Boiling Patterns of Iso-asymmetric Nuclear Matter

1. Boiling Patterns of Iso-asymmetric Nuclear Matter Boiling Patterns of Iso-asymmetric Nuclear Matter J. Tõke University of Rochester • H2O and gentle thermodynamics of open meta-stable systems. • Physics and math behind the limits of thermodynamic (meta-)stability of uniform matter for various ensembles – subtleties of Hessian matrix. • Limits of thermodynamic meta-stability of open (uniform) bulk nuclear matter => volume boiling with formation of bubbles. • Limits of thermodynamic meta-stability of finite nuclear systems => surface boiling without formation of bubbles. • Limits of thermodynamic meta-stability of uniform iso-asymmetric matter => iso-boiling. • Paramount importance of the boiling phenomenon for the understanding of the behavior of highly excited systems => appearance of limiting temperature => a decay mode like no other => rise and fall of m-fragmentation => possible relevance to directed and elliptic flaw.

2. 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 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 => “thermostatic” limit on temperature and a spontaneous loss of liquid as more energy is supplied. For confined systems => gentle thermodynamics of stability is always possible, resulting invariably in stable configurations at all “reasonable” excitations. Boiling shows up only when meta-stability is considered as well and only in some but not other thermodynamic ensembles. The question is: what is it that makes water to lose meta-stability at some point and 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.

3. 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 to see? The reason is insistence of fashionable models on stability within a confining box, sometimes called freezeout volume=> percolation, Ising, Pots, lattice-gas, SMM, MMMC, while the boiling phenomenon 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 right ways to “see” boiling! Right ensemble: Open Microcanonical at zero pressure – matter distribution adjusted to yield maximum 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. Obvious instability against uniform expansion at high excitation such that entropy grows indefinitely with expansion. Boiling (thermal) instability (loss of stability against phase separation ) at somewhat lower excitations - this is what matters.

4. Understanding Thermodynamical 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 space of extensive parameters. Note the obvious ensemble non-equivalence (mono-component): • Entropy for confined microcanonical is a function of two extensive parameters, E and V => thermo-mechanical (spinodal) instability with L-G coexistence in sight. • Entropy for open microcanonical is a function of just one Energy => boiling (pure thermal) instability with no L-G coexistence in sight.

5. 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!!!

6. 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: Obtain equilibrium density 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. Thermal instability (boiling point) where

7. 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.

8. 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. GEMINI • 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”.

9. The entropy surface for open 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.

10. 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 “borrow” via statistical fluctuation energy from the neighboring parts they expand and cool down, rather than heating up. As the “borrower” parts cool down, they now are legally (Second Law of Thermodynamics) entitled for even more heat at the expense of their unsuspecting “lender” neighbors, which actually got hotter as a result of “loaning” energy. The expansion of the now “recipient” bubble continues at the expense of the neighboring “donor” part until the bubble has acquired enough energy to expand on its own resources indefinitely and vaporize into open space. The residue will be left at the boiling temperature.

11. 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:

12. 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..

13. 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 parts 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.

14. 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 negative-definite: ( ) Sylvester: and/or and It turns out that the second condition breaks down before the first one -> thermo-chemical instability sets in before thermal instability and before chemical instability -> distillation. Better: Diagonalize Hessian and inspect eigenvalues. Both must be negative.

15. 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 -> 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. Ground state

16. Distillation of I=0.5 Iso-asymmetric Matter From the origin of the plot to point A: normal thermalized heating of I=0.5 matter. 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

17. CONCLUSIONS • Boiling is the most overlooked phenomenon in nuclear science. • Supported by common sense, but also by solid experimental evidence that has no alternative “plausible” explanation. • The nature of thermodynamic theory is such that rather than seeking further evidence for boiling, one must seek evidence to the contrary, the possible exceptionality of nuclei for not being prone to boiling. • Characteristics of boiling 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 – one must look for an isotropic low-temperature component. • Measure the mass and isospin vs. temperature of the boiling residues. • Not easy, but hey! People study critical exponents in open systems (which is in par with bringing water in an open kettle to 218 atm of pressure, 374oC and 13 times the normal volume)!