#### The Energetic of Particle, Physics tutorial

α-Decay:

α-particles are stable and show definite range when they transverse a medium. For α decay to be possible there is minimum energy requirement.

X (A, Z) → Y(A-Y, A-Z)

AXX→A-4Z-2Y + 4zα + Qα

Where Qα = Δmc2

As mass defect between mi and mf (initial and final masses).

mi = [M(A,Z)] = ZMP + NMN - EBi

mf = [M(A-4, Z-2)] = (Z-2)MP + (N-2)MN - EBf

Mα = [M(4,2)] = 2MP + 2MN - E

Qα = mi - mf - mα

= -EBi + EBf + E

= E + EBf - EBi

And E= 28.3MeV

Qα = (28.3 + ΔEB)MeV

From semi-empirical formula

Binding energy = E(Z, A)

Thus, disintegration energy of nuclei Qα or total energy released in α-decay is provided as

Qα = 28.3 + (2E/2A)2ΔA + (2E/2Z)AΔZ

α emission is not possible if Qα < O that is Qα should be > 0

It has been found that Qα > O for nuclide for which Z > 82

Assume, mass of parents = MP.

Mass of daughter= Md

Mass of α particle= Mα

Velocity of α particle when emitted = Vα

Velocity of record of daughter = Vd

From conservation of momentum

MαVα = MdVd

Total energy = Qα = final kinetic energy - initial kinetic energy

Qα = 1/2MαVα2 + 1/2MdVd2

Vd = MαVα/Md

By substituting

Qα = 1/2MαVα2 + 1/2Md[MαVα/Md]2

Qα = 1/2MαVα2 + 1/2Mα2Vα2/Md

= 1/2MαVα2[1 + Mα/Md]

For small approximation

Mα/Md ≈ 4/(A-4)

Qα = 1/2MαVα2[1 + 4/(A - 4)]

Qα = Eα[4/A-4 + 1]

= Eα[4+A-4/A-4]

Qα = Eα[A/A-4]

As A is large

Qα ≈ Eα

This signifies most of the energy released is carried away by α-particle.

Range of α -Particle:

α - particles are closely ionizing and lose their energies in rapid succession in air or any medium. Number of ions pairs generated per unit length is known as specific ionization (S.I). Mean distance travelled by α -particle before absorption is known as range.

Intersection of α -particle with atoms or molecules of the medium are entirely statistical and thus they don't have same range as in air.

Range, R = 318E3/2

Empirical relation between range of α-particle and disintegration constant is provided by Geiger Nuttall Law

logλ = AlogR + B

As α-particle is tightly bound entity we can declare it pre-exists in nucleus before its emission. For α-particle to come out or go in nucleus it implies it should have energy in neighborhood of potential well of nucleus.

Energy of α-particle generally ranges between 4-8Mev that is far less than what is needed to overcome potential barrier. Naturally it is not possible to understand this as it has no chance of leaving nucleus.

In 1928, George Gamow considered α -particles as matter wave. This signifies that α-particle as finite probability of penetrating wall of thickness where it suffers series of collisions per second.

β Decay:

Decay procedure in which charge of nucleus changes without change in number of nucleons. There are three kinds of β decay:

i) β- decay: e.g.

AZX→AZ+1Y + 0-1β + v ‾

125B→126C + β + v ‾

ii) β+ decay: e.g.

AZX→AZ-1Y + 0-1β + v ‾

127B→126C + β + v ‾

iii) Electron capture or k-capture: Procedure through which nucleus captures orbital electron, most frequently from closest shell to convert a proton to neutron.

AZX+ 0-1e → AZ-1Y + v ‾

74Be + 0-1e → 73li + v ‾

Energetic of β- decay:

AZX→AZ+1Y + β- + v ‾

In terms of nuclear masses:

Q/C2 = Mn(AZX) - Mn(AZ+1Y)-Me

And in terms of atomic masses:

Q/C2 = Ma(AZX) -Ma(Az+1Y)

For β- to be possible: Q>0

β+ Decay:

AZX→AZ-1Y + β-1 + v ‾

Nuclear masses:

Q/C2 = Mn(AZX) - Mn(Az-1Y)-Me

Atomic masses:

Q/C2 = Ma(AZX) - Ma(Az-1Y) - 2Me

Electron capture:

Q/C2 = Ma(AZX) - Ma(Az-1Y)

β-spectrum:

i) Unlike α-rays, spectrum of β-rays occurs continuous that is electrons emitted have different kinetic energies.

ii) It is also an energy transition between two definite energy states.

iii) Mono-energetic β-rays forming line spectrum are expected.

Most of electrons are emitted with only 1/3 of energy. Thus, this makes one to imagine where remaining of 2/3 of maximum energy would have gone to.

As measurements like momentum and angular momentum are not conserved. These recommend that third particle should exist that always accompany β-decay. It was detected to be neutrino (μ).

Neutrino (μ):

1. Carries away energy equal to energy different between observed energy for β-decay and maximum energy of continuous spectrum.

2. To maintain principle of conservation of energy, neutrino was given following properties

i). 0 charge ii).0 mass iii). Moves with speed of light iv). Spin of 1/2(h/2π)

3. Antiparticle of neutrino, (antineutrino) has given properties

i) 0 charge ii) 0 mass iii) Spin of 1/2(h/2π)

γ - Decay:

When the nucleus is in excited gamma rays are emitted and it is brought to ground state. Nucleus is generally left in excited state after emitting either α or β rays then it is de-excitated by emitting gamma rays. Gamma rays are emitted with discrete and definite energies that is indication of nuclear structure. Energy carried away is ΔE = hf

When mean life time of excited nucleus is >10-6 sec., daughter nucleus is said to show nuclear isomerism.

γk and γ are nuclear isomers and are chemically and physically the same. Difference is that γk is more energetic than γ and it finally emits energy as γ ray and returns to ground state. At times, instead of γ ray being emitted, this excess energy of excited nucleus may be transferred to the extra nuclear electron to get it from shell (generally K or L shell). This procedure is known as internal conversion.

Kinetic energy of converted electron is

Ke = ΔE - Be

Be = binding energy of electron

ΔE = Ei - Ef

Usually, due to internal conversion, yield of γ rays in particular decay <100%. Some spikes observed in continuous β- spectrum is generally because of internal conversion process.

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