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EkEk ∆ δ A

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I of rigid Gap parameters from moments of Inertia

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Gilbert and Cameron 0 lnρ E lnρ≈E/T BnBn 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 T G.C. ≈ T Cr pairing = 2∆/3.5

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Level densities, actinides 6

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Luciano G. Moretto Hallmark of 1 st order phase transition in micro-canonical systems? Linear Dependence of Entropy with Energy ! or ρ(E) 0 5 10 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

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

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1.In non magic nuclei Pairing 2.In magic nuclei Shall gap

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Δ= 1.76 T Cr

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T Cr T ∆ ∆0∆0 2 nd order Nearly 1 st order? Fixed energy cost per quasi particle up to criticality : little blocking ? # quasi particle at T Cr Energy at criticality !

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Is this consistent with blocking? ∆ goes down (ε k -λ) goes up Proof: λ=0 g g x x for x=0 E Cr /Q Cr = ½ ∆ 0 for x>0 E Cr /Q Cr ∆ 0

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Superfluid phase gas of independent quasi particles superfluid What fixes the transition temperature? constant entropy per quasi particle Remember Sackur Tetrode

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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”

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1)Get T Cr from Δ=12/A 1/2 1)Write lnρ(E)=S(E)=E/T 2)Shift horizontally by Δ or 2Δ for odd or odd-odd nuclei

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EkEk ∆ δ EkEk Pairing Shell Model δ quasi particles vacuum N slots Entropy/particle

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Entropy/ quasi particle Good enough!!!! 6-7 levels/ quasi particle

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1)The “universal” linear dependence of S=lnρ with E at low energies is a clear cut evidence of a first order phase transition 1) In non magic nuclei the transition is due to pairing. The coexisting phases are a) superfluid; b) ideal gas of quasi particles 2)In magic nuclei the transition is due to the shell gap ……. AD MULTOS ANNOS, ALDO. WITH FRIENDSHIP

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E lnρ Condensation energy Gilbert and Cameron did empirically the match between linear and square root dependence. In so doing they extracted T CR !

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Gilbert and Cameron 0 lnρ E lnρ≈E/T BnBn 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 T G.C. ≈ T Cr pairing = 2∆/3.53

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Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 1 Shell model Notes: 1. The shell model is most useful when applied to closed-shell.

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