Statistical Nuclear Multifragmentation as Generalized Fission

1
Spinodal Vaporization an Overlooked Prompt
Decay Mode of Highly Excited Nuclei and its
Familiar Telltale
J. Tõke, University of Rochester

• Open microcanonical framework of nuclear
  thermodynamics.
• H2O and gentle thermodynamics of open
  meta-stable systems.
• Physics and math behind the limits of
  thermodynamic (meta-)stability of compound nuclei
  subtleties of Hessian matrices.
• Volume boiling with formation of bubbles gt
  prompt spinodal vaporization.
• Surface boiling (without bubbles) gt spinodal
  surface vaporization.
• In iso-asymmetric matter gt distillative
  spinodal vaporization.
• Paramount importance of thermal expansion in
  nuclear thermodynamics at elevated excitations gt
  retardation of statistical decay gt
  phase-transition like scaling of Coulomb
  fragmentation yields gt limit of the validity of
  the concept of the compound nucleus gt boiling
  phenomenon and the appearance of limiting
  temperature. All experimentally verifiable!!!
• SPINODAL VAPORIZATION is BOILING

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Open Microcanonical Framework for Understanding
Decay Modes of Highly Excited Nuclear Systems

• Weisskopff 1937 (no EOS, no thermal expansion,
  valid at lower E)
• Based on the concept of Boltzmanns entropy
• Approximates a metastable system by a system at
  equilibrium within the boundaries set by
  transition states -gt system is assumed to decay
  whenever a transition state is reached via finite
  fluctuations
• Macroscopic configurations populated according to
  their statistical weights given by their
  respective partition functions -gt need only to
  calculate Boltzmanns entropy for (transition)
  configurations of interest.
• For high excitations -gt thermal expansion
  surface diffuseness (EOS)
• Given a (Thomas-Fermi) recipe for evaluating
  configuration entropy, everything follows from
  the fundamental postulate of all microstates
  being equally probable no ad hoc assumptions of
  freezeout volumes, no casual (non-causal)
  expansions, no tricks with EOS, vanishing Coulomb
  interactions, vanishing surface free energies,
  etc., etc
• Kind of art it is not possible to calculate
  entropy for all possible configurations -gt
  requires intuition in figuring out which
  configurations or degrees of freedom might matter
  (affect decay modes).

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Decay Modes etc.

• Generally, decay modes are associated with
  degrees of freedom and the associated
  fluctuations
• Nucleonic degrees of freedom -gt particle
  evaporation
• Shape degrees of freedom -gt binary Coulomb
  fragmentation (fission) at lower excitations,
  multiple Coulomb fragmentation at higher
  excitations. Controlled by surface tension,
  vanishing with increasing excitation energy -gt
  (second-order) phase-transition-like scaling of
  Coulomb fragmentation -gt apparent vanishing of
  Coulomb interaction with increasing excitation
  energy (vide Fishers model) -gt apparent large
  sizes of fragmenting systems (vide ad hoc
  freezeout volume)
• Expansion degree of freedom (heavily
  un(der)appreciated) -gt retardation of statistical
  decays -gt (prompt) spinodal vaporization as a
  definite boiling-point excitation energy per
  nucleon is exceeded. EOS intensive, with
  interesting experimental signatures.
• Surface degrees of freedom (density profile) -gt
  facilitate fragmentation -gt spinodal surface
  vaporization.
• Isospin degree of freedom -gt distillative
  spinodal vaporization

