Download Spinodal vaporization

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 => prompt spinodal vaporization.
• Surface boiling (without bubbles) => spinodal surface vaporization.
• In iso-asymmetric matter => distillative spinodal vaporization.
• Paramount importance of thermal expansion in nuclear thermodynamics at
elevated excitations => retardation of statistical decay => phase-transition like
scaling of Coulomb fragmentation yields => limit of the validity of the concept of
the compound nucleus => boiling phenomenon and the appearance of limiting
temperature. All experimentally verifiable!!!
SPINODAL VAPORIZATION is BOILING
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 Boltzmann’s entropy
•
Approximates a metastable system by a system at equilibrium within the
boundaries set by transition states -> 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 -> need only to
calculate Boltzmann’s entropy for (transition) configurations of interest.
•
For high excitations -> 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 -> requires intuition in figuring out which configurations or
degrees of freedom might matter (affect decay modes).
Decay Modes etc.
Generally, decay modes are associated with degrees of freedom and the
associated fluctuations:
• Nucleonic degrees of freedom -> particle evaporation
• Shape degrees of freedom -> binary Coulomb fragmentation (fission) at
lower excitations, multiple Coulomb fragmentation at higher excitations.
Controlled by surface tension, vanishing with increasing excitation energy
-> (second-order) phase-transition-like scaling of Coulomb fragmentation ->
apparent “vanishing” of Coulomb interaction with increasing excitation
energy (vide Fisher’s model) -> apparent large sizes of fragmenting
systems (vide ad hoc “freezeout” volume)
• Expansion degree of freedom (heavily un(der)appreciated) -> retardation
of statistical decays -> (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) -> facilitate fragmentation
-> spinodal surface vaporization.
• Isospin degree of freedom -> distillative spinodal vaporization
Case of H2O
@1 atm: Tboil = 100oC; Vboil = 1.043L/kg;
Tcrit = 374oC (!); Pcrit=218 atm (!!!); Vcrit=13.5L/kg (!!!);
For open systems => gentle thermodynamics of meta-stability is possible
at temperatures below boiling point only. Life on Earth owes it to the metastability 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 =>
“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.
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 volume=> 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 => 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.
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) –> 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:
(i) Entropy for confined microcanonical system is a function of two extensive
parameters, E and V => thermo-mechanical (spinodal) instability with L-G
coexistence as an outcome.
(ii) Entropy for open microcanonical system is a function of just energy =>
boiling (pure thermal) instability with no L-G coexistence in sight => vapors are
never in equilibrium with the residual liquid.
(iii) No spinodal instability in grand canonical and iso-neutral isobaricisothermal ensembles.
Ensemble nonequivalence of thermodynamic instabilities continuation
(iii) Helmholtz free energy A=A(V,T) – only V extensive => mechanical
(spinodal) instability in canonical systems – ultimately L-G coexistence.
(iv) Gibbs free energy G=G(T,P) – no extensive argument => no spinodal
instability of any kind in isothermal-isobaric system!
(v) Landau potential L=L(T,μ, V) – V is extensive but N is not fixed => 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 => 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!!!
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 ->
bulk properties only.
2. Fundamental strategy -> express the (uniform) configuration entropy as
a function of excitation energy E* and bulk density ρ and then for any
given E* find the bulk density that maximizes entropy.
Start
with:

)2 , a  ao ( ) 2 / 3
o
o
*

eq 1 
Etot
 1  9  8


o 4 
Ebind 
*
S  2 a( Etot
 Ecompr ),
Obtain equilibrium
density
Ecompr  Ebind (1  
Now, study the 1-by-1 Hessian of entropy as a function of solely energy ->
the second derivative of entropy with respect to energy is the sole
eigenvalue and it must be negative – heat capacity must be positive.
2
d S (E)
0
2
dE
Thermal instability (boiling
point) where
d 2S (E)
0
2
dE
Boiling instability in open microcanonical system
(Harmonic Interaction Fermi Gas)
Density drops with increasing energy –
equilibrium thermal expansion ends at
the star => 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.
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 ->
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 metastable - may be visited transiently by homogeneous matter – will
evaporate/condense to end up on a suitable “step” of the “Maxwell ladder”.
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 (E1=E2). 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 Etot>Eboiling
•Demonstrates the fallacy of the very concept of negative heat capacity.
There simply is no way of establishing what the temperature is when
Etot>Eboiling.
•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.
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.
Interacting Fermi-Gas Model for finite systems with
diffuse 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). :
r  Rhalf
 Norm 

1

erf
(

o
2 
2d

)

  
a( Rhalf , d )   


 o
1
3
dv
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:
S ( Rhalf , d , E*)  2 a( Rhalf , d )( E *  Eint ( Rhalf , d ))
Droplet of interacting Fermi liquid with A=100
Half-density radius->thermal
expansion, then “contraction” (?)
Surface diffuseness->thermal
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..
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) => rise and fall of mutifragmentation.
Thermo-Chemical Instability in Iso-asymmetric Matter
Again: self-contained microcanonical system -> volume is adjusted so
as to maximize entropy -> S=S(E,I), where I=(N-Z)/A
S must be concave in all directions -> H(S) must be negativedefinite:
 2S
 E 2
H ( S ( E , I ))   2
 S
 I E
2S 
E I 
2S 
2 
I 
Diagonalize Hessian and inspect eigenvalues. Both must be negative
for the system to be stable.
Instabilities in Iso-asymmetric Bulk Matter; IsospinDependent 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 -> isospin fractionation
and distillation. Mathematically,
one eigenvalue of the Hessian
turns zero to go positive.
Contour plot is of matter equilibrium
density.
May be studied experimentally!!
Distillative boiling of I=0.5 Iso-asymmetric Matter
From the origin of the plot to point
A: normal thermalized heating of
I=0.5 matter.
12
I=0
B
T (MeV)
10
8
A
I=0.5
6
4
IHIFG: cv=-16MeV
cI=23MeV
2
0
0
4
8
12
E*/A (MeV/A)
16
Along the segment AB: boiling off of
iso-rich matter (neutrons) as I
approaches I=0. From point B on,
system stays there, while
subsequent portions of azeotropic
I=0 matter are being boiled off at
the boiling-point temperature TB of
around 11 MeV.
IHIFG – Isospin-dependent Harmonic-Interaction Fermi-Gas Model
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 => 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 => 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.
Congratulations Joe with reaching another
milestone in a remarkable career !!!