Measurements of interaction cross sections for 22 − 35 Na isotopes

Interaction cross sections (σI) for 22−35Na isotopes from the stability line to the vicinity of the neutron drip line have been measured at around 240 MeV/nucleon. The σI for 33−35Na were measured for the first time. Enhancement in cross sections is clearly observed from the systematics for stable nuclei, for isotopes with large mass numbers. From the known values of the nuclear-deformation parameters β2 of 22−31Na, these enhancement can be mainly ascribed to the nuclear deformation. Large enhancement in heavier isotopes suggest that these nuclei are strongly deformed. The root-mean-square (RMS) nuclear matter radii were deduced from the σI by using Glauber-type calculation. Furthermore, a monotonic growth of the neutron-skin thickness has been deduced with increasing neutron number for Na isotopes.


Introduction
The neutron-halo and neutron-skin structures are among the most notable topics in exotic nucei.The observation of large enhancement of nuclear matter radii of unstable nuclei by measuring the σ I , greatly contributed to discovery of these exotic structures [1,2].Therefore, measurement of σ I is a good tool to study exotic nuclei such as neutron-halo and neutron-skin nuclei.Recently, the region of neutron-rich Ne, Na and Mg isotopes around N = 20, known as "island of inversion [3]" has been extensively studied.In this region, in spite of having neutron number around magic number 20, it has been reported that 32 Mg and 34 Mg are strongly deformed [4,5], and 31 Ne has a neutron-halo structure a present address: National Institute of Radiological Sciences, Chiba 263-8555, Japan;e-mail: ssuzuki@nirs.go.jp  [7].In the case of Na isotopes, the previous measurement of σ I at GSI has been done up to 32 Na.In the present work, we have measured σ I for 22−35 Na, including heavier ones 33−35 Na, to study the nuclear structures of Na isotopes located in or beyond the "island of inversion".

Experiment
Experiments were performed by using BigRIPS at RIBF, operated by the RIKEN Nishina Center and the Center for Nuclear Study, University of Tokyo.Figure 1 shows a schematic drawing of the experimental setup.A primary beam of 345-MeV/nucleon 48 Ca with a typical intensity of 100 pnA was used.Secondary beams of 22−35 Na were produced via projectile fragmentation of 48 Ca on Be production targets at the first focal plane (F0).Produced fragments were pre-separated at the first stage of BigRIPS (F1-F3).The Al wedge-shaped degrader was placed at F1.The second stage of BigRIPS (F3-F7) was used to separate and identify the secondary beams.We measured σ I by using the transmission method.The reaction targets were installed in the F5 (Fig. 1).Incident and out-going particles were identified and counted by using detectors that were located upstream side (F3-F5) and downstream side (F5-F7) of the reaction targets, respectively.For particle identification, we used Bρ-TOF-∆E method by using parallel plate avalanche counters (PPAC), plastic-scintilation counters (PL) and ion chambers (IC).These detectors are also shown in Fig. 1.The interaction cross section is deduced from , where t denotes the number of nuclei per unit area in the reaction target.R is the ratio of number of out-going particles divided by that of the incoming particles, and R 0 is the same ratio on a measurement without a reaction target to correct the nuclear reactions in the detectors.EPJ Web of Conferences 03084-p.2

Results and discussions
The σ I for 22−35 Na at 240 MeV/nucleon are plotted as a function of mass number in Fig. 2.
In the present work, we measured the σ I for 33−35 Na for the first time.From mass number 28, the present data deviate from the systematics for stable nuclei as the mass number increases.The solid circles and the closed triangles show present and previous data [2], respectively.The gray line shows systematics for stable nuclei [7].The closed squares show the calculated values of the σ I by using deformation parameters β 2 , on the assumption that the systematics for stable nuclei corresponds with β 2 = 0.The three-dashed lines show the calculated values in the case of β 2 are 0.6, 0.5 or 0.3.
The beam energies of Na isotopes in the middle of reaction target were around 240 MeV/nucleon.The σ I at exact 240 MeV/nucleon were deduced by correcting the energy dependence of nucleon-nucleon interaction cross section (σ NN ).The previous data measured at 950 MeV/nucleon in GSI [2] are also plotted in Fig. 2. Plotted values are also corrected to adjust the beam energy to be 240 MeV/nucleon, by using energy dependence of σ NN .The present data are in good agreement with the previous data within the statistical error except for 22 Na.

