Exercises — Chapter 5: Quantum Mechanics Review

These exercises progress from straightforward applications of the formalism to problems requiring synthesis of multiple concepts. Problems marked with a star () are more challenging. Problems marked with a double star (*) connect to later chapters and may be revisited.

Angular Momentum (Exercises 5.1--5.7)

Exercise 5.1 — Single-Particle Quantum Numbers

For each of the following single-particle states, give the values of $n$ (radial quantum number), $l$ (orbital angular momentum), $j$ (total angular momentum), the parity $\pi = (-1)^l$, and the degeneracy $2j+1$ (number of magnetic substates). This exercise reinforces the spectroscopic notation that is used throughout the nuclear shell model.

(a) $1s_{1/2}$ \ (b) $1p_{3/2}$ \ (c) $1d_{5/2}$ \ (d) $1f_{7/2}$ \ (e) $2s_{1/2}$ \ (f) $1g_{9/2}$

Exercise 5.2 — Allowed $J$ Values

Using the triangle rule, determine all allowed values of the total angular momentum $J$ when coupling the following pairs of angular momenta:

(a) $j_1 = 3/2$ and $j_2 = 5/2$ \ (b) $j_1 = 2$ and $j_2 = 3$ \ (c) $j_1 = 7/2$ and $j_2 = 7/2$ \ (d) $l = 3$ and $s = 1/2$

For each case, verify that $\sum_J (2J+1) = (2j_1+1)(2j_2+1)$.

Exercise 5.3 — Identical Nucleons in the Same Shell

Two identical neutrons occupy the $1f_{7/2}$ shell.

(a) List all allowed values of the total angular momentum $J$, accounting for antisymmetry. \ (b) How many states (counting all $M_J$ substates) exist? Verify your answer by noting that the total number of two-particle states is $\binom{2j+1}{2} = \binom{8}{2} = 28$. \ (c) Which value of $J$ is the ground state and why? (Hint: the pairing interaction favors $J = 0$.)

Exercise 5.4 — L-S to j-j Coupling

Consider two nucleons, each with $l = 1$ and $s = 1/2$.

(a) In L-S coupling, find all possible values of $L$, $S$, and $J$ for the system. List them in spectroscopic notation $^{2S+1}L_J$. \ (b) In j-j coupling, the individual nucleons have $j = 1/2$ or $j = 3/2$. For each combination $(j_1, j_2)$, find the possible $J$ values. \ (c) Show that the total number of states is the same in both coupling schemes. \ (d) Which coupling scheme is more appropriate for nuclear physics, and why?

Exercise 5.5 — Ground State Predictions

Using the shell model filling order (given in Chapter 6, but previewed here: $1s_{1/2}$, $1p_{3/2}$, $1p_{1/2}$, $1d_{5/2}$, $2s_{1/2}$, $1d_{3/2}$, $1f_{7/2}$, ...), predict the ground-state spin and parity $J^\pi$ for:

(a) $^{3}$He ($Z=2$, $N=1$) \ (b) $^{15}$N ($Z=7$, $N=8$) \ (c) $^{17}$O ($Z=8$, $N=9$) \ (d) $^{41}$Ca ($Z=20$, $N=21$) \ (e) $^{5}$He ($Z=2$, $N=3$) — this nucleus is unbound. Does the shell model still predict a $J^\pi$?

Compare your predictions to the evaluated nuclear data (NNDC). For each case, state clearly which nucleon (the "odd" proton or neutron) determines $J^\pi$, and why the even-nucleon contribution is $0^+$.

Exercise 5.5b — Schmidt Magnetic Moments

(*) Using the Schmidt formulas from Section 5.3.6:

$$\mu(j = l + \tfrac{1}{2}) = j \cdot g_l + \frac{1}{2}g_s$$

$$\mu(j = l - \tfrac{1}{2}) = \frac{j}{j+1}\left[(l + \frac{3}{2})g_l - \frac{1}{2}g_s\right]$$

with $g_l^p = 1$, $g_l^n = 0$, $g_s^p = 5.586$, $g_s^n = -3.826$ (in nuclear magneton units):

(a) Calculate the Schmidt value for $^{17}$O ($J^\pi = 5/2^+$, odd neutron in $1d_{5/2}$). The measured value is $\mu = -1.894\ \mu_N$. How well does the Schmidt prediction agree? \ (b) Calculate the Schmidt value for $^{41}$Ca ($J^\pi = 7/2^-$, odd neutron in $1f_{7/2}$). The measured value is $\mu = -1.595\ \mu_N$. \ (c) Calculate the Schmidt value for $^{15}$N ($J^\pi = 1/2^-$, odd proton in $1p_{1/2}$). The measured value is $\mu = -0.283\ \mu_N$. \ (d) Plot the Schmidt lines (the locus of Schmidt values for $j = l + 1/2$ and $j = l - 1/2$ as functions of $j$) for both protons and neutrons. Do the measured magnetic moments fall on, between, or outside the Schmidt lines?

