Determination of the neutron skin thickness from interaction cross section and charge-changing cross section for B, C, N, O, F isotopes

NUCLEAR PHYSICS AND INTERDISCIPLINARY RESEARCH

Determination of the neutron skin thickness from interaction cross section and charge-changing cross section for B, C, N, O, F isotopes

Xiu-Fang Li，

De-Qing Fang，

Yu-Gang Ma

Nuclear Science and Techniques

Vol.27, No.3

Article number 71

Published in print 20 Jun 2016

Available online 14 May 2016

DOI：10.1007/s41365-016-0064-z

52902

The effective neutron and proton root-mean-square (rms) radius of stable and unstable nuclei (^12-15,17B, ^12-20C, ^14-21N, ^16-24O and ^18-21,23-26F) were deduced from the charge-changing cross section, _cc, and the interaction cross sections, _I, by using a statistical abrasion-ablation (SAA) model calculation. The extracted proton radii are in good agreement with the data from the Atomic Data and Nuclear Data Tables within the errors. Furthermore, we can observe that the neutron skin thickness increases monotonously with the increasing neutron number in these isotopes, which is consistent with the systematical trend of theoretical calculations.

unstable nucleistatistical abrasion-ablation modelneutron skin thickness

1 Introduction

Researches on proton and neutron density distributions in nuclei are very important in nuclear physics for the study of neutron skin [1] or neutron halo structure [2]. Usually, at the nuclear surface, an excess of neutrons is qualitatively described as the neutron skin. The neutron halo is defined as a long tail of neutron density plus the excess. With the improvement of measuring methods and the high experimental precision of instruments, the uncertainties of the nuclear charge radius measured by muon-nucleus scattering and elastic electron-nucleus experiments are smaller than 1% [3, 4]. This allows for the deduction of the proton rms radius ( $R_{p} = {〈 r_{p}^{2} 〉}^{1 / 2}$ ) precisely. In contrast, the neutron density distribution in nuclei is very difficult to measure directly. Actually, we lack accurate methods to determine the rms radius of the neutron density ( $R_{n} = {〈 r_{n}^{2} 〉}^{1 / 2}$ ). This indicates that the neutron skin thickness has no precise value either, which is defined as the neutron-proton rms radius difference, ΔR_np = R_n-R_p [5, 6].

Since the mid 1950s, many discussions about the existence of a neutron skin in stable nuclei have occured. At first, there was no evidence for a thick neutron skin in stable nuclei, even though many of stable nuclei have a large neutron excess (N-Z). However, Tanihata et al. measured the neutron skin thickness of ⁸He to be 0.9 fm [7]. A similar observation was made by Suzuki et al. for the neutron skin thickness in Na isotopes [8]. Therefore, further investigation on the neutron skin is important to understand the ground state properties of nuclei, which are far away from the β stability line [9-11]. On the other hand, the research about the neutron skin have an important effect for nuclear structure and nuclear symmetry energy [12-14]. As is well known, the nuclear matter radii of light nuclei are determined by measurements of the interaction cross sections (_I) [15-19] or the reaction cross sections [20, 21]. Although these measurements can give the matter radii, the neutron and the proton radii are still lacking and need to be measured separately.

In Ref. [22], it is proposed to extract the proton rms radii R_p from charge-changing cross sections (_cc). If the nuclear matter rms radii, R_m, are determined from interaction cross section data, the neutron rms radii, R_n, can be deduced by using the relationship between R_p, R_n and R_m. In this work, systematic calculations of proton and neutron rms radii combined with the data of interaction cross sections and charge-changing cross sections for B, C, N, O, and F isotopes have been done by a statistical abrasion-ablation (SAA) model [23-26]. The interaction cross section, _I, and charge-changing cross section, _cc, of ^12-15,17B, ^12-20C, ^14-21N, ^16-24O, and ^18-21,23-26F on a carbon target have been measured at high energies [22, 27]. The neutron skin thicknesses for these isotopes are deduced for the first time by reproducing the measured _I and _cc. Within the errors, most of the extracted neutron skin thickness data is in agreement with the systematics.

