NCERT Textbook: Nuclei | Physics Class 12 - NEET PDF Download

 Page 1


Physics
306
13.1  INTRODUCTION
In the previous chapter, we have learnt that in every atom, the positive
charge and mass are densely concentrated at the centre of the atom
forming its nucleus. The overall dimensions of a nucleus are much smaller
than those of an atom. Experiments on scattering of a-particles
demonstrated that the radius of a nucleus was smaller than the radius
of an atom by a factor of about 10
4
. This means the volume of a nucleus
is about 10
–12
 times the volume of the atom. In other words, an atom is
almost empty. If an atom is enlarged to the size of a classroom, the nucleus
would be of the size of pinhead. Nevertheless, the nucleus contains most
(more than 99.9%) of the mass of an atom.
Does the nucleus have a structure, just as the atom does?  If so, what
are the constituents of the nucleus?  How are these held together? In this
chapter, we shall look for answers to such questions. We shall discuss
various properties of nuclei such as their size, mass and stability, and
also associated nuclear phenomena such as radioactivity, fission and fusion.
13.2  ATOMIC MASSES AND COMPOSITION OF NUCLEUS
The mass of an atom is very small, compared to a kilogram;  for example,
the mass of a carbon atom, 
12
C, is 1.992647 × 10
–26 
kg. Kilogram is not
a very convenient unit to measure such small quantities. Therefore, a
Chapter Thirteen
NUCLEI
2024-25
Page 2


Physics
306
13.1  INTRODUCTION
In the previous chapter, we have learnt that in every atom, the positive
charge and mass are densely concentrated at the centre of the atom
forming its nucleus. The overall dimensions of a nucleus are much smaller
than those of an atom. Experiments on scattering of a-particles
demonstrated that the radius of a nucleus was smaller than the radius
of an atom by a factor of about 10
4
. This means the volume of a nucleus
is about 10
–12
 times the volume of the atom. In other words, an atom is
almost empty. If an atom is enlarged to the size of a classroom, the nucleus
would be of the size of pinhead. Nevertheless, the nucleus contains most
(more than 99.9%) of the mass of an atom.
Does the nucleus have a structure, just as the atom does?  If so, what
are the constituents of the nucleus?  How are these held together? In this
chapter, we shall look for answers to such questions. We shall discuss
various properties of nuclei such as their size, mass and stability, and
also associated nuclear phenomena such as radioactivity, fission and fusion.
13.2  ATOMIC MASSES AND COMPOSITION OF NUCLEUS
The mass of an atom is very small, compared to a kilogram;  for example,
the mass of a carbon atom, 
12
C, is 1.992647 × 10
–26 
kg. Kilogram is not
a very convenient unit to measure such small quantities. Therefore, a
Chapter Thirteen
NUCLEI
2024-25
307
Nuclei
different mass unit is used for expressing atomic masses. This unit is the
atomic mass unit (u), defined as 1/12
th
 of the mass of the carbon (
12
C)
atom. According to this definition
12
 mass of one C atom
1u = 
12
     
26
1.992647 10 kg
12
- ×
=
     
27
1.660539 10 kg
- = × (13.1)
The atomic masses of various elements expressed in atomic mass
unit (u) are close to being integral multiples of the mass of a hydrogen
atom. There are, however, many striking exceptions to this rule. For
example, the atomic mass of chlorine atom is 35.46 u.
Accurate measurement of atomic masses is carried out with a mass
spectrometer, The measurement of atomic masses reveals the existence
of different types of atoms of the same element, which exhibit the same
chemical properties, but differ in mass. Such atomic species of the same
element differing in mass are called isotopes. (In Greek, isotope means
the same place, i.e. they occur in the same place in the periodic table of
elements.) It was found that practically every element consists of a mixture
of several isotopes.  The relative abundance of different isotopes differs
from element to element. Chlorine, for example, has two isotopes having
masses 34.98 u and 36.98 u, which are nearly integral multiples of the
mass of a hydrogen atom.  The relative abundances of these isotopes are
75.4 and 24.6 per cent, respectively.  Thus, the average mass of a chlorine
atom is obtained by the weighted average of the masses of the two
isotopes,  which works out to be
= 
75.4 34.98 24.6 36.98
100
× + ×
=  35.47 u
which agrees with the atomic mass of chlorine.
