what can we learn from nuclear level density? magne guttormsen department of physics and safe...

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

2

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

3

Oslo Cyclotron Laboratory

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

4

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

5

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

6

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

7

Blind test

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

8

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

9

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

€

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

11

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

€

2144Sc 23

€

2145Sc 24

€

2145Sc 24

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

13

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

28

50d5/2

g9/2

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

16

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

19

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

21

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