Chapter 2 — Key Takeaways

Essential Results

The nuclear radius scales as $R = r_0 A^{1/3}$, with $r_0 \approx 1.21$ fm (half-density radius). This encodes density saturation: all nuclei have approximately the same interior density, $\rho_0 \approx 0.17$ nucleons/fm$^3$.
The nuclear surface is diffuse, well described by a Fermi distribution with diffuseness $a \approx 0.54$ fm and constant surface thickness $t \approx 2.4$ fm.
Charge radii (electron scattering, muonic atoms, isotope shifts) measure the proton distribution. Matter radii (hadronic probes) include neutrons. The difference defines the neutron skin, which constrains the nuclear equation of state.
The PREX-2 result for ${}^{208}\text{Pb}$: $\Delta r_{np} = 0.283 \pm 0.071$ fm.

The mass excess $\Delta = M_\text{atom} - Au$ is the standard tabulation format (AME2020).
Modern Penning traps achieve $\delta m / m \sim 10^{-9}$ to $10^{-11}$, corresponding to sub-keV absolute precision.
Separation energies $S_n$, $S_p$, $S_{2n}$, $S_{2p}$ reveal shell closures as sharp drops at magic numbers ($N$ or $Z = 2, 8, 20, 28, 50, 82, 126$).
The drip lines ($S_n = 0$ or $S_p = 0$) define the boundaries of nuclear existence.

All even-even nuclei: $J^\pi = 0^+$ (no exceptions).
Odd-$A$ nuclei: $J^\pi$ determined by the unpaired nucleon's quantum numbers $(\ell, j)$.
Odd-odd nuclei: coupling rules (Nordheim) are unreliable; experiment is essential.

The nuclear magneton: $\mu_N = e\hbar/(2m_p) = 5.051 \times 10^{-27}$ J/T.
Free nucleon $g$-factors: $g_s^p = +5.586$, $g_s^n = -3.826$, $g_\ell^p = 1$, $g_\ell^n = 0$.
Schmidt values give the single-particle prediction; experimental moments fall between the Schmidt lines but rarely on them.
Deviations arise from core polarization, meson exchange currents, and configuration mixing. Empirically, $g_s^\text{eff} \approx 0.7\, g_s^\text{free}$.

$Q > 0$: prolate (elongated). $Q < 0$: oblate (flattened). $Q = 0$ for $J = 0$ or $J = 1/2$ regardless of shape.
Single-particle $Q$ values are small ($|Q_\text{sp}| \lesssim 0.3$ b). Measured values reach 8 b midshell — evidence for collective deformation.
Intrinsic deformation parameter: $\beta_2 \sim 0.05$ (near magic numbers) to $\sim 0.35$ (rare earths/actinides) to $\sim 0.6$ (superdeformed).

Nucleon isospin: $t = 1/2$; $t_3 = +1/2$ (proton), $t_3 = -1/2$ (neutron).
Nuclear isospin: $T_3 = (Z - N)/2$; ground states have $T = |T_3|$.
Mirror nuclei $(Z, N) \leftrightarrow (N, Z)$ test charge independence; energy differences are dominated by Coulomb effects.
The IMME $M = a + bT_3 + cT_3^2$ works to $\lesssim 10$ keV, confirming isospin as an approximate symmetry.

Quantity	Formula
Nuclear radius	$R = r_0 A^{1/3}$, $r_0 \approx 1.21$ fm
Saturation density	$\rho_0 = 3/(4\pi r_0^3) \approx 0.17$ fm$^{-3}$
RMS radius (Fermi)	$R_\text{rms} = \sqrt{3/5}\, R_{1/2}\sqrt{1 + 7\pi^2 a^2/(3R_{1/2}^2)}$
Mass excess	$\Delta = M_\text{atom} - Au$
Binding energy	$B = Z\Delta_H + N\Delta_n - \Delta$
Neutron separation energy	$S_n(A,Z) = B(A,Z) - B(A-1,Z)$
Schmidt moment ($j = \ell + 1/2$)	$\mu = (\ell + g_s/2)\,\mu_N$
Schmidt moment ($j = \ell - 1/2$)	$\mu = \frac{j}{j+1}(j + 3/2 - g_s/2)\,\mu_N$
Spectroscopic $Q$ (single particle)	$Q_\text{sp} = -e_\text{eff}\langle r^2\rangle (2j-1)/[2(j+1)]$
Intrinsic $Q_0$ from $\beta_2$	$Q_0 = (3/\sqrt{5\pi})\,ZR_0^2\,\beta_2$
Isospin third component	$T_3 = (Z - N)/2$

Chapter 3 (Nuclear Force): The charge independence tested here by mirror nuclei is a fundamental property of the nuclear force.
Chapter 4 (SEMF): Mass systematics from this chapter are the data the SEMF must fit.
Chapter 6 (Shell Model): The spins, moments, and separation energy shell signatures we catalogued here are the predictions the shell model must reproduce.
Chapter 8 (Collective Motion): The large quadrupole moments that defeat the single-particle model are explained by collective rotation and vibration.
Chapter 14 (Beta Decay): Isospin selection rules from Section 2.6 govern allowed and forbidden beta transitions.
Chapter 25 (Neutron Stars): The neutron skin of ${}^{208}\text{Pb}$ constrains the same equation of state that determines neutron star radii.