Even the lightest element, hydrogen has three isotopes having masses
1.0078 u, 2.0141 u, and 3.0160 u.  The nucleus of the lightest atom of
hydrogen, which has a relative abundance of 99.985%, is called the
proton.  The mass of a proton is
27
1.00727 u 1.67262 10 kg
p
m
- = = × (13.2)
This is equal to the mass of the hydrogen atom (= 1.00783u), minus
the mass of a single electron (m
e 
= 0.00055 u).  The other two isotopes of
hydrogen are called deuterium and tritium. Tritium nuclei, being
unstable, do not occur naturally and are produced artificially in
laboratories.
The positive charge in the nucleus is that of the protons. A proton
carries one unit of fundamental charge and is stable. It was earlier thought
that the nucleus may contain electrons, but this was ruled out later using
arguments based on quantum theory. All the electrons of an atom are
outside the nucleus. We know that the number of these electrons outside
the nucleus of the atom is Z, the atomic number. The total charge of the
2024-25
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Physics
306
13.1  INTRODUCTION
In the previous chapter, we have learnt that in every atom, the positive
charge and mass are densely concentrated at the centre of the atom
forming its nucleus. The overall dimensions of a nucleus are much smaller
than those of an atom. Experiments on scattering of a-particles
demonstrated that the radius of a nucleus was smaller than the radius
of an atom by a factor of about 10
4
. This means the volume of a nucleus
is about 10
–12
 times the volume of the atom. In other words, an atom is
almost empty. If an atom is enlarged to the size of a classroom, the nucleus
would be of the size of pinhead. Nevertheless, the nucleus contains most
(more than 99.9%) of the mass of an atom.
Does the nucleus have a structure, just as the atom does?  If so, what
are the constituents of the nucleus?  How are these held together? In this
chapter, we shall look for answers to such questions. We shall discuss
various properties of nuclei such as their size, mass and stability, and
also associated nuclear phenomena such as radioactivity, fission and fusion.
13.2  ATOMIC MASSES AND COMPOSITION OF NUCLEUS
The mass of an atom is very small, compared to a kilogram;  for example,
the mass of a carbon atom, 
12
C, is 1.992647 × 10
–26 
kg. Kilogram is not
a very convenient unit to measure such small quantities. Therefore, a
Chapter Thirteen
NUCLEI
2024-25
307
Nuclei
different mass unit is used for expressing atomic masses. This unit is the
atomic mass unit (u), defined as 1/12
th
 of the mass of the carbon (
12
C)
atom. According to this definition
12
 mass of one C atom
1u = 
12
     
26
1.992647 10 kg
12
- ×
=
     
27
1.660539 10 kg
- = × (13.1)
The atomic masses of various elements expressed in atomic mass
unit (u) are close to being integral multiples of the mass of a hydrogen
atom. There are, however, many striking exceptions to this rule. For
example, the atomic mass of chlorine atom is 35.46 u.
Accurate measurement of atomic masses is carried out with a mass
spectrometer, The measurement of atomic masses reveals the existence
of different types of atoms of the same element, which exhibit the same
chemical properties, but differ in mass. Such atomic species of the same
element differing in mass are called isotopes. (In Greek, isotope means
the same place, i.e. they occur in the same place in the periodic table of
elements.) It was found that practically every element consists of a mixture
of several isotopes.  The relative abundance of different isotopes differs
from element to element. Chlorine, for example, has two isotopes having
masses 34.98 u and 36.98 u, which are nearly integral multiples of the
mass of a hydrogen atom.  The relative abundances of these isotopes are
75.4 and 24.6 per cent, respectively.  Thus, the average mass of a chlorine
atom is obtained by the weighted average of the masses of the two
isotopes,  which works out to be
= 
75.4 34.98 24.6 36.98
100
× + ×
=  35.47 u
which agrees with the atomic mass of chlorine.
