week ending PRL 119, 122502 (2017) PHYSICAL REVIEW LETTERS 22 SEPTEMBER 2017 Mirror Charge Radii and the Neutron Equation of State B. Alex Brown Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824-1321, USA (Received 2 June 2017; revised manuscript received 19 August 2017; published 22 September 2017) The differences in the charge radii of mirror nuclei are shown to be proportional to the derivative of the neutron equation of state and the symmetry energy at nuclear matter saturation density. This derivative is important for constraining the neutron equation of state for use in astrophysics. The charge radii of several neutron-rich nuclei are already measured to the accuracy of about 0.005 fm. Experiments at isotope- separator and radioactive-beam facilities are needed to measure the charge radii of the corresponding proton-rich mirror nuclei to a similar accuracy. It is also shown that neutron skins of nuclei with N ¼ Z depend upon the value of the symmetry energy at a density of 0.10 nucleons=fm3. DOI: 10.1103/PhysRevLett.119.122502 The neutron skin is the difference in the root-mean- and the results for the mirror nucleus 52Ni are square (rms) radii of neutrons and protons in a nucleus. It 208 52 0 0 was shown [1,2] that the neutron skin of Pb is propor- Ni;Rn ¼ 3.4956;Rp ¼ 3.5850; and tional to the derivative of the neutron equation of state 0 3 ΔRnp ¼ −0.0894: (EOS) ½E=NðρÞ near ρ ¼ 0.10 nucleons=fm in both non- relativistic and relativistic mean-field models. Further work – 208 The proton radius difference of 0.0889 fm is essentially [3 7] showed that the neutron skins of Pb and other 52 neutron-rich nuclei such as 48Ca are proportional to the equal to the neutron skin in Cr. (The small difference of derivative of the symmetry energy at nuclear matter about 0.0007 fm comes from the 0.13 percent difference of 3 the proton and neutron mass in the kinetic energy operator. saturation density (ρ0 ¼ 0.16 nucleons=fm ): The uncertainty from this is an order of magnitude smaller than the Coulomb effects discussed below.) The shell- L ¼ 3ρ½∂E ðρÞ=∂ρj ; ð1Þ 48 56 sym ρ¼ρ0 model wave functions for the nuclei from Ca to Ni are dominated by the 0f7=2 orbital and I have assumed this E ðρÞ¼½E=AðρÞ − ½E=NðρÞ where sym is the symmetry configuration for the EDF calculations. (This is the con- energy, and ½E=AðρÞ is the symmetric nuclear-matter figuration used for most EDF calculations of the 48Ca EOS. L is important for the extrapolation of the EOS to neutron skin.) The EDF results presented here for charge the lower and higher densities needed for understanding radii and neutron skins should be extended beyond the neutron stars [8–11]. 0f7=2 model, for example, by including pairing in the EDF If one has perfect charge symmetry, then the neutron rms calculations. This may change the final numerical values a radius in a given nucleus is equal to the proton charge little, but not the trends and conclusions made here. radius in its mirror nucleus. This means that the neutron In this Letter I show results from 48 Skyrme functionals. skin in a given nucleus can be obtained from the proton They start from the 12 functionals used in Ref. [13] that are radii of mirror nuclei. In practice, this can be done by among the “best” chosen in Ref. [14] out of several hundred measuring the charge rms radii and then making the from a variety of experimental criteria as well as some relativistic and finite size corrections to deduce the constraints from neutron-star properties. As shown in point-proton rms radii. Charge symmetry is distorted by Table I, they cover a reasonable range of values for the the Coulomb interaction in a way that can be calculated. symmetric nuclear-matter effective mass (mÃ=m ¼ For example, with the Skx Skyrme energy-density 0.70–1.00) and incompressibility (Km ¼ 212−242 MeV) functional (EDF) [12] without the Coulomb interaction, as compared to values extracted from the energy of the 52 the results for Cr for the rms radius of neutrons, Rn, the giant monopole resonances (Km ¼ 217 − 230 MeV) [15]. rms radius of protons, Rp, and the neutron skin, There are four versions of these 12 basic types. The first ΔRnp ¼ Rn − Rp, (in