NYJC EJC 2026 Nuclear Physics Tutorial

9814 H3 Physics (2026) Nuclear Physics Tutorial 1 Atomic Structure 1. (a) Explain what is meant by binding energy of a nucleus. Use the following values of masses to calculate the binding energy per nucleon in a 56 26Fe nucleus. Give your answer in MeV per nucleon. neutron mass = 1.0087 u; proton mass = 1.0073 u; mass of 56 26Fe nucleus = 55.9207 u [6] (b) An approximate formula for the radius R of a nucleus of nucleon number (mass number) A is R ≈ (1.2 x 10 –15) A1/3 Where R is expressed in metres. (i) Use this formula to calculate the approximate radius of a 56 26Fe nucleus. (ii) Assuming the nucleus to be a uniform sphere, estimate the density of nuclear matter in the iron nucleus. (iii) Sketch a graph to show how the density of nuclear matter depends on the nucleon number A of the nucleus. [6] (c) The mass of 1 m3 of iron is 8 x 10 3 kg. There is a general trend for the densities of solid elements to increase with increasing nucleon number. Comment on these two pieces of information in relation to your answers in (b) (ii) and (iii) and suggest an explanation for any difference between the behaviour of the density of solid elements and the density of nuclear matter. [3] 2. (a) The binding energy B of a nucleus may be expressed as B = 931.4 (1.008665 N + 1.007825 Z – M) In this equation, B is expressed in MeV and M is the atomic mass, expressed in u. (i) What do the symbols N and Z represent? Why are they multiplied by the numbers 1.008665 and 1.007825 respectively? (ii) Show how the number 931.4 arises.

9814 H3 Physics (2026) Nuclear Physics Tutorial 2 (b) Figure 2.1 lists the atomic mass M and the binding energy B of some nuclides. Nuclide M / u B / MeV 15 7N 15.00011 115.5 16 8 O 15.99492 127.6 18 8 O 17.99916 139.8 19 9F 18.99841 147.8 23 11Na 22.98777 186.6 24 12Mg 23.98505 198.3 40 18 Ar 39.96238 343.8 40 20 Ca 39.96259 342.0 44 20 Ca 43.95549 380.9 45 21Sc 44.95592 387.8 46 22Ti 45.95263 398.2 Fig. 2.1 (i) In nuclear physics, nuclei with equal nucleon numbers but different proton and neutron numbers are called isobars. Identify two isobars in Fig.2.1. [1] (ii) Although isobars must have the same nucleon number, their atomic masses can differ because of a difference in binding energy. Confirm that, for the two isobars you have identified in (i), the heavier nuclide of the pair has a smaller binding energy than the lighter one. [1] (iii) Would it be possible for the heavier nuclide of a pair of isobars to have a greater binding energy than the lighter one? Explain your answer. [2] (iv) Suppose that a single proton were removed from each of the following nuclei: 168O, 199F, 2412Mg, 4521Sc, 4622Ti. Use information from Fig. 2.1 to calculate the energy required in each case. What trend, if any, can you detect in the values of energy? [4] [speed of light c = 2.9979 x 108 m s–1; elementary charge e = 1.6022 x 10–19 C; atomic mass unit 1 u = 1.6605 x 10–27 kg; electron mass me = 0.000549 u; neutron mass mn = 1.008665 u; proton mass mp = 1.007276 u.]