Even the lightest element, hydrogen has three isotopes having masses
1.0078 u, 2.0141 u, and 3.0160 u.  The nucleus of the lightest atom of
hydrogen, which has a relative abundance of 99.985%, is called the
proton.  The mass of a proton is
27
1.00727 u 1.67262 10 kg
p
m
- = = × (13.2)
This is equal to the mass of the hydrogen atom (= 1.00783u), minus
the mass of a single electron (m
e 
= 0.00055 u).  The other two isotopes of
hydrogen are called deuterium and tritium. Tritium nuclei, being
unstable, do not occur naturally and are produced artificially in
laboratories.
The positive charge in the nucleus is that of the protons. A proton
carries one unit of fundamental charge and is stable. It was earlier thought
that the nucleus may contain electrons, but this was ruled out later using
arguments based on quantum theory. All the electrons of an atom are
outside the nucleus. We know that the number of these electrons outside
the nucleus of the atom is Z, the atomic number. The total charge of the
2024-25
Physics
308
atomic electrons is thus (–Ze), and since the atom is neutral, the charge
of the nucleus is (+Ze). The number of protons in the nucleus of the atom
is, therefore, exactly Z, the atomic number.
Discovery of Neutron
Since the nuclei of deuterium and tritium are isotopes of hydrogen, they
must contain only one proton each.  But the masses of the nuclei of
hydrogen, deuterium and tritium are in the ratio of 1:2:3.  Therefore, the
nuclei of deuterium and tritium must contain,  in addition to a proton,
some neutral matter.  The amount of neutral matter present in the nuclei
of these isotopes, expressed in units of mass of a proton, is approximately
equal to one and two, respectively.  This fact indicates that the nuclei of
atoms contain, in addition to protons, neutral matter in multiples of a
basic unit.  This hypothesis was verified in 1932 by James Chadwick
who observed emission of neutral radiation when beryllium nuclei were
bombarded with alpha-particles (a-particles are helium nuclei, to be
discussed in a later section). It was found that this neutral radiation
could knock out protons from light nuclei such as those of helium, carbon
and nitrogen. The only neutral radiation known at that time was photons
(electromagnetic radiation). Application of the principles of conservation
of energy and momentum showed that if the neutral radiation consisted
of photons, the energy of photons would have to be much higher than is
available from the bombardment of beryllium nuclei with a-particles.
The clue to this puzzle, which Chadwick satisfactorily solved, was to
assume that the neutral radiation consists of a new type of neutral
particles called neutrons.  From conservation of energy and momentum,
he was able to determine the mass of new particle ‘as very nearly the
same as mass of proton’.
The mass of a neutron is now known to a high degree of accuracy. It is
m
n
 = 1.00866 u = 1.6749×10
–27 
kg (13.3)
Chadwick was awarded the 1935 Nobel Prize in Physics for his
discovery of the neutron.
A free neutron, unlike a free proton, is unstable.  It decays into a
proton, an electron and a antineutrino (another elementary particle), and
has a mean life of about 1000s.  It is, however, stable inside the nucleus.
The composition of a nucleus can now be described using the following
terms and symbols:
Z - atomic number = number of protons [13.4(a)]
N - neutron number = number of neutrons [13.4(b)]
A - mass number = Z + N
                              = total number of protons and neutrons [13.4(c)]
One also uses the term nucleon for a proton or a neutron. Thus the
number of nucleons in an atom is its mass number A.
Nuclear species or nuclides are shown by the notation X
A
Z
 where X is
the chemical symbol of the species.  For example, the nucleus of gold is
denoted by 
197
79
Au .  It contains 197 nucleons, of which 79 are protons
and the rest118 are neutrons.
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Physics
306
13.1  INTRODUCTION
In the previous chapter, we have learnt that in every atom, the positive
charge and mass are densely concentrated at the centre of the atom
forming its nucleus. The overall dimensions of a nucleus are much smaller
than those of an atom. Experiments on scattering of a-particles
demonstrated that the radius of a nucleus was smaller than the radius
of an atom by a factor of about 10
4
. This means the volume of a nucleus
is about 10
–12
 times the volume of the atom. In other words, an atom is
almost empty. If an atom is enlarged to the size of a classroom, the nucleus
would be of the size of pinhead. Nevertheless, the nucleus contains most
(more than 99.9%) of the mass of an atom.