Exercise 5.6 — Three Angular Momenta

(*) Couple three angular momenta $j_1 = j_2 = j_3 = 1/2$ (e.g., three spin-$1/2$ particles).

(a) How many total states are there? \ (b) Find all possible values of $J$. \ (c) How many times does each $J$ value appear? (Hint: first couple $j_1 + j_2 = J_{12}$, then couple $J_{12} + j_3 = J$.) \ (d) Relate this to the spin of the triton ($^3$H) or $^3$He.

Exercise 5.7 — Magnetic Substates

A nucleus has ground-state spin $J = 5/2$.

(a) How many magnetic substates $M_J$ exist? \ (b) In a magnetic field, the degeneracy is lifted. If the energy splitting between adjacent substates is $\Delta E = g_J \mu_N B$ (where $g_J$ is the nuclear $g$-factor, $\mu_N$ is the nuclear magneton, and $B$ is the field), draw the energy level diagram. \ (c) For $^{17}$O ($g_J \approx -1.89$ in nuclear magneton units) in a field of $B = 1$ T, estimate the total energy spread across all substates. (Use $\mu_N = 3.15 \times 10^{-8}$ eV/T.)

Clebsch-Gordan Coefficients (Exercises 5.8--5.12)

Exercise 5.8 — CG Coefficients by Hand

Calculate the following Clebsch-Gordan coefficients using the formulas in Section 5.3.2:

(a) $\langle 1\, 0;\, 1/2\, 1/2 | 3/2\, 1/2\rangle$ \ (b) $\langle 1\, 1;\, 1/2\, -1/2 | 3/2\, 1/2\rangle$ \ (c) $\langle 1\, 0;\, 1/2\, 1/2 | 1/2\, 1/2\rangle$ \ (d) $\langle 2\, 1;\, 1/2\, 1/2 | 5/2\, 3/2\rangle$

Verify your answers using a CG table or the code in this chapter's code/ directory.

Exercise 5.9 — Orthogonality of CG Coefficients

The CG coefficients satisfy the orthogonality relation:

$$\sum_{m_1 m_2} \langle j_1\, m_1;\, j_2\, m_2 | J\, M\rangle \langle j_1\, m_1;\, j_2\, m_2 | J'\, M'\rangle = \delta_{JJ'}\delta_{MM'}$$

(a) Verify this explicitly for $j_1 = 1$, $j_2 = 1/2$, $J = 3/2$, $J' = 1/2$, $M = M' = 1/2$. \ (b) Verify the completeness relation:

$$\sum_{J M} \langle j_1\, m_1;\, j_2\, m_2 | J\, M\rangle \langle j_1\, m_1';\, j_2\, m_2' | J\, M\rangle = \delta_{m_1 m_1'}\delta_{m_2 m_2'}$$

for $j_1 = 1$, $j_2 = 1/2$, $m_1 = 0$, $m_2 = 1/2$, $m_1' = 1$, $m_2' = -1/2$.

Exercise 5.10 — Pairing CG Coefficient

Show that $\langle j\, m;\, j\, {-m} | 0\, 0\rangle = (-1)^{j-m}/\sqrt{2j+1}$ for general $j$.

(a) Verify this for $j = 1/2$ by explicit construction. \ (b) Verify for $j = 3/2$ by checking the normalization: $\sum_m |\langle 3/2\, m;\, 3/2\, {-m} | 0\, 0\rangle|^2 = 1$. \ (c) Why is this particular CG coefficient so important in nuclear physics?

Exercise 5.11 — CG Coefficients and Symmetry

(*) Using the symmetry relation $\langle j_1\, m_1;\, j_2\, m_2 | J\, M\rangle = (-1)^{j_1+j_2-J}\langle j_2\, m_2;\, j_1\, m_1 | J\, M\rangle$, prove that:

(a) For two identical particles ($j_1 = j_2 = j$) coupling to $J$, the coefficient $\langle j\, m_1;\, j\, m_2 | J\, M\rangle$ vanishes when $2j - J$ is even (i.e., $J$ is odd for half-integer $j$). \ (b) Show that this means two neutrons in the same $j$-orbit can only couple to $J = 0, 2, 4, \ldots, 2j-1$. \ (c) Count the total number of allowed states for $j = 7/2$ and verify it equals $\binom{8}{2} = 28$.

