Exercises — Chapter 3: The Nuclear Force

Section 3.1–3.2: Properties of the Nuclear Force and NN Scattering

Problem 3.1 — Coulomb Energy in Lead (Warm-up)

Estimate the total Coulomb energy of $^{208}$Pb by treating the nucleus as a uniformly charged sphere of radius $R = r_0 A^{1/3}$ with $r_0 = 1.2$ fm and $Z = 82$ protons. The Coulomb energy of a uniform charge sphere is:

$$E_C = \frac{3}{5} \frac{Z(Z-1)e^2}{4\pi\epsilon_0 R}$$

(a) Calculate $R$ for $^{208}$Pb.

(b) Calculate $E_C$ in MeV. Use $e^2/(4\pi\epsilon_0) = 1.44$ MeV$\cdot$fm.

(c) Compare this to the total binding energy of $^{208}$Pb ($B = 1636.4$ MeV). What fraction of the nuclear binding must the strong force overcome just to compensate the Coulomb repulsion?

Problem 3.2 — Saturation and the Range of the Nuclear Force

(a) If every nucleon in a nucleus of mass number $A$ interacted with every other nucleon with a constant pair interaction energy $\epsilon$, show that the total binding energy would scale as $B \propto A(A-1)/2 \approx A^2/2$ for large $A$, giving $B/A \propto A$.

(b) The experimental binding energy per nucleon saturates at $B/A \approx 8.5$ MeV for $A > 12$. Using the liquid drop analogy, argue that each nucleon interacts with approximately $n$ nearest neighbors. Estimate $n$ by comparing $B/A$ to the pair interaction energy per nucleon.

(c) For a short-range force with range $R_f$ in nuclear matter of density $\rho_0 = 0.16$ fm$^{-3}$, estimate the number of neighbors within range $R_f$. For what value of $R_f$ does this match your estimate from part (b)?

Problem 3.3 — Scattering Lengths and Near-Threshold States

The effective range expansion for $S$-wave scattering is:

$$k \cot\delta_0(k) = -\frac{1}{a} + \frac{1}{2} r_0 k^2$$

(a) Show that for a bound state at energy $E = -\hbar^2 \kappa^2/(2\mu)$, the bound-state condition $k \cot\delta_0 = -\kappa$ (analytically continued to $k = i\kappa$) gives:

$$\kappa \approx \frac{1}{a} - \frac{1}{2} r_0 \kappa^2$$

to leading order in the effective range expansion.

(b) For the deuteron ($a_t = 5.42$ fm, $r_0 = 1.75$ fm), solve the quadratic equation for $\kappa$ and compare to the exact value $\kappa = 0.2316$ fm$^{-1}$ from the binding energy.

(c) For the singlet $np$ channel ($a_s = -23.7$ fm), would a bound state exist? If the scattering length were changed to $a_s = +23.7$ fm (keeping the same $r_0 = 2.75$ fm), what would the binding energy of the virtual state be?

Problem 3.4 — Charge Independence from Mirror Nuclei

The binding energies of $^3$H and $^3$He are $8.482$ MeV and $7.718$ MeV, respectively.

(a) The difference $\Delta B = B(^3\text{H}) - B(^3\text{He}) = 0.764$ MeV is primarily Coulomb in origin. Estimate the Coulomb energy of $^3$He by treating it as a uniform charge sphere with $Z = 2$ and $R = r_0 A^{1/3}$ ($r_0 = 1.2$ fm, $A = 3$). Compare to $\Delta B$.

(b) The remaining discrepancy (if any) between your Coulomb estimate and $\Delta B$ could arise from charge-symmetry breaking in the nuclear force. Estimate the size of this effect as a percentage of the total binding energy.

(c) What would you conclude about charge symmetry if the $nn$ scattering length were $a_{nn} = -18.9 \pm 0.4$ fm and the (Coulomb-corrected) $pp$ scattering length were $a_{pp} = -17.3 \pm 0.4$ fm?

Problem 3.5 — Pauli Exclusion and Allowed Partial Waves

For the $pp$ system ($T = 1$), the generalized Pauli principle requires $(-1)^{L+S+T} = -1$.

(a) List all partial waves ($^{2S+1}L_J$) allowed for $pp$ scattering with $L \leq 3$.

(b) Repeat for the $np$ system in the $T = 0$ channel.

(c) Which partial wave contains the deuteron? Is this state allowed for $pp$ or $nn$? Explain why this is consistent with the nonexistence of the di-proton and di-neutron as bound states.

