Bounds to Binding Energies from Concavity

Bounds to Binding Energies from Concavity

N.P. TobergDr. B.R. BarrettDepartment of Physics,

University of Arizona, Tucson Az 85721 USADr. B.G. Giraud

Service de Physique Théorique,DSM, CE Saclay, F-911191 Gif/Yvette, France

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) IntroductionComplete Binding Energy

Krane, Kenneth. Introductory Nuclear Physics. John Wiley & Sons, Inc., 1988.

AZAaAZZaAaAaB symcsv

23/13/2 )2()1(

NZA

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

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:

6) Sn Staggering from 1st Differences

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 EEEAE

8) Methods (Result of Second Differences for Sn)

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) Methods (pairing suppression for Sn)

Data after pairing correction

11) Methods

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

2)(2 middlenegativemost AA

E

12) Results (Sn)

Bare Data

Pairing Correction (full line)

Parabolic Correction (dashed line)

Second Differences 11 2 AAA EEE

13) Results (Pb)

14) Methods

Sn isotope bindings, irregular line joins bare data;pairing and parabolic corrections give non-connecteddots.

15) Methods (Results for Lead Isotopes)

16) Goal

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

17) Required Parameters

1. Sequence of known isotopic energies surrounding unknown values

2. Empirical value to suppress pairing3. Most negative value after Second

Differences are obtained

18)Results

• Extrapolations

• Interpolations

212 AAA EEE

221

AAA

EEE

19) Extrapolations &Interpolations:

Sn117

Uncorrected Data Corrected DataB.E

A

B.E.

A

20)Results from Interpolation For

Uncorrected (keV) Error (Underbinding)

-996816 1191

Corrected (keV) Error (Underbinding)

-995466 159

Sn117

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

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

Uncorrected (keV) Error (keV)-931957 2605

Corrected (keV) Error (keV)-934657 95

Sn110

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

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

Acknowledgements

• Dr. Bruce Barrett

• Dr. Alex Lisetsky