Exercise 5.12 — Wigner 3j Symbols

Convert the following CG coefficient to a 3j symbol and verify the symmetry properties:

$$\langle 1\, 1;\, 1\, -1 | 1\, 0\rangle = \frac{1}{\sqrt{2}}$$

(a) Write the corresponding 3j symbol. \ (b) Verify that cyclically permuting the columns leaves the 3j symbol unchanged. \ (c) Verify that transposing two columns introduces the phase $(-1)^{j_1+j_2+J}$.

Parity and Selection Rules (Exercises 5.13--5.16)

Exercise 5.13 — Parity Assignments

Determine the parity of the following nuclear states:

(a) A single proton in the $1f_{7/2}$ orbit ($l = 3$) \ (b) A single neutron in the $2s_{1/2}$ orbit ($l = 0$) \ (c) The ground state of $^{16}$O (doubly magic, all shells closed up to $1p_{1/2}$) \ (d) The first excited state of $^{17}$O, if the 9th neutron is promoted from $1d_{5/2}$ to $2s_{1/2}$

Exercise 5.14 — Selection Rules for Gamma Transitions

For each of the following transitions, determine the lowest allowed multipolarities and whether they are electric (E) or magnetic (M):

(a) $2^+ \to 0^+$ \ (b) $3^- \to 2^+$ \ (c) $1/2^- \to 1/2^+$ \ (d) $4^+ \to 2^+$ \ (e) $0^+ \to 0^+$ (this one is special — explain why)

Exercise 5.15 — Forbidden Beta Decay Classification

Classify the following beta transitions by their degree of forbiddenness (allowed, first forbidden, second forbidden, etc.):

(a) $0^+ \to 0^+$ (Fermi transition) \ (b) $0^+ \to 1^+$ (Gamow-Teller transition) \ (c) $0^+ \to 2^-$ \ (d) $3^+ \to 0^+$ \ (e) $1/2^+ \to 1/2^-$

For each, estimate the suppression factor relative to an allowed transition (each degree of forbiddenness suppresses by roughly $10^3$--$10^4$).

Exercise 5.16 — The $0^+ \to 0^+$ Monopole Transition

(*) A nucleus decays from a $0^+$ excited state to a $0^+$ ground state. Show that:

(a) This transition cannot proceed by single-photon emission (why?). \ (b) The transition can proceed by internal conversion (electron emission). \ (c) If the energy exceeds $2m_e c^2 = 1.022$ MeV, the transition can proceed by internal pair creation ($e^+e^-$ emission). \ (d) Give a real nuclear example of a $0^+ \to 0^+$ transition and its primary decay mode. (Hint: $^{16}$O has a $0^+$ state at 6.05 MeV.)

Fermi's Golden Rule (Exercises 5.17--5.21)

Exercise 5.17 — Dimensional Analysis

Verify that Fermi's golden rule $\Gamma = (2\pi/\hbar)|V_{fi}|^2 \rho(E_f)$ has dimensions of inverse time (i.e., a rate).

(a) What are the dimensions of $|V_{fi}|^2$? \ (b) What are the dimensions of $\rho(E_f)$? \ (c) Verify the dimensions of $\Gamma$.

Exercise 5.18 — Photon Density of States

Derive the photon density of states $\rho_\gamma(E_\gamma) = E_\gamma^2 V / (\pi^2 \hbar^3 c^3)$ from first principles:

(a) Start with the number of photon states in a cubic box of side $L$ with momentum between $p$ and $p + dp$. \ (b) Convert to energy using $E = pc$. \ (c) Include the factor of 2 for photon polarization. \ (d) Show that for a 1 MeV photon in a box of side $L = 10$ fm (nuclear size), $\rho_\gamma$ is incredibly small. What does this imply about photon emission rates compared to strong-interaction processes?

Exercise 5.19 — Estimating Gamma Decay Rates

(**) Using Fermi's golden rule, estimate the E1 gamma decay rate for a transition with energy $E_\gamma = 1$ MeV.

