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arXiv:nucl-ex/0103010v1 26 Mar 2001

Isotopic Scaling in Nuclear Reactions

M. B. Tsang1, W.A. Friedman2, C.K. Gelbke1, W.G. Lynch1, G. Verde1, H. Xu1

1National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy,

Michigan State University

2Department of Physics, University of Wisconsin, Madison, WI 53706

(February 8, 2008)

Abstract

A three parameter scaling relationship between isotopic distributions for

elements with Z≤ 8 has been observed that allows a simple description of the

dependence of such distributions on the overall isospin of the system. This

scaling law (termed iso-scaling) applies for a variety of reaction mechanisms

that are dominated by phase space, including evaporation, multifragmentation

and deeply inelastic scattering. The origins of this scaling behavior for the

various reaction mechanisms are explained. For multifragmentation processes,

the systematics is influenced by the density dependence of the asymmetry

term of the equation of state.

Typeset using REVTEX

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The availability of high intensity radioactive beams facilitates the exploration of the

isospin degree of freedom in nuclear reactions. Understanding the connection between the

entrance channel isospin and the isotopic distribution of reaction products is important

for studying the charge asymmetry term of the nuclear equation-of-state[1-3], obtaining

information about charge equilibration[4-6], providing stringent tests for reaction models and

optimizing the production of rare isotopes far from stability. In this letter, we demonstrate

that isotopic distributions for statistical production mechanisms follow scaling laws. We

also find circumstances where the values for the scaling parameters are influenced by the

density dependence of the asymmetry term of the nuclear equation of state, a quantity that

influences many important properties of neutron stars.

The scaling laws in question relate ratios of isotope yields measured in two different

nuclear reactions, 1 and 2, R21(N,Z) = Y2(N,Z)/Y1(N,Z). In multifragmentation events,

such ratios were shown to obey an exponential dependence on the neutron and proton

number of the isotopes characterized by three parameters α,β and C [7]:

R21(N,Z) = C · exp(αN + βZ)(1)

Here we choose the convention that the isospin composition (neutron to proton ratio) of

system 2 is larger than that of system 1. The systematics described by Eq. 1 occur naturally

within the grand-canonical ensemble [7-9]. As shown in Ref. [7], the parameters α and β in

that limit are the differences between the neutron and proton chemical potentials for the two

reactions (i.e., α = ∆µn/T and β = ∆µp/T ) and C is an overall normalization constant.

The accuracy of the iso-scaling described by Eq. 1 can be compactly displayed if one

plots the scaled isotopic ratio,

S(N) = R21(N,Z) · exp(−βZ ) (2)

as a function of N. For all elements, S(N) must lie along a straight line on a semi-log

plot when Eq. 1 accurately describes the experimental data. The data points marked as

”multifragmentation” in Figure 1 show values of S(N) extracted from isotope yields with

1 ≤ Z ≤ 8 measured for multifragmentation events in central124Sn+124Sn and112Sn+112Sn

collisions at E/A= 50 MeV [7]. Selection of central events ensures that the average excitation

energies and temperatures in the participant source should be nearly identical [10]. The

observed iso-scaling is a necessary condition for the applicability of equilibrium models;

such models have described other aspects of these collisions quite well [11].

Rather surprisingly, iso-scaling is also observed for strongly damped binary collisions (16O

induced reactions on two targets232Th and197Au) [12] and evaporative compound nuclear

decay (4He +116Sn and4He +124Sn collisions)[13], for which Grand-Canonical Ensemble

approaches would appear to have little relevance. Our studies suggest that iso-scaling is

obeyed where 1.) both reactions 1 and 2 are accurately described by a specific statistical

fragment emission mechanism and 2.) both systems are at nearly the same temperature.

Indeed for deeply inelastic reactions emitted forward of the grazing angles, (22Ne +232Th

and22Ne +97Zr at θ = 12◦and E/A = 7.9MeV [12]) and for reactions with different tem-

peratures (124Sn+124Sn [7] and4He+124Sn [13]), iso-scaling is violated. Conditions 1 and 2

are met by the three reactions shown in Figure 1. Why iso-scaling is specifically observed in

these cases and what aspects of statistical physics such scaling probes are examined below.

