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Heat Capacities of 56 Fe and 57 Fe

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

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Heat Capacities of 56 Fe and 57 Fe

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  1. Heat Capacities of 56Fe and 57Fe 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 3He beam • ~95% enriched, 3.38mg/cm2, self supporting 57Fe target • Relevant reactions: 57Fe(3He,αγ) 56Fe 57Fe(3He, 3He’γ) 57Fe • 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 → Ex vs. Eγ matrix • Unfolding γ spectra with NaI detector response function • Obtained primary γ spectra by squential subtraction method → P(Ex, Eγ) matrix

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

  7. 167Er(3He,3He’)167Er

  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 of56Fe ●LD obtained from Oslo method O LD obtained from 55Mn(d,n)56Fe reaction discrete levels BSFG LD with von Egidy and Bucurescu parameterization Normalization:

  12. Level density of 56Fe 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 of57Fe ●LD obtained from Oslo method discrete levels BSFG LD with von Egidy and Bucurescu parameterization data point obtained from 58Fe(3He,α)57Fe reaction (A. Voinov, private communication) Normalization:

  14. Level density parameters Isotope a(MeV-1) E1(MeV) σηρ(MeV-1) at Bn 56Fe 6.196 0.942 4.049 0.64 2700±600 57Fe 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: ρo=1 MeV-1

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

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

  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=Tc Tc= 1 – 1.6 MeV

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

  22. Summary and conclusions • A unique technique to extract both ρ(E) and fXL experimentally • Extend ρ(E) data above Ex=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 57Fe is ∆S=0.82kB. • 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…

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