(a) The E1 matrix element for a single proton is approximately $|\langle f|\hat{r}|i\rangle| \sim R_0 A^{1/3}$ where $R_0 \approx 1.2$ fm. For $A = 50$, estimate $|V_{fi}|$. \ (b) Calculate $\rho_\gamma$ for a 1 MeV photon. \ (c) Combine to estimate $\Gamma$ and convert to a half-life. \ (d) Compare your estimate to the Weisskopf single-particle estimate (which you will derive in Chapter 9): $T_{1/2}(\text{E1}) \approx 6.3 \times 10^{-15} A^{-2/3} E_\gamma^{-3}$ seconds (with $E_\gamma$ in MeV).

Exercise 5.20 — Phase Space and Three-Body Decay

(*) A nucleus decays by emitting a particle with energy $E$ (treat as non-relativistic, mass $m$). The density of states for the emitted particle is $\rho(E) = C\sqrt{E}$ where $C$ is a constant.

(a) If the matrix element $|V_{fi}|^2$ is approximately constant over the relevant energy range (a common approximation), show that the energy spectrum of emitted particles is $dN/dE \propto \sqrt{E}$. \ (b) Now consider beta decay, where both an electron (energy $E_e$, momentum $p_e$) and a neutrino (energy $E_\nu = Q - E_e$, momentum $p_\nu$) are emitted. The combined density of states is $\rho \propto p_e^2 \cdot p_\nu^2 = p_e^2(Q - E_e)^2$ (for $m_\nu \approx 0$). Show that the electron energy spectrum is $dN/dE_e \propto p_e^2(Q - E_e)^2 F(Z, E_e)$ where $F(Z, E_e)$ is the Fermi function (Coulomb correction). \ (c) Why does the beta spectrum go to zero at both $E_e = m_e c^2$ (minimum kinetic energy) and $E_e = Q + m_e c^2$?

Exercise 5.21 — Lifetime Estimation

A nuclear excited state decays by two competing processes: gamma emission with rate $\Gamma_\gamma = 10^{12}$ s$^{-1}$ and internal conversion with rate $\Gamma_{IC} = 5 \times 10^{12}$ s$^{-1}$.

(a) What is the total decay rate $\Gamma_{total}$? \ (b) What is the mean lifetime $\tau$ and half-life $t_{1/2}$? \ (c) What is the branching ratio for gamma emission? \ (d) If you measured the energy width of this state, what would it be? (Use the energy-time uncertainty relation $\Delta E \cdot \tau \sim \hbar$.)

WKB and Tunneling (Exercises 5.22--5.26)

Exercise 5.22 — Rectangular Barrier

A particle of mass $m$ and energy $E$ encounters a rectangular barrier of height $V_0 > E$ and width $a$.

(a) Using the WKB formula, show that the tunneling probability is $T \approx \exp(-2\kappa a)$ where $\kappa = \sqrt{2m(V_0 - E)}/\hbar$. \ (b) For a proton ($m = 938.3$ MeV$/c^2$) with energy $E = 5$ MeV tunneling through a barrier of height $V_0 = 30$ MeV and width $a = 5$ fm, calculate $T$. \ (c) How does $T$ change if the energy increases from 5 MeV to 10 MeV? (This illustrates the extreme sensitivity of tunneling to energy.)

Exercise 5.23 — Sommerfeld Parameter

The Sommerfeld parameter $\eta = z_1 z_2 e^2 / (4\pi\epsilon_0 \hbar v)$ characterizes the importance of the Coulomb interaction in a nuclear reaction.

(a) Calculate $\eta$ for proton-proton scattering at $E_{cm} = 1$ MeV. \ (b) Calculate $\eta$ for alpha-$^{208}$Pb scattering at $E_{cm} = 20$ MeV. \ (c) The Coulomb barrier penetrability at low energies is $T \sim e^{-2\pi\eta}$. Calculate $T$ for both cases. \ (d) What does the extreme smallness of $T$ for case (b) imply about the feasibility of superheavy element synthesis?

Exercise 5.24 — Alpha Decay Systematics

(**) The Geiger-Nuttall law (an empirical relation discovered in 1911) states that $\log t_{1/2} \approx a/\sqrt{E_\alpha} + b$, where $E_\alpha$ is the alpha particle kinetic energy and $a, b$ are constants for a given isotopic chain.

