Nuclear Models — Question 10

Question 10

A rotational band tests collective rather than single-particle motion

A simple even-even nuclear rotor has levels $E_J=CJ(J+1)$ for J=0,2,4,6, where $C=\hbar^2/(2\mathcal I)$. The measured $2^+$ excitation energy is .120 MeV.

1. Infer C and predict the $4^+$ and $6^+$ excitation energies.

2. Find the $4\to2$ and $6\to4$ photon energies ignoring recoil, and the ratio $E_4/E_2$ used to test this rotor model.

Question 10 - Solution

A rotational band tests collective rather than single-particle motion

(a) Method and calculation

$$E_2=6C\Rightarrow\boxed{C=.120/6=.0200\,\mathrm{MeV}}.$$$$\boxed{E_4=20C=.400\,\mathrm{MeV},\quad E_6=42C=.840\,\mathrm{MeV}}.$$

(b) Interpretation and further calculation

$$\boxed{E_{\gamma,4\to2}=.280\,\mathrm{MeV},\quad E_{\gamma,6\to4}=.440\,\mathrm{MeV}}.$$$$\boxed{E_4/E_2=20/6=10/3=3.333}.$$

Physical check

The increasing transition spacing reflects J(J+1), unlike a harmonic-vibration ladder with equally spaced quanta. Real rotational bands can deviate as their moment of inertia changes.