Isospin Diffusion in Heavy-Ion Collisions and the Neutron Skin Thickness of Lead
Abstract
The correlation between the thickness of the neutron skin in Pb, and the degree of isospin diffusion in heavy-ion collisions is examined. The same equation of state is used to compute the degree of isospin diffusion in an isospin-depedent transport model and the neutron skin thickness in the Hartree-Fock approximation. We find that skin thicknesses less than 0.15 fm are excluded by the isospin diffusion data.
pacs:
25.70.-z, 21.30.Fe, 21.10.Gv, 24.10.LxI Introduction
The nuclear symmetry energy and its dependence on density holds a unique place in nuclear physics and astrophysics. It is an essential ingredient in understanding many aspects of nuclear physics, from heavy-ion collisions to nuclear structure, and astrophysics, from supernovae and neutron stars to r-process nucleosynthesis (for a recent review see Ref. Steiner et al. (2005)). At the same time, the nuclear symmetry energy is relatively uncertain; it is known to within only about 5 MeV at saturation density (compared to the binding energy which is known to within 1 MeV). The density dependence of the nuclear symmetry energy at both sub-saturation and super-saturation densities is very poorly known.
The symmetry energy is also an important quantity in determining the compressibility of matter. The incompressibility of symmetric nuclear matter at its saturation density fm has been determined to be MeV from nuclear giant monopole resonances Youngblood et al. (1999) and the equation of state (EOS) at densities of has been constrained by measurements of collective flows Gutbrod et al. (1989); Danielewicz and Odyniec (1985); Gustafsson et al. (1984); Welke et al. (1988); Siemens and Rasmussen (1979) in relativistic heavy-ion collisions Danielewicz et al. (2002). However, further progress in determining more precisely both the parameter and the EOS of symmetric nuclear matter is severely hindered by the uncertainties of the symmetry energy Danielewicz et al. (2002); Coló et al. (2004).
The possibility of determining the EOS of nuclear matter from heavy-ion collisions has been discussed for almost 30 years, and indeed impressive progress has been achieved. A number of heavy-ion collision probes of the symmetry energy have been proposed including isospin fractionation Muller and Serot (1995); Li and Ko (1997); Baran et al. (1998); Xu et al. (2000); Tan et al. (2001), isoscaling Tsang et al. (2001); Ono et al. (2003), neutron-proton differential collective flow Li (2000), pion production Li (2002) and neutron-proton correlation function Chen et al. (2003). Determination of the equation of state from heavy-ion data is typically performed through comparisons with transport model simulations Danielewicz et al. (2002); Li et al. (1998); Li and Schröder (2001); Baran et al. (2005); Li et al. (2005). A recent analysis of the symmetry energy comes from isospin diffusion data for reactions involving Sn and Sn from the NSCL/MSU Tsang et al. (2004); Chen et al. (2005). Using an isospin- and momentum-dependent transport model, IBUU04 Li et al. (2004a, b), the NSCL/MSU data on isospin diffusion were found to be consistent with a nuclear symmetry energy of at sub-saturation densities.
The pressure of neutron-rich matter at densities near two-thirds saturation density (which is closely related to the symmetry energy, see Ref. Steiner et al. (2005)) is tightly correlated to the neutron skin thickness, , in Pb Brown (2000); Typel and Brown (2001). The present theoretical uncertainty in the symmetry energy results in a large uncertainty in the theoretical calculations of the skin thickness. Experimental measurements of the skin thickness are also very uncertain, mostly resulting from the difficulty in disentangling the strong interactions between the probe and the neutrons in the nucleus. Values from 0.07 fm to 0.24 fm have been suggested by the data Starodubsky and Hintz (1994); Clark et al. (2003). A determination of the symmetry energy at sub-saturation densities would offer a prediction of skin thickness of lead or vice versa.
