ekek ∆ δ a i of rigid gap parameters from moments of inertia
TRANSCRIPT
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