Problem 3.6 — Phase Shifts and the Repulsive Core

At low energies, the $^1S_0$ $pp$ phase shift is large and positive. At $E_{\text{lab}} \approx 260$ MeV, it passes through zero and becomes negative.

(a) What does a positive phase shift indicate about the interaction? A negative one?

(b) Use the Born approximation relation $\delta_0 \propto \int_0^\infty V(r) r^2 \, dr$ (qualitatively) to argue that the sign change implies the potential changes from net attractive (integrated over all $r$) to net repulsive as higher-energy scattering probes shorter distances.

(c) Estimate the distance of closest approach for two protons at $E_{\text{lab}} = 260$ MeV in a head-on collision (classical turning point). Take $\mu = m_N/2$.

Section 3.3: The Deuteron

Problem 3.7 — Deuteron Bound State: Detailed Calculation

Consider the deuteron in a square well potential with $V_0 = 35$ MeV and $R = 2.1$ fm.

(a) Calculate the interior wavevector $K$ and the exterior decay constant $\kappa$, using $\mu = 469.5$ MeV/$c^2$ and $\hbar c = 197.3$ MeV$\cdot$fm.

(b) Verify that the transcendental equation $K \cot(KR) = -\kappa$ is satisfied (or find the well depth $V_0$ that exactly reproduces $B_d = 2.225$ MeV for $R = 2.1$ fm).

(c) Calculate the probability of finding the nucleon pair outside the well:

$$P_{\text{outside}} = \frac{\sin^2(KR)}{2\kappa} \left/ \left[ \frac{R}{2} - \frac{\sin(2KR)}{4K} + \frac{\sin^2(KR)}{2\kappa} \right] \right.$$

(d) Compare $P_{\text{outside}}$ to the fraction for a typical atomic bound state (e.g., hydrogen ground state probability beyond $2a_0$).

Problem 3.8 — Condition for the Deuteron to Exist

(a) Show that the minimum well depth for a bound state in a square well of radius $R$ is:

$$V_0^{\text{min}} = \frac{\pi^2 \hbar^2}{8\mu R^2}$$

(b) Evaluate $V_0^{\text{min}}$ for $R = 1.5$, $2.0$, $2.1$, and $2.5$ fm.

(c) For a well depth of $V_0 = 35$ MeV, find the minimum $R$ for which a bound state exists.

(d) Show that a second bound state requires $V_0 R^2 > 9\pi^2\hbar^2/(8\mu)$ and verify that no realistic combination of $V_0$ and $R$ supports a second state.

Problem 3.9 — The Deuteron Magnetic Moment

The magnetic moment of the deuteron in a state with $L = 0$ (pure $S$-state) would be:

$$\mu_d(S) = \mu_p + \mu_n = 2.793 - 1.913 = 0.880 \; \mu_N$$

The measured value is $\mu_d = 0.857$ $\mu_N$.

(a) The deviation $\Delta\mu = \mu_d(S) - \mu_d = 0.023$ $\mu_N$ arises from the $D$-state admixture. For a state that is a fraction $P_D$ in the $D$-state ($L = 2$, $S = 1$, $J = 1$) and $P_S = 1 - P_D$ in the $S$-state, show that:

$$\mu_d = P_S(\mu_p + \mu_n) + P_D\left(\mu_p + \mu_n - \frac{3}{2}(\mu_p + \mu_n - \frac{1}{2})\right)$$

and derive the expression for $P_D$ in terms of $\mu_d$, $\mu_p$, and $\mu_n$. (Hint: The $D$-state contribution to the magnetic moment involves the expectation value of $\mu_z$ in the $|L=2, S=1, J=1, M_J=1\rangle$ state; use Clebsch-Gordan coefficients.)

(b) Estimate $P_D$ from the measured $\mu_d$. Why is this only an estimate? (What other effects contribute to $\Delta\mu$?)

Problem 3.10 — The Deuteron Quadrupole Moment

The electric quadrupole moment of the deuteron is $Q_d = 0.2860$ fm$^2$.

(a) Explain why a pure $S$-state ($L = 0$) has $Q = 0$.

(b) The quadrupole moment depends on the $S$-$D$ mixing. For a deuteron with $D$-state probability $P_D$ and the asymptotic normalization of the $S$- and $D$-wave components, the quadrupole moment is approximately:

$$Q_d \approx \frac{1}{\sqrt{50}} \frac{A_S A_D}{\kappa^2}$$

where $A_S$ and $A_D$ are the asymptotic normalization constants and $\kappa$ is the binding wavevector. Using $Q_d = 0.286$ fm$^2$ and $\kappa = 0.2316$ fm$^{-1}$, estimate $A_D/A_S$ and compare to the known value ($\eta = A_D/A_S = 0.0256$).

