Heat capacities of 56 fe and 57 fe
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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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Heat capacities of 56 fe and 57 fe

Heat Capacities of 56Fe and 57Fe

Emel Algin

Eskisehir Osmangazi University

Workshop on Level Density and Gamma Strength in Continuum

May 21-24, 2007


Motivation

Motivation

  • Apply Oslo method to lighter mass region

  • SMMC calculations predict pairing phase transition

  • Astrophysical interest


Cactus silicon telescopes

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


Experimental details

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


Data analysis

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


5 7 fe 3 he 3 he 5 7 fe and 5 7 fe 3 he 56 fe

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


167 er 3 he 3 he 167 er

167Er(3He,3He’)167Er


Brink axel hypothesis

Brink-Axel hypothesis

→ Radiative Strength Function

Least method → ρ(E) and T(Eγ)


Does it work

Does it work?


Normalization

Normalization

Transformation through equations:

Common procedure for normalization:

  • Low-lying discrete states

  • Neutron resonance spacings

  • Average total radiative widths of neutron resonances


Level density of 56 fe

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:


Level density of 56 fe with smmc

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).


Level density of 57 fe

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:


Level density parameters

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.


Entropy

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


Entropy cont

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.


Entropy cont1

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


Entropy and entropy excess

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.


Helmholtz free energy entropy average energy heat capacity

Helmholtz free energy, entropy, average energy, heat capacity

In canonical ensemble

where

- - - - 56Fe

57Fe


Chemical potential

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


Probability density function

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


Summary and conclusions

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


Collaborators

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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