2:15 Nuclear 2019 Done 7 of 8 SPA5302 (2019) Question B4 Page 7 a) The Friedmann equation for the cosmological expansion rate H(t) in terms of the scale factor a(t) is H(t) H (ma+Na+). Give the meaning of each term on the right hand side of this equation, in terms of the matter content of the universe. [5 marks] b) Sketch a rough graph of the behaviour of density versus scale factor of each component. Show that there exists a value of the scale factor a, where matter and radiation have equal contributions to the expansion rate, which occurs at temperature Teq= To 6.5 x 10¹²K, Ազգ where Ho 100 kms¹Mpc1. You may use the fact that To 2.725 K and 2, ≈ 4.2 × 10-³h-² [5 marks] c) Show that at temperatures much higher than T., the radiation temperature scales with time as T(1) ~ 100K (15) 12 Here you may use the fact that 1 Но 9.785 h 3.086 x 1017 h [5 marks] d) The number density of a non-relativistic particle of mass m in equilibrium scales as Neutrinos freeze out at t 1s. Calculate the fraction of neutrons to protons at that time. You may use the fact that kT ~(0.7/t¹/2) MeV with t in seconds. Describe the evolution of the neutron fraction from this point until nucleosynthesis begins. [5 marks] e) Discuss the processes involved in the production of Helium, and indicate how there is a deuterium bottleneck. Helium production starts at 300s. Hence show that the mass fraction of "He to Hydrogen is about 1/4. [5 marks] Page 8 End of Paper - An Appendix of 1 page follows Turn over SPA5302 (2019) Appendix The radius of a nuclei may be approximated by R1.2A1/3 fm. The comi.empirical mass formula (SEME) for the binding energy of a nucleon is 2:15 Nuclear 2019 Done 6 of 8 Turn over Page 6 Question B3 SPA5302 (2019) a) U at rest undergoes a-decay. Write down this reaction (all relevant nuclei are in the appendix). By considering conservation of energy and momentum, calculate the kinetic energy of the a-particle in MeV. [6 marks] b) Describe, with the aid of a diagram, how ẞ-decay led to the discovery of the neutrino. [5 marks] c) The isotopes Plutonium-240 and -241 have activation energies 6.3 MeV and 6.0 MeV re- spectively. Compare these to the excitation energy after absorbing a neutron. Show which will fission from absorption of a thermal neutron. Explain your reasoning. [5 marks] d) In the Sun, protons fuse to form deuterium in the reaction 'H+H H+e+ve. Calculate the Coulomb potential barrier for this reaction. Compare your answer to the central temperature of the Sun, which is about 1.5 x 107 K. With the aid of suitable diagrams, explain 2 processes which allow this reaction to take place. SPA5302 (2019) Question R4 Page 7 [9 marks] E