Marielle Chartier, Roy Lemmon (Visiting Senior Research Fellow), Peter Wigg (PhD student), Zoe Matthews (PDRA)

Understanding the equation of state of asymmetric nuclear matter is one of the key questions of the UK Nuclear Physics Strategy, the 2007 Long Range Plan for Nuclear Science in the USA and the Science Cases for new rare isotope facilities such as FAIR, SPIRAL2 and EURISOL in Europe, FRIB in the USA and RIBF-RIKEN in Japan.

The symmetry energy and the ASY-EOS collaboration

The equation of state (EOS) is a fundamental property of nuclear matter and describes the relationships between the energy, pressure, temperature, density and isospin asymmetry δ=(ρ_n-ρ_p)/ρ for a nuclear system [Dan02]. It can be divided into a symmetric matter contribution that is independent of the isospin asymmetry and an isospin term (also known as the symmetry energy) that is proportional to the square of the asymmetry [Lat01, Lat04]. The EOS of asymmetric nuclear matter is also a quantity of crucial significance in understanding the physics of isolated and binary neutron stars, type II supernovae and neutron star mergers. Experimental information about the EOS can help to provide improved predictions for neutron star observables such as stellar radii and moments of inertia, crustal vibration frequencies [Lat04, Vil04], and neutron star cooling rates [Lat04, Ste05], which are currently being investigated with ground-based and satellite observatories. Strong synergies exist between our research programme and several other STFC programmes in astrophysics which address the physics of neutron stars and gravitational waves, including Advanced LIGO/GEO600, LISA and SKA.

Measurements of isoscalar collective vibrations, collective flow and kaon production in energetic nucleus-nucleus collisions have constrained the equation of state for symmetric matter for densities ranging from saturation density to five times saturation density [Dan02, Fuc06]. However, the EOS of asymmetric matter has comparatively few experimental constraints [Bro00]. Recent measurements of the Giant Monopole [Li07] and Pygmy Dipole [Ki07] Resonances in neutron-rich nuclei, isospin diffusion [Tsa04], neutron/proton emission and flow [Fa06], and fragment isotopic ratios [Tsa01a, Ig06] have provided initial constraints on the density dependence of the symmetry energy at sub-saturation densities. In the near future, refinement of these measurements with both stable and rare isotope beams will provide further stringent constraints at sub-saturation densities. In contrast, virtually no experimental constraints exist on the symmetry energy at supra-saturation densities i.e. ρ~2ρ_o and above. This is the domain with the greatest theoretical uncertainties and the largest impact on the understanding of neutron stars [Lat01, Lat04]. A major challenge in the future is therefore to obtain such experimental constraints.

The international ASY-EOS collaboration (Europe, the USA and Japan), of which we are leading members, has recently been formed to study the EOS of asymmetric nuclear matter. In the next few years, the collaboration intends to exploit the stable and rare isotope beams already available from existing facilities such as GSI, GANIL, MSU and RIBF-RIKEN to study the behaviour of the symmetry energy from sub-saturation densities (ρ~0.5-1ρ_o) to supra-saturation densities (ρ~2ρ_o and above). This will pave the way for studies in the future at new facilities such as FAIR, FRIB and EURISOL.

We are strongly involved in two specific experimental studies within the ASY-EOS collaboration:

• Neutron/proton flow measurements in Au+Au, Zr+Zr, Mo+Mo reactions at 400-800 AMeV at GSI,
• Isospin diffusion measurements in Ca+Ca reactions at 35 AMeV at GANIL.

Studies at supra-saturation densities

The behaviour of the symmetry energy at supra-saturation densities can only be explored in terrestrial laboratories by using relativistic heavy-ion collisions of isospin asymmetric nuclei. The current GSI facility has worldwide unique capabilities for studies in this direction as beam energies of 1-2 AGeV are available which can produce densities of ρ~3ρ during the collision. We wish to exploit this unique capability by performing the first experiments worldwide specifically designed to study the symmetry energy at supra-saturation densities. Ideally one would use rare isotope beams for this work as they introduce the widest possible isospin asymmetry into the reaction. However, the intensities available at the present GSI facility are insufficient and so we will use stable beams for these initial investigations. Once FAIR is available, however, the rare isotope beam intensities will be sufficient for such experiments. FAIR will therefore present an absolutely unique opportunity to pursue these investigations as it will be the only facility worldwide to produce rare isotopes beams with energies up to 1-2 AGeV and with sufficient intensities.

