Nuclei





Composition of Nucleus

 

The atom consists of central nucleus, containing entire positive charge and almost entire mass. According to accepted model the nucleus is composed of protons and neutrons.

The proton was discovered by Rutherford by bombardment of \alpha -particles on nitrogen in accordance with the following equation :

 

Composition of Nucleus

Composition of Nucleus


The superscripts (on the right) denote the mass number and subscripts (on the left) denote the atomic number.

The neutron was discovered by Chadwick by the bombardment of \alpha -particles on beryllium in accordance with:

 

Composition of Nucleus

Composition of Nucleus


[A neutron is neutral (zero charge) and mass number is 1].

The number of protons in a nucleus is called atomic number while the number of nucleons (i.e., protons neutrons) is called the mass number (A). In general mass number > atomic number (except for hydrogen nucleus when A = Z).

Due to being neutral, neutron is used for artificial disintegration.

 

Atomic masses

 

The masses of atoms, nuclei etc are expressed in terms of atomic mass unit (amu) represented by amu or ‘u’. For this mass of C-12 is taken as standard.

1 u = \dfrac{mass \, \, of \, \, carbon - 12 \, \, atom}{12}

 

=1.660565 \times 10^{-27} kg

 

Mass \, \, of \, \, proton ( m_p ) = 1.007276 u

 

Mass of Neutron  ( m_n ) = 1.008665 u

 

Mass of electron  ( m_e ) = 0.000549 u

 

Isotopes, isobars and isotones

 

The nuclides having same atomic number (Z) but different mass number (A) are called isotopes. The nuclides having same mass number (A), but different atomic number (Z) are called isobars.

The nuclides having same number of neutrons are called isotones. A nuclide is represented as Z^X A ,being the symbol of element.

 

Size of Nucleus

 

The size of nucleus is of the order of 10^{-14} m . Most of nuclei are spherical in shape. According to experimental observations, the radius of nucleus of atom of atomic weight ‘A’ is given by:

R = R_0 A^{1 / 3} \, \, \, Where R_0 = 1.2 \times 10^{-15} = 1.2 Fermi

 

Mass Defect and Binding Energy

 

According to Einstein the mass and energy are equivalent i.e., mass can be converted into energy and vice versa. The mass energy equivalence relation is E = mc^2 .

Accordingly 1 Kg mass is equivalent to energy:

= 1 \times ( 3 \times 10^8 ) ^2

 

= 9 \times 10^{16} Joules

 

And 1 amu = \dfrac{1}{6.02 \times 10^{26}} Kg \, \, mass is equivalent to energy 931 MeV.

 

It is observed that the mass of a nucleus is always less than the mass of constituent nucleons (i.e., protons neutrons). This difference of mass is called the mass defect. Let M (Z, A) be the mass of nucleus, m_p = the mass proton and m_n mass of neutron, then the mass defect.

 

\Delta m = Mass \, \, of \, \, nucleons \, \, - Mass \, \, of \, \, nucleus

 

= Z m_p + ( A - Z ) m_n - M ( Z , A )

 

This mass defect is in the form of binding energy of nucleus, which is responsible for binding the nucleons into a small nucleus.

Therefore,

Binding energy of nucleus =( \Delta m ) c^2 ,

And binding energy per nucleon =  \dfrac{ ( \Delta m ) c^2}{A}

 

 

Variation of binding energy per nucleon with mass number ‘A’

 

The graph in figure represents the average binding energy per nucleon in MeV against mass number A. It is observed that the binding energy for nucleon (except  _2 He^4 , _6 C^{12} \, \, and \, \, _8 O^{16} ) rises first sharply, reaches a maximum value 8.6 MeV at A = 56 and then falls slowly, decreasing to 7.6 MeV for elements of higher mass number A = 240.

 

Variation of binding energy

Variation of binding energy


Nuclear Forces

 

The protons and neutrons inside the nucleus are held together by strong attractive forces. These attractive forces cannot be gravitational since forces of repulsion between protons >> attractive gravitational force between protons. These forces are short range attractive forces called nuclear forces. The nuclear forces are strongest in nature, short range and charge independent, therefore the force between proton-proton is same as the force between neutron-neutron or proton-neutron.
Yukawa tried to explain the existence of these forces, accordingly the proton and neutron do not have independent existence between nucleus. The proton and neutron are inter convertible through negative and positive \pi - mesons .

The existence of meson gives rise to meson field which gives rise to attractive nuclear forces.

The mass of \pi -meson = 273 X mass of electron.

 

Radioactivity

 

The phenomenon of spontaneous emission of radiation (  \alpha , \beta , \gamma etc ) by certain nuclei is called radioactivity.

Laws of Radioactive Disintegration:

 

(a) Rutherford – Soddy laws:

 

(i) Radioactivity is nuclear phenomenon. It is independent of all physical and chemical conditions.