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Case of H2O
_at_1 atm Tboil 100oC Vboil 1.043L/kg Tcrit
374oC (!) Pcrit218 atm (!!!) Vcrit13.5L/kg
(!!!) For open systems gt gentle thermodynamics
of meta-stability is possible at temperatures
below boiling point only. Life on Earth owes it
to the meta-stability of water below the boiling
point. Beyond the boiling point, the
meta-stability is lost and a gentle
thermodynamics is not possible. Boiling is a very
common phenomenon not a sensational one. It
must happen and does happen every time one tries.
Hallmark signature of boiling gt thermostatic
limit on temperature and a spontaneous (spinodal)
vaporization of parts of the liquid as more
energy is supplied.
The question is what is it that makes water to
lose meta-stability at some point and to begin
boiling? The reason is the same as for realistic
(open) nuclear systems appearance of thermal
instability, a particular case of spinodal
instability associated with wrong curvature of
the entropy function.
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Case of excited atomic nuclei
Atomic nuclei are inherently open systems,
meta-stable up to certain excitation energy and
inherently subject to boiling, which has
experimentally detectable signatures. So, why has
the boiling phenomenon escaped theoretical
attention when the experimental signatures were
there, since 1988, to see? The reason is
insistence of fashionable models on stability
within a rigid confining box, sometimes called
freezeout volumegt percolation, Ising, Pots,
lattice-gas, SMM, MMMC, while the boiling
phenomenon absolutely relies on an unconstrained
thermal expansion of Wan-der-Waals type liquid
and the expansion-induced cooling. There simply
are so many wrong ways and so few (one?) right
ways to see boiling!
Right ensemble Open Microcanonical at zero
pressure matter distribution adjusted to yield
maximum configuration entropy gt zero
pressure. Conceptually System is confined in the
full (momentum geometrical) phase space by the
hypersurface of transition states (fragmentation
saddle points and particle evaporation barriers)
same as in compound nucleus.
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Understanding Spinodal Instability

• For a system to be stable (necessary and
  sufficient) its characteristic state function
  must have proper curvature be either concave
  (entropy) or convex (free energy, Landau
  potential) in the space of extensive system
  parameters (energy, volume, isospin, number of
  particles) gt Hessian (curvature matrix) of these
  characteristic functions must be either negative
  definite (entropy) or positive definite (free
  energy, Landau potential). If not, spinodal
  instability sets in with different
  phenomenologies for different ensembles.
• Hessian matrix made of second derivatives of a
  function.
• Positive-definite ? all eigenvalues are positive.
• Negative-definite ? all eigenvalues are negative.
• All this means is that the characteristic state
  function must be concave/convex in all possible
  directions in the argument space of extensive
  parameters.
• Note the obvious ensemble non-equivalence
• Entropy for confined microcanonical system is a
  function of two extensive parameters, E and V gt
  thermo-mechanical (spinodal) instability with L-G
  coexistence as an outcome.
• Entropy for open microcanonical system is a
  function of just energy gt boiling (pure thermal)
  instability with no L-G coexistence in sight gt
  vapors are never in equilibrium with the residual
  liquid.
• No spinodal instability in grand canonical and
  iso-neutral isobaric-isothermal ensembles.

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Ensemble nonequivalence of thermodynamic
instabilities - continuation

• (iii) Helmholtz free energy AA(V,T) only V
  extensive gt mechanical (spinodal) instability in
  canonical systems ultimately L-G coexistence.
• (iv) Gibbs free energy GG(T,P) no extensive
  argument gt no spinodal instability of any kind
  in isothermal-isobaric system!
• (v) Landau potential LL(T,µ, V) V is extensive
  but N is not fixed gt no spinodal instability of
  any kind in grandcanonical systems!.
• When considering additionally N-Z asymmetry or
  isospin
• (i) thermo-chemo-mechanical spinodal instability
  in confined microcanonical (L-G).
• (ii) thermo-chemical spinodal instability in open
  microcanonical (no L-G).
• (iii) chemo-mechanical instability in canonical.
• (iv) Pure chemical instability in
  isothermal-isobaric.
• (v) Still no instability of any kind in
  grandcanonical.
• Ensemble equivalence applies to individual
  configurations gt nonequivalence is not
  sensational but trivial for systems that allow
  multiple configurations, also for large systems.
  Nonequivalence does not mean that all are equally
  bad. Good is only microcanonical!!!

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Framework of Harmonic-Interaction Fermi-Gas Model
for Self-Contained System Open Microcanonical
J.T. et al. in PRC 67, 034609 (2003).

• 1. Consider system large enough to justify the
  neglect of surface effects -gt bulk properties
  only.
• 2. Fundamental strategy -gt express the (uniform)
  configuration entropy as a function of excitation
  energy E and bulk density ? and then for any
  given E find the bulk density that maximizes
  entropy.

