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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,

Abstract

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.

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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 [2-6]. 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

112Sn+

112Sn,

112Sn+

124Sn,

124Sn+

112Sn

and

124Sn+

124Sn collisions have been measured [2]. 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].

),(),(),(

12 21

ZNYZNYZNR

=

=C⋅

) exp(

ZN

βα+

, (1)

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) ≈

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F2(N,Z,T), then Equation (1) is obtained, and

n

ρ ˆ =exp(∆µn/ T)= exp(α) and

p

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

),(

21

ZNR

is small and that

),(

21

ZNR

reflects the properties of the

primary source [2]. If true,

),(

21

ZNR

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 [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

),(

21

ZNR

, 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

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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

performing the calculations. We have chosen source sizes corresponding to 75% of the

collision systems

112Sn+

112Sn and

124Sn+

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

around

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. [27] 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

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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 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 [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