisa .

The reaction cross section shift between a neutron-rich nucleus and its stable isotope is defined as $\delta\sigma^{\rm R}_{\rm A_{2},\rm A_{1}}=\sigma^{\rm R}_{\rm A_{2}}-\sigma^{\rm R}_{A_{1}}$. Both $\sigma^{\rm R}_{\rm A_{2}}$ and $\sigma^{\rm R}_{\rm A_{1}}$ are measured on a proton target and at the same energy. From Eq. (1), the reaction cross section shift is given by

[TABLE]

where, $\Delta\chi_{N}({\bm{b}})=\chi_{N_{2}}({\bm{b}})-\chi_{N_{1}}({\bm{b}})$ and

[TABLE]

For the zero range approximation, it reduces to

[TABLE]

It has been shown, in Ref. kaki , that various nuclear density distributions having the same matter radii provide almost the same values of the $p$-nucleus reaction cross section in the energy region 100$A$-800$A$ MeV. Furthermore, charge radii obtained from the isotope-shift technique show that the maximum difference between the measured charge radii of the isotopes of the same element for 4,6,8He, 6,8,9,11Li, 9-11Be, and 24-32Mg are 0.392$\pm$0.019 fm (23$\%$ from that of 4He) mueller07 , 0.3$\pm$0.065 fm (12$\%$ from that of 6Li) sanchez06 , 0.162$\pm$0.03 fm (6$\%$ from that of 9Be) nortershauser09 , and 0.158$\pm$0.008 fm (5$\%$ from that of 24Mg) Yordanov12 , respectively. Also, the proton radii obtained from the charge-changing cross section indicate that the maximum difference between the measured proton radii of the isotopes of the same element for 12-17B, 12-16C, and 12-19C, are 0.35$\pm$0.04 fm (15$\%$ from that of 10B) estrade , 0.09$\pm$0.012 fm,(4$\%$ from that of 12C) tran , and 0.12$\pm$0.08 fm (5$\%$ from that of 12C) kanungo , respectively. As one sees, the maximum difference between the proton radii for the isotopes of the same element is less than 10$\%$ except for He, Li, and B isotopes. It has been shown in Ref. kanungo that the effect of the center-of-mass motion of the halo on the proton radius becomes smaller with increasing mass number. The values of the charge radii, given in adndt , show that the differences between the charge radii of the isotopes of the same element are small. Based on the above discussion, we assume that the contribution of $\Delta\chi_{p}({\bm{b}})$ to the value of the reaction cross section shift is negligible. This means that $\Delta\chi_{p}({\bm{b}})\simeq 0$. Under this assumption, Eq. (2) reduces to

[TABLE]

Note that Eq. (5) does not depend on the proton density of the neutron-rich isotope. In Eq. (5), the neutron and proton density distributions of the stable isotope can be obtained from literature eg. devries ; patterson , while the neutron density distribution of the neutron-rich isotope is the key quantity. To check the validity of the approximation in Eq. (5), the values of $\delta\sigma^{\rm R}_{\rm A_{2},\rm A_{1}}$ calculated using Eq. (5) is compared to the exact values obtained from Eq. (2). Thus, the factor FS is defined as

[TABLE]

and its values are calculated for Be, B, C, O, Ne and Mg isotopes at different projectile energies. In the present calculations, the harmonic oscillator (HO) density badawyJPSJ has been used for Be and B with radii taken from krieger12 and tel (SKM*⋆*), respectively. The densities of Carbon isotopes are taken from Ref. badawy08 . For O, Ne and Mg isotopes, the Hartree-Fock densities have been used ripl . The nucleon-nucleon interaction parameters are taken from Ref. badawy08 .

A sample of the results is given in Fig. 1, showing the FS factor as a function of energy for 11,12Be, 15,17,19B, 16,19,20C, 21,23,24O, 30,31,32Ne and 32,35,37Mg where the reaction cross section shifts are calculated with respect to the stable nuclei 9Be, 10B, 12C, 16O, 20Ne and 24Mg respectively. As can be seen from the figure, the deviation between the values calculated using Eq. (5) and the exact ones calculated using Eq. (2) does not exceed $7\%$ for all the considered isotopes in the considered energy range.

Figure 2 shows the FS factor as a function of the neutron number of all the considered isotopes at the energies 40$A$ MeV and 800$A$ MeV. Confirming that the deviation between the values calculated using Eq. (5) and the exact ones calculated using Eq. (2) does not exceed $7\%$ irrespective of the neutron number.

In order to examine the effect of the increase of the proton radii of the neutron-rich nuclei on the validity of the approximation used in Eq. (5), a model analysis with HO density distribution for 12,19C has been preformed. The configuration of the 12C wave function is assumed to be $(0s_{1/2})^{2}(0p_{3/2})^{4}$ for both protons and neutrons. The HO length parameter is fixed in such a way to reproduce the proton radius, 2.33 fm, extracted from the charge radius. Also, the neutron radius of 19C is fixed to be 3.37 fm badawy08 , with the configuration $(0s_{1/2})^{2}(0p_{3/2})^{4}(0p_{1/2})^{2}(0d_{5/2})^{4}(1s_{1/2})^{1}$. The HO length parameter is increased in order to increase the proton radius of 19C from 2.33 fm by 0.05 fm in each step. Figure 3 shows the relation between the FS factor and the difference in proton radii of 19C and 12C at different energy values. If the proton radius of 19C increases by almost 10$\%$, 20$\%$, 30$\%$ from that of 12C, the FS factor increases by almost 1$\%$ (6$\%$), 3$\%$ (13$\%$), 5$\%$ (20$\%$), respectively at incident energy 40 MeV (800 MeV). From this figure, one notices that the approximation given in Eq. (5) is very effective at energies less than 300$A$ MeV.

