Concept Review Ch.46 The Nucleus

JEE 2008 syllabus topics

Atomic nucleus 46.1;
Alpha, beta and gamma radiations 46.4, 46.6;
Law of radioactive decay 46.4,46.5;
Decay constant;
Half-life and mean life 46.5;
Binding energy and its calculation 46.3;
Fission 46.8, and
fusion processes 46.10;
Energy calculation in these processes.


Properties of Nucleus

A nucleus is made of protons and neutrons.

Studies have shown that average nucleus R of a nucleus may be written as

R = R₀A^(1/3) .. (46.1)

where R₀ = 1.1*10^-15 m ≈ 1.1 fm and A is the mass number

Density within a nucleus is independent of A.

Nuclear Forces

When nucleons are kept at a separation of the order of femtometre (10^-15 m), a new kind of force, called nuclear force starts acting.

Binding energy

If the constituents of a hydrogen atom (a proton and an electron) are brought from infinity to form the atom, 13.6 3V of energy is released. Thus, the binding energy of a hydrogen atom in ground state is 13.6 eV. Also 13.6 eV energy must be supplied to the hydrogen atom in ground state to separate the constituents to large distances.

Similarly, the nucleons are bound together in a nucleus and energy must be supplied to the nucleus to separate the constituent nucleons to large distances. The amount of energy needed to do this is called the binding energy of the nucleus. If nucleons are brought together to form the nucleus from large separation this much energy is released.

It is evident from the above discussion that the rest mass energy of a nucleus is smaller than the rest mass energy of its constituents.



Radioactive Decay

Two main processes by which an unstable nucleus decays are alpha decay and beta decay.

Alpha decay

In alpha decay, the unstable nucleus emits an alpha particle reducing its proton number Z as well as its neutron number N by 2. As the proton number is changed, the element itself is changed and hence the chemical symbol of the residual nucleus is different from that of the original nucleus (Parent nucleus is original nucleus and the resulting nucleus due to decay is called daughter nucleus).

Alpha decay occurs in all nuclei with mass number A>210.

Beta Decay

Beta decay is a process in which either a neutron is converted into a proton or a proton is converted into a neutron.
When a neutron is converted into a proton, an electron and a new particle named antineutrino are created and emitted from the nucleus. The electron emitted from the nucleus is called a beta particle and is denoted by the symbol β^-.


If the unstable nucleus has excess protons than needed for stability, a proton converts itself into a neutron. In the process, a positron and a neutrino are created and emitted from the nucleus.

Positron is represented by e⁺. The neutrino is represented by ν.

When a positron and electron collide, both the particles are destroyed and energy is made available.

The decay which gives beta rays consisting of positrons is called beta plus decay


Electron capture

A nucleus captures one of the atomic electrons, most likely an electron from the K shell, and a proton in the nucleus combines with the electron and converts itself into a neutron. A neutrino is created in process and emitted from the nucleus. So a combination of proton and electron results in neutron and neutrino.

Gamma Decay

When a daughter nucleus is formed due to alpha or beta decay, the nucleus may be at higher energy level compared to its ground or normal state. The electromagnetic radiation emitted in nuclear transitions from higher energy or excited state to ground state is called gamma ray,

Law of radioactive decay

N = N₀e^{- λ t}

where
N = number of active nuclei at time t

N₀ = number of active nuclei at t = 0.

λ = decay constant


-dN/dt = λN

-dN/dt gives the number of decays per unit time and is called the activity (A) of the sample

A = λN

A = A₀e^{- λ t}

Unit of Activity

The activity of a radioactive material is measured in terms of the disintegrations per unit time. Its SI unit is Becquerel which is the same as 1 disintegration/second. It is denoted by the symbol Bq.

However, a large unit Curie is the popular unit used.

1 curie = 3.7*10^10 disintegrations/s.

Curie is represented by the symbol Ci.

The activity per unit mass is called specific activity.


Half life:

The time elapased before half the active nuclei decay is called half-life and is denoted by t_1/2.

t_1/2. = 0.693/ λ

where
λ = decay constant.