This paper is organized as followed: A brief introduction about the statistical abrasion-ablation (SAA) model and the form of density distribution is given in Sect. 2. In Sect. 3, we compare the extracted R_p from _cc with the experimental data measured by other methods. Then, we continue to deduce R_n by the SAA-model. Finally, ΔR_np is obtained from the difference of the neutron and proton rms radii. The conclusions are presented in Sect. 4.

2 Model description

2.1 Statistical abrasion-ablation (SAA) model

A statistical abrasion-ablation model is presented by Brohm et al. [23-26] to describe high energy peripheral nuclear reactions. The model uses the individual nucleon-nucleon collisions happening in the overlap zone of the colliding nuclei to describe the collision process and calculate the abraded neutron and proton numbers. The neutron and proton density distributions in the projectile are treated separately in the SAA model, which allows us to study the dependence of fragmentation cross sections on neutron and proton densities, respectively. The SAA model has been modified to satisfy the reaction in low energy [28].

The total reaction cross section _R in SAA is calculated by using the Glauber method. It is described by the following equations:

σ_{R} = 2 π \int b [1 - T (b)] d b,

(1)

T (b) = \exp [- \sum_{i, j} σ_{i j} \int ρ_{z}^{targ} (s) ρ_{z}^{proj} (| b - s |) d s],

(2)

where b stands for the impact parameter. T(b) is the transmission function. ij denotes the nucleon-nucleon collision cross sections. $ρ_{z}^{targ}$ or $ρ_{z}^{proj}$ is the z-integrated density distribution of the target or projectile.

Following Refs. [29, 30], we take T = T^pTⁿ. Where,

T^{p} (b) = \exp [- (σ_{pp} \int ρ_{p}^{targ} ρ_{p}^{proj} + σ_{np} \int ρ_{n}^{targ} ρ_{p}^{proj})],

(3)

T^{n} (b) = \exp [- (σ_{pn} \int ρ_{p}^{targ} ρ_{n}^{proj} + σ_{nn} \int ρ_{n}^{targ} ρ_{n}^{proj})],

(4)

where T^p is only the proton contribution of the projectile and Tⁿ is only the neutron contribution of the projectile. Therefore, two pure cross sections and a cross term cross section can be defined:

{\tilde{σ}}_{cc} = 2 π \int b [1 - T^{p} (b)] d b,

(5)

{\tilde{σ}}_{- nx} = 2 π \int b [1 - T^{n} (b)] d b,

(6)

σ_{cross} = - 2 π \int b [1 - T^{p} (b)] [1 - T^{n} (b)] d b .

(7)

Obviously, $σ_{R} = {\tilde{σ}}_{cc} + {\tilde{σ}}_{- nx} + σ_{cross}$ . Since inelastic scattering can be neglected at high energies, the reaction cross section is approximate to the interaction cross section, namely _R _I. Moreover, the charge-changing cross section, _cc, plus the neutron-removal cross section, _-nx, is equal to the interaction cross section. Then, the reaction cross section is _R _cc+_-nx. The charge-changing cross section, _cc, isn’t only made up of projectile-proton contributions, but also the cross term, _cross. As shown in Ref. [29], it can be calculated by the following equation,

σ_{cc} = 2 π \int b [1 - T^{p} (b)] ε (E) d b,

(8)

where ε(E) is the correction factor, which is defined as $σ_{cc} / {\tilde{σ}}_{cc}$ , namely the ratio of the experimental _cc data and calculated ${\tilde{σ}}_{cc}$ values, as shown in Fig. 1.

Fig. 1.

Energy dependence relation of the correction factor ε(E). The data represented by solid circles are from Ref. [29] and solid square from Ref. [31]. The dashed line is from a linear fitting for the energy range 100–600 MeV/nucleon from T. Yamaguchi et al. and the solid line from an exponential fitting for the energy range 100–1500 MeV/nucleon.

