Question #43648

Consider a heavy nucleus of mass number $A \gg 1$. A student investigates the scaling laws of the nuclear energy levels using two different models for a nucleon of mass $m$.

\textbf{Model I (Incorrect Bohr-like Model):} The nucleon experiences a long-range attractive central potential $V(r) = -\frac{\alpha A}{r}$, where $\alpha$ is a positive constant. Let the ground state binding energy of the nucleon in this model scale as $A^p$.

\textbf{Model II (Correct Shell Model):} The nucleon is trapped in a 3D isotropic harmonic oscillator potential $V(r) = \frac{1}{2}m\omega^2 r^2 - V_0$, where $V_0$ is a constant well depth. The nucleus is modeled as a Fermi gas where nucleons fill all energy levels up to the Fermi level $N_F \gg 1$. The degeneracy of the $N$-th energy level is $g_N \propto N^2$. The nuclear radius $R$, known to scale as $R \propto A^{1/3}$, is defined as the classical turning point of a nucleon at the Fermi energy $E_F \approx N_F \hbar \omega$. Let the oscillator frequency $\omega$ scale as $A^q$.

Find the values of the exponents $(p, q)$.