2.3 Interconversion of Nucleons Within the Nucleus
19
dropping electrons from the next neighboring orbital. The difference in the energies
of the two orbitals is emitted as X -rays. This process leads to emissions of X -rays
of different energies, because the process of filling of electrons from neighboring
orbitals continues in the atom, until an electron of the uppermost orbital has adjusted
by the process of filling the vacancy created in its next orbital. Therefore, the energy
of X -rays due to electron capture and filling of vacancy may not always be much
different from each other. Thus, the identification of an isotope, which decays by
electron capture, becomes a difficult task.
Very useful information concerning β-decay can be drawn by calculating binding
energy of various isobaric elements (especially for mass number greater than 80),
using an empirical equation mentioned below:
Binding Energy
(MeV) = 14.0 A − 19.3
A − 2Z
A
− 0.585
z
A
1
3
− 13.05 A
2
3 ±
125
A
(2.2)
where A is mass number and Z is proton number. This equation has been developed
by considering various factors, which may have an effect on the binding energy of
a nucleus. In this equation, the first term accounts for short range nuclear forces
between two nuclei. The binding energy of the nucleus decreases accordingly with
the extra number of neutrons, hence the second term is subtracted from the binding
energy value. Increase in repulsive energy between protons is inversely related to
distance, i.e., the radius of the nucleus. Accordingly, the binding energy decreases.
This accounts for the third term. With increase in radius of the nucleus, A
1/3 , which is
proportional to the surface area of nucleus, also increases. In other words, the number
of less tightly bound nucleons near the surface increases accordingly, followed by an
expected decrease in binding energy. This accounts for the 4th term. The last term
considers even–odd character of the number of protons and neutrons present in the
nucleus. This value is positive for even–even nuclei, negative for odd–odd, and zero
for odd–even nuclei.
This equation has been used by the author to calculate the binding energies of
isobaric nuclides of mass number 144. The results are plotted against the neutron
number, giving two parabolas (Fig. 2.2), for odd and even number of neutrons. The
advantage of this calculation is that one can easily decide which of the isobaric
nuclides of 144 mass number will decay by β-decay and which will decay by positron
or electron capture type decay. Moreover, an idea about the magnitude of instability
can also be realized from this curve. For example,
144 Xe 54 is energetically very
unstable and will have the tendency to decay to
144 Cs 55 by β-decay process.
Elements lying on the left-hand side of the curve are unstable to β-decay, while
those present at the beginning of the two parabolas (i.e., before Cerium-144) show a
very sharp change in binding energy; hence the energy available for decay is large.
In other words, these isotopes will decay with the emission of the β-particle having
the maximum E max value as compared to nuclides like
144 La 57 , which will decay to
144 Ce 58 with lower E max energy. Nuclides on the right-hand side of the parabolas are
deficient in neutrons; they decay by positron emission or electron capture processes.
19
dropping electrons from the next neighboring orbital. The difference in the energies
of the two orbitals is emitted as X -rays. This process leads to emissions of X -rays
of different energies, because the process of filling of electrons from neighboring
orbitals continues in the atom, until an electron of the uppermost orbital has adjusted
by the process of filling the vacancy created in its next orbital. Therefore, the energy
of X -rays due to electron capture and filling of vacancy may not always be much
different from each other. Thus, the identification of an isotope, which decays by
electron capture, becomes a difficult task.
Very useful information concerning β-decay can be drawn by calculating binding
energy of various isobaric elements (especially for mass number greater than 80),
using an empirical equation mentioned below:
Binding Energy
(MeV) = 14.0 A − 19.3
A − 2Z
A
− 0.585
z
A
1
3
− 13.05 A
2
3 ±
125
A
(2.2)
where A is mass number and Z is proton number. This equation has been developed
by considering various factors, which may have an effect on the binding energy of
a nucleus. In this equation, the first term accounts for short range nuclear forces
between two nuclei. The binding energy of the nucleus decreases accordingly with
the extra number of neutrons, hence the second term is subtracted from the binding
energy value. Increase in repulsive energy between protons is inversely related to
distance, i.e., the radius of the nucleus. Accordingly, the binding energy decreases.
This accounts for the third term. With increase in radius of the nucleus, A
1/3 , which is
proportional to the surface area of nucleus, also increases. In other words, the number
of less tightly bound nucleons near the surface increases accordingly, followed by an
expected decrease in binding energy. This accounts for the 4th term. The last term
considers even–odd character of the number of protons and neutrons present in the
nucleus. This value is positive for even–even nuclei, negative for odd–odd, and zero
for odd–even nuclei.
This equation has been used by the author to calculate the binding energies of
isobaric nuclides of mass number 144. The results are plotted against the neutron
number, giving two parabolas (Fig. 2.2), for odd and even number of neutrons. The
advantage of this calculation is that one can easily decide which of the isobaric
nuclides of 144 mass number will decay by β-decay and which will decay by positron
or electron capture type decay. Moreover, an idea about the magnitude of instability
can also be realized from this curve. For example,
144 Xe 54 is energetically very
unstable and will have the tendency to decay to
144 Cs 55 by β-decay process.
Elements lying on the left-hand side of the curve are unstable to β-decay, while
those present at the beginning of the two parabolas (i.e., before Cerium-144) show a
very sharp change in binding energy; hence the energy available for decay is large.
In other words, these isotopes will decay with the emission of the β-particle having
the maximum E max value as compared to nuclides like
144 La 57 , which will decay to
144 Ce 58 with lower E max energy. Nuclides on the right-hand side of the parabolas are
deficient in neutrons; they decay by positron emission or electron capture processes.