Nuclei

 1. 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 α-particles on nitrogen in accordance with the following equation:

N714(Nitrogen)+He24(α−particle)→O817Oxygen+H11Proton

The superscripts (on the top) denote the mass number and subscripts (in the base) denote the atomic number. Symbolically a nuclide is written as XZAorZXA, where A is the mass number and Z is the atomic number.

The neutron was discovered by J. Chadwick by the bombardment of α-particles on beryllium in accordance with

Be49(Beryllium)+He24(α−particle)→C613(Carbon)+n01(Neutron)

A neutron is neutral (zero charge) particle and its 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 where A = Z).

Since neutron is neutral, it is used for artificial disintegration.

2. Size of Nucleus

According to experimental observations, the radius of the nucleus of an atom of mass number A is

R = R0A1/3 where R0 = 1.2×10–15 m = 1.2 fm

3. Atomic Masses

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

1 u = massofcarbon–12atom12

= 1.660565×10–27 kg

mass of proton (mp) = 1.007276 u

mass of neutron (mn) = 1.008665 u

mass of electron (me) = 0.000549 u

4. Isotopes, Isobars and Isotones

The nuclides having the same atomic number (Z) but different mass number (A) are called isotopes. The nuclides having the same mass number (A), but different atomic number (Z) are called isobars. The nuclides having the same number of neutrons (A–Z) are called isotones.

5. Nuclear Instability: Radioactivity

Becquerel discovered that some heavy nuclei (A >180 like radium) are unstable and spontaneously decay into other elements by the emission of certain radiations: α, β and γ-radiations. This phenomenon is called radioactivity.

6. Properties of α,β and γ-Radiations

α-particles: (i) α-particles are helium nuclei, so they have positive charge +2e and mass nearly four times the mass of proton.

(ii) On account of positive charge, α-particles are deflected by electric and magnetic fields.

(iii) α-particles have strong ionizing power.

(iv) α-particles have small penetrating power.

(v) α-particles are scattered by metallic foils (eg, gold foils).

(vi) α-particles produce fluorescence in some substances like zinc sulphide.

(vii) α-particles affect photographic plate feebly.

β-particles: (i) β-particles are fast moving electrons.

(ii) The speed of β-particles is very high ranging from 0.3 c to 0.98 c (c = speed of light in vacuum).

(iii) β-particles carry negative charge equal to – e = – 1.6×10–19 C; so they are deflected by electric and magnetic fields opposite to the direction of deflection of α-particles.

(iv) β-particles have small ionising power (100 times smaller than α-particles)

(v) β-particles have large penetrating power (100 times larger than α-particles)

(vi) β-particles cause fluorescence.

(vii) β-rays are similar to cathode rays.

γ-Rays: (i) γ-rays are electromagnetic radiations, of wavelength 0·01 Å.

(ii) γ-rays are neutral, so they are not affected by electric and magnetic fields.

(iii) γ-rays travel in vacuum with the speed of light.

(iv) γ-rays have the highest penetrating power.

(v) γ-rays have the least ionising power.

(vi) γ-rays are similar to X-rays

7. Radioactive Decay Laws

Rutherford-Soddy law

(i) Radioactivity is a 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 α or β-particles along with γ-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 undecayed radioactive atoms present at any instant. If N is the number of undecayed atoms in a radioactive substance at any time t, dN the number of atoms disintegrating in time dt the rate of decay is dNdt so that

–dNdt∝NordNdt=–λN ...(i)

where λ is a constant of proportionality called the decay (or disintegration) constant,

Equation (i) results

N = N0 e–λt …(ii)

where N0 initial number of undecayed radioactive atoms.

8. Radioactive Displacement Laws

(i) When a nuclide emits an α-particle, its mass number is reduced by four and atomic number by two,

i.e., XZA→YZ–2A–4+He24+Energy

(ii) When a nuclide emits a β-particle, its mass number remains unchanged but atomic number increases by one,

i.e., XZA→YZ+1A+β–10+ν−+ Energy,

where ν− is the antineutrino.

The β-particles are not present initially in the nucleus but are produced due to the disintegration of neutron into a proton,

i.e., n01→H11+β–10+ν− (antineutrion)

When a proton is converted into a neutron, positive β-particle or positron is emitted.

H11→n01+β–10+ν (neutrino)

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

9. 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 N0 is the initial number of radioactive atoms present, then in a half life time T, the number of undecayed radioactive atoms will be N0 / 2 and in next half N0 / 4 and so on.

That is t = T (half-life), N=N02

∴ From relation N = N0 e –λT …(i)

we get, N02=N0e–λTore–λT=12 …(ii)

From equations (i) and (ii), we get

NN0=e–λt=(12)t/T …(iii)

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

The half-life T and disintegration constant λ are related as

T=0.6931λ …(iv)

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

i.e., Mean life,

τ=sumoflifetimeofallatomstotalnumberofatoms=1λ …(v)

From equations (iv) and (v), we get

T = 0.6931 τ i.e., T < τ …(vi)

10. 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

A=∣∣dNdt∣∣=∣∣ddt(N0e–λt)∣∣=λN …(vii)

If A0 is the activity at time t =0, then,

A0 = λN0.