Recent theoretical studies [Li07] have suggested various experimental probes which should be sensitive to the behaviour of the symmetry energy at supra-saturation densities. Some of the most promising include neutron/proton ratios and flows [Fam06, Li97, Li06, Li00], π⁺/π^- ratios and flows [Li04, Yon06], K⁺/K^o [Fer06, Lop07] and Σ^-/Σ⁺ [Li05] ratios. We will focus on measurements of the neutron/proton ratios and flows. Measurements of neutron spectra and flows are difficult and very few previous measurements exist [Lei93, Lam94]. However, we are currently laying the groundwork for these studies by performing an analysis of one of the few existing data sets. These are from an experiment studying the reaction ¹⁹⁷Au + ¹⁹⁷Au at 400, 600 and 800 AMeV performed at GSI using the LAND neutron detector and the FOPI forward wall. This is giving us a baseline measurement which we are using to examine the size of the flow effects and design the new experiments. First results for 400 AMeV have been published [Rus11]. The availability of the LAND neutron detector at GSI will make the new proposed measurements possible. LAND can detect high energy neutrons with high efficiency. It can also be used to detect protons and so neutron and proton spectra and flow measurements can be made with the same detector reducing several systematic experimental errors.

An experiment submitted by R. Lemmon and P. Russotto on behalf of the ASY-EOS collaboration was recently run at the GSI accelerator facility in Germany to study neutron and proton collective flow in the reactions Au+Au, Zr+Zr and Ru+Ru at 400 AMeV. This network of reactions will allow double differential flow values (both directed and elliptic) to be constructed. Such an approach reduces the effects of various systematic experimental effects such as detector acceptance on the measurements and increases the sensitivity of the measurements to the symmetry energy. The setup uses the LAND neutron detector together with ancillary detectors to measure the impact parameter and the reaction plane. These include sections of the CHIMERA multi-detector array and a forward multiplicity wall.

Studies at sub-saturation densities

The behaviour of the symmetry energy at sub-saturation densities can be explored using a wide variety of techniques. These include nuclear structure probes such as neutron skins in neutron-rich nuclei, mass measurements and Isobaric Analogue States, E0 and E1 collective modes such as the Giant Monopole Resonance [Li07] and the Pygmy Dipole Resonance [Ki07] in neutron-rich nuclei. Reaction observables such as fragment isotopic distributions, neutron/proton spectra and flows and isospin diffusion can also be used. In the past few years, this has become an active area of research and initial measurements using such probes have provided first constraints on the symmetry energy at these densities. It is expected that more precise measurements using increasingly isospin asymmetric nuclei will become possible as the new generation of rare isotope facilities such as SPIRAL2, FAIR, RIBF and FRIB become available.

Isospin diffusion data currently provides the least model dependent constraints on the symmetry energy from reaction dynamics [Tsa08]. Isospin diffusion investigations involve comparisons between “mixed” collisions of a neutron-rich nucleus and a neutron-deficient nucleus to “symmetric” collisions of two neutron-deficient nuclei or two neutron-rich nuclei. In the absence of diffusion, the isospin asymmetry of the projectile-like fragment after the collision should be independent of the original asymmetry of the target. In the limit of rapid diffusion, the isospin asymmetries of both target and projectile residues come to isospin equilibrium. To isolate isospin diffusion effects, the isospin asymmetry δ of a projectile-like residue can be linearly rescaled to obtain an isospin transport ratio R_i [Tsa04] that has the limiting values of R_i=1 for a neutron-rich projectile on a neutron-deficient target in the absence of isospin diffusion and R_i~0 in the limit of isospin equilibrium. A comparison of experimental values for R_i, extracted from initial experiments at MSU which probed isospin diffusion in the asymmetric ¹¹²Sn + ¹²⁴Sn and ¹²⁴Sn + ¹¹²Sn systems at 50 AMeV [Tsa04], to R_i values calculated with Quantum Molecular Dynamics and Boltzmann-Uhling-Uhlenbeck transport models have provided initial constraints on the symmetry energy at sub-saturation densities [Tsa08]. In the near future, efforts will be made to obtain more stringent constraints on the symmetry energy at sub-saturation density using isospin diffusion at GANIL, MSU and RIKEN. Known factors that limit the accuracy of these constraints include the accuracy of the measured transport ratio R_i and uncertainties concerning the isospin dependencies of the nucleon effective masses and medium modified nucleon-nucleon cross sections. The measured values of R_i are obtained from measured isoscaling parameters whose linear dependence on the asymmetry δ has been demonstrated experimentally [Liu07]. However, the observables do not directly measure the residue asymmetries themselves; this introduces some model dependencies to the theoretical interpretation of these results.

