Chapter 9 Exercises: Electromagnetic Properties and Transitions

Section A: Selection Rules and Multipole Classification (Problems 1–7)

Problem 1: Multipole Assignment Practice

For each of the following transitions, determine the lowest allowed multipole order and type ($E$ or $M$). State whether higher multipoles could compete.

For part (f), explain why single-photon emission is forbidden and identify the alternative de-excitation mechanisms.

Problem 2: The Level Scheme of ${}^{60}$Ni

The following excited states of ${}^{60}$Ni are observed:

(a) For each excited state, list all gamma-ray transitions to lower-lying states that are allowed by selection rules, and give the dominant multipolarity for each.

(b) Which transition in this level scheme would you expect to have the longest lifetime? Explain your reasoning in terms of both multipolarity and energy.

(c) The $3^-$ state at 3194 keV can decay to both $2^+$ states. What multipolarities are allowed for each? Which decay branch do you expect to dominate, and why?

Problem 3: Forbidden Transitions

Explain, using selection rules, why the following transitions cannot occur via single-photon emission:

(a) $0^+ \to 0^+$ (the "$E0$ problem")

(b) $0^- \to 0^+$

(c) $1/2^+ \to 1/2^+$ with $\Delta I = 0$ via pure $E2$

For each case, state what de-excitation mechanism(s) are available.

Problem 4: Mixed Transitions

A transition from a $2^+$ state to a $2^+$ state in an even-even nucleus can proceed by both $M1$ and $E2$ multipolarities.

(a) Verify this using the selection rules.

(b) Define the mixing ratio $\delta$ and show that the total transition rate is $T = T(M1)(1 + \delta^2)$.

(c) In ${}^{152}$Gd, the $2^+_2 \to 2^+_1$ transition has $\delta(E2/M1) = +2.7 \pm 0.3$. What fraction of the total transition intensity is $E2$? What does this tell you about the nuclear structure?

Problem 5: Isospin Selection Rules for $E1$ Transitions

(a) Show that the $E1$ operator, in the long-wavelength limit, can be written as:

$$\hat{\mathcal{O}}(E1) \propto \frac{NZ}{A} \left(\mathbf{r}_p - \mathbf{r}_n\right)$$

where $\mathbf{r}_p$ and $\mathbf{r}_n$ are the center-of-mass coordinates of the proton and neutron systems.

(b) Explain why this operator has isovector character ($\Delta T = 0, 1$; but the $\Delta T = 0$ component vanishes in the long-wavelength limit for $N = Z$).

(c) Use this result to explain why $E1$ transitions are typically $10^{-3}$ to $10^{-5}$ Weisskopf units in nuclei.

Problem 6: Higher Multipoles

For the decay of a $J^\pi = 8^+$ isomeric state to a $J^\pi = 0^+$ ground state (as might occur in a high-spin isomer):

(a) What is the lowest allowed multipole? What type ($E$ or $M$)?

(b) Using the Weisskopf estimate for this multipole, estimate the transition rate for $A = 150$ and $E_\gamma = 2$ MeV.

(c) Convert this rate to a half-life. Compare to the age of the universe ($4.3 \times 10^{17}$ s). Could such an isomer be considered "stable"?

Problem 7: The $\gamma$-Decay of ${}^{137}$Ba

The $11/2^-$ isomeric state in ${}^{137}$Ba at 661.66 keV decays to the $3/2^+$ ground state. (This is the famous 661.66 keV line from ${}^{137}$Cs/${}^{137}$Ba decay.)

(a) What multipolarities are allowed?

(b) The measured half-life is 2.552 minutes. Use the Weisskopf estimate to predict the half-life for the lowest allowed multipole. Compare.

(c) The internal conversion coefficient is $\alpha_K = 0.0916$. What fraction of decays produce a gamma ray? What fraction produce a conversion electron?

Section B: Weisskopf Estimates and $B$ Values (Problems 8–14)

Problem 8: Weisskopf Estimate Calculations

Using the Weisskopf formulas, calculate the single-particle transition rates $T_W$ (in s$^{-1}$) and the corresponding half-lives for:

(a) $E1$, $E_\gamma = 1$ MeV, $A = 50$

(b) $E2$, $E_\gamma = 1$ MeV, $A = 150$

(c) $M1$, $E_\gamma = 0.5$ MeV

(d) $E3$, $E_\gamma = 2$ MeV, $A = 200$

(e) $M4$, $E_\gamma = 0.1$ MeV, $A = 180$

Verify your answers against the Python code weisskopf.py.

