Elements of Modern Physics
336
In this analysis, Coulomb interaction which increases the potential for the
protons has been ignored. Because of the Coulomb interaction, in stable, heavy
nuclei, one finds more neutrons than protons. An important point brought out
by the model is that the Pauli principle would push up the energy of a nucleus
with Z very different from
1 ,
2
A and implies that the least-energy state is the
one with nearly equal number of protons and neutrons.
9.4 WEIZSACKER’S MASS FORMULA
Many of the nuclear properties are critically controlled by the masses or
equivalently the binding energies of the nuclei. For example, a nucleus can
decay only into a set of particles with lower masses, with the difference in the
masses appearing as kinetic energy of the final particles. Here, a semi-empirical
formula for the masses of the nuclei is briefly discussed. This formula is not
only useful in the discussion of the stability of the nuclei but also provides a
useful insight into the physical properties that determine their masses.
The total energy of a nucleus is largely controlled by the short-range nature
of the forces leading to saturation in binding, the Pauli exclusion principle, and
Coulombic repulsion between protons with some finer effects due to pairing of
like-nucleons.
To start with, the major contribution to the mass of the nucleus comes from
the masses of the nucleons.
M 1 = Zm p + (A – Z)m n
(9.51)
Since nuclear forces are of short range, the binding energy of the nucleus is
proportional to the number of nucleons A, so that the mass is reduced by
M 2 = – a 1 A
(9.52)
However, nucleons near the surface are less tightly bound which may be
taken into account by a term
M 3 = a 2 A
2/3
(9.53)
proportional to the surface area. The electrostatic, repulsive interaction may be
incorporated by noting that the electrostatic energy of a sphere of charge Ze is
proportional to Z
2
/R. This brings in a contribution
M 4 = a 3 Z
2
A
–1/3
(9.54)
The Pauli exclusion principle brings in a term
M 5 =
2
4
1
2
Z
A
a
A
−
(9.55)
336
In this analysis, Coulomb interaction which increases the potential for the
protons has been ignored. Because of the Coulomb interaction, in stable, heavy
nuclei, one finds more neutrons than protons. An important point brought out
by the model is that the Pauli principle would push up the energy of a nucleus
with Z very different from
1 ,
2
A and implies that the least-energy state is the
one with nearly equal number of protons and neutrons.
9.4 WEIZSACKER’S MASS FORMULA
Many of the nuclear properties are critically controlled by the masses or
equivalently the binding energies of the nuclei. For example, a nucleus can
decay only into a set of particles with lower masses, with the difference in the
masses appearing as kinetic energy of the final particles. Here, a semi-empirical
formula for the masses of the nuclei is briefly discussed. This formula is not
only useful in the discussion of the stability of the nuclei but also provides a
useful insight into the physical properties that determine their masses.
The total energy of a nucleus is largely controlled by the short-range nature
of the forces leading to saturation in binding, the Pauli exclusion principle, and
Coulombic repulsion between protons with some finer effects due to pairing of
like-nucleons.
To start with, the major contribution to the mass of the nucleus comes from
the masses of the nucleons.
M 1 = Zm p + (A – Z)m n
(9.51)
Since nuclear forces are of short range, the binding energy of the nucleus is
proportional to the number of nucleons A, so that the mass is reduced by
M 2 = – a 1 A
(9.52)
However, nucleons near the surface are less tightly bound which may be
taken into account by a term
M 3 = a 2 A
2/3
(9.53)
proportional to the surface area. The electrostatic, repulsive interaction may be
incorporated by noting that the electrostatic energy of a sphere of charge Ze is
proportional to Z
2
/R. This brings in a contribution
M 4 = a 3 Z
2
A
–1/3
(9.54)
The Pauli exclusion principle brings in a term
M 5 =
2
4
1
2
Z
A
a
A
−
(9.55)