Bounds to Binding Energies from Concavity

1. Bounds to Binding Energies from Concavity N.P. Toberg Dr. B.R. Barrett Department of Physics, University of Arizona, Tucson Az 85721 USA Dr. B.G. Giraud Service de Physique Théorique, DSM, CE Saclay, F-911191 Gif/Yvette, France

2. 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

3. 3) Introduction Complete Binding Energy Krane, Kenneth. Introductory Nuclear Physics. John Wiley & Sons, Inc., 1988.

4. 4) Introduction • Dominant terms define a paraboloid energy surface (concave) • Deviations from concavity can be suppressed ( ) • This work done in the zero temperature limit

5. 5) Methods • Choose a sequence of isotopic energies • To first approximation, assume the equality of differences in neighboring isotope energies:

6. 6) Sn Staggering from 1st Differences

7. 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.

8. 8) Methods (Result of Second Differences for Sn)

9. 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.

10. 10) Methods (pairing suppression for Sn) Data after pairing correction

11. 11) Methods • After p(N,Z) is corrected, a parabolic correction is imposed on each Second Difference.

12. 12) Results (Sn) Bare Data Parabolic Correction (dashed line) Pairing Correction (full line) Second Differences

13. 13) Results (Pb)

14. 14) Methods Sn isotope bindings, irregular line joins bare data; pairing and parabolic corrections give non-connected dots.

15. 15) Methods (Results for Lead Isotopes)

16. 16) Goal • Using a simple approximation in 1st order nuclear theory, quickly obtain upper or lower bounds for unknown isotopic energies.

17. 17) Required Parameters • Sequence of known isotopic energies surrounding unknown values • Empirical value to suppress pairing • Most negative value after Second Differences are obtained

18. 18)Results • Extrapolations • Interpolations

19. 19) Extrapolations &Interpolations: B.E. B.E Corrected Data Uncorrected Data A A

20. 20)Results from Interpolation For

21. 21)Results for Extrapolation of

22. 22)Extrapolations for Ground State Energy of = -934562 keV

23. 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

24. 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

25. Acknowledgements • Dr. Bruce Barrett • Dr. Alex Lisetsky