(c) Is the sign of $Q_d$ (positive) consistent with a prolate or oblate deformation? What does this tell you about the spatial distribution of charge in the deuteron?

Problem 3.11 — RMS Radius of the Deuteron

(a) For the square well deuteron wavefunction with $\kappa = 0.2316$ fm$^{-1}$ and $R = 2.1$ fm, compute the RMS matter radius:

$$\langle r^2 \rangle = \int_0^R r^2 |u_I(r)|^2 dr + \int_R^\infty r^2 |u_{II}(r)|^2 dr$$

where $u_I(r) = A\sin(Kr)$ and $u_{II}(r) = Ce^{-\kappa r}$.

(b) In the "zero-range approximation" (neglecting the interior contribution entirely), show that $\langle r^2 \rangle \approx 1/(2\kappa^2)$ and evaluate numerically.

(c) The measured RMS charge radius of the deuteron is $2.13$ fm, while the matter radius is $1.97$ fm. Why do these differ? (Hint: think about the finite charge radii of the proton and neutron themselves.)

Section 3.4: Meson Exchange and the Yukawa Potential

Problem 3.12 — Yukawa Potential Derivation

Starting from the Klein-Gordon equation with a static point source:

$$(\nabla^2 - \mu^2)\phi = -g\,\delta^3(\mathbf{r})$$

(a) Take the Fourier transform and show that $\tilde{\phi}(\mathbf{q}) = g/(q^2 + \mu^2)$.

(b) Evaluate the inverse Fourier transform by performing the angular integration first (to get $\sin(qr)/(qr)$), then the radial integral using contour integration. Show every step.

(c) Verify that $\phi(r) = (g/4\pi)(e^{-\mu r}/r)$ satisfies the original equation for $r \neq 0$ by direct substitution.

Problem 3.13 — Properties of the Yukawa Potential

(a) The Fourier transform of the Yukawa potential is $\tilde{V}(q) = g^2/(q^2 + \mu^2)$. Show that this is the propagator of a massive scalar particle in momentum space (the Born approximation scattering amplitude).

(b) Calculate the volume integral of the Yukawa potential:

$$\int V(r) \, d^3r = -g^2 \int_0^\infty \frac{e^{-\mu r}}{r} 4\pi r^2 \, dr$$

and express the result in terms of $g$ and $\mu$.

(c) For the pion ($\mu = m_\pi c/\hbar = 0.71$ fm$^{-1}$) with $f_{\pi NN}^2/(4\pi) = 0.08$ and $g^2/(4\pi) = 14.4 f_{\pi NN}^2/(4\pi) \approx 14.4 \times 0.08$, evaluate the Yukawa potential at $r = 1$, $2$, and $3$ fm and compare each to the Coulomb potential at the same distances.

Problem 3.14 — Meson Mass and Force Range

(a) Calculate the Compton wavelength $\lambda_C = \hbar/(mc)$ for the following mesons and state what range of force each mediates:

(b) The nuclear force is attractive at $r \sim 1$--$2$ fm and repulsive at $r < 0.5$ fm. Which meson exchanges are responsible for each regime?

(c) If there existed a meson with mass 50 MeV/$c^2$, how would the nuclear force (and nuclear structure) be different?

Problem 3.15 — Tensor Force from Pion Exchange

The one-pion exchange potential includes a tensor component proportional to:

$$V_T(r) \propto S_{12} \left(1 + \frac{3}{x} + \frac{3}{x^2}\right) \frac{e^{-x}}{x}$$

where $x = r/\lambda_\pi$ with $\lambda_\pi = 1.41$ fm.

(a) Evaluate $V_T(r)$ at $r = 1.0$, $1.5$, $2.0$, and $3.0$ fm (in arbitrary units, just the radial dependence).

(b) Show that $V_T$ is more singular at short distances than the central Yukawa potential ($\sim e^{-x}/x$). What are the implications for nuclear structure calculations? (Why is the tensor force "hard to handle"?)

(c) The tensor operator $S_{12}$ has eigenvalues $+2$ for $M_S = 0$ (spin aligned along the interparticle axis) and $-1$ for $M_S = \pm 1$. Show that the tensor force is attractive in the $M_S = 0$ configuration and repulsive in $M_S = \pm 1$ (for the case where the overall sign of $V_T$ is negative).

