Question

2. “Mirror Nuclei” are a pair of nuclei with the same A, but where the number of neutrons in ’nucleus 1’ is equal to the number of protons in ’nucleus 2’. In other words, switching n ↔ p will switch between the nuclei. In even more words, Z1 = N2 and N1 = Z2. (a) Show that the only terms of the SEMF binding energy that differs between mirror nuclei is the Coloumb term. (b) The isotopes 39K and 39Ca are mirror nuclei. Estimate the mass difference (in MeV) between them. Remember that you also need to account for the mass difference between a (proton+electron) and a neutron.

Answer 1

Binding Energy of nuclei as a function of mass number A and atomic number Z is given by,

where a₁ , a₂ , a₃ , a₄ and a₅ are constants.

First term is volume term, Second term is surface term, third term is Coulomb term , fourth term is asymmetry term and last term is pairing term.

First term and second term are related to mass number. Since mass number is same for mirror nuclei, these terms are identical to mirror nuclei.

fourth term is function of (N-z)² , which is same for mirror nuclei. Hence fourth term is identical to mirror nuclei.

Last term, pairing term is also function of mass number. Hence last term is also identical to mirror nuclei.

Hence the only term that differ for mirror nuclei is, third term i.e. coulomb term

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³⁹K is potassium with mass number A = 39 and atomic number Z = 19

hence ³⁹K has 19 protons and 20 neutrons

⁴⁰Ca is calcium with mass number A = 40 and atomic number Z = 20

hence ³⁹Ca has 20 protons and 19 neutrons

since neutron mass is higher, atomic mass of Calcium is greater than that of potassium

mass difference = ( mass of 1 neutron - mass of 1 proton ) = ( 1.0087 - 1.0073) amu

mass difference = 0.0014 amu = 0.0014 1.674 10^-27 kg = 2.34410^-30 kg