Study of isoscaling with statistical multifragmentation models
M.B. Tsang, C.K. Gelbke, X.D. Liu, W.G. Lynch, W.P. Tan,
G. Verde, H.S. Xu
National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy,
Michigan State University, East Lansing, MI 48824, USA
W. A. Friedman,
Department of Physics, University of Wisconsin, Madison, WI 53706
R. Donangelo, S. R. Souza,
Instituto de Física, Universidade Federal do Rio de Janeiro,
Cidade Universitária, CP 68528, 21945-970 Rio de Janeiro, Brazil
C.B. Das, S. Das Gupta, D. Zhabinsky
Physics Department, McGill University, 3600 University Street, Montreal, Canada H3A 2T8,
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(αN+βZ). Even though the primary
isotope distributions are quite different from the final distributions due to evaporation from
the excited fragments, the values of α and β are not much affected by sequential decays. α is
shown to be mainly sensitive to the proton to 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.
* On leave from the Institute of Modern Physics, Lanzhou, China.
% Research Experience for Undergraduates at Michigan State University, 2000.
Our understanding of nuclear collision mechanisms is obtained from measuring particles
emitted during nuclear collisions. 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 [2-6]. The availability of these data makes it possible to examine systematic
trends exihibited by the isotope distributions .
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 .
Recently, isotope yields from the central collisions of
124Sn collisions have been measured . The ratio of isotope yields from two
different reactions, 1 and 2, R21(N, Z) = Y2(N, Z)/ Y1(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, µn and µp and the temperature
T, plus the individual binding energies, B(N,Z), of the various isotopes [9,10].
Y(N,Z)=F(N,Z,T)exp(B(N,Z)/ T)exp(Nµn/ T+ Zµp/ T) (2)
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 Y2(N,Z)/ Y1(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. F1(N,Z,T) ≈
F2(N,Z,T), then Equation (1) is obtained, and
ρ ˆ =exp(∆µn/ T)= exp(α) and
ρ ˆ = exp(∆µp/ T)=
exp(β) are the relative ratios of the free neutron and free proton densities in the two systems,
where ∆µn and ∆µp are the differences in the neutron and proton chemical potentials. The
empirical observation that this fugacity dependence is respected suggests that the effect of
sequential decays on
is small and that
reflects the properties of the
primary source . If true,
may be an important and robust observable.
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 .
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  and Antisymmetrized
Molecular Dynamics models .
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 . 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  has been
developed to address the secondary decay problem. Each decay from the initial excited
fragment is calculated using tabulated branching ratios when available , or by using the
Hauser-Feshbach formalism , 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. , 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
performing the calculations. We have chosen source sizes corresponding to 75% of the
124Sn, an excitation energy of E
*/ A=6 MeV, and a
breakup density of 1/ 6ρO. The general conclusion of this paper would not change if other
source sizes were used. We characterize the neutron and proton composition of the source by
the neutron to proton ratio, N/ Z or the isospin asymmetry δ=(N-Z)/ A=(N/ Z-1)/ (N/ Z+1).
To examine the effects of secondary decay, the predicted carbon isotope distributions
from SMM-MSU are shown in Figure 1. The primary distributions from a source of A=186,
N/ Z=1.48 are shown as open points joined by a dashed line while the final distributions after
secondary decay are shown as closed circles joined by a solid line in the top panel. The
primary distributions are wide and spread over a large range of neutron-rich nuclei and peak
14C. After sequential decays, the distributions are much narrower and peaked near
12C, more in agreement with experimental observation. Such narrowing of isotope
distributions due to sequential decays has been well established [13,24-26].
It has been suggested in Ref.  that the isotope distributions are sensitive to the proton
and neutron composition of the sources from which the fragments are emitted. To explore
this issue, we eliminate the size effect by changing the charge of the emitting source but
keeping the size constant, i.e. A=186. The carbon isotope distribution after secondary decay
with N/ Z=1.48 (closed circles) and N/ Z=1.24 (open squares) are compared in the bottom
panel of 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 . It suggests that isotope yield distributions
can be used to study properties that reflect the neutron to proton composition of the emitting
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 R21(N, Z) are plotted, as a function of N for the
primary and secondary isotope yields predicted by the SMM-MSU model. We choose A1=168
and Z1=75 (N1/ Z1=1.24, δ1 =0.107) and A2=186, Z2=75 (N2/ Z2=1.48, δ2=0.194) for sources 1 and
2 where Ai and Zi are the mass and charge number of source i. Ratios constructed from
primary (final) yields are plotted in the top (bottom) panel. The open symbols represent
R21(N, Z) of odd-Z elements while the closed symbols are predicted ratios for the even-Z
elements. The ratios of both primary and secondary fragments closely follow the trend
described by Eq. (1); isotopes of the same Z, plotted with the same symbol, lie along lines
with similar slope in the semi-log plots. For comparison, the solid and dashed lines
correspond to the calculations using the best-fit values of α, β and C of Eq. (1) to the
predicted ratios. Since more neutron-rich isotopes are produced from the neutron-rich
system, the slopes of these lines are positive.
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 R21(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 . 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