The structure of nuclei and neutron stars and the properties of heavy-ion collisions are all determined by the same underlying EOS. In particular, both the size of the neutron skin and the degree of isospin diffusion in heavy-ion collisions at intermediate energies are sensitive to the symmetry energy at sub-saturation densities. In this article, we study the correlation between the thickness of neutron skin in Pb and the strength of isospin diffusion using the same set of equations of state for asymmetric nuclear matter. This differs from previous work in that, for the first time, we make a direct connection between observables from nuclear structure and heavy-ion collisions. The transport model IBUU04 was used to predict the isospin diffusion for several equations of state which differ only in the density dependence of the symmetry energy. These equations of state are also used to calculate for Pb in the Hartree-Fock (HF) approximation. The correlation between the and the isospin diffusion provides a more stringent constraint than their individual values on the EOS of neutron-rich matter. Comparisons with the available data are also discussed.
Ii The Equation of State and The Transport Model Calculations of Isospin Diffusion in Heavy-Ion Collisions
Isospin diffusion measures quantitatively the net exchange of isospin contents between the projectile and target nuclei. Using symmetric reactions and as references the degree of isospin diffusion in the asymmetric reaction of can be measured by Rami et al. (2000)
(1) |
where is any isospin-sensitive tracer. In the recent NSCL/MSU experiments with Sn on Sn at a beam energy of 50 MeV/nucleon and an impact parameter about 6 fm, the isospin asymmetry of the projectile-like residue was used as the isospin tracer Tsang et al. (2004). Consistent with the experimental selection, in model analyses the average isospin asymmetry of the Sn-like residue was calculated from nucleons with local densities higher than and velocities larger than the beam velocity in the c.m. frame. Reactions at intermediate energies are always complicated by preequilibrium particle emission and the production of possibly neutron-rich fragments at mid-rapidity, however, the quantity has the advantage of minimizing significantly these effects Tsang et al. (2004).
The NSCL/MSU data were recently analyzed by using the IBUU04 version of an isospin and momentum dependent transport model Li et al. (2004a, b). Here, we recall the ingredients of the model that are most relevant for the present study. In this model one can select to use either the experimental nucleon-nucleon cross sections in free space or reduced in-medium nucleon-nucleon cross sections. Since the later involves several model-dependent assumptions, we have used the free-space cross sections (as in Ref. Chen et al. (2005)). The EOS used in this model is based on the Gogny effective interactions. The potential for a nucleon with isospin ( for neutrons and for protons) and momentum in asymmetric nuclear matter at total density is given by Prakash et al. (1988, 1997); Bombaci (2001); Das et al. (2003)
(2) | |||||
where and denote proton or neutron density with ; and is the isospin asymmetry. The denotes the phase-space distribution function at coordinate and momentum . The corresponding momentum-dependent interaction (MDI) leads to an incompressibility of MeV for the symmetric nuclear matter at saturation density. On the right hand side of Eq.(2), the first four terms with and MeV describe the momentum-independent interaction. The terms with parameters MeV and MeV describe the momentum-dependent interaction of a nucleon of isospin and momentum with like and unlike nucleons in the background fields, respectively. We stress that the momentum dependence in both the isoscalar and isovector potentials is taken into account in a way consistent with that known empirically from nucleon optical potentials Li et al. (2005). With the parameter , where denotes nucleon Fermi momentum at , the isoscalar potential coincides with predictions from the variational many-body theory using inputs constrained by nucleon-nucleon scattering data Wiringa (1988); Li et al. (2005), and the isovector potential also agrees with the momentum dependence of the Lane potential extracted from nucleon-nucleus scattering experiments up to about 100 MeV Hodgson (1994) and (p,n) charge exchange reactions up to about 45 MeV Hoffmann (1972); Li et al. (2005).
The parameter in Eq.(2) is introduced to allow variations in the density dependence of the nuclear symmetry energy , which is defined via the parabolic approximation to the nucleon specific energy in isospin asymmetric nuclear matter Li et al. (1998); Li and Schröder (2001), i.e.,
(3) |
The symmetry energies for three of the models considered here are displayed in Fig. 1 as a function of density. With , matches what is predicted by a Hartree-Fock calculation using the Gogny effective interaction Das et al. (2003). The parameters and are and , respectively. It is important to note that large positive (negative) values of lead to a “higher” (“lower”) symmetry energy at sub-saturation densities, but a lower (higher) symmetry energy at super-saturation densities.