∴ AA0=NN0=e–λt

i.e., A = A0e–λt …(viii)

11. Units of Radioactivity

(1) Curie: It is defined as the activity of radioactive substance which gives 3.7 × 1010 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 106 disintegrations per second.

(3) Becquerel: In SI system the unit of radioactivity is becquerel.

1 becquerel =1 disintegration/second

12. Simple Explanation of α-decay, β-decay and γ-decay

α-emission: A proton in nucleus has a binding energy of nearly 8 MeV; so to come out of a nucleus, it requires an energy of 8 MeV; but such amount of energy is not available to a proton; hence proton as such cannot come out of nucleus on its own. On the other hand, mass of a-particle is subsequently less than the total mass of 2 protons + 2 neutrons. According to Einstein&aposs mass energy equivalence relation, sufficient energy is released in the formation of an α-particle within the nucleus. This energy appears in the form of kinetic energy of α-particle. With this kinetic energy, α-particle hits the wall of nucleus again and again and finally escapes out. The process may be represented as

XZA→YZ–2A–4+He24

(α-particle)

β-emission: b-particles are not the constituents of nucleus, then question is why and how they are emitted by radioactive nucleus. Pauli, in 1932, suggested that at the time of emission of a β -particle, a neutron in nucleus is converted into a proton, a β-particle and an antineutrino. This may be expressed as

n01→H11+β–10+ν−

In general XZA→YZ+1A+β–10+ν−

Antineutrino is a massless and chargeless particle. The energy of the above process is shared by β-particle and antineutrino; that is why the energy of β-particle ranges from 0 to certain maximum value.

γ-emission: When α or β-particle is emitted from a nucleus, the residual nucleus is left in an excited state. The excited nucleus returns to its ground state by the emission of a γ-photon.

Thus γ-photon is emitted either with a-emission or with β-emission.

13. Mass Energy Equivalence Relation

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 = mc2.

Accordingly, 1 kg mass is equivalent to energy

= 1 × (3 × 108)2 = 9 × 1016 joules

and 1 amu = 16.02×1026kgmass

is equivalent to energy 931 MeV.

14. Mass Defect

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 (Z, A) be the mass of nucleus, mn = the mass of proton and mn = mass of neutron, then the mass defect

∆m = Mass of nucleons – Mass of nucleus

= Zmp+(A – Z)mn – Mnucleus

15. Binding Energy per Nucleon

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

∴ Binding energy of nucleus = (∆m) c2

and Binding energy per nucleon =(∆m)c2A

16. Nature of Nuclear Forces

The protons and neutrons inside the nucleus are held together by strong attractive forces. These attractive forces cannot be gravitational since forces on 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 the 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 interconvertible through negative and positive π-mesons, i.e.,

Proton ⇄π+π– Neutron and Neutron →π° Neutron

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

The mass of π-meson = 273 × mass of electron.

17. Nuclear Reaction

When a beam of monoenergetic particles (e.g., α-rays, neutrons etc.) collides with a stable nucleus, the original nucleus is converted into a nucleus of new element. This process is called a nuclear reaction. A typical nuclear reaction is

a + X → Y + b

where a is incident energetic particle, X is target nucleus, Y is residual nucleus and b is outgoing particle. This reaction in compact form is expressed as

X (a, b) Y

In a nuclear reaction mass number, electric charge, linear momentum, angular momentum and total energy are always conserved. The energy of reaction is

Q = (Ma + MX) c2 – (Mb + MY)c2

18. Nuclear Fission

The splitting of heavy nucleus into two or more fragments of comparable masses, with an enormous release of energy is called nuclear fission. For example, when slow neutrons are bombarded on 92U235, the fission takes place according to reaction

U92235+n01(slowneutron)→B56141+Kr3692+3(n01)+200MeV

 

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 Ba141 and Kr92 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 U238 takes place by fast neutrons.

19. 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 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

H12+H12→He14+21.6MeV.

Thus, fusion process occurs at an extremely high temperature and high pressure as in sun where temperature is 107 K.

Remarks:

1. For the fusion to take place, the component nuclei must be brought within a distance of 10–14m. 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 is nuclear fusion. There are two possible cycles:

(a) Proton-proton cycle:

H11+H11→H12+β10+ν. (Neutrino) + Energy

H12+H11→He13+Energy

He23+He23→He24+H11+H11+Energy

Net result is H11+H11+H11+H11→He24+2β10+2ν+Energy (26.7 MeV)

(b) Carbon-nitrogen cycle:

H12+C612→N713+Energy

N713→C613+β10+ν(neutrino)

C613 + H11→N714+(Energy)

N714 + H11→O815+Energy

O815→N715+1β0+ν(neutrino)

N715+H11→C612+He24+Energy

Net result is H11+H11+H11+H11→He24+2β10+2ν+Energy (26.7 MeV)

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

At the sun whose interior temperature is about 2 × 106 K, the proton-proton cycle has more chances for occurrence.