As a first application of the proposed method, the neutron radius of 22C is determined. The 22C nucleus is known as a two-neutron halo type. It has been studied in the three-body model of 20C+n+n predicting a binding energy in the range 0.122-0.489 MeV and a matter radius (neutron radius) 3.61-4.11 fm (3.96-4.58 fm) badawy07 . Accurate experimental data on 22C binding energy is not yet available, so 22C is still attracting attention shulgina18 . The interaction cross section of 22C on a proton target at 40$A$ MeV was measured first by Tanaka et al. tanaka10 , in 2010, to be $\sigma^{\rm R}_{22}$[Exp]=1338 $\pm$274 mb, resulting in the rather large matter radius 5.4 $\pm$ 0.9 fm. On the other hand, Togano et al. togano16 , in 2016, measured the interaction cross section of 22C on a 12C target at almost 240$A$ MeV obtaining a matter radius 3.44 $\pm$ 0.08 fm. Nevertheless, the results of Ref. tanaka10 have been used in the present calculations since the method requires a proton target. The available data of the reaction cross section of p+12C at 40 MeV is for natural Carbon and has the value of 371$\pm$11 mb menet . Thus, the reaction cross section shift for 22C and 12C incident on a proton target at 40$A$ MeV is $\delta\sigma^{\rm R}_{22,12}$[Exp]=967$\pm$285 mb.

Figure 4 shows the reaction cross section shift $\delta\sigma^{\rm R}_{22,12}$ as a function of the neutron radius of 22C. The harmonic oscillator wave function is assumed for 22C, with the neutron configuration for the ground states being $(0s_{1/2})^{2}(0p_{3/2})^{4}(0p_{1/2})^{2}(0d_{5/2})^{6}(1s_{1/2})^{2}$. From the figure, the neutron radius of 22C is found to be 5.2$\pm$ 0.95 fm. Using this value and assuming that the proton radius of 22C is $2.33$ fm (same as 12C), the matter radius of 22C is found to be $4.59$ fm which is consistent with that of Ref. tanaka10 . Consequently, the neutron skin thickness of 22C is evaluated to be $2.87\pm 0.95$ fm. The authors of  nagahisa tried to simultaneously reproduce the two measured cross sections of 22C given in  tanaka10 and  togano16 , they concluded that the simultaneous reproduction of the two measured cross sections of 22C is not feasible.

As a second application, the neutron radius of 14Be will be determined. The 14Be nucleus is also known as a two-neutron halo type with the two-neutron separation energy 1.27$\pm$0.13 MeV wang . The interaction cross sections of 14Be on Carbon and proton targets have been measured and the matter radius was deduced as 3.22$\pm$0.07 fm moriguchi14 , 3.25$\pm$0.11 fm Ilieva12 , and 3.1$\pm$0.15 fm suzuki99 . Also, the charge-changing cross section of 14Be has been measured at 900A MeV on a Carbon target and the proton radius has been deduced to be 2.41$\pm$0.06 Terashima14 .

The reaction cross section of 14Be on a proton target at 41$A$ MeV has been measured to be $712\pm 14$ mb moriguchi14 . On the other hand, the value of the reaction cross section of 9Be on a proton target at 39.7$A$ MeV has been measured to be $398\pm 12$ mb Mcgill74 . Thus, the reaction cross section shift for 14Be and 9Be incident on a proton target at $\simeq$ 40$A$ MeV is $\delta\sigma^{\rm R}_{14,9}$[Exp]=314$\pm$26 mb. Assuming a HO wave function for the ground state of 14Be with neutron configuration $(0s_{1/2})^{2}(0p_{3/2})^{4}(0p_{1/2})^{2}(1s_{1/2})^{2}$, the result of the calculations is shown in Fig. 5. From the figure, the neutron radius of 14Be is found to be 3.36$\pm$0.11 fm, consistent with the results of Ref. enyo15 ; lataz04 . Assuming that the proton radius of 14Be is equal to that of 9Be, i.e. 2.36 fm ahmed17 , one gets for the matter radius of 14Be the value 3.11 fm. On the other hand, using the proton radius of 14Be obtained from the charge-changing cross section, 2.41$\pm$0.06 Terashima14 , one gets the matter radius of 14Be to be 3.12 fm. Both agree well with the radii measured in Refs. moriguchi14 ; Ilieva12 ; suzuki99 . The neutron skin thickness of 14Be is found to be (3.36$\pm$0.11- 2.41$\pm$0.06) $0.95\pm 0.17$ fm.

In summary, a new method has been introduced to determine the neutron radii of neutron-rich nuclei. This method requires measuring the difference between the reaction cross sections of the neutron-rich nucleus and that of its stable isotope on proton targets at the same energy. By applying this method, the neutron radii of 22C and 14Be have been determined. The obtained radii are consistent with the previously published values.