Average life of the nuclei of a material

t_av. = t_1/2/0.693

Properties and Uses of Nuclear Radiation

Alpha Ray
1. Each particle contains two protons and two neutrons. It is a helium nucleus.
2. It is made of positive particles and hence deflected by electric field as well as magnetic field.
3. Its penetrating power is low. Few cm in air also.
4. They travel at large speeds of the order of 10^6 m/s.
5. All particles from a source and decay scheme have the same energy.
6. Alpha rays produce scintillation (flashes of light) when they strike certain fluorescent materials such as barium platinocynide.
7. It causes ionization in gases.




Beta ray

1. It is a stream of electrons. Electrons are created during nuclear transformation.
2. They are negative particles and hence deflected by electric as well as magnetic fields.
3. Penetrating power greater than alpha rays. They can travel several meters in air before its intensity drops down to small values.
4. Ionizing power is less than alpha rays.
5. beta rays also produced scintillation but it is weak.
6. The energy of particles is not uniform as they share energy with antineutrinos. Energy of beta particles varies from zero to a maximum


Beta plus ray

It has all the properties of beta rays or beta negative rays, except that it is made of positively charged particles.

Gamma Ray

1. Gamma ray is an electromagnetic radiation of short wavelength. Its wavelength is shorted than X-rays.
2. Many properties are similar to X-rays.
3. As there is no charge no deflection in electric or magnetic fields.
4. All the photons coming from a particular gamma decay scheme has the same energy.
5. As it is electromagnetic wave, gamma ray travels with the velocity ‘c’ in vacuum.

Energy from the Nucleus

Nuclear energy may be obtained either by breaking a heavy nucleus into two nuclei of middle weight (fission) or by combining two light nuclei to form a middle weight nucleus (fusion).

Reason: The middle weight nuclei are more tightly bound than heavy weight nuclei. When the nucleons of a heavy nucleus regroup in two middle weight nuclei called fragments the total binding energy increases and the rest mass energy decreases. The difference in energy appears as the kinetic energy of the fragments or in some other form.

In the case of fusion, the light weight nuclei are less tightly bound than the middle weight nuclei. Therefore, if two light weight nuclei combine, the binding energy increases and the rest mass decreases. Energy is released in the form of kinetic energy or in some other external form.

Nuclear Fission Process

The rest mass energy of the heavy nucleus represented by E1 is greater than the rest mass energy of the fragments represented by E3. But the energy level of the heavy nucleus is to be increased to E2 to get the fission process started according to classical physics.

But according to quantum mechanics, fission can take place even if no external energy is given. Such a fission process is termed as barrier penetration. The amount of energy created and the time for which it is created through a barrier penetration process are related through Heisenberg uncertainty relation.

∆E. ∆t ≈ h/2 π

Where h is the Planck constant

Barrier penetration is possible but is not easy.

Uranium Fission reactor

The most attractive way of using fission reaction to produce energy is to use ₉₂²³⁶U as the fission material. Natural uranium contain about 99.3% of ₉₂²³⁸U and 0.7% of ₉₂²³⁵U. The technique used is hit the natural uranium sample by slow moving neutrons whose kinetic energy is 0.04 eV (also called thermal neutrons). There is a large probability that a ₉₂²³⁵U nucleus absorbing a slow neutron and forming ₉₂²³⁶U nucleus.

₉₂²³⁶U so formed then fissions into two parts. The two fragments could be different combinations. Two of the possible are combinations are:

₉₂²³⁶U --> ₅₃¹³⁷I + ₃₉⁹⁷Y+2n … (i)

₉₂²³⁶U --> ₅₆¹⁴⁰Ba + ₃₆⁹⁴Kr + 2n … (ii)

During a fission event, two or three neutrons are emitted (we showed both the examples that give 2 neutrons). The average comes out to 2.47.

₅₃¹³⁷I further transforms through beta negative decay into ¹³⁷Xe and emits a neutron to become ¹³⁶Xe.

Such neutrons released are called delayed neutrons.

In each fission event about 200 MeV of energy is released a large part of which appears in the form of kinetic energies of the two fragments.


Chain reaction

Each fission is producing 2.47 neutrons on average or 2 or 3 neutrons. If these neutrons are absorbed by ₉₂²³⁵U nucleus they cause more fissions and the rate of fission can increases in geometrical progression and the entire material is consumed in fission in a small time. On the other hand if the neutrons released in fission escape outside the material no further fission takes place.