Does the nucleus have a structure, just as the atom does?  If so, what
are the constituents of the nucleus?  How are these held together? In this
chapter, we shall look for answers to such questions. We shall discuss
various properties of nuclei such as their size, mass and stability, and
also associated nuclear phenomena such as radioactivity, fission and fusion.
13.2  ATOMIC MASSES AND COMPOSITION OF NUCLEUS
The mass of an atom is very small, compared to a kilogram;  for example,
the mass of a carbon atom, 
12
C, is 1.992647 × 10
–26 
kg. Kilogram is not
a very convenient unit to measure such small quantities. Therefore, a
Chapter Thirteen
NUCLEI
2024-25
307
Nuclei
different mass unit is used for expressing atomic masses. This unit is the
atomic mass unit (u), defined as 1/12
th
 of the mass of the carbon (
12
C)
atom. According to this definition
12
 mass of one C atom
1u = 
12
     
26
1.992647 10 kg
12
- ×
=
     
27
1.660539 10 kg
- = × (13.1)
The atomic masses of various elements expressed in atomic mass
unit (u) are close to being integral multiples of the mass of a hydrogen
atom. There are, however, many striking exceptions to this rule. For
example, the atomic mass of chlorine atom is 35.46 u.
Accurate measurement of atomic masses is carried out with a mass
spectrometer, The measurement of atomic masses reveals the existence
of different types of atoms of the same element, which exhibit the same
chemical properties, but differ in mass. Such atomic species of the same
element differing in mass are called isotopes. (In Greek, isotope means
the same place, i.e. they occur in the same place in the periodic table of
elements.) It was found that practically every element consists of a mixture
of several isotopes.  The relative abundance of different isotopes differs
from element to element. Chlorine, for example, has two isotopes having
masses 34.98 u and 36.98 u, which are nearly integral multiples of the
mass of a hydrogen atom.  The relative abundances of these isotopes are
75.4 and 24.6 per cent, respectively.  Thus, the average mass of a chlorine
atom is obtained by the weighted average of the masses of the two
isotopes,  which works out to be
= 
75.4 34.98 24.6 36.98
100
× + ×
=  35.47 u
which agrees with the atomic mass of chlorine.
Even the lightest element, hydrogen has three isotopes having masses
1.0078 u, 2.0141 u, and 3.0160 u.  The nucleus of the lightest atom of
hydrogen, which has a relative abundance of 99.985%, is called the
proton.  The mass of a proton is
27
1.00727 u 1.67262 10 kg
p
m
- = = × (13.2)
This is equal to the mass of the hydrogen atom (= 1.00783u), minus
the mass of a single electron (m
e 
= 0.00055 u).  The other two isotopes of
hydrogen are called deuterium and tritium. Tritium nuclei, being
unstable, do not occur naturally and are produced artificially in
laboratories.
The positive charge in the nucleus is that of the protons. A proton
carries one unit of fundamental charge and is stable. It was earlier thought
that the nucleus may contain electrons, but this was ruled out later using
arguments based on quantum theory. All the electrons of an atom are
outside the nucleus. We know that the number of these electrons outside
the nucleus of the atom is Z, the atomic number. The total charge of the
2024-25
Physics
308
atomic electrons is thus (–Ze), and since the atom is neutral, the charge
of the nucleus is (+Ze). The number of protons in the nucleus of the atom
is, therefore, exactly Z, the atomic number.
