Challenges on phase transitions and gamma-strength functions

Magne GuttormsenDepartment of Physics, University of Oslo

Workshop onLevel Density andGamma Strength in Continuum, Oslo, May 21 - 24, 2007

• Oslo method• Thermodynamics• Electromagnetic transitions• Challenges

Spin-energy diagram

Spin2-6 ħ

Ex

T = 1 MeV

Yrast line(no levels)

Workshop onLevel Density andGamma Strength in Continuum, Oslo, May 21 - 24, 2007

The first generation method

Workshop onLevel Density andGamma Strength in Continuum, Oslo, May 21 - 24, 2007

F0

F1

F2

F3

€

1

Nsingles

or M

Ncoinc

-

=

F0

G

W

M

M-1

1€

G = w iFii

∑

Gamma-ray multiplicity

Workshop onLevel Density andGamma Strength in Continuum, Oslo, May 21 - 24, 2007

€

M statγ = M tot

γ −M yrastγ

= (E − Eentry ) Eγ >E 0

M totγ = E Eγ

Spin

Spin

E

E

The Brink Axel hypothesis

€

P(E i,Eγ )∝ ρ(E f ) ⋅T (Eγ )

Workshop onLevel Density andGamma Strength in Continuum, Oslo, May 21 - 24, 2007

Does it work?

Workshop onLevel Density andGamma Strength in Continuum, Oslo, May 21 - 24, 2007

Nuclear entropy

€

S(E) = kB lnΩ(E)

Ω(E) = ρ (E) /ρ 0

ρ 0 = 2.2 MeV -1

Workshop onLevel Density andGamma Strength in Continuum, Oslo, May 21 - 24, 2007

canonical - canonical ensemble

Workshop onLevel Density andGamma Strength in Continuum, Oslo, May 21 - 24, 2007

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S(E) = k lnΩ

1

T=∂S

∂E

⎛

⎝ ⎜

⎞

⎠ ⎟N

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Z(T) = Ω(E)e−E / kT

E

∑

E = kT 2 ∂

∂TlnZ

F = −kT lnZ

S = −∂F

∂T

⎛

⎝ ⎜

⎞

⎠ ⎟V

Rare-earth region, case erbium

Workshop onLevel Density andGamma Strength in Continuum, Oslo, May 21 - 24, 2007

Proton number Z = 50Neutron Cooper pairs

Workshop onLevel Density andGamma Strength in Continuum, Oslo, May 21 - 24, 2007

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Structural transition when :

FC (E1) = FC (E2) for a given TC

Linearized free energy :

FC (E) = E −TC ⋅S(E)

J.Lee and J.M.Kosterlitz, Phys. Rev. Lett. 65, 137 (1990)

Workshop onLevel Density andGamma Strength in Continuum, Oslo, May 21 - 24, 2007

€

T(E) =∂S(E)

∂E

⎛

⎝ ⎜

⎞

⎠ ⎟−1

€

T(E) =∂S(E)

∂E

⎛

⎝ ⎜

⎞

⎠ ⎟−1

Workshop onLevel Density andGamma Strength in Continuum, Oslo, May 21 - 24, 2007

Workshop onLevel Density andGamma Strength in Continuum, Oslo, May 21 - 24, 2007

Challenges, new thermodynamic

€

T(E) =∂S(E)

∂E

⎛

⎝ ⎜

⎞

⎠ ⎟−1

Isolated E

€

E(T) = kT 2 ∂

∂TlnZ(T)

HeatbathHeatbath T

S = S0 + S* + S1

Negative Cv(E) and T(E) Smoothed (up to E = 100 MeV)

T

€

ΩR +S = ΩRΩS = ΩR (E0 − E)

= exp S0 +∂S

∂E(−E) +

1

2

∂ 2S

∂E 2 E2 + ...

⎡

⎣ ⎢

⎤

⎦ ⎥

Boltzmann factor

€

e−E /T

€

f (Eγ ) =1

2π

T (Eγ )

Eγ3

€

f (Eγ ) =1

2π

T (Eγ )

Eγ3

Loved child, many names

Workshop onLevel Density andGamma Strength in Continuum, Oslo, May 21 - 24, 2007

• Radiative strength function (RSF)

• Photon strength function (PSF)

• Gamma-ray strength function

Simulation tests of the Oslo method

Workshop onLevel Density andGamma Strength in Continuum, Oslo, May 21 - 24, 2007

Event-by-event datagenerated in Praguewith DICEBOX

Sorted in Oslo into(Ex, E) matrix

First generationprocedure

Axel-Brinkfactorizationinto and f

2 4 6 8

2

5

8

Ex

E

2 4 6 8

2

5

8Ex

E

Results from the blind test

Workshop onLevel Density andGamma Strength in Continuum, Oslo, May 21 - 24, 2007

€

f (Eγ ) =1

2π

T (Eγ )

Eγ3

€

f (Eγ ) =1

2π

T (Eγ )

Eγ3

Radiative strength functions

Workshop onLevel Density andGamma Strength in Continuum, Oslo, May 21 - 24, 2007

€

f py (Eγ ) =1

3π 2h2c 2

σ pyEγΓpy2

(Eγ2 − E py

2 )2 + Eγ2Γpy

2

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f py (Eγ ) =1

3π 2h2c 2

σ pyEγΓpy2

(Eγ2 − E py

2 )2 + Eγ2Γpy

2

Pygmy resonance

Workshop onLevel Density andGamma Strength in Continuum, Oslo, May 21 - 24, 2007

€

π ν

€

B(M1↑) =9hc

32π 2

σΓ

E

⎛

⎝ ⎜

⎞

⎠ ⎟py

= 6.5(15)μN2

€

Pygmy resonance in 172Yb at Eγ = 3.3 MeV

Scissors mode

Workshop onLevel Density andGamma Strength in Continuum, Oslo, May 21 - 24, 2007

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fE1(Eγ ) =1

3π 2h2c 2

0.7σ E1ΓE12 (Eγ

2 + 4π 2T 2)

EE1(Eγ2 − EE1

2 )2

€

fE1(Eγ ) =1

3π 2h2c 2

0.7σ E1ΓE12 (Eγ

2 + 4π 2T 2)

EE1(Eγ2 − EE1

2 )2

Kadmenskii, Markushev and Furman (KMF) model

Giant dipole resonance

Workshop onLevel Density andGamma Strength in Continuum, Oslo, May 21 - 24, 2007

Physics News Update

Unexpected RSF upbend

Workshop onLevel Density andGamma Strength in Continuum, Oslo, May 21 - 24, 2007

Voinov et al., Physical Review Letters,1 October 2004.

Also observed in: Mo, V, Sc, Ni, Ti (3He, 3He) and (3He, 4He) reactions

Upbend in iron

Workshop onLevel Density andGamma Strength in Continuum, Oslo, May 21 - 24, 2007

Temperature dependence?

Workshop onLevel Density andGamma Strength in Continuum, Oslo, May 21 - 24, 2007

€

96Mo( 3He, 3He′ γ )96Mo

€

×31

€

×32

€

×33

Subatomic and Astrophysics Division Annual Meeting 2007

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Bn = Eγ1 + Eγ 2Verifications

Budapest Research Reactor(n, ) reaction

INPP, Ohio University(d,n) reaction

Challenges

-canonical or canonical ensemble - or another theory?

Critical T, how to interpret and measure?

Single quasi-particle entropy

Workshop onLevel Density andGamma Strength in Continuum, Oslo, May 21 - 24, 2007

Understand the difference in quantities observed in different experiments Pygmy resonance GEDR-tail match Low energy -strength

Spin distribution Parity asymmetry Thermalization