there is a net gain in binding energy.

Therefore, fission of heavy nuclei is

again energetically feasible. It is left as an exercise to show that fusion of
heavy nuclei is not feasible.

1.6 NUCLEAR ANGULAR MOMENTUM

Total angular momentum of the nucleus taken about its own axis, is easily
measurable. But we cannot say much about the individual contributions of
protons and neutrons, as their motion is very complex. One may attempt to build
a vector model of the individual particles, which is similar to the vector model of
the atom. To visualize the vector model, let us discuss the quantum numbers
associated with the individual nucleons. The state of particular nucleon is
described by its wave function, i.e. solution of its wave equation, which is
characterized in terms of quantum numbers: Principal Quantum Number (n)
Each bound particle is associated with principle quantum number n, which can
take only positive integral values, n = 1, 2, 3, …. In Coulomb field, first-order
term for the total energy of the state is characterized by n, i.e. radial quantum
number (Higher order terms include contributions from fine and hyperfine
structures). This is not true in non-Coulomb fields such as rectangular potentials,
Yukawa potentials, etc., in which nuclear particles are bound. In such potentials,
the principal quantum number is a sum of radial (n) and orbital () quantum
numbers, i.e. n = n + .
Orbital Quantum Number () Orbital angular momentum quantum number () can
take on positive integral values, i.e. 0, 1, 2, … (n – 1). The magnitude of orbital
angular momentum is . Various states are respectively designated as s,
p, d, f, g, h, …, corresponding to = 0, = 1, = 2, = 3, = 4,
= 5,…
Orbital Magnetic Quantum Number (m) Orbital magnetic quantum number is
the component of in the specified direction such as that of an applied magnetic
field. It can take (2 + 1) possible values starting from – to + differing by unity.
Spin Quantum Number (s)
Spin quantum number (s) has value 1/2 for proton, neutron and electron, which
follows from Fermi–Dirac statistics and thereby obey the Pauli exclusion
principle. The magnitude of spin angular momentum is .
Magnetic Spin Quantum Number (ms) Magnetic spin quantum number is the
component of s in the specified direction say that of the applied magnetic field.
It can take 2s + 1 values from –s to +s. For particles with s = 1/2,
ms = 1/2 and –1/2.

Total Angular Momentum Quantum Number ( ) It is the vector sum of the

orbital ( ) and spin ( ) angular momenta, i.e. = + . The magnitude of the
total angular momentum is . For particles with s = 1/2, only two values
of j are permitted as j = + 1/2 and j = – 1/2. If = 0, only one value of j is
allowed as j = 1/2.
Radial Quantum Number (n) As stated earlier, in a non-Coulomb field, the
principle quantum number n does not represent the energy of the state. In the
radial wave equation solution, a radial quantum number n arises. n represents the
number of radial nodes in the wave function and can have values n = 1, 2, 3, …
Total Angular Momentum
It is the vector sum of the individual angular momenta of the constituent
nucleons and is represented by J or I. It is also called nuclear spin and its
magnitude is given by . The value of J can be calculated in two
different ways depending upon the type of coupling between angular momenta
of the nucleons. The two types of coupling are: (i) L–S coupling.
(ii) j–j coupling.
(i) L–S coupling: In this there is negligibly weak coupling between the orbital
and spin angular momenta of the individual nucleons. Orbital angular
momenta of all the nucleons couple together to give a resultant total
angular momentum L. Similarly, the spin angular momenta of all the
nucleons couple together to give resultant total spin angular momentum S.

Then the resultant L and S couple strongly to give the total nuclear spin J, i.e. J
= L + S.
 (ii) j–j coupling: In this scheme orbital (i) and spin (si) angular momenta of
individual nucleons couple together to give resultant angular momentum
(ji), or

Value of J for nuclei having even number of protons and even number of
neutrons is always zero. For nuclei having odd Z and even N or even Z and
odd N have half integer nuclear spin (J). All odd Z and odd N nuclei have
integral spin (J).

