Bounds to Binding Energies from Concavity N.P. Toberg Dr. B.R. Barrett Department of Physics,...
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Bounds to Binding Energies from Concavity
N.P. TobergDr. B.R. BarrettDepartment of Physics,
University of Arizona, Tucson Az 85721 USA
Dr. B.G. GiraudService de Physique Théorique,
DSM, CE Saclay, F-911191 Gif/Yvette, France
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1) INTRODUCTION
• Search for 1st order approximation to isotope binding energy (Upper and Lower Bounds)
• Exploit properties of the quadratic terms in nuclear binding energy formula
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3) Introduction
Complete Binding Energy
Krane, Kenneth. Introductory Nuclear Physics. John Wiley & Sons, Inc., 1988.
A
ZAaAZZaAaAaB symcsv
23/13/2 )2(
)1(
NZA
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4) Introduction
i. Dominant terms define a paraboloid energy surface (concave)
ii. Deviations from concavity can be suppressed ( )
iii. This work done in the zero temperature limit
),(,),(,3/2 ZNpZNsA
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5) Methods
• Choose a sequence of isotopic energies
• To first approximation, assume the equality of differences in neighboring isotope energies:
121 AAAA EEEE
SnSnSnEEEor 1131151142:
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6) Sn Staggering from 1st Differences
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7)Methods
• To estimate curvature, look at second differences :
• This is analogous to taking the second derivative of the energy with respect to the atomic number A.
11 2
AAA EEEA
E
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8) Methods (Result of Second Differences for Sn)
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9)Methods
• Second Differences showcase alternating signs due to pairing effects from even isotopes.
• We suppress pairing by looking at the general trend and adding an appropriate constant energy to each even isotope. This will affect each number in SD’s.
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10) Methods (pairing suppression for Sn)
Data after pairing correction
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11) Methods
• After p(N,Z) is corrected, a parabolic correction is imposed on each Second Difference.
2)(2 middlenegativemost AA
E
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12) Results (Sn)
Bare Data
Pairing Correction (full line)
Parabolic Correction (dashed line)
Second Differences 11 2 AAA EEE
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13) Results (Pb)
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14) Methods
Sn isotope bindings, irregular line joins bare data;pairing and parabolic corrections give non-connecteddots.
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15) Methods (Results for Lead Isotopes)
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16) Goal
• Using a simple approximation in 1st order nuclear theory, quickly obtain upper or lower bounds for unknown isotopic energies.
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17) Required Parameters
1. Sequence of known isotopic energies surrounding unknown values
2. Empirical value to suppress pairing
3. Most negative value after Second Differences are obtained
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18)Results
• Extrapolations
• Interpolations
212 AAA EEE
221
AAA
EEE
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19) Extrapolations &Interpolations:
Sn117
Uncorrected Data Corrected DataB.E
A
B.E.
A
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20)Results from Interpolation For
Uncorrected (keV) Error (Underbinding)
-996816 1191
Corrected (keV) Error (Underbinding)
-995466 159
Sn117
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21)Results for Extrapolation of Sn117
Uncorrected (keV) (From higher masses) Error (Overbinding)
-998467 2842
Corrected (keV) Error (Overbinding)
-996967 1342
Uncorrected (keV) (From lower masses) Error (Overbinding)
-998243 2618
Corrected (keV) Error (Overbinding)
-996743 1118
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22)Extrapolations for Ground State Energy of = -934562 keV
Uncorrected (keV) Error (keV)
-931957 2605
Corrected (keV) Error (keV)
-934657 95
Sn110
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23)Results
• Extrapolation for with uncorrected data gives an over-binding of keV
• The same extrapolation for with corrected data gives an over-binding of keV
Pb179
Pb179
keVEPb
3101378179 3103
3101
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24) Conclusions
• Future work is needed to expand this technique for both N & Z as variables
• Development of algorithms to quickly process energy sequences is in development
• High temperature limit gives estimates of partition functions
• Predicative ability greatly enhanced by introducing pairing suppression and by favoring of parabolic terms in binding energy formula
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Acknowledgements
• Dr. Bruce Barrett
• Dr. Alex Lisetsky