In Ref. [29], the relation of the correction factor with the energy (ε(E)) is fitted by a linear function while the energy changes from 100 to 600 MeV/nucleon, as shown by the dashed line in Fig. 1. The data represented by solid circles is from Ref. [29] and the solid squares are from Ref. [31]. However, the energies of the _cc for B, C, N, O, and F isotopes in our work are close to 1000 MeV/nucleon. So it’s necessary to get the correction factor, ε(E), at high energies. From Fig. 1, we can see that if the energy in the linear correction factor, ε(E), is directly extended to high energy, such as 800 MeV/nucleon or 1200 MeV/nucleon, there is a big deviation with the data. Considering the weak energy dependence of the cross sections for energies above 600 MeV/nucleon, it’s more reasonable to fit energy dependence of the correction factor, ε(E), by an exponential function. The fitted result is shown by the solid line. The expression of the exponential function is as follows:

ε (E) = 1.107 + 0.01191 \times \exp (1.444 - 0.004623 E) .

(9)

2.2 Density distribution

For the proton and neutron density distribution in the projectile, we use the two-parameter Fermi-type function. According to the droplet model [25, 26], the density, ρi(r), at a distance r in the SAA-model is given by

ρ_{i} (r) = \frac{ρ_{i}^{0}}{1 + \exp (\frac{r - C_{i}}{t_{i} / 4.4})}, i = n, p,

(10)

C_{i} = R_{i}^{sur} [1 - {(0.413 t_{i} / R_{i}^{sur})}^{2}], i = n, p,

(11)

R_{i}^{sur} = R_{0 i} + (Z / A) D, i = n, p,

(12)

D = \frac{2}{3} R_{0 i} [\frac{A - 2 Z}{A} - δ] / B_{s}, i = n, p,

(13)

where $ρ_{i}^{0}$ denotes the normalization parameter of the neutrons’ (i = n) or protons’ (i = p) density distribution; ti stands for the diffuseness parameter; Ci is the neutron or proton half-density radius; $R_{i}^{sur}$ is called the equivalent neutron or proton sharp-surface radius; and R_0i is the separate effective neutron or proton sharp-radii.

Here, we introduce the neutron and proton radii constant, r_0i, instead of the former matter radii constant, r₀, to the R_0i, namely,

R_{0 i} = r_{0 i} A^{1 / 3} (1 + ϵ), i = n, p .

(14)

Parameters δ, Bs, and ϵ are determined by Ref. [26]. As for the matter radii constant, we assume the relationship $r_{0}^{2} = \frac{N}{A} r_{0 n}^{2} + \frac{Z}{A} r_{0 p}^{2}$ . From the perspective of calculation accuracy and convenience, we chose directly to change the neutron radii constant, r_0n, instead of r₀ to reproduce _I. Now we can use the _cc to get r_0p and use _I to get r_0n. The calculation details are described in the following.

3 Calculation and discussion

The _cc for ^12-15,17B, ^12-20C, ^14-21N, ^16-24O, and ^18-21,23-26F were measured by Chulkov et al. at 930±44 MeV/u on a carbon target [27]. A summary for the interaction cross section _I on Be, C, and Al-targets has been presented by Ozawa et al. [22]. For consistency, only the _cc and _I data on the C-target are used.

3.1 The proton rms radii

From Eq. (8), the charge-changing cross section is mainly determined by the proton density distribution of projectile ρp and ε(E) from Eq. (9). However, from Eq. (10), ρp is mainly determined by the proton radius constant, r_0p.

In our work, by adjusting the input parameter r_0p, the experimental _cc will be reproduced. Once the r_0p is fixed, we can get the the proton rms radius, R_p, from ρ_p.

The extracted proton rms radii are compared with the data given by the Atomic Data and Nuclear Data Tables [32], as shown in Fig. 2.

Fig. 2.

