Mononuclear Caloric Curve

1. Mononuclear Caloric Curve LGS - WU ACS (3/13/05) IU – V.V. (10/29/05)

2. Mononuclear Caloric Curve: DATA

3. Mononuclear Caloric Curve: DATA + Fermi Gas 1) Relation of Thermal energy (U) to Temperature (T) and (as for e- in metals) => so Phase transition?

4. Mononuclear Caloric Curve: Idea This model • is designed to create a better description of the (mononuclear) density of states appropriate for statistical (kinetic) decay calculations. • takes into account, in a plausible fashion, both expansion and many-body correlations. • provides a recalibration of the mononuclear (thermodynamics) expectations. Principle results: • One should expect a plateau in the mononuclear CC. 2) The idea of the CN is “expanded” (pun intended) to higher E* 3) But in terms of the real world we must confront VV and then we find …..

5. Entropy… of a Fermi drop 1) Relation of Thermal energy U to Temperature or and 2) With no PT’s Entropy is a function of the “thermal” energy. Must subtract E required for expansion (as one does for collective rotation.) 3) As always, the statistical T is With models for “a” and exp => calc S(U) => T(U) 4) The local-density approximation lead to a simple expression for “a”. a  to the density of SP states (g)

6. Mononuclear Caloric curve: Overview Expansion + Effective mass logic ~ S. Shlomo, J. Natowitz, et al. Numerical scheme for little “a” Calculation with m* logic ….“LDA” Barranco and Treiner Prakash, Wambach and Ma Nuclear expansion by maximizing entropy (Toke et al.) Mononuclear (pseudo) plateau Sobotka et al.

7. Mononuclear “Caloric Curve”: Intellectual input • The nuclear expansion is determined by maximization of the entropy. 1st pass: Self-similar expansion ( c ) - 2nd pass: add surface diffuseness (b,c) • Entropy from simple FG expression. (No clustering) 3.The level density constant is determined by a local-density approximation. 4.Many-body correlations enter through effective mass factors, accounts for .. a) the FR (standard effective mass for NM) [mk < 1 , mk => 1 as  => 0 ] b) the surface [mw > 1 (sur), mw => 1 as ’ => 0 ] 5. The thermal energy is the difference between the total excitation energy and the energy required for expansion. 6. The expansion energy is determined either by: a) the simple form suggested by Friedman (EES) b) that suggested by Myers and Swiatecki (TF) c) DFT (light – a la’ van der Waals) 7. Statistical Temperature from differentiation of entropy w.r.t. E*: 1/T = (dS/dE*)v

8. Family of drop shape profiles Native radial profile Self-similar expansion 1/c = linear expansion b = surface diffusness r(r) [#/fm3] r r (fm) Gaussian Derivative

9. Need for m* Designer fix to get single-particle E’s = > • ~ Corrects for momentum dependence of interaction. • Can be viewed as a non-locality in SPACE (a r-p “thing”) • It is caused by the “finite range” of the interaction. • If you want a simple local “potential” picture you must correct for this simplification. • The math

10. Many-body correlations via Effective mass“k-mass” – (mk) and “omega-mass” – [mω]momentum dependent interaction“single-particle” to surface coupling Accounts for: the true E dependence of the potential or (through a FT) the t-nonlocality of the true potential. Needed for: a) “multiparticle” Density of States or its log the Entropy => mω > 1 b) reduced spectroscopic strength near the Fermi surface. Can be viewed as: a coupling of collective (surface) excitations with “single-particle” excitations + The result of SRC Accounts for: the true p dependence of the potential or (through FT) the r-nonlocality of the true interaction Needed for: a) “single-particle” energies - SE solutions of LOCAL potential problem. mk < 1 In principle depends on density [r] and asymmetry [d =(n-z)/A]. HOWEVER As an “mk”<1is needed to get SP E’s… This factor REDUCES the density of both SP and MP states.