To address some of the uncertainties in the present measurements and to provide complimentary data to the Sn measurements, an experiment was performed at GANIL in 2007 to study isospin diffusion in peripheral asymmetric ⁴⁰Ca + ⁴⁸Ca, ⁴⁸Ca + ⁴⁰Ca systems at an incident energy of 35 AMeV. Symmetric ⁴⁰Ca + ⁴⁰Ca, ⁴⁸Ca + ⁴⁸Ca systems were also studied in the same experiment to provide reference measurements. The experiment was performed using the VAMOS spectrometer coupled to the INDRA 4? array. This experiment is an important step in using the technique of isospin diffusion as the projectile residues were directly measured, unlike in the previous experiments, using the VAMOS spectrometer. Comparisons to calculations using BUU and QMD models will be used to extract constraints on the symmetry energy at sub-saturation densities.

References

[Bro00] B.A. Brown, Phys. Rev. C 43, R1513(1991)
[Dan02] P. Danielewicz, R. Lacey, W.G. Lynch, Science 298, 1592 (2002).
[Dem96] J.F. Dempsey et al., Phys. Rev. C 54 (1996) 1710.
[Fam06] M.A. Famiano et al., Phys. Rev. Lett. 97, 052701 (2006).
[Fer06] G. Ferini, et al., PRL 97, 202301 (2006).
[Fuc06] C. Fuchs, Prog. Part. Nucl. Phys. 56 (2006) 1.
[Hua96] M.J. Huang et al., Phys. Rev. Lett. 77 (1996) 3739.
[Igl06] J. Iglio et al., Phys. Rev. C 74, 024605 (2006).
[Kli07] A. Klimkiewicz et al, Phys. Rev. C 76, 051603 (2007).
[Lat01] J.M. Lattimer, M. Prakash, Ap. J. 550, 426 (2001).
[Lat04] J.M. Lattimer, M. Prakash, Science 304, 536 (2004).
[Lei93] Y. Leifels et al., Phys. Rev. Lett. 71 (1993) 963.
[Lam94] D. Lambrecht et al., Z. Phys. A 350 (1994) 115.
[Li97] B.A. Li, C.M. Ko, Z. Ren, Phys. Rev. Lett. 78, 1644 (1997).
[Li00] B.-A. Li, Phys. Rev. Lett. 85 (2000) 4221.
[Li04] B.-A. Li, Nucl. Phys. A 734 (2004) 593c.
[Li05] Q. Li et al., Phys. Rev. C 71 (2005) 054907.
[Li06] B.-A. Li et al., Phys. Lett. B 634 (2006) 378.
[Li07] T. Li et al., Phys. Rev. Lett. 99, 162503 (2007).
[Liu07] T.X. Liu, et al., Phys. Rev. C 76 (2007) 034603.
[Lop07] X. Lopez, et al., Phys. Rev. C 75, 011901 (2007).
[Rus11] P. Russotto et al., Phys. Lett. B 697, 471 (2011).
[Ste05] A.W. Steiner, M. Prakash, J.M. Lattimer, P.J. Ellis, Phys. Rep. 411, 325 (2005).
[Tsa01] M.B. Tsang, W.A. Friedman, C.K. Gelbke, W.G. Lynch, G. Verde, H.S. Xu, Phys. Rev. Lett. 86, 5023 (2001).
[Tsa02] M.B. Tsang et al., Phys. Rev. C 66 (2002) 044618.
[Tsa04] M.B. Tsang, et al., Phys. Rev. Lett. 92, 062701 (2004).
[Tsa08] M.B. Tsang, et al., MSU Preprint, November 2008.
[Tso95] K. Tso et al., Phys. Lett B 361 (1995) 25.
[Vil04] A. R. Villarreal and T. E. Strohmayer, ApJ, 614, L121 (2004).
[Yon06] G.-C. Yong, B.-A. Li, L.-W. Chen, Phys. Rev. C 73 (2006) 034603.