units of fm) are three versions are those from Ref. [13] where the isovector properties of the functionals were chosen to fit arbitrarily 52 208 Cr;Rn ¼ 3.5844;Rp ¼ 3.4961; and fixed values for the neutron skin of Pb of 0.16, 0.20, and 0.24 fm. The result from this work was that the neutron ΔR ¼ 0 0882; 3 np . EOS at a density of 0.10 neutrons=fm was found to be 0031-9007=17=119(12)=122502(5) 122502-1 © 2017 American Physical Society week ending PRL 119, 122502 (2017) PHYSICAL REVIEW LETTERS 22 SEPTEMBER 2017 TABLE I. Properties of the Skyrme functionals from Ref. [13]. symmetry energy in Fig. 1. As found previously The power of the density-dependent term in the Skyrme func- the correlation between ΔRnp and L is high. There is a 1 þ σ K tional is . m is the symmetric nuclear-matter incompress- similar high correlation between ΔRnp and the slope of the ibility and mÃ=m is the effective mass. symmetry energy at a lower density of 0.10 nucleons=fm3 Ã as found in Ref. [13]. Name σ Km (MeV) [m =m] The result for the neutron skin of 52Cr is plotted versus KDE0v1 s3 1=6 216 0.79 the mirror radius difference between the protons in 52Ni and NRAPR s6 0.14 225 0.85 52 Ska25 s7 0.25 219 0.99 Cr in Fig. 2. These were obtained with the 48 functionals, Ska35 s8 0.35 244 1.00 but with the Coulomb potential turned off. As expected, the SKRA s9 0.14 212 0.79 points on this plot lie on a straight line. The values of the SkT1 s10 1=3 242 0.97 points are clustered according to the corresponding results SkT2 s11 1=3 242 0.97 for the neutron skin of 208Pb as labeled in the figure. SkT3 s12 1=3 241 0.98 With the addition of the Coulomb interaction the Skx 1=6 SQMC750 s15 228 0.71 results are (in units of fm) SV-sym32 s16 0.30 237 0.91 1=6 52 SLy4 s17 224 0.70 Cr;Rn ¼ 3.605;Rp ¼ 3.562; and SkM* s18 1=6 218 0.78 ΔRnp ¼ 0.043; ½E=Nð0.10Þ¼11.4ð1.0Þ MeV. The symmetry energy at and this density is E ð0.10Þ¼25.5ð1.0Þ MeV. In addition, sym 52 ;R0 ¼ 3 523;R0 ¼ 3 674; for this Letter, I add a fourth version where the neutron skin Ni n . p . and 208 0 of Pb was constrained to be 0.12 fm. For all of these the ΔRnp ¼ −0.151; Ã neutron effective mass was fixed to be mn=m ≈ 0.90.In 0 Ref. [16] it was found that a reasonable variation in the with Rp − Rp ¼ 0.112 fm. There is an asymmetry in the neutron effective mass had a small effect on the neu- neutron skin due to the fact that the Coulomb interaction tron skin. pushes out the density of the protons relative to neutrons. The results for the neutron skins of 48Ca and 208Pb for The results for the 48 Skyrme functionals are shown in these 48 functionals are plotted versus the slope of the Fig. 3. The linear correlation becomes distorted. This is due to the self-consistent competition between the Coulomb 0.22 interaction and the symmetry potential in the EDF calcu- 48 0.20 (a) Ca lations. The effective Coulomb interaction used in all of these EDF calculations reproduces the binding energy 0.18 (fm) np 0.16 0.11 R Δ 0.14 no Coulomb 0.10 0.24 0.24 0.20 208 0.09 0.22 (b) Pb 0.20 Cr) (fm) 0.08 0.16 52 0.18 ( (fm) np np R 0.12 R 0.16 Δ 0.07 Δ 0.14 0.06 0.12 0.06 0.07 0.08 0.09 0.10 0.11 0.10 52 − 52 0.00 20.0 40.0 60.0 80.0 100. Rp( Ni) Rp( Cr) (fm) L (MeV) FIG. 2. Difference in the proton rms radii of the mirror nuclei FIG. 1. Neutron skins for 48Ca and 208Pb for 48 Skyrme 52Ni and 52Cr compared to the neutron skin of 52Cr. Results are functionals plotted versus the slope of the symmetry energy, shown for the 48 Skyrme functionals discussed in the text but L. The colors correspond to the constraints made for the neutron without the Coulomb interaction. The points are labeled accord- skin of 208Pb: 0.12 fm (red), 0.16 fm (orange), 0.20 fm (green), ing to their respective values for the neutron skin of 208Pb ranging and 0.24 fm (blue). The lines are to guide the eye. from 0.12 to 0.24 fm. 122502-2 week ending PRL 119, 122502 (2017) PHYSICAL REVIEW LETTERS 22 SEPTEMBER 2017 0.07 0.010 54 0.000 (a) Fe 0.06 with Coulomb (fm) -0.01 np 0.05 R Δ -0.02 Cr) (fm) Cr) 0.04 52 ( np (b) 54Ni − 54Fe R 0.07 Δ 0.03 (fm) 0.06 0.02 ch 0.09 0.10 0.11 0.12 0.13 0.14 R Δ 0.05 52 − 52 Rp( Ni) Rp( Cr) (fm) 0.04 0.00 20.0 40.0 60.0 80.0 100.