 

(ii) The disintegration is random and spontaneous. It is a matter of chance for any atom to disintegrate first.

 

(iii) The radioactive substances emit \alpha \, \, or \, \, \beta -particles along with \gamma -rays. These rays originate from the nuclei of disintegrating atom and form fresh radioactive products.

 

(iv) The rate of decay of atoms is proportional to the number of un-decayed radioactive atoms present at any instant. If N is the number of un-decayed atoms in a radioactive substance at any time t, dN the number of atoms disintegrating in time dt, the rate of decay is \dfrac{dN}{dt} so that,

- \dfrac{dN}{dt} \propto N \, \, \, or \, \, \, \dfrac{dN}{dt} = - \lambda N \cdots Equation \, \, 1

 

Where \lambda is a constant of proportionality called the decayed disintegration constant,

Equation (1) results,

N = N_0 e^{- \lambda t} \cdots Equation \, \, 2

 

Where N_0 = initial number of un-decayed radioactive atoms.

 

(b) Displacement Laws:

 

(i) When a nuclide emits \alpha -particie, its mass number is reduced by four and atomic number by two,

I.e.

 

Displacement Law

Displacement Law


(ii) When a nuclide emits a  \beta -particles, its mass number remains unchanged but atomic number increases by one.

Displacement Law

Displacement Law


Where \overline{v} is the antineutrino.

The \beta – particles is not present initially in the nucleus but is produced due to dis-integration of neutron into a proton,

I.e.

_0 n^1 \rightarrow _1 H^1 + _{-1} \beta ^0 + \overline{v} ( antineutrino )

 

When a proton is converted into a neutron, positive \beta – particles or positron is emitted.

_1 H^1 \rightarrow _0 n^1 + _1 \beta ^0 + v (neutrino )

 

(iii) When a nuclide emits a gamma photon, neither the atomic number nor the mass number changes.

 

 

Half-life and Mean life

 

 

The half-life period of a radioactive substance is defined as the time in which one-half of the radioactive substance is disintegrated. If N_0 is initial number of radioactive atoms present; then in a half-life time T, the number of un-decayed radioactive atoms will be N_0 / 2 and in next half N_0 / 4 and so on.

That is t = T ( half-life), N = \dfrac{N_0}{2}

\therefore From \, \, relation \, \, N = N_0 e^{- \lambda t} \cdots Equation \, \, 1

 

We get,

 

\dfrac{N_0}{2} = N_0 e^{- \lambda T}

 

Or \, \, \, e^{- \lambda T} = \dfrac{1}{2} \cdots Equation \, \, 2

 

From equation 1 and 2, we get,

\dfrac{N}{N_0} = e^{- \lambda t} = ( \dfrac{1}{2} ) ^{ \dfrac{t}{T}} \cdots Equation \, \, 3

 

Equation (3) is the basic equation for the solution of half-life problems of radioactive elements.

The half-life ‘T’ and disintegration constant \lambda are related as:

T = \dfrac{0.6931}{ \lambda} \cdots Equation \, \, 4

The mean life of a radioactive substance is equal to the sum of life times of all atoms divided by the number of all atoms,

I.e.

Mean life,

\tau = \dfrac{sum \, \, of \, \, life \, \, times \, \, of \, \, all \, \, atoms}{Total \, \, number \, \, of \, \, atoms} = \dfrac{1}{ \lambda} \cdots Equation \, \, 5

From equation 4 and 5, we get:

T = 0.6931 \tau \, \, I \, E \, \, T < \tau \cdots Equation \, \, 6

 

Activity of radioactive substance

 

The activity of a radioactive substance means the rate of decay (or the number of disintegrations / sec).

This is denoted by:

Activity of radioactive substance

Activity of radioactive substance


If A_0 is the activity at time t=0, then,

A_0 = \lambda N_0 \therefore \dfrac{A}{A_o} = \dfrac{N}{N_0} = e^{ - \lambda t}

I.e.

A = A_0 e^{- \lambda t} \cdots Equation \, \, 8

 

Units of radioactivity

 

(1) Curie: It is defined as the activity of radioactive substance which gives 3.7 X 10^{10}  disintegration/sec which is also equal to the radioactivity of 1 g of pure radium.

 

(2) Rutherford: It is defined as the activity of radioactive substance which gives rise to 10^6 disintegrations per second.

 

(3) Becquerell: In S.I. system the unit of radioactivity is Becquerell.

1 Becquerell = 1 disintegration/sec

 

Mass-Energy Relation

 

According to Einstein mass and energy are inter convertible through relation E = mc^2 .

This is called mass-energy equivalence relation. Accordingly 1 Kg \, \, mass = 9 \times 10{16} Joule and 1 u = 931 Mev.

Positron: It is antiparticle of electron. It has same mass but opposite charge as that of electron. It was discovered by Anderson.