Problem 9: The First $2^+$ State in Even-Even Nuclei

The measured $B(E2; 0^+ \to 2^+_1)$ values and $2^+_1$ energies for several even-even nuclei are:

(a) Convert each $B(E2)$ value to Weisskopf units. (You will need to calculate $B_W(E2)$ for each mass number.)

(b) Identify which nuclei show single-particle behavior and which show collective behavior.

(c) Calculate the intrinsic quadrupole moment $Q_0$ for ${}^{238}$U using the rotational model formula:

$$B(E2; 0^+ \to 2^+) = \frac{5}{16\pi} e^2 Q_0^2$$

and convert $Q_0$ to the deformation parameter $\beta_2$ using $Q_0 = \frac{3}{\sqrt{5\pi}} Z R_0^2 \beta_2$.

Problem 10: Rotational Band Systematics

In the ground-state rotational band of ${}^{174}$Hf, the following energies and $B(E2)$ values are measured:

(a) Verify that the energy spacings follow $E(I) \propto I(I+1)$ and extract the moment of inertia $\mathcal{J}$.

(b) Using the rotational model, calculate the expected ratios $B(E2; I \to I-2) / B(E2; 2^+ \to 0^+)$ and compare to the measured ratios.

(c) The Clebsch-Gordan coefficients $\langle IK20 | (I-2)K \rangle^2$ for $K=0$ give:

$$\frac{B(E2; I \to I-2)}{B(E2; 2^+ \to 0^+)} = \frac{\langle I\,0\,2\,0 | (I-2)\,0 \rangle^2}{\langle 2\,0\,2\,0 | 0\,0 \rangle^2}$$

Calculate these ratios for $I = 4, 6, 8$ and compare to experiment.

Problem 11: Weisskopf Estimate Derivation

Derive the Weisskopf estimate for $B_W(E2)$ step by step:

(a) Starting from $\hat{\mathcal{O}}(E2) = e r^2 Y_{2\mu}(\hat{r})$, evaluate the radial integral $\langle r^2 \rangle$ for a uniform density distribution inside radius $R$.

(b) Show that this gives $B_W(E2) = \frac{1}{4\pi}\left(\frac{3}{5}\right)^2 R^4 e^2$.

(c) Plug in $R = 1.2 A^{1/3}$ fm and evaluate numerically for $A = 100$.

(d) Now derive $B_W(M1) = \frac{10}{\pi} \mu_N^2$ and evaluate numerically.

Problem 12: The Suppression of $E1$ in Even-Even Nuclei

In ${}^{208}$Pb, the strongest known $E1$ transition is the $3^-_1 \to 2^+_1$ transition with $B(E1) \approx 1.3 \times 10^{-3}$ W.u.

(a) Calculate the Weisskopf estimate $T_W(E1)$ for this transition ($E_\gamma = 2.615$ MeV).

(b) The measured lifetime of the $3^-$ state is $\tau = 30$ fs. Calculate the measured $T(E1)$ and hence $B(E1)$ in W.u.

(c) Explain the factor of $\sim 1000$ suppression relative to the Weisskopf estimate in terms of the isospin structure of the $E1$ operator.

Problem 13: Competing Decay Modes

A nuclear state at excitation energy $E^* = 2$ MeV in a $Z = 72$, $A = 178$ nucleus (${}^{178}$Hf) can in principle decay by either: - $\gamma$-ray emission (to a state 0.5 MeV lower in energy) - Internal conversion

(a) If the transition is $E2$, estimate $T_\gamma$ using the Weisskopf estimate.

(b) Using the BrIcc tables, the $K$-shell internal conversion coefficient for an $E2$ transition of energy 0.5 MeV in hafnium is $\alpha_K \approx 1.5$, and the total $\alpha \approx 2.2$. Calculate the total de-excitation rate $T_\text{total}$.

