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Study of Isoscaling with Statistical Multifragmentation Models

Different statistical multifragmentation models have been used to study isoscaling, i.e. the factorization of the isotope ratios from two reactions, into fugacity terms of proton and neutron number, R21(N,Z)=Y2(N,Z)/Y1(N,Z)=C*exp(a*N+b*Z). Even though the primary isotope distributions are quite different from the final distributions due to evaporation from the excited fragments, the values of a and b are not much affected by sequential decays. a is shown to be mainly sensitive to the proton and neutron composition of the emitting source and may be used to study isospin-dependent properties in nuclear collisions such as the symmetry energy in the equation of state of asymmetric nuclear matter.


I. Introduction
Our understanding of nuclear collision mechanisms is obtained from measuring particles emitted during nuclear collisions [1]. The importance of the isotopic degree of freedom to obtain information about charge equilibration and the charge asymmetry dependent terms of the nuclear equation-of-state has prompted recent measurements of isotope distributions beyond Z=2 [2][3][4][5][6]. The availability of these data makes it possible to examine systematic trends exihibited by the isotope distributions [7].
Ideally, primary fragments should be detected right after emission in order to extract information about the collisions. However, the time scale of a nuclear reaction (10 -20 s [5][6]) is much shorter than the time scale for particle detection (≥10 -10 s) and most particles decay to stable isotopes in their ground states before being detected. It is therefore important to study model predictions of both primary and secondary isotope distributions [8].
Recently, isotope yields from the central collisions of 112 [2]. The ratio of isotope yields from two different reactions, 1 and 2, R 21 (N, Z) = Y 2 (N, Z)/Y 1 (N, Z), is found to exhibit an exponential relationship as a function of the isotope neutron number N, and proton number, Z [2,7].
where C is the normalization factor, α and β are empirical parameters.
Equation (1) can be derived from the primary isotope yields assuming that at breakup the system may be approximated by an infinite equilibrated system and employing the Grand Canonical Ensemble. In this case, predictions for the observed isotopic yield are governed by both the neutron and proton chemical potentials, µ n and µ p and the temperature T, plus the individual binding energies, B(N,Z), of the various isotopes [9,10].
The factor F(N,Z,T) includes information about the secondary decay from both particle stable and particle unstable states to the final ground state yields. If the main difference between system 1 and 2 is the isospin [2,9,10], then the binding energy terms in Eq.
(2) cancel out in the ratio of Y 2 (N,Z)/Y 1 (N,Z). If one further assumes that the influence of secondary decay on the yield of a specific isotope is similar for the two reactions, i.e. F 1 (N,Z,T) ≈ F 2 (N,Z,T), then Equation (1) is obtained, and n ρ =exp(∆µ n /T)= exp(α) and p ρ = exp(∆µ p /T)= exp(β) are the relative ratios of the free neutron and free proton densities in the two systems, where ∆µ n and ∆µ p are the differences in the neutron and proton chemical potentials. Furthermore, Eq. (1) allows one to extrapolate isotope yields over a wide range of the reacting systems from the measurements of a few selected isotopes [7].
Since the Grand Canonical limit is strictly valid only for statistical fragment production in an infinite dilute equilibrated system, it is important to study the validity of the scaling behavior of Eq. (1) with more realistic models. In this paper, we demonstrate that the isoscaling property of Eq. (1) is also predicted by three additional statistical models, the microcanonical and canonical Statistical Multifragmentation Models as well as the Expanding Emission Source (EES) model. In all three of them, isoscaling is affected only slightly by sequential decays, and α and β are mainly sensitive to the proton to neutron composition of the emitting source. In a future paper, we will discuss predictions of non-equilibrium transport models such as the Boltzmann-Nordheim-Vlasov [11] and Antisymmetrized Molecular Dynamics models [12].

