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Triune Pairing Revelation. Luciano G. Moretto & Augusto Macchiavelli. e ven-odd mass differences . Critical temperatures from level densities. S uperfluid momen ts of inertia. Anomalous Quasi Particle Spectrum . E k. ∆. Ground State Masses. Hence even odd mass differences . δ.

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slide1

Triune Pairing Revelation

Luciano G. Moretto

&

Augusto Macchiavelli

even-odd

mass differences

Critical temperatures

from level densities

Superfluid moments

of inertia

slide2

Anomalous Quasi Particle Spectrum

Ek

Ground State Masses

Hence even odd mass differences

δ

A

slide3

Anomalous Moments of Inertia in rotational nuclei

I

of rigid

Gap parameters from moments of Inertia

slide4

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

slide7

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

can a thermostat have a temperature other than its own

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!
slide9

What causes the phase transition?

In non magic nuclei Pairing

In magic nuclei Shall gap

slide12

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 ?

slide13

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

slide14

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

slide16

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”

slide18

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

slide19

Spectra with “any” gap

Ek

Ek

δ

Pairing

Shell Model

quasi particles vacuum N slots

δ

Entropy/particle

slide21

Let us compare….

Entropy/ quasi particle

Good enough!!!!

6-7 levels/ quasi particle

slide23

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

slide25

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 !

slide28

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