11. Effective mass - summary m* Expansion: Returns mk to 1 (as ρ => 0) Returns mw to 1 (as ρ’ => 0) r Heating: Returns mw to 1. (as collective surface modes die.) r

12. Expansion Energy BENM(c,d) Eexpansion NUC BE Tot asymmetric EES Coul symmetric  Expansion  Expansion r Compression => Myers and Swiatecki EoSsimilar to Friedman (EES) form Coulomb modified=> expansion E = eb(1 - r)2 for small density reduction

13. Mononuclear “Caloric Curve”Results for:expansion ( c ) wo m* b= 8,and with m* for b= 8 and b= 6

14. The “go of it” - Level Density parameter (a) a (#/MeV) Remember a  SP “level density” at Fermi surface = g  (MeV/amu) • mk provides +ve feedback on expansion! • b) mw drops and initially pulls down “a” then • c) expansion and mk take over and increase “a”

15. First-pass conclusions • This is a repackaging of what Natowitz and Shlomo have been saying. • Its advantage over the previous work is that the approach is straightforward. The mathematics is well posed and the solution predicts: an ~ plateau in the caloric curve for a mononucleus resulting from a) reduction in density and b) the destruction of collective effects. • At some point in the plateau, the multi-fragmented (MF) density of states exceeds the mononuclear value. Where this happens depends on the volume used for MF. HOWEVER as VMF > VMN, the MN can (still) be considered meta stable. HOWEVER there MUST be a transient delay (PS exploration time.) • The logic is simple AND restricted to the time window for which an equilibrated mononucleus is reasonable. However as the Temperature saturates so do the decay rates.

16. GREATER Context DFT landmarks: 1873 – J.D. van der Waals, Thesis DFT - light 1964 – P. Hohenberg and W. Kohn [PR 136B, 864 (1964)] Real DFT • – D. Mermin’s [PR 137, A1441 (1965)] Finite T DFT Generally appreciated that the gradient expansion does NOT converge for induced dipole (u~r-6). See Rowlinson and Widom, Molecular theory of Capillarity, but (Oρ)2 works damn well! On the other hand, expansion justified for pair potential = Yukawa. Nuclear (EDF) 1973 – R. Lombard [Ann. Phys 77,380 (1973)]. (Oρ)2 ,(Oδ)2 1985 – J. Bartel, M. Brack and M. Durand NP A445, 263 (1985) (Oρ)2,4 and Wigner-Kirkwood expansions + m* (formally!) 1990 – P. Gleissel, M. Brack, J. Meyer and P. Quentin, Ann Phys. 197, 205 (1990)] Approach: Paramatierized shape + variational => GR and validity of Lepdoderous (DM) expan.  FHelm [ρ], KE[ρ], Entropy [ρ], Jso [ρ]  However m*(r,E*) is not accounted for. (In principle YES in practice NO. 2005 – include mkmw decomposition (in a simplified model)

17. Binding Energy On beta stability for ( b = c = 1) “EDFT” - light 1. Energy of “matter drop” BE(MeV/amu) 2. Coulomb Energy (exchange also) 3. Gradient2 Correction (gc parameter) 4. Sum with 3D Coulomb integral A (amu) Agrees with LDM to ~ 4 ppt from A = 10-250

18. Moving on: Eexp(b,c)b: surface => y c: bulk expansion => x

19. Entropy(per nucleon) for  = 2,3,4,5 MeVs(c,b) for m* = 1

20. 2D analysis flatter than 1D analysis, which is flatter than FG

21. But can this model reproduce Vic’s fragment energies?CalculateK = B(b,c)/B(1,1)Compare Keq to Vic’sK = experimental Barrier reductionsANS = NO !

22. Should the coefficient of the gradient term be temperature dependent ? bg2(T,ρ) = ? Formal view: Second view: WEAK! The cost of a gradient is: In terms of the direct correlation function: Surface tension iso T compressibility Miscibility gap.

23. Second-pass Conclusions • The S analysis implies a “normal-mode” with about 2/3th surface (b) and 1/3rd bulk (c) character. • In this model, “Multifragmentation” is then a fluctuation in this mode, creating a “surface” region unstable to cluster formation. SUFACE not VOLUME • A plateau in a Caloric Curve ≠ Phase Transition Precursor to PT – Yes but ….. In fact, why bother talking about PT at all? Does PT logic really help you understand “the go of it” in the meso world. I think not. 4. BUT the fragment energies indicate the EQ model is not sufficient as Kexp < K(eq). “Dynamical” expansion is indicated. EES is such a model (although it is completely misunderstood and advertised.)

24. Mononuclear CC