(c) What fraction of the de-excitations produce gamma rays? What fraction produce $K$-shell conversion electrons?

Problem 14: Sensitivity to Nuclear Radius

The Weisskopf estimate for $B_W(E\lambda)$ depends on the nuclear radius as $R^{2\lambda}$.

(a) Calculate how the Weisskopf estimate for $E2$ changes if $r_0$ is changed from 1.2 fm to 1.25 fm for $A = 150$.

(b) Why is it important to use a consistent value of $r_0$ when expressing experimental $B$ values in Weisskopf units?

(c) Some authors define $B_W(E\lambda)$ with $r_0 = 1.0$ fm, others with $r_0 = 1.2$ fm, and still others with $r_0 = 1.25$ fm. A measured $B(E2) = 500$ $e^2$fm$^4$ for $A = 150$: express this in W.u. for each choice of $r_0$ and show the variation.

Section C: Internal Conversion and Lifetime Measurements (Problems 15–20)

Problem 15: Conversion Electron Spectroscopy

The $2^+_1 \to 0^+_1$ transition in ${}^{152}$Sm has $E_\gamma = 121.8$ keV.

(a) Calculate the kinetic energies of conversion electrons from the $K$, $L_I$, $L_{II}$, and $L_{III}$ shells of samarium ($B_K = 46.83$ keV, $B_{L_I} = 7.74$ keV, $B_{L_{II}} = 7.31$ keV, $B_{L_{III}} = 6.72$ keV).

(b) The measured conversion coefficients are $\alpha_K = 0.65$, $\alpha_{L_I} = 0.085$, $\alpha_{L_{II}} = 0.055$, $\alpha_{L_{III}} = 0.038$. Calculate the total conversion coefficient and the fraction of decays that produce a gamma ray.

(c) This transition is known to be pure $E2$ from angular correlation measurements. Calculate the $K/L$ ratio and compare it to the theoretical value for $E2$ in samarium from BrIcc tables ($\alpha_K/\alpha_L \approx 3.7$ for $E2$).

Problem 16: Isomeric Transitions

The $9/2^+$ isomeric state in ${}^{73}$Ge at 13.3 keV has a half-life of 0.499 s and decays to the $1/2^-$ ground state.

(a) What is the multipolarity of this transition?

(b) Using the Weisskopf estimate, calculate the expected half-life for this multipolarity at $E_\gamma = 13.3$ keV and $A = 73$.

(c) The internal conversion coefficient at this low energy is very large ($\alpha_K = 2920$). What fraction of decays produce a gamma ray?

(d) Why is this state a good example of a "nuclear isomer"?

Problem 17: DSAM Lifetime Measurement

In a DSAM experiment on ${}^{56}$Fe, the $2^+_1$ state at 847 keV is populated in a reaction where the residual ${}^{56}$Fe recoils into a gold backing with initial velocity $v_0/c = 0.02$.

(a) Calculate the maximum Doppler shift at $\theta = 0°$.

(b) The measured attenuation factor is $F(\tau) = 0.85$. If the stopping time in gold for this recoil is estimated at $\tau_s \approx 0.5$ ps, estimate the lifetime $\tau$ of the $2^+_1$ state. (Use the simple approximation $F(\tau) \approx \tau_s / (\tau + \tau_s)$ for exponential slowing.)

(c) The actual measured lifetime is $\tau = 8.1 \pm 0.5$ ps. What does the discrepancy with your simple estimate tell you about the slowing-down model?

Problem 18: Plunger (RDM) Measurement

In an RDM experiment, a $4^+$ state in ${}^{110}$Cd is populated and the recoiling nucleus has velocity $v/c = 0.025$. The ratio of shifted to total intensity is measured at several target-stopper distances:

(a) Plot $\ln R$ vs. $d$ and extract the lifetime $\tau$ from the slope. (Recall: $R(d) = e^{-d/(v\tau)}$.)

(b) Convert the lifetime to a $B(E2)$ value assuming a pure $E2$ transition. The transition energy is 658 keV.

(c) Express your $B(E2)$ in Weisskopf units for $A = 110$. Is this transition single-particle or collective?