Section 3.5–3.6: Modern Potentials and Three-Nucleon Forces

Problem 3.16 — Operator Structure of the Nuclear Force

The most general NN potential consistent with rotational invariance, parity, time-reversal, and the Pauli principle can be written:

$$V = V_C(r) + V_\sigma(r)\,\boldsymbol{\sigma}_1\cdot\boldsymbol{\sigma}_2 + V_T(r)\,S_{12} + V_{LS}(r)\,\mathbf{L}\cdot\mathbf{S} + V_{L2}(r)\,L^2 + V_{LS2}(r)\,(\mathbf{L}\cdot\mathbf{S})^2$$

each multiplied by $(1 + c_\tau \,\boldsymbol{\tau}_1\cdot\boldsymbol{\tau}_2)$.

(a) How many independent radial functions does this imply? (Count the isospin factor.)

(b) The Argonne $v_{18}$ potential has 18 operator components. Explain where the extra terms (beyond the 12 from part (a)) come from. (Hint: charge-dependent and charge-symmetry-breaking terms.)

(c) Why must the tensor operator $S_{12}$ vanish for $S = 0$ (spin singlet) states? Show this explicitly by evaluating $S_{12}$ on the $|S=0, M_S=0\rangle$ state.

Problem 3.17 — Three-Nucleon Force and the Triton

(a) The Argonne $v_{18}$ potential predicts $B(^3\text{H}) = 7.62$ MeV. The experimental value is $8.482$ MeV. What is the underbinding in MeV and as a percentage of the total binding?

(b) When the Urbana IX three-nucleon force (with two adjustable parameters, fitted to $B(^3$H$)$ and the density of nuclear matter) is added, the $^4$He binding energy is predicted as $28.34$ MeV versus the experimental $28.296$ MeV. Comment on the accuracy of this prediction. Why is the $^4$He result more impressive than the $^3$H result (which was fitted)?

(c) Explain qualitatively why three-nucleon forces are relatively more important in neutron-rich nuclei. (Hint: consider the isospin structure of the dominant Fujita-Miyazawa term.)

Problem 3.18 — The Coester Band

When nuclear matter is calculated with various realistic NN potentials (without three-nucleon forces), the saturation point (equilibrium density $\rho_0$ and energy per nucleon $E/A$) lies on a roughly linear trajectory in the $(\rho_0, E/A)$ plane — the "Coester band."

(a) The empirical saturation point is $\rho_0 = 0.16$ fm$^{-3}$ and $E/A = -16$ MeV. Typical NN-only calculations give $\rho_0 \approx 0.25$--$0.30$ fm$^{-3}$ and $E/A \approx -18$ to $-22$ MeV. On a sketch of the $(\rho_0, E/A)$ plane, mark the empirical point and the approximate Coester band.

(b) Three-nucleon forces provide net repulsion at high density. Qualitatively, explain how adding a density-dependent repulsive contribution shifts the saturation point from the Coester band toward the empirical value.

(c) Why is the saturation of nuclear matter an important test of the nuclear force? What would it mean for astrophysics (neutron stars) if the nuclear force did not produce saturation?

Problem 3.19 — The Oxygen Drip Line

With two-nucleon forces only, the neutron-rich oxygen isotopes $^{25}$O through $^{28}$O are predicted to be bound. Experimentally, $^{24}$O is the last bound oxygen isotope.

(a) The drip line is defined by $S_n = 0$, where the one-neutron separation energy is $S_n(Z,N) = B(Z,N) - B(Z,N-1)$. Explain why $S_n < 0$ means the nucleus is unbound.

(b) Three-nucleon forces provide additional repulsion in neutron-rich systems. Explain qualitatively how this could make $^{25-28}$O unbound while leaving $^{24}$O bound. (Think about the isospin dependence of the 3NF.)

(c) This result was a major success of chiral EFT with consistent two- and three-nucleon forces (Otsuka, Suzuki, Holt, et al., 2010). Why is the oxygen anomaly a more stringent test of three-nucleon forces than the triton binding energy?

Synthesis and Computational Problems

Problem 3.20 — Fine-Tuning in Nuclear Physics

The singlet $np$ scattering length is $a_s = -23.7$ fm, indicating a near-threshold virtual state. The triplet channel ($a_t = +5.42$ fm) has the deuteron.

(a) If the nuclear force were 2% stronger (increasing $V_0$ by 2%), estimate the new binding energy of the deuteron. (Use the square well model with $R = 2.1$ fm.)

