heat capacities of 56 fe and 57 fe emel algin eskisehir osmangazi university workshop on level...
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![Page 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](https://reader035.vdocuments.site/reader035/viewer/2022062718/56649e795503460f94b79a4d/html5/thumbnails/1.jpg)
Heat Capacities of 56Fe and 57Fe
Emel AlginEskisehir Osmangazi University
Workshop on Level Density and Gamma Strength in
Continuum
May 21-24, 2007
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Motivation
• Apply Oslo method to lighter mass region
• SMMC calculations predict pairing phase transition
• Astrophysical interest
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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
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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
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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
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57Fe(3He,3He’)57Fe and 57Fe(3He,α)56Fe
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167Er(3He,3He’)167Er
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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
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Does it work?
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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
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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
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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).
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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
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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.
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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
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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
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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
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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.
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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
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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
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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()()(
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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
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Collaborators
U. Agvaanluvsan, Y. Alhassid, M. Guttormsen, G.E. Mitchell,
J. Rekstad, A. Schiller, S. Siem, A. Voinov
Thank you for listening…