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We first examine the strongly damped collisions, where iso-scaling is reasonably well

respected at low incident energies (E/A < 10MeV ) and at relatively backward angles [12,14],

i.e., when equilibrium is established between the orbiting projectile and target. In such cases,

the isotopic yields follow the ”Qgg-systematics”[12,14], and can be approximated by

Y (N,Z) ∝ exp((MP+ MT− M′

P− M′

T)/T)(3)

where MP and MT are the initial projectile and target masses, and M′

final masses of the projectile- and target-like fragment. Here, T can be interpreted as the

temperature. Eq. 3 reproduces the systematics shown in Fig. 1. To show why this is so,

we have expanded the nuclear binding energy contributions to the masses in Taylor series

in N and Z. Expressing explicitly only the leading order terms that depend on N and Z,

we obtain a relatively accurate leading order approximation to Eq. 3 :

Pand M′

Tare the

R21∝ exp[(−∆sn· N − ∆sp· Z)/T]. (4)

where ∆snand ∆spare the differences of the neutron and proton separation energies for

the two compound systems. Thus, the difference in the average separation energies in Eq.

4 plays a corresponding role to the difference in chemical potentials in the grand canonical

approach, an intriguing result when one considers that µ ≈ −s in the low temperature limit

[15]. The straightforward dependence of Eq. 4 on temperature suggests that it may provide

information relevant to the temperatures achieved in strongly damped collisions. The data

of Figure 1 imply a temperature of 2.7 MeV, not inconsistent with values derived from

alternative analyses [12,14].

Next we consider the yields from processes involving the formation of a composite system

and its subsequent decay via the evaporation of different isotopes. Corresponding scaled

isotopic ratios for fragments detected at backward laboratory angles (θ = 160o) in4He+116

Sn and4He +124Sn collisions at E/A = 50MeV [13] are shown in Figure 1, next to the

label ”evaporation”.

To explore the factors that govern the relevant evaporation rates, we utilize the formalism

of Friedman and Lynch [16] which invokes statistical decay rates derived from detailed

balance [17]. When the yields are dominated by emission within a particular window of

source-mass or source-temperature, the relative yields of a fragment with neutron number

N and proton number Z are directly related to the instantaneous rates

dn(N,Z)/dt ∝ T2· exp(−Vc/T + N · f∗

−{BE(Ni,Zi) − BE(Ni− N,Zi− Z) − BE(N,Z)}/T)

n/T + Z · f∗

p/T

(5)

where Vcgives the Coulomb barrier, the terms f∗

to the free energy per neutron (proton), BE is the binding energy and Niand Ziidentify

the neutron and proton numbers of the parent nucleus.

Applying Eq. 5 to the calculations of R21for two systems at the same temperature, we

find that the binding energies of the emitted fragments cancel and the systematics shown

in Fig. 1 can be reproduced. To understand why, we again expand the binding energy of

the residue with neutron number Ni− N and proton number Zi− Z to leading order in a

Taylor series to obtain:

n(f∗

p) represent the excitation contribution

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R21(N,Z) ∝ exp[{(−∆sn+ ∆f∗

n) · N + (−∆sp+ ∆f∗

p+ e∆Φ(Zi− Z)) · Z}/T] (6)

where Φ(Z) is the electrostatic potential at the surface of a nucleus with neutron and proton

number N and Z. Aside from the second order term from the electrostatic potential, which

is small for the decay of large nuclei, all factors in the exponent are proportional to either N

or Z, consistent with 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. Using the functions of ∆snand ∆f∗

one finds that a fixed temperature of about 3.7 MeV is required in Eq. 6, to obtain the

experimental value of α = 0.6. Running a full evaporation chain, using the procedure of

Ref. [16], provides an average fragment emission temperature of about 3.3 MeV. These

temperature values are comparable to those extracted by other techniques [13,18].