Also the neutrons releases are at fast neutrons with kinetic energies of 2 MeV where as slow neutrons of kinetic energy 0.04 eV are required to transform ₉₂²³⁵U into ₉₂²³⁶U. The neutrons slow down due to collisions within the material. But neutrons of kinetic energy 1-100 eV can be absorbed by ₉₂²³⁸U and a different fission reaction occurs.

Design of Nuclear reactor

A nuclear reactor is designed to have the fission reaction in a controlled manner controlling for the above two alternatives. One a fast chain reaction and the other termination of the reaction or unwanted fission reaction.

In the specially designed reactor uranium in the form cylindrical rods is arranged in a regular pattern and this pattern is filled with a low-Z material such as heavy water (D₂O), graphite or beryllium etc. These materials act as moderator. After the fission reaction neutrons escape into the moderator as the fission reaction occurs at the surface of cylindrical uranium rods. Due to collisions with deuteron molecules (heavy water) after 25 collisions, these neutrons slow down to thermal neutron energies. The distance between uranium rods is so adjusted that by the time neutrons reach the next rod, their energy is reduced to the required 0.04 eV. Also out of the 2 or three neutrons produced only one will reach the uranium rod and the other are absorbed. Moveable cadmium rods are placed between the uranium rods to control the reaction further. Cadmium is good absorber of neutrons. Hence if cadmium rods are totally inserted into the space between uranium rods, the fission reaction will totally stop. When they are retraced fission reaction can start once again. Some coolant liquid such as water at high pressure is passed through the reactor core area housing uranium rods, moderator and cadmium rods and its absorbs the heat produced in the fission and this heat is used to prepare steam from water.

As ₉₂²³⁵U is used up only ₉₂²³⁸U will remain in the uranium rod and then it needs to be changed.


Breeder Reactors

₉₂²³⁸U can capture neutrons and become
₉₂²³⁹U. On B radiation it becomes
₉₃²³⁹Np. On beta radiation it becomes
₉₄²³⁹Pu.


₉₄²³⁹Pu when hit by neutron becomes ₉₄²⁴⁰Pu, which is a fissionable material. Thus if out of the 2.47 neutrons produced on average in fission reaction, one neutron is absorbed by ²³⁸U we produce fuel equivalent to what is consumed. Such a reactor is called breeder reactor.


Nuclear Fusion

For he light nuclei to come together with in a distance of 1 fm (femtometer), we need a temperature of the order of 10^9 K. At that temperature electrons are completely detached from atoms and only nuclei remain. It is called plasma. In Sun, the temperature is 1.5*10^7 K and fusion is taking place. So fusion can take place due to barrier penetration process at 10^7 K.

Fusion in Laboratory

The major problem on earth for fusion reaction is holding plasma at high temperature for extended period of time.

.Lawson criterion for fusion reactor

In order to get an energy output greater than the energy input, a fusion reactor should achieve

n τ >10^14 s/cm³

where
n = the density of the interacting particles

τ = confinement time

the quantity n τ in s/cm³ is called Lawson number

Tokamak Design

In this design, the deuterium plasma is contained in a toroidal region by specially designed magnetic field. The directions and magnitudes of the magnetic field are so managed in the toroidal space that whenever a charge plasma particle attempts to go out qv×B force tends to push it back into the toroidal volume.

With such designs, confinement of the plasma has been achieved for short duration of few microseconds.

A large fusion machine known as Joint European Torus (JET) is designed to achieve fusion energy on this principle.

At the Institute of Plasma Research (IPR) Ahmedabad, a small machine named Aditya is functioning on the Tokamak design. This machine is being used to study properties of plasma.

Inertial Confinement

It is an alternate method of confinement of plasma. A small solid pellet is made that contains deuterium and tritium. Intense laser beams are directed on the pellet from many directions on all the sides. The laser vaporizes the pellet converting it into plasma and then compresses it. The density increases by 10^3 to 10^4 time the initial density and temperature raises to high values. Fusion occurs in these conditions. In this method also so far, confinement for very small duration is only achieved.

Research into fusion energy is going on. The source of fuel is water only and water is abundant in oceans. Also these reactions do not result in radioactive emissions like that of fission.