Heat Capacities of 56Fe and 57Fe

Emel AlginEskisehir 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

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

• 45 MeV 3He beam• ~95% enriched, 3.38mg/cm2, self

supporting 57Fe target• Relevant reactions:

57Fe(3He,αγ) 56Fe57Fe(3He, 3He’γ) 57Fe

• Measured γ rays in coincidence with particles

• Measured γ rays in singles

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

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

167Er(3He,3He’)167Er

P(E i,E)(E f )T (E)

Brink-Axel hypothesis

)( Ef XL → Radiative Strength Function

)(2)( 12 EfEET XL

XL

L

Least method → ρ(E) and T(Eγ)2

Does it work?

Normalization

Transformation through equations:

Common procedure for normalization:• Low-lying discrete states• Neutron resonance spacings• Average total radiative widths of neutron

resonances

)()exp()(~

)()](exp[)(~

ETEBET

EEEEAEE xxx

Level density of 56Fe

● 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:

BSFG

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 57Fe ● 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:

BSFG

Level density parameters

Isotope a(MeV-1) E1(MeV) σ η ρ(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 densityin order to extract the thermodynamic quantities.

EntropyIn microcanonical ensemble entropy S is given by

→ multiplicity of accessible states at a given

E

One drawback:

We have level density not state density

)(ln)( EkES B

)(E

I

IEIE ),()12()(

22 2

)1(exp

2

12)(),(

IIIEIE

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.

212 I4/1E

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

oEE /)()(

0)0( TS

Entropy and entropy excess

Strong increase in entropy atEx=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 givesT=1.5 MeV for 56FeT=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

VV

V

T

ETC

TSFTE

T

FTS

TZTTF

)(

)(

)(

)(ln)(

- - - - 56Fe 57Fe

In canonical ensemble

E

TEEETZ )/exp()()( where

Chemical potential μ

n

FT

)(

n: # of thermal particles not coupled in Cooper pairs

Typical energy cost for creating a quasiparticle is -∆ which is equal to the chemical potential:

oddeven

evenodd

FF

FF

n

F

1

at T=Tc

Tc= 1 – 1.6 MeV

Probability density function

)(

)/exp()()(

TZ

TEEEpT

where Z(T) is canonical partition function:

Recall critical temperatures:T=1.5 MeV for 56FeT=1.2 MeV for 57Fe

The probability that a system at fixed temperature has an excitation energy E

E

TEEETZ )/exp()()(

Summary and conclusions

• A unique technique to extract both ρ(E) and f XL 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

U. Agvaanluvsan, Y. Alhassid, M. Guttormsen, G.E. Mitchell,

J. Rekstad, A. Schiller, S. Siem, A. Voinov

Thank you for listening…