(a) Using the WKB tunneling formula for the Coulomb barrier, show that $\ln T \approx -2\pi\eta + (\text{terms that vary slowly with } E_\alpha)$. \ (b) Show that $\eta \propto 1/\sqrt{E_\alpha}$, and therefore $\log T \propto -1/\sqrt{E_\alpha}$. \ (c) Since $t_{1/2} \propto 1/T$ (to a good approximation), derive the Geiger-Nuttall law from the WKB result. \ (d) The measured alpha energies and half-lives for some Po isotopes are:

Plot $\log_{10}(t_{1/2}/\text{s})$ vs. $1/\sqrt{E_\alpha}$ and verify the approximate linearity.

Exercise 5.25 — Stellar Fusion: The Gamow Peak

(**) For proton-proton fusion at the center of the Sun ($T = 1.5 \times 10^7$ K):

(a) Calculate the Coulomb barrier height $V_C = e^2/(4\pi\epsilon_0 R)$ where $R \approx 1.2$ fm. Express in keV. \ (b) Calculate the average thermal energy $\langle E \rangle = \frac{3}{2}k_B T$. Express in keV. \ (c) Calculate the Gamow energy $E_G = (b k_B T / 2)^{2/3}$ where $b = \pi e^2 \sqrt{2m_p} / (4\pi\epsilon_0 \hbar) \approx 22.7$ MeV$^{1/2}$. \ (d) What fraction of the Coulomb barrier height is $E_G$? Comment on the role of quantum tunneling in stellar energy production. \ (e) Calculate the tunneling probability at the Gamow peak energy.

Exercise 5.26 — WKB Validity

(*) The WKB approximation requires that the potential varies slowly compared to the local de Broglie wavelength: $|dV/dx| \ll p^2/(\hbar m)$ or equivalently $|d\lambda/dx| \ll 1$.

(a) For the Coulomb potential $V(r) = Z_1 Z_2 e^2/(4\pi\epsilon_0 r)$, show that the WKB validity condition becomes $r \gg \hbar/p$, i.e., the distance must be much larger than the de Broglie wavelength. \ (b) For an alpha particle with $E = 5$ MeV near the nuclear surface ($r \approx 7$ fm), estimate $\lambda_{dB}$ and assess WKB validity. \ (c) Where does the WKB approximation break down, and how are "connection formulas" used to patch the solution across the turning points?

Identical Particles and Antisymmetrization (Exercise 5.27)

Exercise 5.27 — Slater Determinant for $^4$He

(*) $^4$He has 2 protons and 2 neutrons. In the simplest shell model, all four nucleons occupy the $1s_{1/2}$ orbit.

(a) List the four single-particle states, including spin and isospin quantum numbers. \ (b) Write the Slater determinant for the $^4$He ground state (symbolically — do not expand the $4\times4$ determinant). \ (c) What is the total spin $J$, total isospin $T$, and parity $\pi$ of this state? \ (d) Explain why $^4$He is exceptionally stable (its binding energy per nucleon, 7.07 MeV, exceeds that of its neighbors). Connect this to the Pauli principle and the shell structure.

Exercise 5.27b — Antisymmetric States of the $A = 6$ System

(*) Consider the $A = 6$ nuclei: $^6$He ($Z=2$, $N=4$), $^6$Li ($Z=3$, $N=3$), and $^6$Be ($Z=4$, $N=2$). These form an isospin triplet ($T = 1$) and singlet ($T = 0$ for $^6$Li).

(a) $^6$He has $Z = 2$ (closed $s$-shell for protons) and $N = 4$. In the simplest shell model, the 4 neutrons fill $1s_{1/2}$ (2 neutrons) and $1p_{3/2}$ (2 neutrons). If the two $p_{3/2}$ neutrons couple to $J = 0$ (the pairing prediction), what is the predicted ground state of $^6$He? Compare to the measured value ($0^+$). \ (b) $^6$Li has one proton and one neutron outside the $^4$He core. The proton is in $1p_{3/2}$ and the neutron is in $1p_{3/2}$. Since they are not identical particles, the Pauli restriction does not apply directly. What are the allowed $J$ values? The measured ground state is $1^+$. Can you explain this with L-S coupling (hint: the $T = 0$, $S = 1$, $l = 0$ state)? \ (c) Why is $^6$He a "halo nucleus" (Chapter 10) while $^6$Li is not? What role does the Pauli principle play in this difference?

Density of States (Exercises 5.28--5.29)

Exercise 5.28 — Nuclear Level Density

The Bethe formula for the nuclear level density is $\rho(E^*) \approx C \exp(2\sqrt{aE^*}) / (E^*)^{5/4}$ where $a \approx A/8$ MeV$^{-1}$.

