Chapter Thirteen��NUCLEI

Prasad K R

PGT PHYSICS

JNV PERIYE KASARAGOD

KERALA

ATOMIC NUCLEUS

1. Composition of Nucleus
2. Atomic Number, Mass Number and Atomic Mass Unit
3. Radius of the Nucleus and Nuclear Density
4. Mass Energy Relation and Mass Defect
5. Binding Energy and Binding Energy per Nucleon
6. Binding Energy Curve and Inferences
7. Nuclear Forces and Meson Theory
8. Radioactivity and Soddy’s Displacement Law
9. Rutherford and Soddy’s Laws of Radioactive Decay
10. Radioactive Disintegration Constant and Half-Life Period
11. Units of Radioactivity
12. Nuclear Fission and Fusion

Composition of Nucleus:

Every atomic nucleus except that of Hydrogen has two types of particles – protons and neutrons. (Nucleus of Hydrogen contains only one proton)

Proton is a fundamental particle with positive charge 1.6 x 10^-19 C and mass 1.67 x 10^-27 kg (1836 times heavier than an electron).

Neutron is also a fundamental particle with no charge and mass 1.675 x ^10-27 kg (1840 times heavier than an electron).

​

Atomic Number (Z):

The number of protons in a nucleus of an atom is called atomic number.

​

Atomic Mass Number (A):

The sum of number of protons and number of neutrons in a nucleus of an atom is called atomic mass number.

A = Z + N

Atomic Mass Unit (amu):

Atomic Mass Unit (amu) is (1 / 12)th of mass of 1 atom of carbon.

1 12

12 6.023 x 10²³

1 amu = x g

= 1.66 x 10^-27 kg

Size of Nucleus:

Nucleus does not have a sharp or well-defined boundary. However, the radius of nucleus can be given by

R = R₀ A^⅓

where R₀ = 1.2 x 10^-5 m is a constant which is the same for all nuclei and

A is the mass number of the nucleus.

Radius of nucleus ranges from 1 fm to 10 fm. Nuclear Volume, V = (4/3) π R³ = (4/3) π R₀³ A

V α A

Nucleus Density:

Mass of nucleus, Nuclear Volume,

M = A amu = A x 1.66 x 10^-27 kg V = (4/3) π R³ = (4/3) π R₀ A

3

4 22

=

x x (1.2 x 10^-15)³ A m³

3 7

= 7.24 x 10^-45 A m³

Nucleus Density, ρ = M / V = 2.29 x 10¹⁷ kg / m³

Discussion:

1. The nuclear density does not depend upon mass number. So, all the nuclei possess nearly the same density.
2. The nuclear density has extremely large value. Such high densities are found in white dwarf stars which contain mainly nuclear matter.
3. The nuclear density is not uniform throughout the nucleus. It has maximum value at the centre and decreases gradually as we move away from the centre of the nucleus.
4. The nuclear radius is the distance from the centre of the nucleus at which the density of nuclear matter decreases to one-half of its maximum value at the centre.

Mass Defect:

It is the difference between the rest mass of the nucleus and the sum of the masses of the nucleons composing a nucleus is known as mass defect.

Δm = [ Zm_p + (A – Z) m_n ] - M

Mass defect per nucleon is called packing fraction.

Binding Energy:

It is the energy required to break up a nucleus into its constituent parts and place them at an infinite distance from one another.

B.E = Δm c²

Nuclear Forces:

They are the forces between p – p, p – n or n – n in the nucleus. They can be explained by Meson Theory.

This force is nuclear force (strongest force in nature). It is strong enough to overcome the repulsion between the (positively charged) protons and to bind both protons and neutrons into the tiny nuclear volume.

​

Binding Energy per Nucleon:

It is the binding energy divided by total number of nucleons. It is denoted by B

B = B.E / Nucleon = Δm c² / A

​

​

Binding Energy Curve:

0 20 40

80 100 120 140 160 180 200 220 240

Mass Number (A)

Average B.E per Nucleon (in MeV)

6

7

5

1

4

3

2

9

8.8

8

Region of maximum stability

Fission

Fusion

.

1

56 ₆₀

Li⁷ Li⁶

_Be11

_C12

He⁴

_F19 _N14

Be⁹

_O16

Ne

20

_Al27

_Cl35

_Ar40

_Fe56

_Mo98

_Xe124

_Xe136

Xe

130

_As75

_Sr86

Cu

63

_W182

_Pt208

_U235

_U238

Pt

194

_H1

_H2

_H3

He³

Special Features:

1. Binding energy per nucleon of very light nuclides such as ₁H² is very small.
2. Initially, there is a rapid rise in the value of binding energy per nucleon.
3. Between mass numbers 4 and 20, the curve shows cyclic recurrence of peaks corresponding to ₂He⁴, ₄Be⁸, ₆C¹², ₈O¹⁶ and ₁₀Ne²⁰. This shows that the

B.E. per nucleon of these nuclides is greater than those of their immediate meighbours. Each of these nuclei can be formed by adding an alpha particle to the preceding nucleus.