Discovery of Neutron
Since the nuclei of deuterium and tritium are isotopes of hydrogen, they
must contain only one proton each.  But the masses of the nuclei of
hydrogen, deuterium and tritium are in the ratio of 1:2:3.  Therefore, the
nuclei of deuterium and tritium must contain,  in addition to a proton,
some neutral matter.  The amount of neutral matter present in the nuclei
of these isotopes, expressed in units of mass of a proton, is approximately
equal to one and two, respectively.  This fact indicates that the nuclei of
atoms contain, in addition to protons, neutral matter in multiples of a
basic unit.  This hypothesis was verified in 1932 by James Chadwick
who observed emission of neutral radiation when beryllium nuclei were
bombarded with alpha-particles (a-particles are helium nuclei, to be
discussed in a later section). It was found that this neutral radiation
could knock out protons from light nuclei such as those of helium, carbon
and nitrogen. The only neutral radiation known at that time was photons
(electromagnetic radiation). Application of the principles of conservation
of energy and momentum showed that if the neutral radiation consisted
of photons, the energy of photons would have to be much higher than is
available from the bombardment of beryllium nuclei with a-particles.
The clue to this puzzle, which Chadwick satisfactorily solved, was to
assume that the neutral radiation consists of a new type of neutral
particles called neutrons.  From conservation of energy and momentum,
he was able to determine the mass of new particle ‘as very nearly the
same as mass of proton’.
The mass of a neutron is now known to a high degree of accuracy. It is
m
n
 = 1.00866 u = 1.6749×10
–27 
kg (13.3)
Chadwick was awarded the 1935 Nobel Prize in Physics for his
discovery of the neutron.
A free neutron, unlike a free proton, is unstable.  It decays into a
proton, an electron and a antineutrino (another elementary particle), and
has a mean life of about 1000s.  It is, however, stable inside the nucleus.
The composition of a nucleus can now be described using the following
terms and symbols:
Z - atomic number = number of protons [13.4(a)]
N - neutron number = number of neutrons [13.4(b)]
A - mass number = Z + N
                              = total number of protons and neutrons [13.4(c)]
One also uses the term nucleon for a proton or a neutron. Thus the
number of nucleons in an atom is its mass number A.
Nuclear species or nuclides are shown by the notation X
A
Z
 where X is
the chemical symbol of the species.  For example, the nucleus of gold is
denoted by 
197
79
Au .  It contains 197 nucleons, of which 79 are protons
and the rest118 are neutrons.
2024-25
309
Nuclei
The composition of isotopes of an element can now be readily
explained.  The nuclei of isotopes of a given element contain the same
number of protons, but differ from each other in their number of neutrons.
Deuterium, 
2
1
H , which is an isotope of hydrogen, contains one proton
and one neutron.  Its other isotope tritium, 
3
1
H , contains one proton and
two neutrons.  The element gold has 32 isotopes, ranging from A =173 to
A = 204.  We have already mentioned that chemical properties of elements
depend on their electronic structure.  As the atoms of isotopes have
identical electronic structure they have identical chemical behaviour and
are placed in the same location in the periodic table.
All nuclides with same mass number A are called isobars.  For
example, the nuclides 
3
1
H and 
3
2
He are isobars.  Nuclides with same
neutron number N but different atomic number Z, for example 
198
80
Hg
and 
197
79
Au , are called isotones.
13.3  SIZE OF THE NUCLEUS
As we have seen in Chapter 12, Rutherford was the pioneer who
postulated and established the existence of the atomic nucleus. At
Rutherford’s suggestion, Geiger and Marsden performed their classic
experiment: on the scattering of a-particles from thin gold foils. Their
experiments revealed that the distance of closest approach to a gold
nucleus of an a-particle of kinetic energy 5.5 MeV is about 4.0 × 10
–14 
m.
The scattering of a-particle by the gold sheet could be understood by
Rutherford by assuming that the coulomb repulsive force was solely
responsible for scattering. Since the positive charge is confined to the
nucleus, the actual size of the nucleus has to be less than 4.0 × 10
–14 
m.
If we use a-particles of higher energies than 5.5 MeV, the distance of
closest approach to the gold nucleus will be smaller and at some point
the scattering will begin to be affected by the short range nuclear forces,
and differ from Rutherford’s calculations. Rutherford’s calculations are
based on pure coulomb repulsion between the positive charges of the a-
particle and the gold nucleus. From the distance at which deviations set
in, nuclear sizes can be inferred.
By performing scattering experiments in which fast electrons, instead
of a-particles, are projectiles that bombard targets made up of various
elements, the sizes of nuclei of various elements have been accurately
measured.
