Question B3 a) Explain the Geiger-Nuttal rule for a-decay, illustrating your answer with a graph. b) Sketch the potential for a-decay, assuming it can be modelled as a pre-formed particle inside the daughter nucleus. On this sketch, illustrate a typical tunnelling wave-function, which has an energy Q which is less than the potential for R < r < b, where r is the radius from the centre of the daughter nucleus, and r = R and r = b are the radii where the potential has the value Q. The transmission probability is P = e-2G where the Gamow factor is c) 2m G h² √ √V(r) – Q dr, where m = ma is the mass of the a-particle. The potential for a-decay in terms of a daughter nucleus with Z protons and mass number A is V(r) = 2e2Z Απερι From this, show that the Gamow factor is x Z/VQ. Hence show that the decay constant > for a-decays approximately obeys Ꮓ Inxx const. + Relate the answer you obtain to that in part (a). (You may use the approximation 1 r - 1 dr ~ √√√b for b≫ R.) b Consider two nuclei which might undergo fusion. Assuming that the potential between nuclei 1 and 2 is a coulomb potential, and that Q = 1mv², where m = m1m2/(m₁ + m2) is the reduced mass here and in the definition of G above, and v is the relative velocity of the nuclei before fusion, show that G= EG 2mv2 where EG = 2mc²(TαZ1Z2)² where a = e²/4πhe 1/137 is the fine structure constant. Two protons collide with a relative kinetic energy of 1 keV. Calculate the probability of barrier penetration.

Question B3 a) Explain the Geiger-Nuttal rule for a-decay, illustrating your answer with a graph. b) Sketch the potential for a-decay, assuming it can be modelled as a pre-formed particle inside the daughter nucleus. On this sketch, illustrate a typical tunnelling wave-function, which has an energy Q which is less than the potential for R < r < b, where r is the radius from the centre of the daughter nucleus, and r = R and r = b are the radii where the potential has the value Q. The transmission probability is P = e-2G where the Gamow factor is c) 2m G h² √ √V(r) – Q dr, where m = ma is the mass of the a-particle. The potential for a-decay in terms of a daughter nucleus with Z protons and mass number A is V(r) = 2e2Z Απερι From this, show that the Gamow factor is x Z/VQ. Hence show that the decay constant > for a-decays approximately obeys Ꮓ Inxx const. + Relate the answer you obtain to that in part (a). (You may use the approximation 1 r - 1 dr ~ √√√b for b≫ R.) b Consider two nuclei which might undergo fusion. Assuming that the potential between nuclei 1 and 2 is a coulomb potential, and that Q = 1mv², where m = m1m2/(m₁ + m2) is the reduced mass here and in the definition of G above, and v is the relative velocity of the nuclei before fusion, show that G= EG 2mv2 where EG = 2mc²(TαZ1Z2)² where a = e²/4πhe 1/137 is the fine structure constant. Two protons collide with a relative kinetic energy of 1 keV. Calculate the probability of barrier penetration.