Problem 19: Electronic Timing with LaBr$_3$

A LaBr$_3$:Ce detector pair is used to measure the lifetime of the $4^+_1$ state in ${}^{138}$Nd via a $6^+ \to 4^+ \to 2^+$ cascade. The centroid difference between prompt and delayed time spectra is $\Delta C = 45 \pm 5$ ps.

(a) The prompt response function has a centroid shift correction of $\Delta C_\text{prompt} = 3$ ps. What is the corrected lifetime?

(b) If the $4^+ \to 2^+$ transition energy is 454 keV and the transition is pure $E2$, calculate $B(E2; 4^+ \to 2^+)$.

(c) Compare to the rotational model prediction if $B(E2; 2^+ \to 0^+) = 1500$ $e^2$fm$^4$ for this nucleus.

Problem 20: The $E0$ Transition in ${}^{72}$Se

${}^{72}$Se exhibits shape coexistence, with an oblate $0^+_1$ ground state and a prolate $0^+_2$ state at 937 keV.

(a) Explain why this transition cannot proceed by gamma-ray emission.

(b) The transition proceeds by internal conversion. The $E0$ strength parameter $\rho^2(E0) = 0.083$. This parameter is defined as:

$$\rho^2(E0) = \frac{1}{e R^2} |\langle 0^+_f \| \hat{\mathcal{O}}(E0) \| 0^+_i \rangle|^2 / (2I_i+1)$$

Interpret this value physically: what does it tell you about the difference in charge radii between the two $0^+$ states?

(c) At 937 keV, the transition can also proceed by $e^+e^-$ pair creation (internal pair conversion). Estimate the ratio of pair conversion to $K$-shell conversion for $Z = 34$.

Section D: Coulomb Excitation and Detector Arrays (Problems 21–25)

Problem 21: Safe Coulomb Excitation

You plan a Coulomb excitation experiment on ${}^{208}$Pb using a ${}^{58}$Ni beam.

(a) Calculate the distance of closest approach $d_\text{min}$ for a head-on collision at $E_\text{cm} = 250$ MeV.

(b) The sum of nuclear radii (with a safety margin) is $R_1 + R_2 + 5 \approx 17$ fm. Is the Coulomb excitation "safe" at this energy?

(c) The adiabaticity parameter is $\xi = a \cdot E_\gamma / (\hbar v)$ where $a = d_\text{min}/2$ and $v$ is the relative velocity at the distance of closest approach. Calculate $\xi$ for the $2^+_1$ excitation in ${}^{208}$Pb ($E_\gamma = 4.086$ MeV). Is Coulomb excitation efficient for this state?

Problem 22: Coulomb Excitation at FRIB

At FRIB, a beam of ${}^{32}$Mg at 100 MeV/nucleon impinges on a ${}^{197}$Au target. The $2^+_1$ state in ${}^{32}$Mg at 885 keV is Coulomb-excited.

(a) Calculate $\beta = v/c$ for the ${}^{32}$Mg projectile.

(b) At this velocity, the gamma ray from the $2^+ \to 0^+$ de-excitation of ${}^{32}$Mg is Doppler-shifted. Calculate the observed gamma-ray energy at $\theta_\text{lab} = 30°$ and $\theta_\text{lab} = 150°$ (use the relativistic Doppler formula).

(c) The energy spread due to the finite opening angle of a single GRETINA crystal ($\Delta\theta \approx 5°$) would broaden the line. Estimate the Doppler-broadened linewidth at $\theta = 30°$. Then estimate the linewidth after position-sensitive tracking reduces the effective opening angle to $\Delta\theta \approx 1°$.

Problem 23: Systematics of $B(E2; 0^+ \to 2^+_1)$

Using the data provided in the transition_rates.py code (or the table in Problem 9, extended):

(a) Plot $B(E2)$ in W.u. vs. neutron number $N$ for even-even nuclei with $Z = 50$-$54$ (Sn, Te, Xe, Ba). Identify the shell closure at $N = 82$.

(b) For the tin isotopes ($Z = 50$), the $B(E2)$ values show a parabolic trend from ${}^{102}$Sn to ${}^{130}$Sn with a maximum near mid-shell. Explain this pattern using the seniority scheme.