A comparison of the rms proton radius Rp for nuclei ¹²C, ¹³C, ¹⁴C, ¹⁴N, ¹⁵N, ¹⁶O, ¹⁷O, ¹⁸O, and ¹⁹F. The solid squares are measured data taken from the Atomic Data and Nuclear Data Tables [32] and the open circles are the calculated results by SAA-model.

In Fig. 2, the data represented by solid squares is determined from the updated charge radii, R_c, of the Atomic Data and Nuclear Data Tables in 2013 [32]. Using the relation $R_{p}^{2}$ = $R_{c}^{2} - 0.64$ , we can get the corresponding R_p from the Atomic Data and Nuclear Data Tables. The open circles represent the extracted R_p values from _cc using the SAA-model. From Fig. 2, we can see that the calculations using the SAA-model are nearly consistent with data from the Atomic Data and Nuclear Data Tables within the error bars.

3.2 The neutron rms radii

Due to the limited R_c of B, C, N, O, and F isotopes from the Atomic Data and Nuclear Data Tables, only a few proton radii are compared in Fig. 2. The R_p of more isotopes, including ^12-15,17B, ^12-20C, ^14-21N, ^16-24O, and ^18-21,23-26F, are deduced using the same method.

After the proton radii constant, r_0p, is fixed, the neutron radii constant, r_0n, is adjusted to reproduce the experimental _I data. In the SAA-model, the _cc is mainly determined by the proton radii constant, r_0p. Therefore, when the r_0p is fixed, even though r_0n is changeable in the process of fitting _I, the _cc value has little change, less than 0.7%. The values of the neutron and proton rms radii extracted by the SAA-model are shown in the Fig. 3.

Fig. 3.

(Color online) The rms neutron radius R_n and proton radii radius R_p calculated by SAA for the nuclei ^12-15,17B,^12-20C,^14-21N,^16-24O and ^18-21,23-26F. R_n is represented by squares and R_p by circles.

In Fig. 3, the data represented by squares is R_n and circles is R_p. The proton rms radii of ¹²B has a large error because the measured _cc for ¹²B has a large error. And the neutron rms radii of ¹⁶N has a large error because the measured _I for ¹⁶N has a large error. On the other hand, we can see that with the growth of the mass number in each isotope chain, the neutron rms radii increase slowly. While the corresponding proton rms radii is almost constant with the mass number increasing.

When the neutron number nearly equals to the proton number, namely the neutron excess (N-Z) is close to zero, the neutron-proton rms radius difference becomes small, which indicates proton and neutron density distribution in the stable nuclei are almost the same. As the (N-Z) increases, the difference between R_n and R_p becomes large. Combining the analysis above, it is illustrated that neutron excess influences R_n more than R_p.

3.3 The neutron skin thickness

In Fig. 4, the neutron skin thickness determined from the difference between the neutron and proton rms radii is plotted with the separation energy difference between proton and neutron (S_p - S_n). We can see that ΔR_np of B, C, N, O and F is strongly connected with (S_p - S_n), i.e. the neutron skin thickness increases with an increase in S_p - S_n. Such a correlation has been predicted by Tanihata et al. using the RMF model [7] and is shown by the shadow area in Fig. 4. The neutron skin thicknesses determined by the method presented in this work agree well with the theoretical systematics.

Fig. 4.

The relation between the proton and neutron separation energy difference and the neutron skin thickness. The shadow area shows the calculated correlation for various isotopes, ranging from helium up to lead [7].

4 Conclusion

In summary, we have extracted the neutron rms radii, R_n, and proton rms radii, R_p, of ^12-15,17B, ^12-20C, ^14-21N, ^16-24O, and ^18-21,23-26F from interaction cross sections and change-changing cross sections with the SAA model. The extracted proton radii with the SAA model are in good agreement with the data. The neutron radii, R_n, increases faster than R_p, which stays almost constant. Furthermore, we have shown an increase in the neutron skin ΔR_np in the neutron-rich isotopes of B, C, N, O, and F. The extracted results are consistent with the systematics. It indicates the feasibility of the method for extracting proton and neutron radii from _cc and _I data.

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