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…