Nuclear Physics Problems 2.3. “Mirror Nuclei” Are Pairs of Nuclei in Which

2.3. “Mirror nuclei” are pairs of nuclei in which the proton number in one equals the neutron number in the other and vice versa. The simplest examples are provided by odd-A nuclei with one odd nucleon. In one of the mirror nuclei the charge is Z = (A + 1)/2 and the neutron number is N = (A − 1)/2; whilst in the other the 13 charge is Z = (A − 1)/2 and the neutron number N = (A + 1)/2. Examples are 7 N 13 31 31 and 6 C or 16S and 15P. The nuclei in these pairs differ from each other only in that a proton in one is exchanged for the neutron in the other. There is strong reason to believe that the force between nucleons does not differentiate between neutrons and protons; consequently, the “nuclear part” of the binding energy of two mirror nuclei must be the same. Hence, the mass difference of two mirror nuclei can be due only to the difference between the proton mass and the neutron mass, and the different Coulomb energies of the two. This can be used to find the radius of the two nuclei (assumed to be the same), thus:

1. Show that the electrostatic potential energy (Coulomb energy) of a uniformly 2 2 charged sphere of radius R and charge Ze is 3Z e /(20π0R).

21 21 2 2. The nucleus 11Na is more massive than its “mirror partner” 10Ne by 4.02 MeV/c . What is the diﬀerence between their Coulomb energies? Use this to estimate a value for R. (The neutron-proton mass diﬀerence is 1.29 MeV/c2.)

2.5. Deﬁne the terms atomic mass unit, mass defect, and binding energy of a nucleus. Show that 1u (atomic mass unit) = 931.50 MeV/c2. (The mass of a 12C atom is 1.99267 × 10−26 kg.) 181 The binding energy of 73 Ta is 1454 MeV. What is the mass defect of this nucleus? (Mass defects of neutron and proton are 0.008667u and 0.007279u respectively.) 2.6. The semi-empirical mass formula (SEMF) for nuclear masses is Z2 (A − 2Z)2 M(Z,A) = Zm + (A − Z)m − a A + a A2/3 + a + a + δ p n V s c A1/3 A A P where mp and mn are the masses of the free proton and neutron, and the coeﬃcients 2 aV , as, ac, aA have the values (in MeV/c ) 15.56, 17.23, 0.697, 23.285 respectively.

1. What expression does this formula give for the binding energy per nucleon of the nucleus (A, Z)? 2. Explain brieﬂy the physical basis of the various terms in the SEMF. How does the last term δP depend on A and Z? 3. Use the calculation you did in question 2.3(1) to predict a value for the coeﬃcient 1/3 ac, assuming that the nuclear radius is given by R = 1.24 × A fm.

135 4. Use the SEMF to calculate the binding energy of the nucleus 56 Ba. Given 135 that the atomic mass of 56 Ba is 134.904553u, the mass of the hydrogen atom is 1.007825u, and the mass of the neutron is 1.008667u, make a second estimate 135 of the binding energy of 56 Ba. Can you suggest reasons for any discrepancy between your two results?

1 2.7. The Fermi Gas model of the nucleus assumes that the nucleus is a spherical “box” of radius R, and that the neutrons and protons are gases of fermions which (in the nuclear ground state) are “at absolute zero”—i.e., they ﬁill up the lowest energy levels. Given that√ the density of levels for spin-half fermions of mass m in a volume V is V (2m)3/2 /(2π2h¯3) (this includes the spin degeneracy factor), show that in a nucleus containing Z protons and N neutrons, the Fermi energy of the proton gas is

h¯2 9πZ 2/3 EF = 2 2mr0 4A

1/3 where R = r0A , and m is the nucleon mass (assumed the same here for neutrons and protons). Write down a similar expression for the Fermi energy of the neutron gas. Calculate the total energy of the proton gas and the neutron gas. Now consider the case where Z and N add to a fixed total A (take A to be even). Show that the total energy E of the nucleons has a minimum when N = Z = A/2. By expanding E about this minimum value, and retaining only the quadratic term in (Z − A/2), derive an expression for the coefficient aA in the SEMF. Evaluate aA, taking r0 to be 1.24 fm. How does it compare with the value quoted in question 6? 3.6. Explain what the “shell model” of the nucleus is. What experimental evidence is there for it? What would the first five “magic numbers” be if nucleons were assumed to move in a three-dimensional harmonic oscillator potential? What additional component in the potential is required to explain the observed magic numbers? Show how this extra term accounts for the first five observed magic numbers. 3.7. Use the shell model to predict the spin and parity of the ground states of the 15 20 27 87 167 195 following nuclei: 7 N, 10Ne, 12Mg, 38Sr, 68 Er, and 80 Hg. Explain any assumptions you make. 1 − + 1 + 9 + 7 + 1 − The observed spin-parities are 2 , 0 , 2 , 2 , 2 , and 2 . Comment on any discrep- ancies. 38 Can the spin-parity of 19K be predicted by the shell model?