Start with
Obtain equilibrium density
Now, study the 1-by-1 Hessian of entropy as a
function of solely energy -gt the second
derivative of entropy with respect to energy is
the sole eigenvalue and it must be negative
heat capacity must be positive.
Thermal instability (boiling point) where
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Boiling instability in open microcanonical
system(Harmonic Interaction Fermi Gas)
Density drops with increasing energy
equilibrium thermal expansion ends at the star gt
spontaneous expansion.

Thermal expansion reduces the rate of growth of T
and eventually causes T to start dropping with E
Low latent heat.
Entropy is first a concave function of E and
then turns convex. Unlike the convex intruder
in boxed systems, here the extruder stays
convex to the end guaranteeing no L-G coexistence.
To better see the convexity, a linear function
subtracted from the entropy function above.
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Isotherms in Harmonic-Interaction Fermi Gas Model
For large systems Open microcanonical possible
only within the green segment. All rich nuclear
thermodynamics is right here. Boiling
Increasing energy at zero pressure causes thermal
expansion and, first, crossing of subsequent
isotherms with increasing indices -gt temperature
first raises. After passing the boiling point
temperature decreases.

• Under the L.G. coexistence curve only two-phase
  system possible in the long run. In the
  confined ensembles, only the long-run stable
  systems matter.
• IMPORTANTLY Space between the spinodal and
  coexistence boundaries is meta-stable - may be
  visited transiently by homogeneous matter will
  evaporate/condense to end up on a suitable step
  of the Maxwell ladder.

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The entropy surface for open hypothetical
bi-phase HIFG
S-Suniform
Two equal-A parts considered with varying split
of the total excitation energy between them
Etot
(E1-E2)/Etot

• Up to the boiling point, the system has maximum
  entropy for uniform configuration (E1E2). It
  fluctuates around uniform distribution.
• Beyond the boiling point, there is no maximum. In
  actuality, the system has no chance to ever reach
  uniformity for EtotgtEboiling
• Demonstrates the fallacy of the very concept of
  negative heat capacity. There simply is no way of
  establishing what the temperature is when
  EtotgtEboiling.
• Note that one never calculates the system S
  (impractical), only S for configurations of
  interest. But it is the system S that defines T,
  p, etc. Configuration entropy may approximate
  well the system entropy in some domains but does
  not do so in some other domains of interest.

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Phenomenology of volume boiling

• As excitation energy is raised, the matter
  expands and heats up by increasing temperature
  the expansion reduces the rate of the T increase.
  When the energy is raised above the boiling-point
  energy, thermal instability sets in, such that
  when parts of the system manage to accept (via
  infinitesimally small statistical) fluctuations
  energy from the neighboring parts they expand
  thermally and cool down, rather than heating up.
  As the acceptor parts cool down, they now
  extract (Second Law of Thermodynamics) even more
  heat from the neighboring parts (which may have
  actually got hotter as a result of donating
  energy). The expansion of the bubble continues
  at the expense of the neighboring donor parts
  until the bubble has acquired enough energy to
  expand on its own resources indefinitely and thus
  vaporize into open space. The residue will be
  left at the boiling temperature.

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Interacting Fermi-Gas Model for finite systems
withdiffuse surface domain

• Express the entropy as a function of total
  excitation energy E and parameters of the matter
  distribution half-density radius Rhalf and
  (Süssmann) surface diffuseness d.
• For any given E find the density profile that
  maximizes entropy.
• Now entropy is a function of solely E.

Assume error-function type of matter density
distribution and calculate little-a from a
(Thomas-Fermi) integral (J.T. and W.J. Swiatecki
in N.P. A372 (1981) 141).
Calculate interaction energy Eint(Rint,d) by
folding the binding energy as a function of
matter density (medium EOS was used) with the
density profile and a smearing gaussian
emulating the finite range of nuclear
interactions. Then, calculate entropy as
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Droplet of interacting Fermi liquid with A100
Half-density radius-gtthermal expansion, then
contraction (?)
Surface diffuseness-gtthermal expansion of the
surface domain
Expansion is not self-similar.
Central density first decreases (decompression)
and, then the trend reverses (?)
Pressure in the bulk decreases as a result of
reduction in surface tension. Then increases (?)
The caloric curve features a maximum now at
around 5 MeV/A, followed by the domain of
negative heat capacity.