9814 H3 Physics (2026) Nuclear Physics Tutorial 3 Radioactivity 3. The radionuclide 60 27 Co is used in radiotherapy. It has a half -life of 5.27 years and upon disintegration, emits two raysγ , one of energy 1.17 MeV and the other of energy 1.33 MeV. This nuclide is prepared by bombarding a suitable target (a stable nuclide) with an appropriate projectile particle. Possible projectiles are neutrons 1 0 n and deuterons 2 1d . (a) Write down three reactions involving the target nuclei 63 29 Cu , 62 28Ni and 59 27 Co with one of the projectiles by which 60 27 Co might be produced. [3] (b) In radiotherapy treatment, it is necessary to determine the amount of energy absorbed from the radiation. Calculate the energy flux (the amount of energy passing normally through unit area per unit time) at a distance 1.50 m from a point source containing 950 μg of 60 27 Co . Assume that the source radiates equally in all directions. [8] 4. (a) A rubidium nucleus 87 37Rb may emit a single particleβ and decay to form a stable strontium nucleus. The decay constant of this reaction is 4.50 x 10–19 s–1. (i) Write down a nuclear equation representing the decay of the rubidium nucleus. [1] (ii) Explain, in terms of probability, what is meant by the decay constant of a radioactive nucleus. [1] (iii) Explain why the radioactive decay of a sample containing a large number of 87 37Rb nuclei should obey an exponential law. Write down the relation between the number nRb of 87 37Rb nuclei at time t, the number no at the start of the decay, and the decay constant λ . [3] (iv) Hence, show that at time t, the total number nsr of stable strontium nuclei formed as a result of the decay is given by nsr = nRb (eλt – 1) [3]

9814 H3 Physics (2026) Nuclear Physics Tutorial 4 (b) The decay of this rubidium nucl ide is used as a method of dating rocks. Values of nsr and nRb are obtained by analysis of rock samples of fixed mass. For rocks of a particular type from a certain location, the results tabulated in Fig. 4.1 were obtained. sample 1 2 3 4 5 nsr 7.16 x 1018 7.50 x 1018 7.72 x 1018 8.18 x 1018 8.50 x 1018 nRb 3.10 x 1018 9.50 x 1018 14.0 x 1018 22.9 x 1018 29.1 x 1018 Fig. 4.1 i) Explain how, by plotting a graph of nsr, against nRb, you could investigate whether these data are consistent with these different rock samples being formed at the same time. [2] ii) Draw this graph. Suggest an explanation for the existence of a non- zero intercept. [6] iii) Use your graph to estimate the age of this type of rock in this location. Give your answer in Ma, where 1 Ma = 1 million years = 3.16 x 1013 s [3] 5. A radionuclide A decays to nuclide B. Nuclide B is unstable and decays into a stable nuclide C. The half-lives of A and B are approximately equal. At time t = 0, the number of nuclei of A is No and the numbers of nuclei of both B and C are zero. Sketch labelled graphs showing the variation with time of the numbers of nuclei of A, B and C. Explain the form of each graph. [6] 6. The abundance of Uranium-238 in naturally occurring uranium minerals on Earth is 99.28%. This means that there are 99.28 atoms of Uranium-238 for every 100 atoms of all uranium isotopes. The abundance of Uranium-235 is 0.72%. Assuming an equal amount of each isotope was present at the time of the formation of the Earth’s crust, estimate the age of the Earth. Decay constants: Uranium-238, 15.5 x 10-11, year–1; Uranium-235, 98.5 x 10-11, year–1. [4]

9814 H3 Physics (2026) Nuclear Physics Tutorial 5 Answers 1. a) Binding energy per nucleon = 8.84 MeV b) i) 4.6 x 10–15 m ii) 2.3 x 1017 kgm–3 2. b) iv) 12.10 MeV (for 16 8 O ); 8.00 MeV (19 9F ); 11.70 MeV ( 24 12Mg); 6.90 MeV ( 45 21Sc ); 10.40 MeV ( 48 22Ti ) 3. Energy flux = 5.62 x 10–4 Wm–2 4. b) iii) 362 Ma 6. 5.9 x 109 years

9814 H3 Physics (2026) Nuclear Physics Tutorial Solutions Suggested Solutions to H3 Tutorial on Nuclear Physics Atomic structure 1. (a) Nuclear binding energy is the energy equivalent of the mass defect of a nucleus. It is the energy required to separate all the nucleons of a nucleus to infinity. The Fe nucleus has 26 protons (atomic number Z) and 30 neutrons (mass number A – atomic number Z). The total mass of 26 protons and 30 neutrons is: × +× =26 1.0073 30 1.0087 56.4508u uu 27 28 mass defect 56.4508 55.9207 0.5301 0.5301 1.66 10 8.7797 10 kg muu u − − ∆= − = = ×× = × 2 28 8 2 11 11 19 6 binding energy (8.7997 10 )(3.00 10 ) 7.9197 10 J 7.9197 10 1.6 10 10 494.98 MeV E