It has been found that a nucleus of mass number A has a radius
R = R
0 
A
1/3
(13.5)
where R
0
 = 1.2 × 10
–15 
m (=1.2 fm; 1 fm = 10
–15 
m). This means the volume
of the nucleus, which is proportional to R
3
 is proportional to A. Thus the
density of nucleus is a constant, independent of A,  for all nuclei. Different
nuclei are like a drop of liquid of constant density. The density of nuclear
matter is approximately 2.3 × 10
17
 kg m
–3
. This density is very large
compared to ordinary matter, say water, which is 10
3
 kg m
–3
. This is
understandable, as we have already seen that most of the atom is empty.
Ordinary matter consisting of atoms has a large amount of empty space.
2024-25
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Physics
306
13.1  INTRODUCTION
In the previous chapter, we have learnt that in every atom, the positive
charge and mass are densely concentrated at the centre of the atom
forming its nucleus. The overall dimensions of a nucleus are much smaller
than those of an atom. Experiments on scattering of a-particles
demonstrated that the radius of a nucleus was smaller than the radius
of an atom by a factor of about 10
4
. This means the volume of a nucleus
is about 10
–12
 times the volume of the atom. In other words, an atom is
almost empty. If an atom is enlarged to the size of a classroom, the nucleus
would be of the size of pinhead. Nevertheless, the nucleus contains most
(more than 99.9%) of the mass of an atom.
Does the nucleus have a structure, just as the atom does?  If so, what
are the constituents of the nucleus?  How are these held together? In this
chapter, we shall look for answers to such questions. We shall discuss
various properties of nuclei such as their size, mass and stability, and
also associated nuclear phenomena such as radioactivity, fission and fusion.
13.2  ATOMIC MASSES AND COMPOSITION OF NUCLEUS
The mass of an atom is very small, compared to a kilogram;  for example,
the mass of a carbon atom, 
12
C, is 1.992647 × 10
–26 
kg. Kilogram is not
a very convenient unit to measure such small quantities. Therefore, a
Chapter Thirteen
NUCLEI
2024-25
307
Nuclei
different mass unit is used for expressing atomic masses. This unit is the
atomic mass unit (u), defined as 1/12
th
 of the mass of the carbon (
12
C)
atom. According to this definition
12
 mass of one C atom
1u = 
12
     
26
1.992647 10 kg
12
- ×
=
     
27
1.660539 10 kg
- = × (13.1)
The atomic masses of various elements expressed in atomic mass
unit (u) are close to being integral multiples of the mass of a hydrogen
atom. There are, however, many striking exceptions to this rule. For
example, the atomic mass of chlorine atom is 35.46 u.
Accurate measurement of atomic masses is carried out with a mass
spectrometer, The measurement of atomic masses reveals the existence
of different types of atoms of the same element, which exhibit the same
chemical properties, but differ in mass. Such atomic species of the same
element differing in mass are called isotopes. (In Greek, isotope means
the same place, i.e. they occur in the same place in the periodic table of
elements.) It was found that practically every element consists of a mixture
of several isotopes.  The relative abundance of different isotopes differs
from element to element. Chlorine, for example, has two isotopes having
masses 34.98 u and 36.98 u, which are nearly integral multiples of the
mass of a hydrogen atom.  The relative abundances of these isotopes are
75.4 and 24.6 per cent, respectively.  Thus, the average mass of a chlorine
atom is obtained by the weighted average of the masses of the two
isotopes,  which works out to be
= 
75.4 34.98 24.6 36.98
100
× + ×
=  35.47 u
which agrees with the atomic mass of chlorine.
Even the lightest element, hydrogen has three isotopes having masses
1.0078 u, 2.0141 u, and 3.0160 u.  The nucleus of the lightest atom of
hydrogen, which has a relative abundance of 99.985%, is called the
proton.  The mass of a proton is
27
1.00727 u 1.67262 10 kg
p
m
- = = × (13.2)
This is equal to the mass of the hydrogen atom (= 1.00783u), minus
the mass of a single electron (m
e 
= 0.00055 u).  The other two isotopes of
hydrogen are called deuterium and tritium. Tritium nuclei, being
unstable, do not occur naturally and are produced artificially in
laboratories.