• Thermodynamic instability of the surface profile
  boiling of the surface. All curves meaningless
  above the boiling point..

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Phenomenology of surface boiling

• As excitation energy is raised, the matter
  expands and heats up by increasing temperature
  the expansion reduces the rate of the T increase.
  The surface domain is more weakly bound and
  expands at a somewhat higher rate the expansion
  is not self-similar. When the boiling-point
  excitation energy is reached, parts of the
  surface domain begin expanding at the expense of
  their neighboring pars and cooling down while
  expanding. Then these sections of the surface
  expand even further eventually diffusing away
  into open space. What is left behind is a
  meta-stable residue at boiling-point temperature.
  In the modeling, the surface boiling occurs at
  significantly lower temperature than the volume
  boiling and consistent with experimentally
  observed limiting temperatures.

• Boiling is an obvious decay mode of highly
  excited open systems with definite and distinct
  experimental signatures - limiting temperature
  of the meta-stable residue, vapors at lower
  temperature than the residue, isotropic escape of
  the vapors, relatively low latent heat of
  boiling.
• Higher the starting energy, more matter is
  vaporized leaving less for Gemini and for
  statistical Coulomb fragmentation a.k.a.
  multifragmentation (including binary fission) gt
  rise and fall of mutifragmentation.

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Thermo-Chemical Instability in Iso-asymmetric
Matter
Again self-contained microcanonical system -gt
volume is adjusted so as to maximize entropy -gt
SS(E,I), where I(N-Z)/A
S must be concave in all directions -gt H(S) must
be negative-definite
Diagonalize Hessian and inspect eigenvalues. Both
must be negative for the system to be stable.
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Instabilities in Iso-asymmetric Bulk Matter
Isospin-Dependent Harmonic-Interaction Fermi-Gas
Model
Loss of stability against uniform expansion
Loss of stability against uniform boiling (onset
of negative heat capacity)
Growth line of the spinodal instability
eigenvector of the Hessian.
The final frontier of meta-stability the onset
of thermo-chemical instability -gt isospin
fractionation and distillation. Mathematically,
one eigenvalue of the Hessian turns zero to go
positive. May be studied experimentally!!
Contour plot is of matter equilibrium density.
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Distillative boiling of I0.5 Iso-asymmetric
Matter
From the origin of the plot to point A normal
thermalized heating of I0.5 matter.
Along the segment AB boiling off of iso-rich
matter (neutrons) as I approaches I0. From point
B on, system stays there, while subsequent
portions of azeotropic I0 matter are being
boiled off at the boiling-point temperature TB of
around 11 MeV.
IHIFG Isospin-dependent Harmonic-Interaction
Fermi-Gas Model
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CONCLUSIONS

• Spinodal vaporization or boiling is (arguably)
  the most overlooked phenomenon in nuclear
  science.
• Thermal expansion is both, the blessing and the
  curse for the concept of the compound nucleus gt
  first it extends the life of the C.N., then
  brings it to an end, and then again, helps a
  metastable residue to persist and undergo
  statistical multifragmentation, etc. Makes the
  life of a compound nucleus rich and worth living
  gt discreet charm of thermodynamics.
• Supported by common sense, but also by solid
  experimental evidence that has no alternative
  plausible explanation.
• Characteristics of spinodal vaporization are
  functions of EOS, asy-EOS, and the range of
  nucleon-nucleon interaction and theory tells what
  these functions are.
• Tempting to study EOS via identifying the
  boiling residues.
• Certainly worth trying to identify boiling
  vapors and determine their temperature
  interesting signatures.
• Measure the mass and isospin vs. temperature of
  the boiling residues.

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• Congratulations Joe with reaching another
  milestone in a remarkable career !!!

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