(a) For $^{56}$Fe ($A = 56$), calculate $\rho$ at excitation energies $E^* = 5, 10, 15, 20$ MeV. (You may ignore the prefactor $C$ and just compute relative values.) \ (b) At approximately what excitation energy does the average level spacing become comparable to the experimental energy resolution of a typical gamma-ray detector ($\sim$ 2 keV for HPGe)? This is the energy above which individual levels can no longer be resolved. \ (c) Why is the nuclear level density important for compound nucleus reactions (preview of Chapter 18)?

Exercise 5.29 — Density of States for Beta Decay

(**) In beta decay, the combined density of states for the electron and neutrino (the "statistical rate function" or Fermi integral) determines the energy spectrum of emitted electrons.

(a) For an electron with total energy $E_e$ (including rest mass) and a massless neutrino, the number of available final states in the energy interval $dE_e$ is proportional to:

$$dN \propto p_e E_e (Q + m_e c^2 - E_e)^2 F(Z, E_e)\, dE_e$$

where $p_e = \sqrt{E_e^2 - (m_e c^2)^2}/c$ is the electron momentum and $F(Z, E_e)$ is the Fermi function (Coulomb correction). Explain the physical origin of each factor. \ (b) At what electron kinetic energy does the spectrum peak? (Neglect the Fermi function for simplicity.) \ (c) The endpoint of the beta spectrum occurs at $T_e^{\max} = Q$. At this energy, the neutrino has zero energy. Why does the spectrum go to zero at the endpoint? \ (d) How would the spectrum change if the neutrino had a nonzero mass $m_\nu$? What observable would change near the endpoint? (This is the principle behind the KATRIN experiment.)

Synthesis Problems (Exercises 5.30--5.32)

Exercise 5.30 — Complete $J^\pi$ Assignment

(**) $^{39}$K has $Z = 19$ and $N = 20$.

(a) $N = 20$ is magic. What is the neutron contribution to $J^\pi$? \ (b) For the protons, fill the shell model levels up to 19 protons. What orbit does the 19th proton occupy? \ (c) Predict the ground-state $J^\pi$ of $^{39}$K. \ (d) The first excited state of $^{39}$K has $J^\pi = 1/2^+$ at 2.52 MeV. What single-particle transition could produce this state? \ (e) What multipolarity gamma transition connects the first excited state to the ground state? Is it E or M?

Exercise 5.31 — Connecting the Tools

(**) A $^{60}$Co source ($J^\pi = 5^+$, $t_{1/2} = 5.27$ years) is used for gamma-ray calibration.

(a) $^{60}$Co beta decays to $^{60}$Ni. The dominant beta branch (99.9%) goes to the $4^+$ state at 2.506 MeV in $^{60}$Ni. Is this beta transition allowed or forbidden? What type (Fermi or Gamow-Teller)? \ (b) The $4^+$ state decays to the $2^+$ state at 1.333 MeV by gamma emission. What is the multipolarity? \ (c) The $2^+$ state decays to the $0^+$ ground state. What is the multipolarity? \ (d) Using the Weisskopf estimate $T_{1/2}(\text{E2}) \sim 7.3 \times 10^{-8} A^{-4/3} E_\gamma^{-5}$ s (with $E_\gamma$ in MeV), estimate the half-lives of both gamma transitions. Are the excited states "long-lived" or "short-lived"? \ (e) Why are $^{60}$Co gamma-ray sources used for food irradiation and cancer therapy? (Consider the gamma-ray energies and the half-life.)

Exercise 5.32 — Order of Magnitude Estimates

(*) Nuclear physics spans many orders of magnitude. Estimate the following using the formulas from this chapter:

(a) The assault frequency of an alpha particle inside a heavy nucleus ($v \sim 10^{22}$ fm/s, $R \sim 7$ fm). \ (b) The tunneling probability for the alpha decay of $^{210}$Po ($E_\alpha = 5.41$ MeV, $Z_\text{daughter} = 82$). \ (c) The ratio of E1 to E3 gamma transition rates for the same transition energy ($E_\gamma = 1$ MeV) in a nucleus with $A = 100$. (Use $\Gamma \propto (R/\lambda_\gamma)^{2\lambda}$.) \ (d) The nuclear level density in $^{208}$Pb at the neutron separation energy ($S_n = 7.4$ MeV).