NUCLEAR AND PARTICLE PHYSICS-5

Range-energy relationship for a-particle

Geiger Empirical relation R = aE^3/2            E is in Mev,              R is in meter

    E = bR^2/3                            a= 0.315 ´ 10^-2,    b = 2.12 ´ 10²

     The velocity vµÖE       \v ³ µ E ^3/2      R = c v ³  where c has same value in all cases v ³ = 1.03 ´ 10²⁷R

 Knowing R, one can calculate energy (E) or velocity (v) of a-particle emitted by the source. Geiger empirical relation is found to be applicable for a-particles having range 3 to 7 cms in air. R is found to be proportional to v^3/2 for lower energies whereas for higher energies, it is proportional to v⁴ .

 

 Geiger-Nuttall law

Empirical relation between range R of a-particles and the decay constant l of the emitting nuclei known as Geiger-Nuttall law. Log l = A+B log R Where slope B is almost same but A has different values for different series

Theory of a-decay a-particle emission Potential energy of a nucleus and an a-particle as a function of their separation     U(r) = 2𝑍𝑒² /𝑟

           For r = 3 ´ 10^-12 cm, the potential energy is 9 Mev and increases to some maximum value for smaller values of r. This repulsive potential prevents a-particles from entering the nucleus and form a potential barrier. Close to the nucleus and inside, the coulomb potential must break down and be replaced by attractive potential.

The interaction between the nucleus and the a-particles in the region of uncertainty is represented by a constant attractive potential U₀ exerted over a distance r₀ called effective radius of the nucleus. To escape from the nucleus, the particle must have a kinetic energy at least as greater as the energy at the maximum of the potential energy curve. Similarly, the a- particle approaching nucleus from outside could penetrate only if it has enough energy to overcome the potential barrier.

The maximum value must be greater than 9 Mev since a-particles from Th C’ are scattered by uranium. Uranium nucleus emits a-particles with an energy of about 4 Mev. Classically it is difficult to understand the emission of a-particles from the nucleus.

β particles and β decay

                In 𝜷 ⁻ decay, a nucleus with (Z, A) emits a negatively charged electron and being transformed to Nucleus (Z+1, A). Fermi formulated the theory of 𝛽 decay using Pauli’s neutrino concept.

                                  n → p + 𝑒 ⁻ + ̅ν_e ;    ̅ν_e is anti-neutrino. The production of a positron emission (𝛼, 𝑛) reaction has been demonstrated by Curie and Joliot.

 In 𝜷 + decay, a proton bound in the nucleus converted to a bound neutron with the emission of positron and neutrino.

                p→ n + 𝑒 + + 𝜗_𝑒

      𝛽 ⁺ decay:    ^A_ZX  → ^A_Z-1Y+ 𝛽 ⁺ + 𝜗_𝑒

     𝛽 ⁻ decay: ^A_ZX  → ^A_Z+1Y+ 𝛽 ⁻ + ̅ν_e

 Electron capture:  In the electron capture process, a Proton bound in the nucleus changes to a bound neutron by capturing one of the atomic electron (shortly from the K-shell) with an emission of a neutrino.

                p + e → n + 𝜗_𝑒

            ^A_ZX_N + e (𝛽 ⁻) ^A_Z-1Y_N+1 + 𝜗_𝑒

K electrons are nearest to the nucleus and the probability of their being captured by Proton is large and hence the process is called K electron capture. The vacancy created by the K-shell is filled by the electrons from higher energy states i.e. L, M, N resulting in x-ray emission. The emitted electron in 𝛽 decay is not an orbital electron. The 𝛽 particle energy spectrum is a continuous spectrum extending from a minimum and attaining a maximum and then falls to zero at a certain energy called as endpoint energy.

 

 

 

 For beta emitters, the value of Emax varies from 0.25MeV to 2.15MeV.

   Pauli postulated a particle called antineutrino which is emitted in the electron emission process. It has (1) zero charges, (2) intrinsic spin -1/2, (3) zero rest mass.

In positron emission and electron capture, the particle emitted is neutrino which has (1) zero charges (2) intrinsic spin 1/2, and (3) zero rest mass.

Dirac's relativity theory shows that every particle with S = 1/2 spin has its antiparticle. The electron has its antiparticle positron, proton and anti-proton, neutron and antineutron.

     Neutrino 𝜗 has a spin 𝑆_𝜗 anti-parallel to its momentum 𝑃_𝜗.

Antineutrino 𝜗̅  has a spin 𝑆_𝜗̅ parallel to its momentum 𝑃_𝜗̅.

 Explanation: The continuous nature of the 𝛽 decay spectrum is unexpected because the beta transition connects two states of definite energy. Pauli proposed that in beta decay, 𝐸_𝛽 is the energy of the beta particle, 𝐸₀ is the end point energy, the energy associated with the neutrino is 𝐸_𝑛 = 𝐸₀ - 𝐸_𝛽 i.e. in each beta decay, the disintegration energy is shared in a continuous manner by 𝜗, 𝛽 particle and the recoil nucleus.

 

 Based on Pauli's neutrino hypothesis, Fermi developed the theory for 𝛽 decay to obtain the continuous energy spectrum as well as decay constant for beta emission. Fermi postulated that the electron and antineutrino are created at the time of 𝛽 emission. The 𝛽 ⁻ particle and neutrino belong to leptons. Leptons are defined by leptonic number 1 and for anti lepton it is –1.

 For nucleons (proton and neutron), l = 0

For electron and neutrino, l = 1

For positron and anti neutrino, l= -1

 𝛽 ⁻ decay:    n → p + 𝛽 ⁻ + 𝜗̅

                             l = 0    0   1    -1

𝛽 ⁺ decay:      p → n + 𝛽 ⁺ + 𝜗

                  l = 0     0     -1      1

  EC process:

             p + 𝛽 ⁻ → n + 𝜗

          l = 0    1      0     1