Page 1
1
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.
Page 2
2
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) ≈
Page 3
3
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
Page 4
4
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
Page 5
5
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
Page 6
6
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. The
histograms represent calculations for the carbon (upper panel) and oxygen (lower panel)
isotope distributions which use the Hauser Feshbach decay formalism [19] and take into
account all the experimental structural information up to Z=15. Closed points joined by
dashed lines are the isotope distributions when the Hauser Feshbach formalism is used with
the experimental structural information up to Z =10 only [13]. In both cases, decays of
heavier fragments not calculated via the Hauser Feshbach approach are calculated with the
Weisskopf formalism and liquid-drop binding energies [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 (A1=168, Z1=75 and A2=186, Z2=75) are calculated with the more
widely used SMM code of Botvina [14-17]. 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
predicted by Eq. (1) but the heavier isotopes (Z≥5) in the final yield ratios (bottom panel) are
not as well behaved. This can be attributed to the simplified sequential decay treatments
used. The best fit parameters of Eq. (1) are listed in the figure. Predictions from the best fit
parameters are plotted as dashed and solid lines. The lines do not describe the predicted
Page 7
7
ratios after sequential decay well (lower panel). However, the fitted values for α are little
altered by sequential decays while the fitted values for β are changed greatly.
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, A1=168, Z1=75 and A2=186, Z2=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
2⋅exp(-Vc/ T+N⋅fn
*/ T+Z⋅fp
*/ T-B/ T) (3)
where Vc gives the Coulomb barrier, and the terms fn
* and fp
* represent the excitation
contributions to the free energy per neutron and proton, respectively. The factor B=BE(Ni,Zi)-
Page 8
8
BE(Ni-N,Zi-Z)-BE(N,Z) reflects the separation energy associated with the removal of the
isotope (N,Z) from the parent nucleus, here denoted by the subscript ''i''.
When constructing R21(N,Z), some terms, such as the binding energy of the emitted
isotope, BE(N,Z), cancel out in the ratio, simplifying the analysis of the dependence of
R21(N,Z) on N and Z. To use what remains of the N and Z dependence of the separation
energy term B, we expand the differences in the binding energies of the residues with
neutron number Ni-N and proton number Zi-Z in a Taylor series as follows:
BE(N2-N, Z2-Z) - BE(N1-N, Z1-Z) ≈a·N+b·Z+c·N
2+ d·Z
2+e·N·Z (4)
Where a, b, c, d and e are coefficients of the Taylor series. Empirically, the coefficients, c,
d, and e of the higher terms in Z
2, N
2 and ZN are surprisingly small. One can approximate the
binding energy difference with the two leading order terms that depend on the difference in
the proton and neutron separation energies between the two systems, 1 and 2 i.e. a=∆sp,
b=∆sn. Assuming for simplicity that the residues for systems 1 and 2 have the same charge,
R21(N,Z) can be written as follows:
R21(N,Z) ∝ exp[{(-∆sn+∆fn
*)⋅N+(-∆sp+∆fp
*+e∆Φ(Zi-Z))⋅Z}/ T] (5)
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
∆sn. The symmetry energy takes the form:
Esym=Csym(N-Z)
2/ A= Csym(A-2Z)
2/ A (6)
Page 9
9
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
α=∆sn/ T≈4Csym[(Z1/ A1)
2- (Z2/ A2)
2]/ T (7)
This dependence leads to a non-linear dependence on Ni and Zi and a linear relationship
between (Z2/ A2)
2 and α for a fixed system 1. In the liquid drop model, Csym takes the value of
23.4 MeV [32]. In the EES model, the symmetry energy term, Csym, must be extrapolated to
sub-saturation density as the system expands, i.e., Csym is density dependent. Measurements
of R21(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
(Z2/ A2) 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,
which are the average slopes of the lines in the semi-log plot of isotope and isotone yield
ratios respectively as shown in Figs. 2, 4 and 5. Since sequential decay does not affect the
scaling parameters strongly, we confine our exploration to the influence of the parameters on
the primary distributions.
For these studies we use the statistical multifragmentation model (SMM-McGill) [27] that
uses recursive techniques to shorten the time needed for a canonical calculation. We have
compared the predictions of this canonical approach to the microcanical model of ref. [13];
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 ρο/ 3 in the left panel, and fixed temperatures of 4, 5 and 6 MeV in the
right panel. The same systems, A1=168, Z1=75 and A2=186, Z2=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.