II. Microcanonical Statistical Multifragmentation model
To explore the effect of secondary decays on ) , ( , we first employ a detailed sequential decay simulation to de-excite primary fragments created in the microcanonical statistical multifragmentation model [13]. Such models have been used successfully to describe fragment multiplicity distributions, charge distributions, mean kinetic energies, and mean transverse energies of the emitted particles from multifragmentation processes [14,15]. However, the most commonly used Statistical Multifragmentation Model (SMM) [16,17] contains only a schematic treatment of the sequential decays of excited fragments and does not include much of the nuclear structure information needed to describe the secondary decay of hot primary fragments. A new improved sequential decay algorithm [13] has been developed to address the secondary decay problem. Each decay from the initial excited fragment is calculated using tabulated branching ratios when available [18], or by using the Hauser-Feshbach formalism [19], when such information is unavailable. Aside from incorporating empirical information on the binding energies of the nuclei, the new algorithm includes accurate structural information such as the discrete bound states and resonant states for nuclei up to Z=15 [13,20]. This new sequential decay algorithm is coupled to the SMM code of ref. [21], which was chosen mainly for the ease of incorporating the sequential decays of the primary fragments. This newly modified SMM code is referred as SMM-MSU in this article. The physics results should be similar if other SMM codes are used.
As the primary goal of this article is to understand the general behavior of various models, we will refrain from fitting data by varying model parameters. Instead, we will use previous studies as a guide [22,23] and choose reasonable and consistent parameters in  Figure 1. As expected, more neutron rich isotopes (A >12) are produced from the neutron richer system, while the opposite is true for the proton-rich isotope yields. This trend is consistent with experimental observation [2]. It suggests that isotope yield distributions can be used to study properties that reflect the neutron to proton composition of the emitting sources. Figure 1 illustrates an important point that the isospin effects on isotope yields are much reduced by sequential decays. The differences between the final isotope yields from two systems with different isospin asymmetry are much less than those between primary and final isotope distributions. It is thus important to search for observables such as relative isotope ratios, which cancel out some of the effects of sequential decays, binding energy etc.
on isotope productions.
In Figure 2, the relative isotope ratios R 21 (N, Z) are plotted, as a function of N for the primary and secondary isotope yields predicted by the SMM-MSU model.  More importantly, the slopes are similar for all elements before and after sequential decay. This result seems surprising considering the big difference between the primary and secondary distributions shown in the top panel of Figure 1, but it corroborates the assumption that R 21 (N, Z) is not very sensitive to sequential decays and justifies the empirical approach of Eq. (2) to approximate the effect of sequential decays by a constant multiplicative factor for reactions with similar excitation energy and temperature [2]. The exponential dependence on Z in Eq. (1) suggests that the vertical spacing between adjacent elements should be the same. However this latter requirement is not strictly observed in the predicted results, especially for the final yield ratios. The solid and dashed lines in the upper panel show the best fits of equation 1 with α=0.40, β=-0.50. The scaling parameters extracted after secondary decays in the bottom panel is the same for the neutron slope parameter, α=0.40, but the proton slope parameter β=-0.41 is different, which may indicate the importance of Coulomb effects [28].
For oxygen isotopes, the agreement between predicted ratios after sequential decays and the best fit lines is not very good. This discrepancy may be an artifact from the sequential algorithm used. The current secondary decay code which has structural information for nuclei up to Z=15 may not be reliable for secondary yields with large Z. The effect of incomplete structural information on sequential decays is illustrated in Figure 3.  [19]. While the yields for the carbon isotopes are similar with both decay tables, the yields for the neutron rich oxygen isotopes are quite different. Sequential decay calculations with more complete structure information predict more yields for neutron-rich oxygen isotopes. This indicates that sequential charged particle decay plays an important role in producing neutron-rich isotopes and that structure information is relevant to such calculations.
