What can we learn from nuclear level density?

Magne GuttormsenDepartment of Physics and SAFE

University of Oslo

Workshop on Statistical Nuclear Physics and Applications in Astrophysics and Technology, July 8-11 2008, Athens, Ohio

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Fermi gas and beyond

●Protons and neutrons

●The Pauli principle

●Interacting particles

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ρ(E) =π

12

exp(2 aE)

a1/ 4 E 5 / 4

Fermi gas (Hans Bethe 1936):

Parameters: a

Workshop on Statistical Nuclear Physics and Applications in Astrophysics and Technology, July 8-11 2008, Athens, Ohio

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Oslo Cyclotron Laboratory

Workshop on Statistical Nuclear Physics and Applications in Astrophysics and Technology, July 8-11 2008, Athens, Ohio

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167Er(3He,3He’)167Er

Spin

Ex

T = 1 MeV

Particle-gamma coincidences

Workshop on Statistical Nuclear Physics and Applications in Astrophysics and Technology, July 8-11 2008, Athens, Ohio

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The Oslo method

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P(E i,Eγ )∝T (Eγ ) ⋅ρ (E f )

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λ =2π

hM if

2⋅ρ(E f )

Workshop on Statistical Nuclear Physics and Applications in Astrophysics and Technology, July 8-11 2008, Athens, Ohio

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Simulations and extractionGenerated in Prague:Event-by-event datawith DICEBOX

Sorted in Oslo:(Ex, E) matrix

First generationprocedure

Factorizationinto ρ and f

2 4 6 8

2

5

8

Ex

E

2 4 6 8

2

5

8Ex

E

Workshop on Statistical Nuclear Physics and Applications in Astrophysics and Technology, July 8-11 2008, Athens, Ohio

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Blind test

Workshop on Statistical Nuclear Physics and Applications in Astrophysics and Technology, July 8-11 2008, Athens, Ohio

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Level density and entropy

Yb172

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S(E) = kB lnΩ(E)

Ω(E) = ρ (E) /ρ 0

ρ 0 = 2.2 MeV -1

Workshop on Statistical Nuclear Physics and Applications in Astrophysics and Technology, July 8-11 2008, Athens, Ohio

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Modeling level density

Cooper pair Broken pair

1 state 25 states

●Odd-even mass differences

●Ground-state spin J = 0

●Abrupt increase at Ex = 2∆

Workshop on Statistical Nuclear Physics and Applications in Astrophysics and Technology, July 8-11 2008, Athens, Ohio

10A simple model for level density

- Combining all possible proton and neutron configurations

- Nilsson single-particle energy scheme

- BCS quasi-particles

j

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Single quasi - particle energy :

eqp = esp − λ( )2

+ Δ2

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Total energy :

Eqp Ωπ ,Ων( ) = eqp ′ Ω π( )′ Ω π , ′ Ω ν{ }

∑ + eqp ′ Ω ν( ) + V ′ Ω π , ′ Ω ν( )

Workshop on Statistical Nuclear Physics and Applications in Astrophysics and Technology, July 8-11 2008, Athens, Ohio

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Nilsson level scheme

Model parameters: = 0.066 = 0.32 = 0.23€

2144Sc 23 and 21

45Sc 24

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For 2144Sc 23 :

1p 1n

1p 3n

1p 5n

1p 7n

3p 1n

3p 3n

3p 5n

5p 1n

5p 3n

7p 1n

20

Workshop on Statistical Nuclear Physics and Applications in Astrophysics and Technology, July 8-11 2008, Athens, Ohio

12Scandium, shape coexistence

Level densities

Number of broken pairs

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2144Sc 23

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2144Sc 23

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2145Sc 24

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2145Sc 24

Workshop on Statistical Nuclear Physics and Applications in Astrophysics and Technology, July 8-11 2008, Athens, Ohio

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Parity asymmetry

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2144Sc 23

€

2145Sc 24

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α =ρ+ −ρ−

ρ+ + ρ−

U. Agvaanluvsan, G.E. Mitchell, J.F. Shriner Jr., Phys. Rev. C 67, 064608 (2003)

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α ≈0.02 for

ρ(J =1/2,J = 3/2)

Workshop on Statistical Nuclear Physics and Applications in Astrophysics and Technology, July 8-11 2008, Athens, Ohio

14Iron, blocking of neutron pairs

Workshop on Statistical Nuclear Physics and Applications in Astrophysics and Technology, July 8-11 2008, Athens, Ohio

15Molybdenum, approaching N=50

93Mo = 0.0894Mo = 0.2595Mo = 0.3496Mo = 0.4697Mo = 0.6598Mo = 0.87

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50d5/2

g9/2

Workshop on Statistical Nuclear Physics and Applications in Astrophysics and Technology, July 8-11 2008, Athens, Ohio

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Tin, proton shell gap Z=50

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T(E) =∂S(E)

∂E

⎛

⎝ ⎜

⎞

⎠ ⎟−1

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T(E) =∂S(E)

∂E

⎛

⎝ ⎜

⎞

⎠ ⎟−1

Workshop on Statistical Nuclear Physics and Applications in Astrophysics and Technology, July 8-11 2008, Athens, Ohio

17Tin, dominance of neutron pairs

Workshop on Statistical Nuclear Physics and Applications in Astrophysics and Technology, July 8-11 2008, Athens, Ohio

18Ytterbium, dominance of proton pairs

Proton and neutron orbitals+/- 8 MeV above Fermi surface: 44(p) + 58(n) = 102 orbitals

Workshop on Statistical Nuclear Physics and Applications in Astrophysics and Technology, July 8-11 2008, Athens, Ohio

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Proton or neutron pairs

pn

pn

p

n

n

p

86% n

14% p

57% n

43% p

Nuclear level density reveals

● Entropy and thermodynamics (Tc, CV)

● Breaking Cooper pairs

● Parity asymmetry

● Shell gaps

● Shape coexistence

New position in Oslo, Professor of Physics!

Workshop on Statistical Nuclear Physics and Applications in Astrophysics and Technology, July 8-11 2008, Athens, Ohio

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Extensivity in nuclei

S = S*

S = S* + S1

S1

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Ideal gas :

S(N) = N lnc + 52 N − N lnN

S(N +1) ≠ S(N) + S1