Discovering the Nucleus — Question 10

Question 10

A diffuse positive charge cannot produce the same short-range barrier

Compare a point charge Ze with the same charge uniformly spread over a sphere of radius R. A projectile has charge q and $K=qZe/(4\pi\varepsilon_0)>0$. Inside the sphere, take $U(r)=K(3-r^2/R^2)/(2R)$, with zero potential at infinity.

1. Find the largest potential barrier in the diffuse model and the condition for a head-on projectile to pass through its center.

2. For $K=144\,\mathrm{MeV\,fm}$ and atomic-scale $R=10^5\,\mathrm{fm}$, compare the diffuse barrier with a 6.00 MeV projectile.

Question 10 - Solution

A diffuse positive charge cannot produce the same short-range barrier

(a) Method and calculation

$$\frac{dU}{dr}=-Kr/R^3\Rightarrow\boxed{U_{\max}=U(0)=3K/(2R)}.$$$$\boxed{E>3K/(2R)}\quad\text{permits passage through the center in this ideal model}.$$

(b) Interpretation and further calculation

$$\boxed{U_{\max}=216/10^5=.00216\,\mathrm{MeV}=2.16\,\mathrm{keV}}.$$$$\boxed{E/U_{\max}=6.00/.00216=2778}.$$

Physical check

Concentrating positive charge into a tiny nucleus allows a much larger Coulomb barrier than distributing it throughout the atom. This comparison isolates the central idea behind large-angle scattering evidence.