To explore the influence of different sequential decay schemes on isoscaling, the same systems described above (A 1 =168, Z 1 =75 and A 2 =186, Z 2 =75) are calculated with the more widely used SMM code of Botvina [14][15][16][17]. This version of SMM has a simplified description of secondary decay [16,17]; excited light fragments (A<16) undergo fermi breakup while heavier fragments decay by evaporating light nuclei. Figure 4 shows the isotope ratios before and after the sequential decays. The primary yield ratios (upper panel) show the trends as

III. Expanding Emitting Source model
In this section, we examine the Expanding Evaporating Source (EES) model [29] which provides an alternative description of multifragmentation. The EES model utilizes a rate equation formula similar to the evaporation formalism. The emission rate of fragments with 3≤Z≤20 is enhanced when the residue expands to sub-saturation density. Within the context of this model, α can be described analytically and provide some physics insight regarding the symmetry energy [7]. Figure 5 shows the relative isotope ratios predicted for multifragmentation processes by the EES model [29] for the systems, A 1 =168, Z 1 =75 and A 2 =186, Z 2 =75. Even though chemical potentials are not a theoretical ingredient of the EES model, the predicted isotope ratios display isoscaling similar to Equation (1). As in the case of Botvina's SMM calculations, isoscaling is more rigorously observed by the primary yield. Some of the deviations from isoscaling obtained with the final yields may be caused by inaccuracies in the treatment of sequential decays. For example, the EES model includes structural information mainly for low mass nuclei and no information about the unstable particle states for any but the lightest nuclei. Even so, the scaling parameters obtained before and after sequential decays are not very different.
To understand the origin of isoscaling in the EES approach, we must examine the EES fragment emission rate. Similar to the formalism of Friedman and Lynch [30], statistical decay rates in the EES model are derived from detailed balance following the Weisskopf model [31]. When the relative rates are dominated by emission within a particular window of source-mass or source-temperature, the relative yields are directly related to the instantaneous rates dn(N,Z)/dt ∝T where ∆Φ(Z) is the difference between electrostatic potential at the surface of residue 1 and residue 2. ∆f * is the differences in free energy for the two systems. Aside from the second order term from the electrostatic potential, which is small for the decay of large nuclei, all terms in the exponent of Eq. 5 are proportional to either N or Z, resembling Eq. (1). The corresponding scaling parameters α and β are functions of the separation energies, the Coulomb potential and small contributions from the free excitation energies.
In general, the contribution from free energy is found to be much smaller than the contribution from the separation energy. This is particularly true for systems of comparable mass and energy but different N/Z ratio. Moreover, the volume, surface, and Coulomb contributions to the separation energy largely cancel if the masses of the parent nuclei are similar, leaving the difference in symmetry energies alone as the dominant contribution to ∆s n . The symmetry energy takes the form: The change in neutron separation energy between the two systems can be approximately obtained by taking the derivatives in Eq. (6) with respect to N to obtain This dependence leads to a non-linear dependence on N i and Z i and a linear relationship between (Z 2 /A 2 ) 2 and α for a fixed system 1. In the liquid drop model, C sym takes the value of 23.4 MeV [32]. In the EES model, the symmetry energy term, C sym , must be extrapolated to sub-saturation density as the system expands, i.e., C sym is density dependent. Measurements of R 21 (N,Z) may thus probe the density dependence of the symmetry energy as discussed in Ref. [7].

IV. Canonical Model
To explore the relationship between the neutron and proton composition of the source (Z 2 /A 2 ) and α in the statistical fragmentation models, we must perform calculations with different sources. To simplify the discussions, we will use the two fitting parameters, α and β, the two approaches provide similar predictions for the observables presented below. There is also a similarity between both approaches and the predictions of the Grand Canonical ensemble [26,33].
Canonical model predictions for the temperature and density dependences of α are shown in the left and right panels of Figure 6, respectively. The calculations assume a fixed freeze-out density of ρ ο /3 in the left panel, and fixed temperatures of 4, 5 and 6 MeV in the right panel. The same systems, A 1 =168, Z 1 =75 and A 2 =186, Z 2 =75, are used. Isospin effects decrease with increasing T. There is a significant sensitivity to temperature at low temperature, but both the sensitivity to temperature and the overall isospin effect diminish at very high temperature. On the other hand, α is less sensitive to the breakup density.