4. After A = 20, there is a gradual increase in B.E. per nucleon. The maximum

value of 8.8 MeV is reached at A = 56. Therefore, Iron nucleus is the most stable.

5. Binding energy per nucleon of nuclides having mass numbers ranging from 40 to 120 are close to the maximum value. So, these elements are highly stable and non-radioactive.
6. Beyond A = 120, the value decreases and falls to 7.6 MeV for Uranium.
7. Beyond A = 128, the value shows a rapid decrease. This makes elements beyond Uranium (trans – uranium elements) quite unstable and radioactive.
8. The drooping of the curve at high mass number indicates that the nucleons are more tightly bound and they can undergo fission to become stable.
9. The drooping of the curve at low mass numbers indicates that the nucleons can undergo fusion to become stable.

Some important features of the nuclear binding force are given below:

1. The nuclear force is much stronger than the Coulomb force acting between charges or the gravitational forces between masses. The nuclear binding force has to dominate over the Coulomb repulsive force between protons inside the nucleus. This happens only because the nuclear force is much stronger than the coulomb force. The gravitational force is much weaker than even Coulomb force.
2. (ii) The nuclear force between two nucleons falls rapidly to zero as their distance is more than a few femtometres. This leads to saturation of forces in a medium or a large-sized nucleus, which is the reason for the constancy of the binding energy per nucleon. A rough plot of the potential energy between two nucleons as a function of distance is shown in the figure given below. The potential energy is a minimum at a distance r0 of about 0.8 fm. This means that the force is attractive for distances larger than 0.8 fm and repulsive if they are separated by distances less than 0.8 fm.
3. (iii) The nuclear force between neutron-neutron, proton-neutron and proton-proton is approximately the same. The nuclear force does not depend on the electric charge.

Radioactivity:

• Rutherford and Soddy’s Laws of Radioactive Decay:
• The disintegration of radioactive material is purely a random process and it is merely a matter of chance. Which nucleus will suffer disintegration, or decay first can not be told.
• The rate of decay is completely independent of the physical composition and chemical condition of the material.
• The rate of decay is directly proportional to the quantity of material actually present at that instant.As the decay goes on, the original material goes on decreasing and the rate of decay consequently goes on decreasing.

Lead Box

Radioactive substance

α

β

γ

-

-

-

-

-

-

-

-

-

-

-

+

+

+

+

+

+

+

+

+

+

Radioactivity is the phenomenon of emitting alpha, beta and gamma radiations spontaneously.

Soddy’s Displacement Law:

1.

_ZYA _Z-2YA-4

α

2.

_ZYA

Z+1^YA

β

3.

Z Z

_YA _YA

(Lower energy)

γ

If N is the number of radioactive atoms present at any instant, then the rate of decay is,

dt

_- dN _{α N}

dN

dt

or -

= λ N

N

where λ is the decay constant or the disintegration constant.

Rearranging,

dN

= - λ dt

Integrating,

log_e N = - λ t + C

where C is the integration constant.

If at t = 0, we had N₀ atoms, then

log_e N₀ = 0 + C

log_e N - log_e N₀ = - λ t

or

or log_e (N / N₀) = - λ t

N

_{= e}- λt

N

0

or

N = N₀ e^{- λ t}

No. of atoms (N)

N₀

Time in half lives

N₀/2

N₀/4 N₀/8

N₀/16 ₀

T 2T 3T 4T

Radioactive Disintegration Constant (λ):

N

According to the laws of radioactive decay, dN

= - λ dt

N

If dt = 1 second, then

dN

= - λ

Thus, λ may be defined as the relative number of atoms decaying per second.

​

Again, since N = N₀ e^{- λ t}

And if, t = 1 / λ, then

N = N₀ / e

0

N

or =

N e

1

Thus, λ may also be defined as the reciprocal of the time when N / N₀ falls to 1 / e.

Half – Life Period:

Half life period is the time required for the disintegration of half of the amount of the radioactive substance originally present.

If T is the half – life period, then

N₀ 2

N 1

_{= e} - λ T

_e λ T

= 2

= (since N = N₀ / 2)

λ T = log_e 2 = 0.6931

T =

λ

0.6931

T

0.6931

or λ =

Time t in which material changes from N₀ to N:

t = 3.323 T log₁₀ (N₀ / N)

Number of Atoms left behind after n Half – Lives:

N = N₀ (1 / 2)ⁿ N = N₀ (1 / 2)^t/T

or

Units of Radioactivity:

1. The curie (Ci): The activity of a radioactive substance is said to be one curie if it undergoes 3.7 x 10¹⁰ disintegrations per second.