The positive charge in the nucleus is that of the protons. A proton
carries one unit of fundamental charge and is stable. It was earlier thought
that the nucleus may contain electrons, but this was ruled out later using
arguments based on quantum theory. All the electrons of an atom are
outside the nucleus. We know that the number of these electrons outside
the nucleus of the atom is Z, the atomic number. The total charge of the
2024-25
Physics
308
atomic electrons is thus (–Ze), and since the atom is neutral, the charge
of the nucleus is (+Ze). The number of protons in the nucleus of the atom
is, therefore, exactly Z, the atomic number.
Discovery of Neutron
Since the nuclei of deuterium and tritium are isotopes of hydrogen, they
must contain only one proton each.  But the masses of the nuclei of
hydrogen, deuterium and tritium are in the ratio of 1:2:3.  Therefore, the
nuclei of deuterium and tritium must contain,  in addition to a proton,
some neutral matter.  The amount of neutral matter present in the nuclei
of these isotopes, expressed in units of mass of a proton, is approximately
equal to one and two, respectively.  This fact indicates that the nuclei of
atoms contain, in addition to protons, neutral matter in multiples of a
basic unit.  This hypothesis was verified in 1932 by James Chadwick
who observed emission of neutral radiation when beryllium nuclei were
bombarded with alpha-particles (a-particles are helium nuclei, to be
discussed in a later section). It was found that this neutral radiation
could knock out protons from light nuclei such as those of helium, carbon
and nitrogen. The only neutral radiation known at that time was photons
(electromagnetic radiation). Application of the principles of conservation
of energy and momentum showed that if the neutral radiation consisted
of photons, the energy of photons would have to be much higher than is
available from the bombardment of beryllium nuclei with a-particles.
The clue to this puzzle, which Chadwick satisfactorily solved, was to
assume that the neutral radiation consists of a new type of neutral
particles called neutrons.  From conservation of energy and momentum,
he was able to determine the mass of new particle ‘as very nearly the
same as mass of proton’.
The mass of a neutron is now known to a high degree of accuracy. It is
m
n
 = 1.00866 u = 1.6749×10
–27 
kg (13.3)
Chadwick was awarded the 1935 Nobel Prize in Physics for his
discovery of the neutron.
A free neutron, unlike a free proton, is unstable.  It decays into a
proton, an electron and a antineutrino (another elementary particle), and
has a mean life of about 1000s.  It is, however, stable inside the nucleus.
The composition of a nucleus can now be described using the following
terms and symbols:
Z - atomic number = number of protons [13.4(a)]
N - neutron number = number of neutrons [13.4(b)]
A - mass number = Z + N
                              = total number of protons and neutrons [13.4(c)]
One also uses the term nucleon for a proton or a neutron. Thus the
number of nucleons in an atom is its mass number A.
Nuclear species or nuclides are shown by the notation X
A
Z
 where X is
the chemical symbol of the species.  For example, the nucleus of gold is
denoted by 
197
79
Au .  It contains 197 nucleons, of which 79 are protons
and the rest118 are neutrons.
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309
Nuclei
The composition of isotopes of an element can now be readily
explained.  The nuclei of isotopes of a given element contain the same
number of protons, but differ from each other in their number of neutrons.
Deuterium, 
2
1
H , which is an isotope of hydrogen, contains one proton
and one neutron.  Its other isotope tritium, 
3
1
H , contains one proton and
two neutrons.  The element gold has 32 isotopes, ranging from A =173 to
A = 204.  We have already mentioned that chemical properties of elements
depend on their electronic structure.  As the atoms of isotopes have
identical electronic structure they have identical chemical behaviour and
are placed in the same location in the periodic table.
All nuclides with same mass number A are called isobars.  For
example, the nuclides 
3
1
H and 
3
2
He are isobars.  Nuclides with same
neutron number N but different atomic number Z, for example 
198
80
Hg
and 
197
79
Au , are called isotones.
