Discovering the Nucleus — Question 9

Question 9

Matter-wave resolution tests whether a nucleus can be resolved

For a nonrelativistic alpha particle use $pc=\sqrt{2mc^2E}$, $mc^2=3727\,\mathrm{MeV}$ and $\hbar c=197.3\,\mathrm{MeV\,fm}$. Define a reduced-wavelength resolution scale $\ell=\hbar/p$.

1. Calculate ell for E=5.00 MeV.

2. Find the kinetic energy for $\ell=2.00\,\mathrm{fm}$, and check whether the nonrelativistic approximation remains reasonable.

Question 9 - Solution

Matter-wave resolution tests whether a nucleus can be resolved

(a) Method and calculation

$$pc=\sqrt{2(3727)(5.00)}=193.1\,\mathrm{MeV}.$$$$\boxed{\ell=\hbar c/(pc)=1.022\,\mathrm{fm}}.$$

(b) Interpretation and further calculation

$$\boxed{E=(\hbar c)^2/[2mc^2\ell^2]=1.306\,\mathrm{MeV}}.$$$$\boxed{E/(mc^2)=3.503\times10^{-4}\ll1}.$$

Physical check

The full de Broglie wavelength is 2 pi times ell. Short wavelength alone does not ensure close approach; the Coulomb barrier also restricts which radii are sampled.