Triune Pairing Revelation

1. Triune Pairing Revelation Luciano G. Moretto & Augusto Macchiavelli even-odd mass differences Critical temperatures from level densities Superfluid moments of inertia

2. Anomalous Quasi Particle Spectrum Ek ∆ Ground State Masses Hence even odd mass differences δ A

3. Anomalous Moments of Inertia in rotational nuclei I of rigid Gap parameters from moments of Inertia

4. Memories…. Gilbert and Cameron lnρ≈E/T lnρ E 0 Bn Low energy level counting …..exponential? Neutron resonances ……………. 1 point Higher energy………………………..Fermi gas Global Solution : matching a Fermi Gas to an exponential dependence Away from shells TG.C. ≈ TCr pairing = 2∆/3.5

6. Level densities, actinides

7. Universal 1stOrder Low Energy Phase Transition in Atomic Nuclei Luciano G. Moretto Hallmark of 1st order phase transition in micro-canonical systems? Linear Dependence of Entropy with Energy! or ρ(E) 5 10 0 E (MeV) This is universally observed in low energy nuclear level densities T is the micro-canonical temperature characterizing the phase transition Energy goes in, Temperature stays the same

8. Can a “thermostat” have a temperature other than its own? ? T = Tc = 273K or 0 ≤ T ≤ 273K • Is T0 just a “parameter”? • According to this, a thermostat, can have any temperature lower than its own!

9. What causes the phase transition? In non magic nuclei Pairing In magic nuclei Shall gap

11. Δ= 1.76 TCr

12. BCS Phase Transition ∆0 2nd order ∆ TCr T Nearly 1st order? # quasi particle at TCr Energy at criticality ! Fixed energy cost per quasi particle up to criticality : little blocking ?

13. Pairing: Fixed Energy cost/ quasi particle up to TCR ! Is this consistent with blocking? ∆ goes down (εk-λ) goes up Proof: g x λ=0 x g for x=0 ECr/QCr= ½ ∆0 for x>0 ECr/QCr ∆0

14. 1st order phase transition implies two phases Superfluid phase gas of independent quasi particles superfluid What fixes the transition temperature? constant entropy per quasi particle Remember SackurTetrode

15. Entropy / Quasi Particle

16. Testing the picture: Even-Odd horizontal shift…. should be compared with even-odd mass differences b) Relationship between the above shift and the slope 1/T c) Vertical shift or ″entropy excess”

18. Low energy level densities for nuclei away from shells vademecum for beginners……….. Get TCr from Δ=12/A1/2 Write lnρ(E)=S(E)=E/T Shift horizontally by Δ or 2Δ for odd or odd-odd nuclei

19. Spectra with “any” gap Ek Ek δ ∆ Pairing Shell Model quasi particles vacuum N slots δ Entropy/particle

21. Let us compare…. Entropy/ quasi particle Good enough!!!! 6-7 levels/ quasi particle

23. Conclusions The “universal” linear dependence of S=lnρ with E at low energies is a clear cut evidence of a first order phase transition In non magic nuclei the transition is due to pairing. The coexisting phases are a) superfluid; b) ideal gas of quasi particles In magic nuclei the transition is due to the shell gap ……. AD MULTOS ANNOS, ALDO. WITH FRIENDSHIP

25. Low Energy Level Densities lnρ E Condensation energy Gilbert and Cameron did empirically the match between linear and square root dependence. In so doing they extracted TCR !

28. Memories…. Gilbert and Cameron lnρ≈E/T lnρ E 0 Bn Low energy level counting …..exponential? Neutron resonances ……………. 1 point Higher energy………………………..Fermi gas Global Solution : matching a Fermi Gas to an exponential dependence Away from shells TG.C. ≈ TCr pairing = 2∆/3.53