Page 10
10
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, ∆µn= α⋅T and ∆µp=
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 ∆µn and ∆µp
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 ˆ =exp(α) and (N2/ Z2) ratio of the system 2, has been observed [2] over the range of (N2/ Z2)
from 1.24 to 1.48. To explore this issue within the context of Eq. (7), we kept our reference
system (reaction 1) fixed at A1=168, Z1=75 and performed calculations on systems with
different (N2/ Z2) values. The results are shown in the right panel of Figure 8. Four groups of
calculations are performed by either keeping source size constant at A2=186 (solid circles), or
A2=124 (open circles) or by keeping the charge of the source constant at Z2=75 (closed
squares) or Z2=50 (open squares). In all cases, the slope parameters α are dependent mainly
on N2/ Z2 or equivalently on the isospin asymmetry δ2=(N2/ Z2-1)/ (N2/ Z2+1), of system two
and independent of its charge number and source size. The experimental linear relationship
between
n
ρ ˆ and (N2/ Z2) is observed only within a narrow range of (N2/ Z2) from 1.2 to 1.5 as
demonstrated in the right panel of Figure 8. Over a larger range of (N2/ Z2), there is a concave
ρ ˆ which is especially noticeable at small N2/ Z2≤ 1. The relationship between curvature in
n
α and N2/ Z2 in the right panel is best described by α=3.0−15.21/(1+N2/ Z2)
2 (solid line).
Page 11
11
The SMM model describes an instantaneous multifragmentation process rather than a
sequential binary breakup process. Interestingly, the linear relationship between
α and (Z2/ A2)
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 Csym=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
µn and sn are closely related as expected. At high excitation energy, the role of multifragment
decay configurations become important. There is no direct connection between µn to sn 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 R21(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 R21(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 R21(N,Z)
provides an opportunity to study isotopic observables that are related to the primary
fragmentation process. This may provide access to the early stages of the fragmentation
process where there may be sensitivity to the symmetry terms of the equation of state, which
directly influence the neutron to proton ratios of the intermediate emission source.
This work is supported by the National Science Foundation under Grant No. PHY-95-
28844, PHY-96-05140, INT-9908727 and contract No. 41.96.0886.00 of MCT/ FINEP/ CNPq
(PRONEX).
REFERENCES
Page 12
12
1. S. Das Gupta, A.Z. Mekjian and M.B. Tsang, Adv. Nucl. Phys. 26 (in press)
2. H.S. Xu, M.B. Tsang, T.X. Liu, X.D. Liu, W.G. Lynch, W.P. Tan and G. Verde, L.
Beaulieu, B. Davin, Y. Larochelle, T. Lefort, R.T. de Souza, R. Yanez, V.E. Viola, R.J.
Charity, L.G. Sobotka; Phys. Rev. Lett. 85, 716 (2000).
3. H. Johnston, T. White, J. Winger, D. Rowland, B. Hurst, F. Gimeno-Nogues, D. O'Kelly
and S.J. Yennello, Phys. Lett. B371, 186 (1996), R. Laforest et al., Phys. Rev. C59 (1999)
2567 and refs. therein and references therein.
4. J. Brzychczyk et al., Phys. Rev. C47, 1553 (1993).
5. V.V. Volkov, Phys. Rep. 44, 93, (1978) and references therein.
6. C.K. Gelbke et. al., Phys. Rep. 42, 311 (1978).
7. M.B. Tsang, W.A. Friedman, C.K. Gelbke, W.G. Lynch, G. Verde, and H. Xu Phys. Rev.
Lett. (in press).
8. H. Xi, W.G. Lynch, M.B. Tsang, and W.A. Friedman, Phys. Rev. C 59, 1567 (1999).
9. S. Albergo S. Costa, Costanzo, Rubbino, Nuovo Cimento A 89, 1 (1985).
10. J. Randrup and S.E. Koonin, Nucl. Phys. A 356, 223 (1981).
11. A.B. Larionov, A.S. Botvina, M. Colonna, and M Di Toro, Nucl. Phys. A. 658, 375
(1999) and M. Colonna, private communications.
12. Yoshiharu Tosaka, Akira Ono, and Hisashi Horiuchi, Phys. Rev. C 60, 064613 (1999)
and A. Ono, private communications
13. S.R. Souza, W.P. Tan, R. Donangelo, C.K. Gelbke, W.G. Lynch, and M.B. Tsang, Phys.
Rev. C62, 064607 (2000).
14. M. D’Agostino, G. J. Kunde, P. M. Milazzo, J. D. Dinius, M. Bruno, N. Colonna, M. L.
Fiandri, C. K. Gelbke, T. Glasmacher, F. Gramegna, D. O. Handzy, W. C. Hsi, M.
Huang, M. A. Lisa, W. G. Lynch, P. F. Mastinu, C. P. Montoya, A. Moroni, G. F.
Peaslee, L. Phair, R. Rui, C. Schwarz, M. B. Tsang, G. Vannini, and C. William, Phys.
Lett. B 371, 175 (1996).
15. C. Williams, W. C. Lynch, C. Schwarz, M.B. Tsang, W.C. Hsi, M.J. Huang, D.R.
Bowman, J. Dinius, C. K. Gelbke, D.O. Handzy, G.J. Kunde, M.A. Lisa, C.F. Peaslee, L.
Phair, A. Botvina, M.C. Lemaire, S.R. Souza, C. Van Buren, R.J. Charity, L.G. Sobotka,
U. Lynen, I. Pochodzalla, H. Sann, W. Trautmann, D. Fox, R.T. de Souza, and N.
Carlin, Phys. Rev. C55, R2132 (1997).