An example of the results from the transport calculations, taken from Ref. Chen et al. (2005) is given in Fig. 2 for the MDI EOS with . The time evolution of is given for this equation of state. The evolution of the average central density in units of the saturation density is also given. It is clear that the late-time evolution of for the MDI equation of state matches the data obtained from MSU. The central density at these times is quite small, indicating clearly that is sensitive to the symmetry energy at sub-saturation densities.
Iii The EOS Fitting Procedure and Properties of Pb
Because the MDI EOS in the form above does not easily avail itself to the calculation of the structure of lead nuclei we choose to fit the EOS to a Skyrme model and use a Skyrme-Hartree-Fock code to calculate corresponding lead nucleus. This procedure has been used successfully to predict the neutron skin based on the Akmal, et. al. EOS Akmal et al. (1998) in Ref. Steiner et al. (2005). The four MDI equations of state with different values of were fit to Skyrme models constrained so values for the binding energy () and charge radius () for lead which were within 2% of the experimental values. For the equation of state with , the results of the fit are displayed in Fig. 3. The similar nature of the Skyrme and MDI parameterizations allows for a very close fit. For all of the four fits employed, the slope, , and the curvature of the symmetry energy and the final properties of lead nuclei in the Hartree-Fock approximation are given in Table 1. Also given in the table are the experimental values.
The effective mass for the original MDI EOS and the new Skyrme model are not as close, for example, the effectve mass at saturation density for the MDI EOS with is 0.67, while for the Skyrme fit, it is 0.77. However, the neutron skin thickness is more sensitive to the bulk energetics than the effective mass. As a demonstration, a new Skyrme fit to the MDI EOS with with a lower effective mass of 0.65 only lowers the skin thickness by 0.01 fm.
x | (MeV) | (fm) | (fm) | L (MeV) | (MeV) |
---|---|---|---|---|---|
-2 | 7.8598 | 5.517 | 0.38 | 153.0 | 276.3 |
-1 | 7.8702 | 5.520 | 0.28 | 107.4 | 94.1 |
0 | 7.8584 | 5.547 | 0.22 | 62.1 | -88.6 |
1 | 7.8719 | 5.522 | 0.16 | 16.4 | -270.4 |
Exp. | 7.8675 | 5.503 | 0.07-0.24 |
Iv Isospin Diffusion versus the Neutron-Skin of Pb
Now we turn to the correlation between the degree of isospin diffusion and the size of neutron-skin in Pb. Do we expect a correlation between these two seemingly different observables? As it has been discussed in detail in Refs. Horowitz and Piekarewicz (2001, 2002), the size of neutron skins in heavy nuclei is determined by the difference in pressures on neutrons and protons . The latter is propotional to , which measures the stiffness of the symmetry enrgy at saturation density. As shown in Table 1, the symmetry energy becomes more stiff (for densities larger than the saturation density) as decreases from 1 to , and the size of the neutron skin in Pb increases. Alternatively, one can understand this effect by noting that the the symmetry energy is lower (e.g., with ) at subsaturation densities the cost of creating a difference between the neutron and proton densities is small and the skin thickness is large. As the symmetry energy increases with larger values of , the neutron skin thickness decreases accordingly.
It is also well known that the degree of isospin diffusion in heavy-ion collisions depends sensitively on the stiffness of the symmetry energy Shi and Danielewicz (2003); Tsang et al. (2004); Li (2004); Chen et al. (2005); Baran et al. (2005). A correlation between the degree of isospin diffusion in heavy-ion collisions and the size of neutron skins in heavy nuclei using the same EOS is thus expected. Of course, it is understood that this correlation is not universal as both the isospin diffusion and size of neutron skins are system dependent. However, an examination of this correlation for any system and simultaneous comparisons with the corresponding data may provide a more stringent constraint on the underlying EOS. Shown in Fig. 4 is our analysis of this correlation for the isospin diffusion in Sn+Sn and the neutron skin in Pb. The available data on both quantities are also included in the graph. There are some remaining systematic uncertainties that are difficult to estimate which are not addressed here (see Ref. Li et al. (2004a) for more details).
