C Page 6 2:11 5 of 8 Nuclear Done SPA5302 (2020) Question B2 a) Discuss the limitations of the semi-empirical mass formula. Page 5 [6 marks] b) Give an overview of the shell model of the nucleus and the observed properties of nuclei that the shell model aims to explain. Include in your discussion the nuclear potential, an- gular momentum and spin, and energy level splitting. Illustrate your answer with detailed sketches and examples as necessary. (Shells are found in the Appendix.) [12 marks] c) If we apply a magnetic field in the -direction to a nucleus with an unpaired proton, then the unpaired proton with orbital angular momentum L, spin S, and total angular momentum J will give a contribution to the -component of the magnetic moment, (5.58(SJ)+(LJ)); (J2) JPN where is a constant. Show that ¸ 5.58(j(j + 1) + s(s + 1) − I(l + 1)) + (j(j + 1) + I(l + 1) − s(s + 1)) ; 2jj+1) JPN where I, s and j are the corresponding quantum numbers of the orbital angular momentum L, spin S, and total angular momentum J. Question B3 [7 marks] Turn over SPA5302 (2020) Page 4 2:11 4 of 8 Nuclear Done Turn over SPA5302 (2020) SECTION B Answer TWO questions from Section B Question B1 a) The semi-empirical mass formula (SEMF) for the binding energy of a nucleon is B(Z, A) ay A-a8A2/3-ac Z(Z-1) A1/3 (A-22)2 QA +8(Z, A). A Give a discussion of the physical origin of the last 2 terms in this expression, and explain the makeup of each of these two terms. [10 marks] b) Consider the mass function along isobars A const. Sketch the typical form of this mass function in the case of A odd, and the case of A even. Indicate on your diagrams how 8-decay and electron capture enter into this picture. De- scribe how this leads the valley of stability. [8 marks] c) Consider a heavy nucleus which splits into two identical daughter nuclei. Ignoring the pairing and asymmetry terms in the semi-empirical mass formula, derive a condition on Z/A for this fission process to release energy. Hence, give an approximate lower bound on the mass number for a nuclei to release energy through fission. (You may assume Z(Z-1) = Z².) SPA5302 (2020) Question B2 a) Discuss the limitations of the semi-empirical mass formula. C [7 marks] Page 5 [6 marks] b) Give an overview of the shell model of the nucleus and the observed properties of nuclei