13.3  SIZE OF THE NUCLEUS
As we have seen in Chapter 12, Rutherford was the pioneer who
postulated and established the existence of the atomic nucleus. At
Rutherford’s suggestion, Geiger and Marsden performed their classic
experiment: on the scattering of a-particles from thin gold foils. Their
experiments revealed that the distance of closest approach to a gold
nucleus of an a-particle of kinetic energy 5.5 MeV is about 4.0 × 10
–14 
m.
The scattering of a-particle by the gold sheet could be understood by
Rutherford by assuming that the coulomb repulsive force was solely
responsible for scattering. Since the positive charge is confined to the
nucleus, the actual size of the nucleus has to be less than 4.0 × 10
–14 
m.
If we use a-particles of higher energies than 5.5 MeV, the distance of
closest approach to the gold nucleus will be smaller and at some point
the scattering will begin to be affected by the short range nuclear forces,
and differ from Rutherford’s calculations. Rutherford’s calculations are
based on pure coulomb repulsion between the positive charges of the a-
particle and the gold nucleus. From the distance at which deviations set
in, nuclear sizes can be inferred.
By performing scattering experiments in which fast electrons, instead
of a-particles, are projectiles that bombard targets made up of various
elements, the sizes of nuclei of various elements have been accurately
measured.
It has been found that a nucleus of mass number A has a radius
R = R
0 
A
1/3
(13.5)
where R
0
 = 1.2 × 10
–15 
m (=1.2 fm; 1 fm = 10
–15 
m). This means the volume
of the nucleus, which is proportional to R
3
 is proportional to A. Thus the
density of nucleus is a constant, independent of A,  for all nuclei. Different
nuclei are like a drop of liquid of constant density. The density of nuclear
matter is approximately 2.3 × 10
17
 kg m
–3
. This density is very large
compared to ordinary matter, say water, which is 10
3
 kg m
–3
. This is
understandable, as we have already seen that most of the atom is empty.
Ordinary matter consisting of atoms has a large amount of empty space.
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310
 EXAMPLE 13.2
Example 13.1 Given the mass of iron nucleus as 55.85u and A=56,
find the nuclear density?
Solution
m
Fe
 = 55.85,    u = 9.27 × 10
–26
  kg
Nuclear density = 
mass
volume
 = 
26
15 3
9.27 10 1
56 (4 /3)(1.2 10 )
- - ×
×
p ×
 = 2.29 × 10
17 
kg m
–3
The density of matter in neutron stars (an astrophysical object) is
comparable to this density. This shows that matter in these objects
has been compressed to such an extent that they resemble a big nucleus.
13.4  MASS-ENERGY AND NUCLEAR BINDING ENERGY
13.4.1  Mass – Energy
Einstein showed from his theory of special relativity that it is necessary
to treat mass as another form of energy. Before the advent of this theory
of special relativity it was presumed  that mass and energy were conserved
separately in a reaction. However, Einstein showed that mass is another
form of energy and one can convert mass-energy into other forms of
energy, say kinetic energy and vice-versa.
Einstein gave the famous mass-energy equivalence relation
E = mc
2
(13.6)
Here the energy equivalent of mass m is related by the above equation
and c is the velocity of light in vacuum and is approximately equal to
3×10
8
 m s
–1
.
Example 13.2 Calculate the energy equivalent of 1 g of substance.
Solution
Energy, E = 10
–3 
 × ( 3 × 10
8
)
2 
J
     E = 10
–3 
× 9 × 10
16 
= 9 × 10
13 
J
Thus, if one gram of matter is converted to energy, there is a release
of enormous amount of energy.
Experimental verification of the Einstein’s mass-energy relation has
been achieved in the study of nuclear reactions amongst nucleons, nuclei,
electrons and other more recently discovered particles. In a reaction the
conservation law of energy states that the initial energy and the final
energy are equal provided the energy associated with mass is also
included. This concept is important in understanding nuclear masses
and the interaction of nuclei with one another. They form the subject
matter of the next few sections.
13.4.2  Nuclear binding energy
In Section 13.2 we have seen that the nucleus is made up of neutrons
and protons.  Therefore it may be expected that the mass of the nucleus
is equal to the total mass of its individual protons and neutrons.  However,
 EXAMPLE 13.1
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