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:
|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:
BE is proportional to the number of nucleons in the nucleus
corresponding to 14.1 A
BE decreases as the surface increases, - 13 A2/3
BE decreases due to the Coulomb interaction of protons,
- 0.6 Z2A-1/3,
BE decreases as the N/Z ratio deviates from the stable
valley as indicated by the term - 20 (A - 2 Z)2
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
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
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
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.