Page 61 - Radiochemistry and nuclear chemistry

50 Radiochemistry and Nuclear Chemistry

correct, however. By bombarding nuclei with very high energy electrons or protons (up to
_> 1 GeV) and measuring the scattering angle and particle energy, the charge and matter
density near the surface of the irradiated nucleus can be studied. These experiments have
led to the conclusion that nuclei do not possess a uniform charge or matter distribution out
to a sharp boundary, but rather are fuzzy as indicated by the s-shaped curves in Figure 3.4.
With an atomic number greater than 20 it has been found that a uniform charge and mass
density exists over a short distance from the center of the nucleus, and this core is
surrounded by a layer of decreasing density which seems to have a constant thickness of
--2.5 fm independent of mass number. In a bismuth nucleus, for example, the density
remains relatively constant for approximately 5 fm then decreases steadily to one-tenth of
that value in the next 2 fm (Fig. 3.4). It has also been found that not all nuclei are
spherical, some being oblate and others prolate around the axis of rotation.
Despite the presence of this outer layer of decreasing density and the nonspherical
symmetry, for most purposes it is adequate to assume a constant density nucleus with a
sharp boundary. Therefore, use is made of the radius equation (3.7) in which the r o value
may be assumed to be 1.4 fm. Using this relationship, we can calculate the radius of 4~
to be r = 1.4 x 10- 15 x 40 1/3 = 4.79 fro, and for 209 Bi to be 8.31 fro. These values are
indicated in Figure 3.4. For 8~ a similar calculation yields 6.0 fro, while for 238U the
radius calculated is 8.7 fro. From these calculations we see that the radius does not change
dramatically from relatively light nuclei to the heaviest.

3.6. Semiempirical mass equation

In preceding sections we have learned that the size as well as the total binding energy of
nuclei are proportional to the mass number. These characteristics suggest an analogy
between the nucleus and a drop of liquid. In such a drop the molecules interact with their
immediate neighbors but not with other molecules more distant. Similarly, a particular
nucleon in a nucleus is attracted by nuclear forces only to its adjacent neighbors. Moreover,
the volume of the liquid drop is composed of the sum of the volumes of the molecules or
atoms present since these are nearly incompressible. Again, as we learned above, this is
similar to the behavior of nucleons in a nucleus. Based on the analogy of a nucleus to a
droplet of liquid, it has been possible to derive a semiempirical mass equation containing
various terms which are related to a nuclear droplet.
Let us consider what we have learned about the characteristics of the nuclear droplet. (a)
First, recalling that mass and energy are equivalent, if the total energy of the nucleus is
directly proportional to the total number of nucleons there should be a term in the mass
equation related to the mass number. (b) Secondly, in the discussion of the neutron/proton
ratios we learned that the number of neutrons could not become too large since the
discrepancy in the energy levels of the neutron and proton play a role in determining the
stability of the nucleus. This implies that the binding energy is reduced by a term which
allows for variation in the ratio of the number of protons and neutrons. (c) Since the
protons throughout the nucleus experience a mutual repulsion which affects the stability of
nucleus, we should expect in the mass equation another negative term reflecting the
repulsive forces of the protons. (d) Still another term is required to take into account that
the surface nucleons, which are not completely surrounded by other nucleons, would not
be totally saturated in their attraction. In a droplet of liquid this lack of saturation of surface