(b) Would the singlet channel then have a bound state? Estimate its binding energy using the effective range expansion.

(c) If both the singlet and triplet $np$ channels were bound, the di-proton ($pp$, $T = 1$, $^1S_0$) would also be nearly bound (by charge independence). Discuss the consequences for Big Bang nucleosynthesis: if di-protons existed as stable (or long-lived) states, how would the primordial $pp$ reaction rate change? (This is discussed further in Chapter 24.)

Problem 3.21 — Comparison of Nuclear Potentials

Consider three models of the central part of the nuclear force:

1. Square well: $V(r) = -V_0$ for $r < R$, $0$ otherwise
2. Gaussian: $V(r) = -V_0 \exp(-r^2/a^2)$
3. Yukawa: $V(r) = -V_0 (\exp(-\mu r)/(\mu r))$

(a) For each potential, adjust the depth $V_0$ to reproduce the deuteron binding energy ($B_d = 2.225$ MeV) with a range parameter of your choice (but consistent with $R_{\text{eff}} \approx 2$ fm). This requires solving the radial Schrodinger equation numerically for the Gaussian and Yukawa cases.

(b) Compare the wavefunctions: plot $u(r)$ for all three potentials on the same graph.

(c) Calculate the RMS radius for each. How sensitive is $r_{\text{rms}}$ to the choice of potential shape?

Problem 3.22 — Effective Range Expansion: Deriving the Parameters

Starting from the radial Schrodinger equation for $L = 0$ with a general short-range potential:

(a) Define the regular solution $u_0(r)$ at zero energy ($k = 0$) by $u_0(0) = 0$, $u_0'(0) = 1$ and the asymptotic form $u_0(r) \to 1 - r/a$ for $r \to \infty$. Show that the scattering length $a$ is the intercept of the asymptotic straight line with the $r$-axis.

(b) The effective range is defined as:

$$r_0 = 2\int_0^\infty \left[ u_0^2(r) - \left(1 - \frac{r}{a}\right)^2 \right] dr$$

where the integrand vanishes outside the range of the potential. Evaluate $r_0$ for the square well potential and compare to the range $R$.

(c) Why are only two parameters ($a$ and $r_0$) needed to characterize low-energy scattering? Under what conditions does the effective range expansion break down?

Problem 3.23 — Dimensional Analysis of the Nuclear Force

(a) The only fundamental scales in the nuclear force problem are $\hbar c = 197.3$ MeV$\cdot$fm, $m_\pi c^2 = 138$ MeV, and $m_N c^2 = 939$ MeV. Form the natural length scale $\hbar/(m_\pi c)$ and energy scale $m_\pi^2 c^2/(2m_N)$ (the "pion recoil energy"). Evaluate both numerically.

(b) Express the deuteron binding energy $B_d = 2.225$ MeV in units of the pion recoil energy. Is the deuteron "natural" or "fine-tuned" in these units?

(c) The nucleon-nucleon scattering lengths ($a_s = -23.7$ fm, $a_t = 5.42$ fm) are both much larger than the natural length scale $1/m_\pi \approx 1.4$ fm. This is a sign of fine-tuning. In effective field theory language, what does an unnaturally large scattering length imply about the proximity of a bound or virtual state to threshold?

Problem 3.24 — Numerical: Phase Shift from a Square Well

Write a program (or solve by hand for selected energies) to compute the $S$-wave phase shift $\delta_0(E)$ for nucleon-nucleon scattering in a square well potential with $V_0 = 35$ MeV and $R = 2.1$ fm.

(a) The phase shift is given by:

$$\delta_0 = \arctan\left(\frac{k}{K} \frac{K R \cos(KR) \cdot (\text{continued})}{...}\right)$$

More precisely, match the interior and exterior solutions at $r = R$ to show:

$$\tan\delta_0 = \frac{k \tan(K'R) - K' \tan(kR)}{K' + k\tan(kR)\tan(K'R)}$$

where $K' = \sqrt{2\mu(E + V_0)}/\hbar$ and $k = \sqrt{2\mu E}/\hbar$.

(b) Plot $\delta_0(E)$ for $E = 0$ to $350$ MeV. At what energy does $\delta_0$ pass through zero?

(c) Compare qualitatively to the experimental $^1S_0$ phase shift. Does the simple square well capture the correct trend?