1 curie = 3.7 x 10¹⁰ disintegrations / second

2. The rutherford (Rd): The activity of a radioactive substance is said to be one rutherford if it undergoes 10⁶ disintegrations per second.

1 rutherford = 10⁶ disintegrations / second

3. The becquerel (Bq): The activity of a radioactive substance is said to be one becquerel if it undergoes 1 disintegration per second.

​

1 becquerel = 1 disintegration / second

1 curie = 3.7 x 10⁴ rutherford = 3.7 x 10¹⁰ becquerel

Chain Reaction:

n = 1

N = 1

n = 2

N = 9

n = 3

N = 27

Neutron (thermal) ₀n¹

Uranium

_92U235

Barium

Krypton

_56Ba141

_36Kr92

n = No. of fission stages N = No. of Neutrons

N = 3ⁿ

Nuclear Fission:�Nuclear fission is defined as a type of nuclear disintegration in which a heavy nucleus splits up into two nuclei of comparable size accompanied by a release of a large amount of energy.���

₀n¹ + ₉₂U²³⁵ → (₉₂U²³⁶) → ₅₆Ba¹⁴¹ + ₃₆Kr⁹² +3₀n¹ + γ (200 MeV)

Chain Reaction:

n = 1 n = 2

N = 1 N = 9

n = 3

N = 27

Critical Size:

For chain reaction to occur, the size of the fissionable material must be above the size called ‘critical size’.

A released neutron must travel minimum through 10 cm so that it is properly slowed down (thermal neutron) to cause further fission.

If the size of the material is less than the critical size, then all the neutrons are lost.

If the size is equal to the critical size, then the no. of neutrons produced is equal to the no. of neutrons lost.

If the size is greater than the critical size, then the reproduction ratio of neutrons is greater than 1 and chain reaction can occur.

Nuclear Fusion:

Nuclear fusion is defined as a type of nuclear reaction in which two lighter nuclei merge into one another to form a heavier nucleus accompanied by a release of a large amount of energy.

Energy Source of Sun:

Proton – Proton Cycle:

_1e0

₁H¹ + ₁H¹ → ₁H² +

₁H¹ + ₁H² → ₂He³

₂He³ + ₂He³ → ₂He⁴

+ 2 ₁H¹

+ 0.4 MeV

+ 5.5 MeV

+ 12.9 MeV

+

_1H1

+

_1e0

Energy Source of Star:

Carbon - Nitrogen Cycle:

+

+

_1H1

_1H1

+

_1e0

+ γ (energy) (positron)

+ γ (energy)

+ γ (energy) (positron)

_6C12

_7N13

_6C13

_7N14

_8O15

_7N15

+

_1H1

→

→

→

→

→

→

_7N13

_6C13

_7N14

_8O15

_7N15

_6C12

+ ₂He⁴ + γ (energy)

End of Atomic Nucleus

NUCLEAR REACTOR

Nuclear reactors have huge complex structures and great care has to be exercised in designing them. The basic principle of a nuclear power plant is very simple and analogous to any power plant. The heat liberated in fission is used to produce steam at high pressure and high temperature by circulating a coolant, say water, around the fuel. (In a coal fired station, coal is burnt to produce steam. Since one fission event generates about 7 ×105 times more energy than that produced in burning one atom of carbon, we can cut down on emission of greenhouse gases substantially by switching over to nuclear energy. However, there are some complex social and political issues with global dimensions that will ultimately decide our ultimate nuclear energy options.)

The steam runs a turbine–generator system to produce electricity. (In research reactors, the heat is discharged into a river or sea. You many have heard about Bhahha Atomic Research Centre at Trombay, Mumbai or Indira Gandhi Atomic Research Centre at Kalpakkam. The heat generated by the research reactors at these centres is discharged into the Arabian sea and the Bay of Bengal, respectively.)

A reactor core, where fission takes place resulting in release of energy. It has fuel rods (embedded in a moderator in a thermal reactors), and control rods to maintain the chain reaction at the desired level. Coolant is circulated to remove the heat generated in fission. Usually, heavy water or ordinary water are used as coolants and cadmium or boron are used for control rods.

A reflector is put next to the core to stop neutron leakage from the core.

The whole assembly is placed inside a vessel, called pressure vessel. Usually, a few inches thick stainless steel is used for this purpose.

A thick shield is provided to protect the scientists and other personnel working around the reactor from radiations coming from the reactor core. It is usually in the form of a thick concrete wall.

All nuclear reactors consist of:

The entire structure is placed inside a reactor building. It is air tight and is maintained at a pressure slightly less than the atmospheric pressure so that no air leaks out of the building.

​

The heat generated inside the reactor core of a reactor due to fission is removed by circulating a coolant. The heated coolant is made to give up its heat to a secondary fluid, usually water in a heat exchanger. This generates steam, which is used to drive turbine-generator system to produce electricity in a power plant and discharged into a river/lake/sea in a research reactor.