Effect of nuclear deformation on σ I
In Fig. 2, the calculated values of σ I taking into account the quadrupole deformation are shown by the closed squares.The deformation parameters β 2 were deduced from the experimental data of quadrupole moment and reduced transition probability B(E2).Therefore, the calculated values are plotted only for isotopes whose β 2 are known.These calculated values were deduced by using simple geometrical equation of interaction radii, σ I = π[R I ( 12 C)+R I ( A Na)] 2 , where R I ( 12 C) is obtained from the σ I of 12 C + 12 C at 240 MeV/nucleon [8], . The R 0 is determined so as to reproduce the σ I for 25 Na whose β 2 is almost zero.The tendency of calculated values reproduces the present data very well.Moreover, the present data suggest that the deformation tends to be larger for isotopes with larger mass numbers.

Neutron-skin thickness of Na isotopes
We deduced the root-mean-square (RMS) nuclear matter radii (r m ) from present data by using Glauber-type calculation.The density of projectile nucleus is assumed to be Fermi-type, and the diffuseness parameter is fixed to the value of 23 Na extracted from elastic electron scattering [10].In Fig. 3, the mass-number dependence of the RMS nuclear matter radii is shown.The calculated values of Relativistic Mean Field (RMF) theory by Geng et al. [11] are also plotted for comparison.The present data are in good agreement with these calculated values except for A = 30, 31.In this RMF theory, the RMS nuclear matter radii of 30,31 Na were calculated by assuming that β 2 of these nuclei are almost zero.This result suggests that the shell structures of nuclei located in "island of inversion" are different from those of spherical nucleus.
In Fig. 3, the RMS nuclear proton radii (r p ) are also shown.The RMS nuclear proton radii are deduced from the data of electron-scattering and of isotope-shift measurements [12].For 32−35 Na, we estimate the RMS nuclear charge radii by extrapolating from the known values [13], because of the lack of experimental data for these nuclei.Figure 4 shows the estimated neutron-skin thickness, where the horizontal axis denotes the difference of the separation energies (S p − S n ).In order to deduce the neutron-skin thickness, we used the following equation, r2 n , where rn is the RMS nuclear neutron radius.The previous results [2] are also plotted in Fig. 4. The present data are consistent with the previous data.The neutron-skin thickness from the RMF calculation by Geng et al. [11] are also plotted for comparison.Present data are in good agreement with these values in lighter mass region.However in heavier mass region, our results are smaller than the RMF calculation.It could be due to our overestimation of the RMS nuclear charge radii for heavier isotopes.The data in the whole mass range roughly show a linear dependence on S p − S n , which is in agreement with the RMF calculation.Figure 3.The rm deduced from present data (solid circles) and the rp (solid triangles) [12].The dotted line shows the rp estimated by the empirical formula for neutron-rich region [13].The open squares and diamonds show the results of RMF calculation for the rm and rp [11], respectively.[2].The closed squares show the RMF calculation values [11].

Figure 1 .
Figure 1.A schematic drawing of the experimental setup.The dipole magnets are indicated as D1-D6.The focal planes are represented as F0-F7.The parallel plate avalanche counters (PPAC), ion chambers (IC) and plastic-scintilation counters (PL) were used to obtain the information of Bρ, TOF (Time Of Flight) and ∆E, respectively.

3 Figure 2 .
Figure 2. The observed mass-number dependence of the σ I for Na isotopes on C targets.The solid circles and the closed triangles show present and previous data[2], respectively.The gray line shows systematics for stable nuclei[7].The closed squares show the calculated values of the σ I by using deformation parameters β 2 , on the assumption that the systematics for stable nuclei corresponds with β 2 = 0.The three-dashed lines show the calculated values in the case of β 2 are 0.6, 0.5 or 0.3.
These experimental results also confirm that the magic number 20 vanishes in this region.Recently, we have measured σ I for Ne isotopes including this region, to suggest the neutron-halo structures of 29 Ne and 31 Ne from the observed enhancement in σ I