Pair production: When a photon of high energy (greater than 1.02 MeV) approaches a heavy nucleus, its energy is converted into mass and a pair of particles electron and positron is produced. This phenomenon is called pair production.

Pair Annihilation: When an electron and positron come near each other, their whole mass is converted into energy in the form of two photons. These photons travel in opposite directions to conserve momentum. This phenomenon is called pair annihilation.

 

 

Nuclear Fission

 

The splitting of heavy nucleus into two or more fragments of comparable masses, with an enormous release or energy is called nuclear fission. For example when slow neutron are bombarded on _{92} U^{235} , the fission takes place according to reaction,

 _{92} U^{235} + _0 n^1 \rightarrow _{56} Ba^{141} + _{36} Kr^{92} + 3 ( _o n^1 ) + 200 MeV

In nuclear fission the sum of masses before reaction is greater than the sum of masses after reaction, the difference in mass being released in the form of fission energy.

 

Remarks:

1. It may be pointed out that it is not necessary that in each fission of uranium, the two fragments are Ba^{141} \, \, and \, \, Kr^{92} are formed but they may be any stable isotopes of middle weight atoms. The most probable division is into two fragments containing about 40% and 60% of the original nucleus with the emission of 2 or 3 neutrons per fission.

2. The fission of U^{238} takes place by fast neutrons.

 

 

Chain reaction

 

If on the average more than one of the neutrons produced in each fission are capable of causing further fission, the number of fission taking place at successive stages goes on increasing at a rapid rate, giving rise to self-sustained sequence of fission known as chain reaction. The chain reaction takes place only if the size of the fissionable material is greater than a certain size called the critical size. There are two types of chain reactions.

 

(1) Uncontrolled chain reaction: In this process the number of fissions in a given interval on the average goes on increasing and the system will have the explosive tendency. This forms the principle of atom bomb.

 

Controlled chain reaction

 

In this process the number of fissions in a given interval is maintained constant by absorbing a desired number of neutrons. This forms the principle of nuclear reactor, consisting of the following parts:

(i) Fuel: This fuel is U^{235} \, \, or \, \, Pu^{239} \, \, or \, \, U_{233}

(ii) Moderator: A moderator is a suitable material to slow down neutrons produced in the fission. The best choices as moderators are heavy water ( D_2 0 ) and graphite (C).

(iii) Controller: To maintain the steady rate of fission, the neutron absorbing material known as controller is used. The control rods are made of cadmium or boron-steel.

(iv) Coolant: To remove the considerable amount of heat produced in the fission process, suitable cooling fluids, known as coolants are used. The usual coolants are water, carbon-dioxide, air etc.

(v) Reactor shield: The intense neutrons and gamma radiations produced in nuclear reactors are harmful for human body. To protect the workers from these radiations, the reactor core is surrounded by concrete, wall, called the reactor shield.

 

 

Nuclear Fusion

 

The phenomenon of combination of two or more light nuclei to form a heavy nucleus with release of enormous amount of energy is called the nuclear fusion. The sum of masses before fusion is greater than the sum of masses after fusion, the difference in mass appearing as fusion energy.

For example, the fusion of two deuterium nuclei into helium is expressed as

_1 H62 + _1 H^2 \rightarrow _2 H^4 + 21.6 MeV

Thus fusion process occurs at extremely high temperature and high pressure just at sun where temperature is 10^7 K .

 

Remarks:

1. For the fusion to take place, the component nuclei must be within a distance of 10^{-14} m m. For this they must be imparted high energies to overcome the repulsive force between nuclei. This is possible when temperature is enormously high.

2. The principle of hydrogen bomb is also based in nuclear fusion.

3. The source of energy of sun and other star’s is nuclear fusion.

 

There are two possible cycles:

(1) Proton – proton cycle:

 

Proton – proton cycle

Proton – proton cycle


 

(2) Carbon cycle:

Carbon cycle

Carbon cycle


Carbon cycles

Carbon cycles


The proton-proton cycle occurs at relatively lower temperature as compared to carbon cycle which has greater efficiency at higher temperature.

At the sun whose interior temperature is about 2 \times 10^6 K , the proton-proton cycle has more chances for occurrence.

 

 

Nuclear Holocaust

 

The estimate of aftereffect of an atomic or nuclear explosion is called the nuclear holocaust. If a fusion bomb (causing the fusion of isotopes of hydrogen, deuterium and tritium) explodes; then the nuclear holocaust will not only destroy every form life on earth but will also make this planet (earth) unfit for life for all times. The radioactive waste will hang like a cloud in earth’s atmosphere and will absorb sun’s radiations, thus causing a long nuclear winter.

On August 6, 1945 USA dropped an atom bomb on Hiroshima (Japan) which produced an explosion equivalent to 20,000 tons of TNT and the entire population of that place was either killed or seriously affected.



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