week ending PRL 119, 262501 (2017) PHYSICAL REVIEW LETTERS 29 DECEMBER 2017
T. Aumann,1,2,* C. A. Bertulani,3,1 F. Schindler,1 and S. Typel1,2 1Institut für Kernphysik, Technische Universität Darmstadt, Schlossgartenstraße 9, D-64289 Darmstadt, Germany 2GSI Helmholtzzentrum für Schwerionenforschung, Planckstraße 1, D-64291 Darmstadt, Germany 3Department of Physics and Astronomy, Texas A&M University-Commerce, Commerce, Texas 75429-3011, USA (Received 30 September 2017; published 28 December 2017) An experimentally constrained equation of state of neutron-rich matter is fundamental for the physics of nuclei and the astrophysics of neutron stars, mergers, core-collapse supernova explosions, and the synthesis of heavy elements. To this end, we investigate the potential of constraining the density dependence of the symmetry energy close to saturation density through measurements of neutron-removal cross sections in high-energy nuclear collisions of 0.4 to 1 GeV=nucleon. We show that the sensitivity of the total neutron- removal cross section is high enough so that the required accuracy can be reached experimentally with the recent developments of new detection techniques. We quantify two crucial points to minimize the model dependence of the approach and to reach the required accuracy: the contribution to the cross section from inelastic scattering has to be measured separately in order to allow a direct comparison of experimental cross sections to theoretical cross sections based on density functional theory and eikonal theory. The accuracy of the reaction model should be investigated and quantified by the energy and target dependence of various nucleon-removal cross sections. Our calculations explore the dependence of neutron-removal cross sections on the neutron skin of medium-heavy neutron-rich nuclei, and we demonstrate that the slope parameter L of the symmetry energy could be constrained down to 10 MeV by such a measurement, with a 2% accuracy of the measured and calculated cross sections.
The knowledge of the equation of state (EOS) of neutron- and charge radii . Although well-calibrated forces rich nuclear matter is fundamental for understanding the describe ground-state properties of nuclei and their exci- properties of neutron stars, such as their masses, radii, and tations satisfactorily, they exhibit a wide scatter in the L gravitational-wave signatures, as well as the mechanisms of parameter in a range of almost 0 to 150 MeV . core-collapse supernovae and neutron-star mergers [1–7]. In recent years, two nuclear observables have been Although the sizes of atomic nuclei differ by many orders of identified to potentially provide tight constraints on L if magnitude compared to neutron stars, their bulk properties accurately determined. These are the neutron-skin thickness and limits of stability, as well as the characteristics of Δrnp of neutron-rich nuclei and the ground-state dipole collisions among heavy ions, are governed by the same polarizability αD. The connection between Δrnp and proper- fundamental interaction. The atomic nucleus thus represents ties of the neutron EOS has first been pointed out and the natural and only environment to investigate nuclear matter quantified in Ref. . A clear relation between the in the laboratory, and constraints on the EOS can be obtained derivative of the neutron EOS close to saturation density from measurements of bulk properties of neutron-rich nuclei. 208 and Δrnp of Pb calculated with both relativistic and The EOS of asymmetric nuclear matter is usually E ðρÞ nonrelativistic models has been obtained [12,13].This characterized by the symmetry energy sym with its implies that a precise determination of the skin thickness value J and slope L ¼ 3ρ0δE ðρÞ=δρj at saturation sym ρ0 would provide constraints on the density dependence of density ρ0. In particular, the latter quantity is very poorly the neutron-matter EOS or, equivalently, on the slope constrained experimentally. This is apparent when inspect- parameter L of the symmetry energy. A similar relation is – ing the various interactions in Hartree-Fock (HF) [8 11] observed with the dipole polarizability, as pointed out first – and relativistic mean-field (RMF) [12 15] models which in Ref. . First measurements of this observable have have been adjusted to the properties of nuclei, e.g., masses been performed, and the most precise value so far has been extracted for 208Pb, where the measurements at the Research Center for Nuclear Physics were analyzed together with the Published by the American Physical Society under the terms of world data set . Including a result for the neutron-rich the Creative Commons Attribution 4.0 International license. 68 L Further distribution of this work must maintain attribution to nucleus Ni , the corresponding range of the slope lies the author(s) and the published article’s title, journal citation, between 20 and 66 MeVaccording to the analysis performed and DOI. in Ref. . One would need to measure the polarizability