It is interesting to note that if one were able to constrain the temperature and density with experimental information, the connection between α and the N/Z ratio of the fragmenting system could be used to constrain the latter quantity. This sensitivity is useful to constrain the N/Z of the fragmentating sub-system (prefragment) if it is modified by the preequilibrium emission prior to breakup. Transport calculations predict that the relative neutron vs. proton preequilibrium emission may be sensitive to the density dependence of the asymmetry term of the nuclear equation of state [34]. If so, charge and mass conservation implies that observables sensitive to the N/Z of the prefragment may provide constraints on the density dependence of this asymmetry term [20].
The temperature dependence of the difference in chemical potentials, ∆µ n = α⋅T and ∆µ p = β⋅T, is shown in the left and right panel of Figure 7, respectively. If the change in the chemical potentials for the two systems as a function of temperature were the same, then ∆µ n and ∆µ p would be constant. Instead, we see a decrease in the differences between the chemical potential, with increasing temperature. Interestingly, there is a break in the slope at T=5 MeV. There is currently no satisfactory explanation for such a break. Further studies are needed.
Experimentally, a nearly linear relation between the "relative free neutron density", n ρ =exp(α) and (N 2 /Z 2 ) ratio of the system 2, has been observed [2] over the range of (N 2 /Z 2 ) from 1.24 to 1.48. To explore this issue within the context of Eq.  Figure 8. Over a larger range of (N 2 /Z 2 ), there is a concave curvature in n ρ which is especially noticeable at small N 2 /Z 2 ≤ 1. The relationship between α and N 2 /Z 2 in the right panel is best described by α=3.0−15.21/(1+N 2 /Z 2 ) 2 (solid line).
The SMM model describes an instantaneous multifragmentation process rather than a sequential binary breakup process. Interestingly, the linear relationship between α and (Z 2 /A 2 ) 2 predicted by the EES model in Eq. (7), is also evident in the SMM calculations as shown in left panel of figure 8. If T is taken to be 5 MeV, we obtain C sym =19.2 MeV as compared to the liquid drop value of 23.4 MeV. Such relationship is perhaps not so surprising at low excitation energy, where recent SMM model calculations [13] indicate that µ n and s n are closely related as expected. At high excitation energy, the role of multifragment decay configurations become important. There is no direct connection between µ n to s n a priori. It is thus intriguing to see that the linear relationship of Eq. (1) is preserved and that Eq. (7) is valid even at high excitation energies. Such dependence probably signals the importance of the symmetry energy as the dominant contribution to α in the SMM model.
Indeed, if the symmetry terms to the binding energies of the nuclei are turned off in the SMM and EES calculations, the isoscaling behavior observed in Figure 2, 4 and 5 will disappear.

V. Summary
We have calculated the isotope distributions from Z=1 to Z=8 particles using different multifragmentation models. The simple factorization of R 21 (N,Z) into the neutron and proton "fugacity" terms has been demonstrated by all the models studied in this article. The relative isotope ratios are not affected very much by the sequential decays, so in these statistical models R 21 (N,Z) reflects the isotope yield ratios of the primary fragments. The isotope distributions are determined mainly by the isospin asymmetry of the emitting source and to a lesser extent the temperature of the system. For statistical models, it appears that R 21   : Predicted (symbols) relative isotope ratios, R 21 (N,Z), of Eq. 1 for the two systems, A 1 =168, Z 1 =75 and A 2 =186, Z 2 =75 using the SMM_MSU code [13,20]as a function of N obtained from the primary isotope yields (upper panel) and the final yields after sequential decays (lower panel). Solid and dashed lines are best fits to Equation 1 using the predicted ratios. Figure 3: Differential multiplicities at θ CM =90° for carbon (top panel) and oxygen (bottom panel) isotopes as a function of the mass number. Closed points are predictions if the sequential decay information from Ref. [13] where the sequential decay table truncates at Z=13, is used. Histograms are predictions when the structure information in the structural information table of Ref [13] is extended to Z=15 [20].      7). In the right panel, the relative free ndensity, n ρ , is plotted as a function of the N 2 /Z 2 ratio of the source. A linear relationship is observed over the range of N 2 /Z 2 =1.24 and 1.48, similar to the experimental results. However, over a wider range, the dependence of n ρ on N 2 /Z 2 is far from linear.