Page 62 - Radiochemistry and nuclear chemistry

Nuclear Mass and Stability 51

forces gives rise to the effect of surface tension. Consequently, the negative term in the
mass equation reflecting this unsaturation effect should be similar to a surface tension
expression. (e) Finally, we have seen that nuclei with an even number of protons and
neutrons are more stable than nuclei with an odd number of either type of nucleon and that
the least stable nuclei are those for odd numbers of both neutrons and protons. This
odd-even effect must also be included in a mass equation.
Taking into account these various factors, we can write a semiempirical mass equation.
However, it is often more useful to write the analogous equation for the mass defect or
binding energy of the nucleus, recalling (3.5). Such an equation, first derived by C. F. von
Weizs,~cker in 1935, would have the form:

EB(MeV) = avA - a a (N- Z)2[A - a c Z 2[All3 - as A2/3 • asia 3/4 (3.8)

The first term in this equation takes into account the proportionality of the energy to the
total number of nucleons (the volume energy); the second term, the variations in neutron
and proton ratios (the asymmetry energy); the third term, the Coulomb forces of repulsion
for protons (the Coulomb energy); the fourth, the surface tension effect (the surface
energy). In the fifth term, which accounts for the odd-even effect, a positive sign is used
for even proton-even neutron nuclei and a negative sign for odd proton-odd neutron nuclei.
For nuclei of odd A (even-odd or odd-even) this term has the value of zero. Comparison
of this equation with actual binding energies of nuclei yields a set of coefficients; e.g.

a v = 15.5, a a = 23, a c = 0.72, a s = 16.8, a~ = 34

With these coefficients the binding energy equations (3.2) and (3.5) give agreement within
a few percent of the measured values for most nuclei of mass number greater than 40.
When the calculated binding energy is compared with the experimental binding energy,
it is seen that for certain values of neutron and proton numbers, the disagreement is more
serious. These numbers are related to the so-called "magic numbers", which we have
indicated in Figure 3.1, whose recognition led to the development of the nuclear shell
model described in a later chapter.

3.7. Valley of O-stability
If the semiempirical mass equation is written as a function of Z, remembering that N =
A - Z, it reduces to a quadratic equation of the form

E B = a Z 2 + b Z + c + d/A 3/4 (3.9)

where the terms a, b and c also contain A. This quadratic equation describes a parabola for
constant values of A. Consequently, we would expect that for any family of isobars (i.e.
constant A) the masses should fall upon a parabolic curve. Such a curve is shown in Figure
3.5. In returning to Figure 3.1, the isobar line with constant A but varying Z cuts diagonally
through the line of stable nuclei. We can picture this as a valley, where the most stable
nuclei lie at the bottom of it (cf. Figs. 3.1 and 3.5), while unstable nuclei lie up the valley