The Expanding Evaporating Source (EES) model [19] provides an alternative descrip-

tion of multifragmentation. Within the context of that model, additional insights can be

obtained. The EES model utilizes a formula for the particle emission rates which is formally

identical to that of Eq. 5 but can differ significantly in its predictions because the residue

may expand to sub-saturation density [20]. In this circumstance, the term enclosed in brack-

ets ”{}”containing three binding energies in Eq. 5 may vanish or become negative, enhancing

the emission rate of fragments with 3 ≤ Z ≤ 20. Detailed examination reveals that ∆f∗

Eq. 6 is usually much smaller than ∆sn, and the volume, surface, and Coulomb contribu-

tions to ∆snlargely cancel, leaving the asymmetry energy term, Sym(ρ)·(N −Z)2/A, alone

as the dominant contribution to α. For simplicity, we assume a power law dependence for

Sym(ρ), i.e. Sym(ρ) = Csym· (ρ/ρ0)γwhere γ is a variable and Csym= 23.4 MeV is the

conventional liquid drop model constant [21].

For illustration, we have performed calculations for the decay of the composite systems

found in124Sn +124Sn and112Sn +112Sn collisions assuming, for simplicity, initial systems

of (Ztot, Atot) of (100, 248) and (100, 224), respectively, initial thermal excitation energies of

E∗

fragments are emitted from these systems as they expand from an initial density ρ/ρ0= 1 to

ρ/ρ0= 0.1. The left panel of Figure 2 shows two iso-scaling functions, S(N), calculated with

the predicted isotope yields for 3 ≤ Z ≤ 6 for γ = 1 and γ = 0. In spite of a rather complex

interplay of expansion and fragment emission, the EES model predictions still exhibit an

approximate scaling. Some of the calculated deviations from iso-scaling may, in fact, be

due to the rather schematic treatment of the Coulomb barrier penetration [22] and to the

incomplete sequential decay information used in the model. Values for α, shown in the right

panel, are determined from the slopes of best fit of the lines to the predicted iso-scaling

functions.

For large values of γ, the asymmetry term, Csym· (ρ/ρ0)γdecreases more rapidly with

density and becomes negligible as the residue expands. At low density where fragments are

predominately emitted [19], the difference in isotopic yields from the two reactions 1 and 2

become smaller, resulting in flatter scaling functions and smaller values of α. (In the extreme

case of identical yields from the two systems, S(N) becomes a horizontal line corresponding

to α = 0.) The dot-dash line in the right panel of Fig. 2 joining the solid points shows the

EES prediction that α decreases with increasing γ values. The multifragmentation data in

Fig. 1 can be fairly well reproduced by γ ≈0.6.

When the emission process ends at ρ = 0.1ρ0, a low density residue (LDR) may remain.

nfrom Friedman and Lynch [16],

nin

ther= 9.5MeV, and initial collective radial expansion energies of EColl/A = 2.5MeV. The

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The final fate of this residue is not predicted by the EES model. A potential ambiguity may

result if this residue is large and if its eventual disintegration produces a significant fraction

of fragments with Z ≥ 3. Calculations using the Statistical Multifragmentation Model [23]

with the mass and energy of the final LDR predict α-values that increase with γ (dashed line)

— opposite to the trend predicted for emission during the expansion. When γ decreases,

the N/Z of the LDR from the two reactions,124Sn +124Sn and112Sn +112Sn become

more similar, resulting in smaller α-values – a trend also predicted from isospin dependent

transport theory [24]. This accounts for the behavior of the dashed line in the right panel

of Figure 2. If high energy fragments are emitted primarily from the expanding system, and

low-energy fragments come from the instantaneous disintegration of a low-density residue,

then the results in Fig. 2 suggest that the predicted difference in the iso-scaling for low and

high energy fragments should be observable for γ < 0.8.

In summary, we have observed a scaling between isotopic distributions which allows

a simple description of the dependence of such distributions on the overall isospin of the

measured systems in terms of three parameters, α,β and C. This scaling seems to apply to a

broad range of statistical fragment production mechanisms, including evaporation, strongly

damped binary collision, and multifragmentation. We have shown how this systematics

arises within models frequently applied to such processes. In one such model, the EES

model of multifragmentation, we find that the iso-scaling parameters are sensitive to the

density dependence of the asymmetry term of the EOS.

This work was supported by the National Science Foundation under Grant Nos. PHY-

95-28844 and PHY-96-05140.

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