Problem 3.25 — The Nuclear Force in Stars

(a) The nuclear force determines the maximum mass of a neutron star. If the nuclear force had no repulsive core, neutron stars could collapse to zero radius. Explain why the repulsive core prevents this. (We return to this in Chapter 25.)

(b) If the nuclear force were long-ranged ($\sim 1/r$, like gravity), nuclear matter would not saturate. What would happen to the density of heavy nuclei? To the density of neutron star cores?

(c) The three-nucleon force stiffens the equation of state of dense matter. A stiffer equation of state supports a more massive neutron star. The observed maximum neutron star mass is $\sim 2.0 M_\odot$ (PSR J0348+0432, Antoniadis et al., 2013). What does this observation tell us about the three-nucleon force at high density?

Problem 3.26 — The $\sigma_1 \cdot \sigma_2$ Operator

The spin-spin operator $\boldsymbol{\sigma}_1 \cdot \boldsymbol{\sigma}_2$ has eigenvalues that depend on the total spin $S$.

(a) Using $\mathbf{S} = \frac{1}{2}(\boldsymbol{\sigma}_1 + \boldsymbol{\sigma}_2)$ and $S^2 = S(S+1)$, show that:

$$\boldsymbol{\sigma}_1 \cdot \boldsymbol{\sigma}_2 = 2S(S+1) - 3$$

(b) Evaluate this for $S = 0$ (singlet) and $S = 1$ (triplet).

(c) A spin-spin potential $V_\sigma(r) \boldsymbol{\sigma}_1 \cdot \boldsymbol{\sigma}_2$ contributes differently to singlet and triplet channels. If $V_\sigma(r) < 0$ (attractive), which channel is more attractive? By what factor?

(d) Similarly, show that $\boldsymbol{\tau}_1 \cdot \boldsymbol{\tau}_2 = 2T(T+1) - 3$ and evaluate for $T = 0$ and $T = 1$.

Problem 3.27 — Isospin Formalism and the Nuclear Force

(a) Write the $pp$, $nn$, and $np$ states in terms of isospin eigenstates $|T, T_z\rangle$. Which states are pure $T = 1$? Which is a mixture of $T = 0$ and $T = 1$?

(b) The nuclear force in the $^3S_1$ channel binds the deuteron (an $np$ state with $T = 0$). Using the generalized Pauli principle ($L + S + T$ = odd), explain why the $^3S_1$ state cannot exist for $pp$ or $nn$.

(c) The $^1S_0$ channel ($T = 1$) is accessible to $pp$, $nn$, and $np$. If charge independence holds exactly, all three scattering lengths should be equal (after Coulomb correction). The measured values are $a_{pp} = -17.3$ fm, $a_{nn} = -18.9$ fm, $a_{np}(T=1) = -23.7$ fm. Compute the fractional deviations and discuss what they tell us about the breaking of charge independence.

Problem 3.28 — Energy Scales in the Deuteron

Compare the energy scales relevant to the deuteron:

(a) The binding energy: $B_d = 2.225$ MeV.

(b) The potential well depth: $V_0 \approx 35$ MeV. What is the ratio $B_d/V_0$?

(c) The kinetic energy of the zero-point motion: estimate as $T \sim \hbar^2/(2\mu R^2)$ for $R = 2.1$ fm. The bound state exists because $V_0$ slightly exceeds $T$.

(d) The Coulomb energy between two protons at the typical deuteron separation ($r \sim 4$ fm): $V_C = e^2/(4\pi\epsilon_0 r)$. Compare this to $B_d$. Why doesn't Coulomb repulsion unbind the deuteron? (Trick question — think carefully about the deuteron's composition.)

(e) The pion mass energy: $m_\pi c^2 = 140$ MeV. This sets the energy scale of the mediating boson. Why is the binding energy so much smaller than the mediator mass?

Problem 3.29 — The Nuclear Force and the Periodic Table

(a) Using the fact that the nuclear force is short-ranged and approximately charge-independent, explain qualitatively why light stable nuclei have $N \approx Z$.

(b) The Coulomb energy of a nucleus scales approximately as $Z^2/A^{1/3}$. For $A = 200$, estimate the ratio of Coulomb energy to nuclear binding energy. Why does this ratio increase with $A$?

(c) Explain why heavy nuclei require $N > Z$ for stability, connecting the nuclear force properties (charge independence, short range) to the valley of stability on the chart of nuclides.

(d) The heaviest element with a stable isotope is $^{209}$Bi ($Z = 83$, $N = 126$). Beyond this, all nuclei are radioactive. Explain this in terms of the competition between the nuclear force and the Coulomb force.