From Eq. 1, we can see that implies the projectile-like residue has the same isospin asymmetry as the projectile Sn without any net exchange of isospin asymmetry with the target Sn. This is also often referred as the complete isospin transparency. A value of zero for implies that a complete mixing between the projectile and target indicating the establishment of an isospin equilibrium during the reaction.
The NSCL/MSU data indicates that the mixing is about 50%. Unlike the relation between and the neutron skin thickness, the relation between and the parameter is non-monotonic. This is because the is directly related to the symmetry potential which depends on both the density and momentum as discussed in detail in Ref. Chen et al. (2005). Transport model calculations with are closest to the experimental data, although and are not necessarily excluded within the current uncertainties of both the experimental data and model calculations.
Upon comparing with previous estimates for , the connection with the moderate amount of isospin diffusion displayed in heavy-ion reactions suggests that is on the upper range of the present experimental uncertainty. Only skin thicknesses larger than about 0.2 fm are consistent with the transport model calculations. Skin thicknesses lower than 0.15 fm appear to be excluded. However, the isospin dependence of the in-medium nucleon-nucleon cross sections may affect the isospin diffusion. It was shown analytically that the degree of isospin diffusion depends on both the symmetry energy and the neutron-proton scattering cross section Shi and Danielewicz (2003). Because of the momentum-dependence of the single nucleon potential of eq. 2, not only the values of the n-n and p-p cross sections are reduced and different from each other compared to their free-space values, the ratio is also modified from their free-space value. Whie the reduction of in-medium n-n and p-p scattering cross sections is expected to have little effect on the value of , the reduced in-medium n-p scattering cross sections leads to smaller isospin diffusion. To reproduce the NSCL/MSU data a larger symmetry energy at sub-saturation densities is then needed. The use of in-medium cross sections will, in effect, decrease the preferred neutron skin thicknesses for the isospin diffusion data. An investigation of this effect is currently underway and the results will be reported elsewhere.
Although isospin diffusion in heavy-ion collisions primarily constrains the symmetry energy at sub-saturation densities, it is useful to check to ensure that the information from heavy-ion collisions is not inconsistent with neutron star observations. Neutron star radii are sensitive probes of the symmetry energy at higher density Lattimer and Prakash (2001). We solve the Tolman-Oppenheimer-Volkov equations for the MDI EOS with , and . The maximum masses (in order of decreasing ) are 1.92, 2.08, and 2.13 and the radii of the maximum mass stars are 10.1, 11.2, and 11.8 km. In addition to being consistent with the compactness constraint given by the measurement of the redshift of EXO0748-676 in Ref. Cottam et al. (2002), this equation of state is also consistent with the recent neutron star radius measurement in Ref. Gendre et al. (2003).
Recently, Ref. Todd-Rutel and Piekarewicz (2005) has also explored the impact of the symmetry energy on the neutron skin thickness in lead. Using an relativistic mean-field model they construct an equation of state which faithfully describes the giant monopole and dipole resonances in nuclei. Their model suggests a neutron skin thickness of 0.21 fm in Pb which is consistent with our constraints from isospin diffusion in heavy-ion collisions.
V Summary
In summary, we examined the correlation of the degree of isospin diffusion in heavy-ion collisions and the size of neutron skin in Pb using the same EOS within the IBUU04 transport model and the HF approach. We found that neutron skin thicknesses less than 0.15 are disfavored by the isospin diffusion data. Reduction of the experimental and theoretical uncertainties will likely lead to an important constraint on the symmetry energy at sub-saturation density.
We would like to thank Lie-Wen Chen and Madappa Prakash for helpful discussions. This work was supported in part by the DOE under grant no. DOE/W-7405-ENG-36 (for AWS) and by the NSF under grant no. PHY-0243571, PHY-0354572 and the NASA-Arkansas Space Grant Consortium Award ASU15154 (for B.-A. Li)
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