Nuclear Models — Question 2

Question 2

Assembling a uniformly charged nuclear sphere gives its Coulomb energy

A nuclear model spreads total charge $Q=Ze$ uniformly through radius R. Build it from spherical shells, with $Q(r)=Qr^3/R^3$, $dq=3Qr^2dr/R^3$ and $dU=Q(r)dq/(4\pi\varepsilon_0r)$.

1. Integrate the Coulomb assembly energy.

2. Evaluate it for Z=28 and R=4.80 fm using $e^2/(4\pi\varepsilon_0)=1.44\,\mathrm{MeV\,fm}$, and state its effect on binding.

Question 2 - Solution

Assembling a uniformly charged nuclear sphere gives its Coulomb energy

(a) Method and calculation

$$U_C=\frac{3Q^2}{4\pi\varepsilon_0R^6}\int_0^Rr^4dr.$$$$\boxed{U_C=\frac35\frac{Z^2e^2}{4\pi\varepsilon_0R}}.$$

(b) Interpretation and further calculation

$$\boxed{U_C=\frac{3(28)^2(1.44)}{5(4.80)}=141.12\,\mathrm{MeV}}.$$$$\boxed{B_C=-U_C}\quad\text{because electrostatic repulsion reduces binding}.$$

Physical check

This continuum expression uses Z squared. Replacing it by Z(Z-1) approximately excludes individual-proton self contributions in a discrete-particle mass formula.