Nuclides - semi-empirical equation for binding energy

Based on the liquid-drop model, Seeger proposed a equation to systematize the binding energies of nuclides. He considered several factors, and derived the coefficients using the observed masses of nuclides.

The binding energy, BE, for a nuclide with mass number A and atomic number Z has been modified and simplified over the years. One of the form used is as follows: the form:

BE(A,Z) = 14.1 A - 13 A2/3 - 0.6 Z2A-1/3 - 20 (A - 2 Z)2 A-1 + Eo The binding energy depends on several factors reflected in each term:

  1. BE is proportional to the number of nucleons in the nucleus corresponding to 14.1 A

  2. BE decreases as the surface increases, - 13 A2/3

  3. BE decreases due to the Coulomb interaction of protons, - 0.6 Z2A-1/3,

  4. BE decreases as the N/Z ratio deviates from the stable valley as indicated by the term - 20 (A - 2 Z)2 A-1

  5. A special value is used to adjust for pairing of nucleons. For odd-even or even-odd nuclides, Eo = 0. For even-even nuclide, Eo has a positive value and for odd-odd nuclide, Eo has a negative value. This adjustment accounts for the high stability of even-even nuclides, and low stability of odd-odd nuclide.

    Binding energy of isobars

    Isobars have the same mass number A but different atomic number. Binding energy varies with atomic number. The stable isobar ideally has the maximum binding energy. Mathematically, a formula can be derived for the atomic number of the stable isobar Zs. Zs = A (1 + 0.0071 A-1/3) (1.983 + 0.016 A2/3)-1 For example, for mass number 123, it can be shown that the atomic number for the stable isobar is 51.8. This is in agreement with the observed result.

    For a lighter nuclide with A = 57, the above formula gives Zs = 32. However, the stable isobar with mass 57 is iron, Z = 26. The error is very large.

    The merit of the semi-empirical formula

    Theorization is an important step in science. A theory is a catalyst for progress. The semi-empirical equation for binding energy is based on the liquid drop model for nuclear structures, and the coefficients are derived by using a large number of data. Thus, the equation is the result of semi-empirical approach.

    E-mail: cchieh@uwaterloo.ca