Triune Pairing RevelationLuciano G. Moretto

&Augusto Macchiavelli

even-oddmass differences

Critical temperatures from level densities

Superfluid momentsof inertia

Anomalous Quasi Particle Spectrum

Ek

∆

22)( kkE

kkE

Ground State Masses

21

12

A

Hence even odd mass differences

δ

A

k

Anomalous Moments of Inertia in rotational nuclei

I )1()(2

)(2

IIIE

%60%50)( of rigid

Gap parameters from moments of Inertia

Memories….Gilbert and Cameron

0

lnρ

E

lnρ≈E/TaEln

Bn

Low energy level counting …..exponential?Neutron resonances ……………. 1 pointHigher 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

Universal 1st Order 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 !

T

EKEES )( or )exp()(

T

EE

ρ(E)

0 5 10E (MeV)

This is universally observed in low energy nuclear level densitiesT 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?

•

•

• Is T0 just a “parameter”?

•

• According to this, a thermostat, can

have any temperature lower than its

own!

Z T dE E e E T T0T

T0 TeS0

E eS eS0

E

T0

S S0 QT

S0 E

T0

T = Tc = 273Kor

0 ≤ T ≤ 273K ?

What causes the phase transition?

1. In non magic nuclei Pairing

2. In magic nuclei Shall gap

Δ= 1.76 TCr

BCS Phase Transition

TCr T

∆

∆0

2nd order

Nearly 1st order?

CrCr gTQ 2ln4

22

20 32

1CrCr gTgE

002ln16

53.3

Cr

Cr

Q

E

Fixed energy cost per quasi particle up to criticality : little blocking ?

# quasi particle at TCr

Energy at criticality

!

Pairing: Fixed Energy cost/ quasi particle up to TCR !

Is this consistent with blocking?

22)( kkE

∆ goes down (εk-λ) goes up

Proof:

λ=0

g

g

x

x

for x=0 ECr/QCr= ½ ∆0

for x>0 ECr/QCr ∆0

21CrCr QE

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 Sackur Tetrode

)/ln(#)3

4ln(

3

3

clequasipartistatesNh

p

N

VNS

Entropy / Quasi Particle

0

00

2

53.3

Q

T

QS

Cr

76.12

53.3

Q

S

Testing the picture:a) 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”

MeV

A

A2

1

12

MeVT53.3

2

76.12

53.3

Q

S

Low energy level densities for nuclei away from shells vademecum for beginners………..

1) Get TCr from Δ=12/A1/2

2) Write lnρ(E)=S(E)=E/T

3) Shift horizontally by Δ or 2Δ for odd or odd-odd nuclei

Ek

∆

22)( kkE

kkE

δ

Ek

k k

Pairing 22)( kkE Shell Model kkE

Spectra with “any” gap

δquasi particles vacuum N slots

qpE nET nN NnS ln

(E) eE lnN

eE

T Nn

Sln

Entropy/particle

T

lnN

Let us compare….

Entropy/ quasi particle

Sn

lnN

53.3

2

ln

NT

76.1ln NGood enough!!!!

6-7 levels/ quasi particle

0

00

2

53.3

Q

T

QS

Cr

76.12

53.3

Q

S

Conclusions

1) The “universal” linear dependence of S=lnρ with E at low energies is a clear cut evidence of a first order phase transition

2) In non magic nuclei the transition is due to pairing. The coexisting phases are a) superfluid; b) ideal gas of quasi particles

3) In magic nuclei the transition is due to the shell gap

……. AD MULTOS ANNOS, ALDO.WITH FRIENDSHIP

Low Energy Level Densities

E

lnρ

202

1 gC

Condensation energy

Gilbert and Cameron did empirically the match between linear and square root dependence.

In so doing they extracted TCR !

Memories….Gilbert and Cameron

0

lnρ

E

lnρ≈E/TaEln

Bn

Low energy level counting …..exponential?Neutron resonances ……………. 1 pointHigher energy………………………..Fermi gas

Global Solution : matching a Fermi Gas to an exponential dependence

Away from shells TG.C. ≈ TCr pairing = 2∆/3.53