1. Heat Capacities of 56 Fe and 57 Fe Emel Algin Eskisehir Osmangazi University Workshop on Level Density and Gamma Strength in Continuum May 21-24, 2007

2. Motivation Apply Oslo method to lighter mass region SMMC calculations predict pairing phase transition Astrophysical interest

3. Cactus Silicon telescopes 28 NaI(Tl) detectors 2 Ge(HP) detectors 8 Si(Li) ∆E-E particle detectors (thicknesses: 140μm and 3000 μm) at 45° with respect to the beam direction

4. Experimental Details 45 MeV 3 He beam ~95% enriched, 3.38mg/cm 2, self supporting 57 Fe target Relevant reactions: 57 Fe( 3 He,αγ) 56 Fe 57 Fe( 3 He, 3 He’γ) 57 Fe Measured γ rays in coincidence with particles Measured γ rays in singles

5. Data analysis Particle energy → initial excitation energy (from known Q value and reaction kinematics) Particle-γ coincidences → E x vs. E γ matrix Unfolding γ spectra with NaI detector response function Obtained primary γ spectra by squential subtraction method → P(E x, E γ ) matrix

6. 57 Fe( 3 He, 3 He’) 57 Fe and 57 Fe( 3 He,α) 56 Fe

7. 167 Er( 3 He, 3 He’) 167 Er

8. Brink-Axel hypothesis → Radiative Strength Function Least method → ρ(E) and T(Eγ)

9. Does it work?

10. Normalization Transformation through equations: Common procedure for normalization: Low-lying discrete states Neutron resonance spacings Average total radiative widths of neutron resonances

11. Level density of 56 Fe ● LD obtained from Oslo method O LD obtained from 55 Mn(d,n) 56 Fe reaction discrete levels BSFG LD with von Egidy and Bucurescu parameterization Normalization:

12. Level density of 56 Fe with SMMC ● LD obtained from SMMC ◊ LD obtained from Oslo method * Discrete level counting --- LD of Lu et al. (Nucl. Phys. 190, 229 (1972).

13. Level density of 57 Fe ● LD obtained from Oslo method discrete levels BSFG LD with von Egidy and Bucurescu parameterization data point obtained from 58 Fe( 3 He,α) 57 Fe reaction (A. Voinov, private communication) Normalization:

14. Level density parameters Isotope a(MeV -1 ) E1(MeV) σ η ρ(MeV -1 ) at B n 56 Fe 6.196 0.942 4.049 0.64 2700±600 57 Fe 6.581 -0.523 3.834 0.38 610±130 BSFG is used for the extrapolation of the level density in order to extract the thermodynamic quantities.

15. Entropy In microcanonical ensemble entropy S is given by → multiplicity of accessible states at a given E One drawback: We have level density not state density

16. Entropy, cont. Spin distribution usually assumed to be Gaussian with a mean of σ: spin cut-off parameter In the case of an energy independent spin distribution, two entropies are equal besides an additive constant.

17. Entropy, cont. Here we define “pseudo” entropy based on level density: Third law of thermodynamics: Entropy of even-even nuclei at ground state energies becomes zero: ρ o =1 MeV -1

18. Entropy and entropy excess Strong increase in entropy at E x =2.8 MeV for 56 Fe E x =1.8 MeV for 57 Fe Breaking of first Cooper pair Linear entropies at high E x Slope: dS/dE=1/T Constant T least-square fit gives T=1.5 MeV for 56 Fe T=1.2 MeV for 57 Fe Critical T for pair breaking Entropy excess ∆S=S( 57 Fe)-S( 56 Fe) Relatively constant ∆S above E x ~ 4 MeV: ∆S=0.82 k B.

19. Helmholtz free energy, entropy, average energy, heat capacity - - - - 56 Fe 57 Fe In canonical ensemble where

20. Chemical potential μ n: # of thermal particles not coupled in Cooper pairs Typical energy cost for creating a quasiparticle is -∆ which is equal to the chemical potential: at T=T c T c = 1 – 1.6 MeV

21. Probability density function where Z(T) is canonical partition function: Recall critical temperatures: T=1.5 MeV for 56Fe T=1.2 MeV for 57Fe The probability that a system at fixed temperature has an excitation energy E

22. Summary and conclusions A unique technique to extract both ρ(E) and f XL experimentally Extend ρ(E) data above E x =3 MeV (where tabulated levels are incomplete) Step structures in ρ(E) indicate breaking of nucleon Cooper pairs Experimental ρ(E) → thermodynamical properties Entropy carried by valence neutron particle in 57 Fe is ∆S=0.82k B. Several termodynamical quantities can be studied in canonical ensemble S shape of the heat capacities is a fingerprint for pairing transition More to come from comparison of experimental and SMMC heat capacities

23. Collaborators U. Agvaanluvsan, Y. Alhassid, M. Guttormsen, G.E. Mitchell, J. Rekstad, A. Schiller, S. Siem, A. Voinov Thank you for listening…