16. J. P. Bondorf, A.S. Botvina, A.S. Iljinov, N. Mishustin, and K. Sneppen, Phys. Rep. 257,
133 (1995)
17. S. Botvina, I.N. Mishustin, M. Begemann-Blaich, J. Hubele, G. Imme, I. Iori, P. Kreutz,
G. Kunde, W.D. Kunze, V. Lindenstruth, U. Lynen, A. Moroni, W.F.J. M?ller, C.A.
Oglivie, J. Pochodzalla, G. Raciti, T. Rubehn, H. Sann, A. Sch?ttauf, W. Seidel, W.
Trautmann, and A. Wörner, Nucl. Phys. A584, (4) 737-756 (1995).
18. F. Ajzenberg-Selove, Nucl. Phys. A392 (1983) 1; A413 (1984) 1; A433 (1985) 1; A449
(1985) 1; A460 (1986) 1.
19. W. Hauser and H. Feshbach, Phys. Rev. 87, 366 (1952).
20. W. Tan et al, preprint, MSUCL-1198 (2001) and http:/ / xxx.lanl.gov/ abs/ nucl-
ex/ 0104017.
21. J.P. Bondorf, R. Donangelo, I.N. Mishustin, C.J. Pethick, and H. Schulz, Nucl. Phys. A
443, 321 (1985), J.P. Bondorf, R. Donangelo, I.N. Mishustin, and H. Schulz, Nucl. Phys.
A 444, 460 (1985)
Page 13
13
22. G.J. Kunde, S. J. Gaff, C. K. Gelbke, T. Glasmacher, M. J. Huang, R. Lemmon, W. G.
Lynch, L. Manduci, L. Martin, and M. B. Tsang W. A. Friedman J. Dempsey, R. J.
Charity, and L. G. Sobotka D. K. Agnihotri, B. Djerroud, W. U. Schröder, W. Skulski, J.
Tõke, and K. Wyrozebski, Phys. Rev. Lett. 77, 2897 (1996).
23. R. Bougault et al., LPC preprint, LPCC 97-04 (1997).
24. J. Barrette et al., Nucl. Phys. A299, 147 (1978).
25. J.C. Steckmeyer et al., Nucl. Phys. A500, 2 (1989).
26. Scott Pratt, Wolfgang Bauer, Christopher Morling, and Patrick Underhill, Phys. Rev. C
63, 034608 (2001).
27. A. Majumder and S. Das Gupta, Das Gupta, Phys. Rev. C 61, 034603 (2000); Scott Pratt
and Subal Das Gupta, Phys. Rev. C 62, 044603 (2000)
28. J. Toke, W. Gawlikowicz, and W. U. Schroeder, Phys. Rev. C 63, 24606 (2001).
29. W.A. Friedman, Phys. Rev. Lett. 60, 2125 (1988); and Phys. Rev. C42, 667 (1990).
30. W.A. Friedman and W.G. Lynch, Phys. Rev. C28, 16 (1983).
31. V. Weisskopf, Phys. Rev, 52, 295 (1937)
32. Aage Bohr and Ben R. Mottelson, Nuclear Structure, (W.A. Benjamin Inc., New York,
1998), Vol II.
33. S. Das Gupta et al., to be published.
34. Bao-An Li, Phys. Rev. Lett. 85, (2000) 4221.
FIGURE CAPTIONS:
Figure: Differential multiplicities at θCM=90° for carbon isotopes as a function of the mass
number of the isotope. Top panel: primary yields are denoted by open points
connected by the dashed lines while the solid points joined by solid lines denote the
yield after sequential decays (see text for details). Bottom panel: Carbon isotope yields
for two systems with different isospin asymmetries, closed circles for δ=0.194,
N/ Z=1.48 and open squares for δ=0.107, N/ Z=1.24. The source size is kept constant at
A=N+Z=186.
Figure 2: Predicted (symbols) relative isotope ratios, R21(N,Z), of Eq. 1 for the two systems,
A1=168, Z1=75 and A2=186, Z2=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 θCM=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].
Figure 4: Predicted (symbols) relative isotope ratios, R21(N,Z), for the same systems as in
Figure 2, using the SMM code of Ref. [17].
Figure 5: Predicted (symbols) relative isotope ratios, R21(N,Z), for the same systems as in
Figure 2, using the EES code of Ref. [29].
Page 14
14
Figure 6: Temperature (left panel) and density (right panel) dependence of the scaling
parameter, α. The sources used are the same as those in Figure 2.
Figure 7: Temperature dependence of the difference in neutron (left panel) and proton (right
panel) chemical potentials obtained from the canonical SMM calculations.
Figure 8: α as a function of (Α2/Ζ2)2 for four calculations with constant source size (solid and
open circles) or constant charge (solid and open squares). The linear relationship
shown in the left panel follows Eq. (7). In the right panel, the relative free n-
density,
n
ρ ˆ , is plotted as a function of the N2/ Z2 ratio of the source. A linear
relationship is observed over the range of N2/ Z2=1.24 and 1.48, similar to the
experimental results. However, over a wider range, the dependence of
far from linear.
n
ρ ˆ on N2/ Z2 is
Page 15
Page 16
Page 17
Page 18
Page 19
Page 20
Page 21
Page 22