On pressure versus density dependence in statistical multifragmentation models
ABSTRACT We show that the statistical multifragmentation model with the standard parameterization of the free volume predicts a constant pressure as density approaches the normal nuclear density. This is contrary to the Raduta&Raduta result obtained by disregarding the center of mass constraint. It is demonstrated that in finite nuclear systems the partitions with small number of fragments play an important role.
arXiv:nucl-th/0203059v1 21 Mar 2002
On pressure versus density dependence in statistical
A.S. Botvinaa,band I.N. Mishustinc,d,e
aGesellschaft f¨ ur Schwerionenforschung, D-64291 Darmstadt, Germany
bInstitute for Nuclear Research, 117312 Moscow, Russia
cInstitute for Theoretical Physics, Goethe University, D-60054 Frankfurt/Main, Germany
dNiels Bohr Institute, DK-2100 Copenhagen, Denmark
eKurchatov Institute, Russian Research Center, 123182 Moscow, Russia
We show that the statistical multifragmentation model with the standard param-
eterization of the free volume predicts a constant pressure as density approaches the
normal nuclear density. This is contrary to the Raduta&Raduta result obtained by
disregarding the center of mass constraint. It is demonstrated that in finite nuclear
systems the partitions with small number of fragments play an important role.
PACS: 25.70.Pq, 24.10.Pa, 24.60-k, 05.70.Fh
In recent years much efforts have been directed to investigating thermodynamical prop-
erties of finite nuclear systems (see e. g. refs. ). In the paper  the authors study
the phase diagram of finite systems with a statistical model designed for describing nuclear
multifragmentation. The authors’ model follows the theoretical prescriptions developed ear-
lier in refs. [3–5] and based on the assumption that fragments are formed at a low density
freeze-out stage. Consequently, the thermodynamical characteristics calculated within such
kind of models have adequate physical meaning at low densities only, e.g. ρ<
(ρ0≈0.15fm−3is the normal nuclear density), when individual fragments are singled out
from the surrounding nuclear matter [3, 4]. However, the authors of ref.  have tried to
apply their model for the higher densities ρ → ρ0by removing the physical assumption that
fragments do not overlap. They have found that the model predicts a Van der Waals kind
of phase diagram. In particular, the pressure P → ∞ at ρ → ρ0, as seen from Fig. 3 of
ref. . In this comment we want to demonstrate that this result can be misleading, even
under assumption of overlapping fragments. Within the standard statistical treatment of
finite systems, when the conservation laws are properly implemented, the pressure always
remains finite, even in the limit ρ → ρ0.
In the statistical models [2–5] the volume (density) affects the partition probabilities via
so called free volume Vfdetermining the phase space available for the translational motion of
fragments. It differs from the actual physical volume of the system V because of the finite size
of the fragments and fragment-fragment interaction. In the standard canonical description
1A comment on ”Investigating the phase diagram of finite extensive and nonextensive systems” by Al.H.
Raduta and Ad.R. Raduta, Phys. Rev. Lett. 87, 202701 (2001)
of multifragmentation the free volume enters in the partition probabilities through the factor
the thermal wavelength of nucleons at temperature T. The exponent (N − 1) comes from
the integration over momenta and coordinates of fragments under constraints that total
momentum and center of mass position are fixed for all partitions. These constraints are
crucial for finite systems in contrast to the thermodynamic limit (N → ∞ ,V → ∞ ,N/V =
const), where this factor is usually taken as (Vf/λ3
system with the number of nucleons A0, the free energy can be represented as
T)N−1, where N is the number of fragments in the partition and λT= 2π¯ h/√mNT is
T)N. In the canonical ensemble, for a finite
F = −T · ln
where CNis the volume-independent weight of N-fragment partitions which is influenced by
combinatorial factors as well as by internal excitation of fragments and their interaction. For
simplicity, below we disregard the Coulomb interaction in the system. In the microcanonical
ensemble the structure of the statistical weights with respect to the volume will be practically
the same, and thus our conclusions concerning the volume dependence of the pressure in the
limit Vf→ 0 will not change. Though the weights CNcan significantly vary from partition
to partition, they are finite for any finite system. Therefore, the behavior of the free energy
in the limit Vf→ 0 is similar for different assumptions on the fragments’ relative motion in
the freeze-out volume. To illustrate our point we use the excluded volume approximation
Vf = V − V0, where V0 = A0/ρ0 is the normal nuclear volume of all fragments. In this
approximation one gets straightforwardly:
P = −∂F
N=2(N − 1)VN−2
Now it is obvious that in the limit V → V0the pressure goes to a constant, P → T·(C2/C1) =
const. It is interesting to note that a similar behavior for this kind of model was also found
in the thermodynamic limit . In this case constant parts of pressure isotherms appear in
the coexistence region of liquid and gaseous phases. And the liquid phase is represented by
an infinite cluster which in a finite system would correspond to the compound-like nucleus.