1.7 NUCLEAR MOMENTS

In this section, we discuss magnetic dipole moment of the nucleus, Bohr
magneton, nuclear magneton and spin g-factor. An expression for electric
quadrupole moment, which arises due to non-spherical charge distribution in the
nucleus, has been derived. Prolate and oblate shapes of the nuclei have been
discussed.
1.7.1 Magnetic Dipole Moment
Magnetic dipole moment of the nucleus arises due to the motion of charged
particles. Orbital and spin angular momenta of protons produce magnetic field
within the nucleus. This field can be described in terms of resultant magnetic
dipole moment located at the centre of the nucleus.
Particle having a charge q and mass m circulates with speed v in a circular
orbit of radius r. If it has a time period t, then the current i associated with the
charge q is
Now magnetic dipole moment m for current i around an area of the loop A is
given as m = i A A = pr2
Substituting the values of i and A, we have

Considering the case of a single electron atom and taking q to be electronic

charge e and multiplying and dividing the above equation by electron mass m,
we have

where = mvr. Since and m are vector quantities, so in vector notation, the above
equation is

where g is a factor called gyromagnetic ratio or g-factor.

As stated earlier, J has the units of is dimensionless. Using this fact, Eq.
(1.26) can be written as

The constant m0 is called Bohr magneton and is taken as the unit to measure
magnetic moments for atoms
For nuclei, we define magnetic moments in terms of the nuclear magneton mN,
which is defined a

which is smaller by a factor than the Bohr magneton.

Equation (1.25) tells us that However, the nucleus has neutrons and
protons. They possess spin angular momentum in addition to orbital angular
momentum. This spin gives an addition contribution to total magnetic moment
of the nucleus. The spin angular momentum contribution is written as

where gs is known as spin g-factor. Its value for proton is 5.586 and for neutron
it is –3.826. Orbital angular momentum contribution to magnetic moment
appears only from the protons, thus in Eq. (1.26) g = 1 for protons and g = 0 for
neutrons.
Total nuclear magnetic dipole moment for the nucleus m is then given by
where g is the gyromagnetic ratio of the nucleus, and Magnetic dipole
moment of odd Z and even N or even Z and odd N nuclei is due to unpaired
single nucleon.
1.7.2 Electric Quadrupole Moment This property arises due to non-spherical
charge distribution in the nucleus. It is worthnoting that for a nucleus of
spherical shape, neither dipole moment exists nor quadrupole moment exists
because centre of mass of the nucleus coincides with centre of charge of the
nucleus.
In non-spherical nuclei, assume that centre of mass is at the origin of the
nucleus and centre of charge is not at the origin. The potential at any point P
situated outside the nucleus at a distance r from the origin is given by

where charge e is located at a distance r from the origin.

Now,
|r – r| = [r2 + (r)2 – 2rr cos q]1/2
where q is the angle between as shown in Figure 1.5. Substituting |r – r|
in Eq. (1.32), we get

…
Figure 1.5 Non-spherical charge distribution. The nuclear charge e is located at a distance r
from the origin and the potential at point P outside the nucleus at a distance r from the origin.

Since r is much smaller than r, we can carry out a binomial expansion in Eq.
(1.33) and obtain

In this equation, the first term on the right-hand side represents the
contribution to potential from single charge, the second term represents the
contribution due to a dipole, whose electric dipole moment is er cos q and the
third term represents the contribution due to electric quadrupole whose
quadrupole moment Q is given as

If Q > 0, then nucleus is prolate in shape, as shown in Figure 1.6. In prolate

shape, when two out of three principal axes are equal and the third (unequal)
axis is longer than the other two axes, the nucleus will have a shape like that of a
fully inflated football. More extreme examples of prolate shapes are cigar or a
hot dog.
 Figure 1.6 Prolate shape of the nucleus.

If Q < 0, then nucleus is oblate in shape, as shown in Figure 1.7. When two
out of three principal axes are equal and the third (unequal) axis is shorter than
the other two axes, the nucleus will have a shape like that of a pumpkin. More
extreme examples of oblate shapes are burger or discus. Q = 0 represents
spherical shape.

Figure 1.7 Oblate shape of the nucleus.

1.8 WAVE MECHANICAL PROPERTIES

Efforts were made to construct a theoretical model of the nucleus on the basis of
classical physics, but they all failed. It was finally realized that classical physics
(Newtonian mechanics, Maxwell theory of electromagnetism) can not explain
atomic and subatomic phenomena. For instance, classical electromagnetic
theory, where electron, being a charged particle should radiate energy while
moving around the nucleus, which would eventually collapse in the atom.
Similarly, classical physics failed to explain other phenomena, like black body
radiations, photoelectric effect, atomic spectra, Compton effect, etc. Initially,
semiclassical–semiquantum ideas like Bohr atomic model, were developed,
which could only pave the way to develop full quantum theory in two forms
called wave mechanical (1927) and matrix mechanics (1925).
The fundamental idea of wave mechanics is based on the wave nature of
matter. The dual wave and particle nature of matter is expressed by means of a