Some Nuclear Properties — Question 6

Question 6

Separation energy measures the cost of removing the last neutron

Two isotopes of the same element have supplied atomic masses $M(A)=99.950000\,\mathrm u$ and $M(A-1)=98.945000\,\mathrm u$. Use $m_n=1.008665\,\mathrm u$ and $1\,\mathrm u\,c^2=931.5\,\mathrm{MeV}$.

1. Find the one-neutron separation energy of isotope A.

2. If its total binding energy is 850 MeV, determine the binding energy of isotope A-1 and explain why the separation energy differs from B/A.

Question 6 - Solution

Separation energy measures the cost of removing the last neutron

(a) Method and calculation

$${}^AX\to{}^{A-1}X+n:\quad\boxed{S_n=[M(A-1)+m_n-M(A)]c^2}.$$$$\boxed{S_n=.003665(931.5)=3.414\,\mathrm{MeV}}.$$

(b) Interpretation and further calculation

$$S_n=B(A)-B(A-1)\Rightarrow\boxed{B(A-1)=850-3.414=846.586\,\mathrm{MeV}}.$$$$\boxed{S_n\text{ is a finite difference of total binding energies, not their average per nucleon}}.$$

Physical check

The electron count is unchanged when a neutron is removed, so atomic masses are suitable. A positive separation energy means spontaneous neutron removal is energetically forbidden from the ground state.