We believe that the main reason of the P → ∞ behavior obtained in ref.  is in
disregarding the center of mass constraint2. The unconstrained integration over coordinates
of all fragments in the freeze-out volume results in a factor proportional to VN
and (2) we take VN
, we get formally P ∼ 1/Vfat V → V0, i.e. P → ∞. One
can easily see that Vfreeadopted in ref.  (their eq. (4)) gives the same limiting behavior
For illustration, in Fig. 1 we show the phase diagram obtained with the canonical calcula-
tions for partitioning a one-component system containing A0=100 nucleons into all possible
fragments characterized by the mass number A only. Contributions of all 190569292 par-
titions were calculated directly, in oder to achieve the best accuracy . We adopt the
liquid-drop description for individual fragments disregarding the Coulomb interaction and
f. If in eqs. (1)
f instead of VN−1
2Because of this and other approximations used in the model , depicting it as ”exact” in comparison
with other models does not seem justified.
of one component system with A0=100 nucleons calculated within the canonical ensemble. Solid
lines are calculations taking into account the center-of-mass constraint in the fragmentation sta-
tistical weights, dashed lines correspond to a similar calculation but without this constraint. The
temperatures (in MeV) are given by numbers at the curves.
Pressure versus free volume (in units of normal nuclear volume) in multifragmentation
internal excitation of fragments. The statistical weight of a partition with N fragments with
individual multiplicities NAis taken as
where B(A) = aV · A − aS· A2/3is the liquid-drop binding energy of fragment A with
parameters aV ≈ aS≈ 16 MeV. In this case the pressure is P = T ·(?N?−1)/Vf, where ?N?
is the mean fragment multiplicity. It is seen that at large Vfthe phase diagram is consistent
with expectations for a gas system. Here the system disintegrates into many fragments and
the fragment mass distribution falls off nearly exponentially with A. By decreasing Vf we
move into region where the partitions with low fragment multiplicity dominate. The mass
distribution turns into ”U-shape”-like one, which is associated with the phase transition.
Namely in this region the pressure has a slight ”backbending” (at low temperatures) and then
approaches a constant. Formally this behavior at small volumes sets in because probabilities
of the channels with N > 1 are suppressed by factors ∝ VN−1
partition with N=1) dominates in this case, however, it does not contribute to the pressure.
This is a trivial consequence of the conservation laws: the compound nucleus is at rest in
its center of mass frame. For N > 1 only relative momenta and positions of fragments have
We have also simulated the effect of non-conservation of the center-of-mass by introducing
an additional factor ∝ Vf in the weights of all partitions in the formula (3). This effect is
shown by dashed lines in Fig. 1, corresponding to the pressure P = T ·?N?/Vf. This means
that compound nucleus can ”move” and “exert a pressure” in the freeze-out volume, and
as a result P → ∞ at Vf → 0. Moreover, in this case the phase diagram looks like a Van
der Waals one, that in fact does not correspond to the physical content of the model. Even
in the region of large volumes relevant for the model this effect can considerably change
many thermodynamical characteristics, such as the critical temperature or the caloric curve
at constant pressure.
This example shows that phase diagrams of finite systems depend sensitively on physical
assumptions adopted in the model. In particular, in the study of a liquid-gas type phase
transition a careful treatment of the partitions with small multiplicities becomes extremely
important. However, in order to investigate realistically the phase diagram at ρ >
is necessary to introduce a really new physics in the statistical models, e.g. instead of the
picture of individual fragments surrounded by the nucleon gas to consider the bubbles of the
nuclear gas inside the nuclear matter.
. The compound nucleus (a
 H. Xi et al., Z. Phys. A359, 397 (1997); M. D’Agostino et al., Nucl. Phys. A650, 329
(1999); V.E. Viola et al., Nucl. Phys. A681, 267c (2001); R.P. Scharenberg et al., Phys.
Rev. C64, 054602 (2001).
 Al.H. Raduta and Ad.R. Raduta Phys. Rev. Lett. 87, 202701 (2001).
 J.P. Bondorf, A.S. Botvina, A.S. Iljinov, I.N. Mishustin and K. Sneppen. Phys. Rep.
257, 133 (1995).
 D.H.E. Gross, Rep. Progr. Phys. 53, 605 (1990).
 S.E. Koonin and J. Randrup, Nucl. Phys. A474, 173 (1987).
 K.A. Bugaev, M.I. Gorenstein, I.N. Mishustin and W. Greiner. Phys. Rev. C62, 044320,
 A.S. Botvina, A.D. Jackson and I.N. Mishustin, Phys. Rev. E62, R64, (2000).