Subatomic Physics

502 Liquid Drop Model, Fermi Gas Model, Heavy Ions
In Section 5.3, nuclear mass measurements were introduced, and in Section 5.4
some basic features of nuclear ground states were mentioned. Fig. 5.20 represents
a plot of the stable nuclei in a NZ plane. We return to the nuclear masses here,
and we shall describe their behavior in more detail than in Chapter 5. Consider a
nucleus consisting of A nucleons, Z protons, and N neutrons. The total mass of
such a nucleus is somewhat smaller than the sum of the masses of its constituents
because of the binding energy B which holds the nucleons together. For bound
states, B is positive and represents the energy that is required to disintegrate the
nucleus into its constituent neutrons and protons. B is given by
B
c 2 = Zmp + Nmn − mnuclear(Z, N). (16.1)
Here, mnuclear(Z, N) is the mass of a nucleus with Z protons and N neutrons. It
is customary to quote atomic and not nuclear masses and use atomic mass units
(see Eq. (5.23)). In terms of the atomic mass m(Z, N), the binding energy can be
written as
B
c 2 = ZmH + Nmn − m(Z, N). (16.2)
A small term due to atomic binding effects is neglected in Eq. (16.2); mH is the mass
of the hydrogen atom. The difference between the atomic rest energy m(Z, N)c 2
and the nucleon or mass number times uc 2 is called the mass excess (or mass defect),
∆=m(Z, N)c 2 − Auc 2 . (16.3)
Comparison between Eqs. (16.2) and (16.3) shows that −∆ andB measure essentially
the same quantity but differ by a small energy. Tables usually list ∆ because
it is the quantity that follows from mass-spectroscopic measurements. The average
binding energy per nucleon, B/A, is plotted in Fig. 16.1. The binding energy curve
exhibits a number of interesting features:
1. Over most of the range of stable nuclei, B/A is approximately constant and of
the order of 8–9 MeV. This constancy results from the saturation of nuclear
forces discussed in Section 14.5. If all nucleons inside a nucleus were within
each other’s force range, the total binding energy would be expected to increase
proportionally to the number of bonds or approximately proportionally
to A 2 . B/A would then be proportional to A.
2. B/A reaches its maximum in the region of iron (A ≈ 60). It drops off slowly
toward large A and more steeply toward small A. Thisbehavioriscrucial
for the synthesis of the elements and for nuclear power production. One
consequence is that the abundance of elements around iron is especially large.
Also if a nucleus of say, A = 240, is split into two roughly equal parts, the
binding of the two parts is stronger than that of the original nuclide, and
energy is liberated.