0031-9007=17=119(26)=262501(5) 262501-1 Published by the American Physical Society week ending PRL 119, 262501 (2017) PHYSICAL REVIEW LETTERS 29 DECEMBER 2017 Z Z with better than 5% uncertainty for neutron-rich nuclei to 2 P 2 T PpðbÞ¼ dzd sρpðs;zÞ exp −σppZT d sρpðb − s;zÞ reach the precision achieved for stable nuclei. Z The experimental determination of Δrnp is rather 2 T challenging—in particular, for short-lived neutron-rich − σpnNT d sρn ðb − s;zÞ ; ð2Þ nuclei, where this effect becomes most pronounced. The supposedly cleanest probe to obtain information on Δrnp is where σpp and σnp are the proton-proton (Coulomb- electron scattering. The lead radius experiment (PREX) , removed) and proton-neutron total cross sections, obtained to be performed at Jefferson Lab, will measure the parity- from a fit of experimental data in the energy range of 10 to violating asymmetry for 208Pb at a relatively low momentum 5000 MeVas in Eqs. (1) and (2) of Ref.  (see Fig. 1). The transfer q, which is related to the neutron radius .The projectile (target) proton (neutron) densities are given by PðTÞ intended PREX precision for the final production run of ρ for the proton and neutron point densities in the 3% nðpÞ for the asymmetry translates into an uncertainty of projectile and in the target, respectively. They are normalized Δrnp of 0.06 fm, and a constraint on L to 40 MeV . R d3rρPðTÞðrÞ¼1 P In this Letter, we propose to use total neutron-removal so that nðpÞ . The expression for n is similar n ↔ p cross sections σΔN in high-energy nuclear collisions (0.4 to to Eq. (2) with the replacement .Asthenucleon- 1 GeV=nucleon), with secondary beams of neutron-rich nucleon cross sections are taken from experiment, the only nuclei with hydrogen and carbon targets as an alternative input parameters in this model are the nuclear proton and method. We will show that σΔN is rather sensitive to the neutron densities, which can be directly taken from density neutron-skin thickness and to the slope parameter L.The functional theory and tested in comparison with the exper- constraint will be derived similarly as for the measurement imental cross sections. We will concentrate on the total of the asymmetry at one momentum transfer q in the case of neutron-removal, σΔN, charge-changing, σΔZ, and reaction σ ¼ σ þ σ PREX,namelybyrelatingσΔN calculated on the basis of cross sections, R ΔN ΔZ. These are obtained from a proton and neutron point densities from density functional sum of all corresponding fragments using Eqs. (1) and (2). theory (DFT) with the corresponding L parameter of the We choose the neutron-rich part of the tin isotopic chain respective functional. The scatter of theory points will for our investigations and concentrate on the reactions 12 12 provide an estimate of the model dependence of such an Sn þ C first. For C, we adopt the density derived in a analysis. Following the same analysis as discussed in model-independent analysis of elastic electron scattering 2 Ref. , we have concluded that this method could up to high q using the Fourier Bessel expansion  potentially constrain L to 10 MeV if experiments could and extrapolated with a Whittaker function for very large provide the related observable with the corresponding radii. The rms radius is taken as the quoted best value with accuracy. The obvious advantage is the abundant number 2.478(9) fm . We assume the same densities for protons of events one can accumulate in facilities using hadronic and neutrons. collisions. This opens a new window of opportunity for 50 future experiments in high-energy radioactive beam facilities pp with the purpose to reveal the neutron skin of stable and 45 unstable nuclear isotopes. 40 np σ L Δr (mb) 35 Before discussing the sensitivity of ΔN to and np, σ we briefly introduce our reaction model. In high-energy 30 collisions, the Glauber multiple scattering method has been 25 shown to be a reliable theoretical model to calculate the 880 removal of nucleons [23,24]. The cross section for the ðZ; NÞ ðZ ;N Þ 860 production of a fragment from a projectile P P 840 due to nucleon-nucleon collisions is given by 820 (mb) R