(c) Compare the Sn $B(E2)$ systematics to those of the Sm isotopes ($Z = 62$). Why does Sm show a much more dramatic jump near $N = 88$-$90$?

Problem 24: Angular Correlation Analysis

In a $\gamma$-$\gamma$ angular correlation measurement of the $4^+ \to 2^+ \to 0^+$ cascade in ${}^{60}$Ni:

(a) Both transitions are pure $E2$. Write the angular correlation function $W(\theta) = 1 + a_{22} P_2(\cos\theta) + a_{44} P_4(\cos\theta)$ and look up or calculate $a_{22}$ and $a_{44}$ for the $4$-$2$-$0$ cascade.

(b) Plot $W(\theta)$ vs. $\theta$ from $0°$ to $180°$.

(c) What is $W(90°)/W(0°)$? This ratio is often used as a quick diagnostic for the cascade type.

(d) If the $2^+ \to 0^+$ transition were mixed $M1 + E2$ with $\delta = 0.5$, how would the angular correlation change? (Qualitative answer is sufficient.)

Problem 25: GRETINA Resolution

A GRETINA experiment measures gamma rays from a nucleus moving at $v/c = 0.35$.

(a) Calculate the Doppler-shifted energy for a 1.0 MeV transition observed at $\theta = 30°$ (relativistic formula).

(b) With a conventional Ge array ($\Delta\theta \approx 8°$), calculate the Doppler-broadened energy resolution (FWHM) by differentiating the Doppler formula.

(c) With GRETINA tracking ($\Delta\theta \approx 1°$), recalculate the resolution.

(d) The intrinsic resolution of a Ge detector at 1 MeV is about 2 keV. At what $\beta$ does the Doppler broadening (with tracking) become comparable to the intrinsic resolution?

Challenge Problems

Problem C1: The Scissors Mode

The "scissors mode" in deformed nuclei is an $M1$ excitation at $E_x \approx 3$ MeV, interpreted as the proton and neutron deformed bodies oscillating in a scissor-like motion about their common symmetry axis.

(a) In ${}^{156}$Gd, the summed $B(M1)$ strength to states between 2.5 and 4 MeV is $\sum B(M1\uparrow) = 3.1\,\mu_N^2$. Express this in Weisskopf units. Is this collective?

(b) The geometrical model predicts $\sum B(M1\uparrow) \propto N_p N_n \delta^2 / A$, where $N_p$ and $N_n$ are the number of valence protons and neutrons (measured from the nearest closed shell) and $\delta$ is the deformation. For ${}^{156}$Gd, $N_p = 14$, $N_n = 10$, $\delta \approx 0.3$. Evaluate this formula with the proportionality constant $\frac{3}{4\pi} \mu_N^2$.

(c) Why is this mode called the "scissors mode"? Draw a schematic of the proton and neutron density oscillation.

Problem C2: Model-Independent Sum Rules

The energy-weighted sum rule (EWSR) for $E1$ transitions is:

$$\sum_f E_\gamma B(E1; 0 \to f) = \frac{9}{4\pi} \frac{NZ}{A} \frac{e^2 \hbar^2}{2m}$$

(a) Evaluate the right-hand side numerically for ${}^{208}$Pb.

(b) Most of this strength is concentrated in the Giant Dipole Resonance (GDR) at $E_\text{GDR} \approx 13.5$ MeV in ${}^{208}$Pb. Estimate the effective $B(E1)$ value of the GDR.

(c) Express this $B(E1)$ in Weisskopf units. Comment on the collective nature of the GDR.

Problem C3: Lifetime of the Hoyle State

The $0^+_2$ state in ${}^{12}$C at 7.65 MeV (the Hoyle state, essential for stellar carbon production) decays primarily by $E0$ transition to the $0^+$ ground state.

(a) Why can't this state decay by $E2$ to the ground state?

(b) The Hoyle state has a very small $\gamma$-decay branch via $2^+ \to 0^+$ ($\Gamma_\gamma / \Gamma = 4.1 \times 10^{-4}$) and decays mostly by $\alpha$ emission via ${}^{8}$Be. The total width is $\Gamma = 9.3$ eV. Calculate the radiative width $\Gamma_\gamma$ and the $\gamma$-ray partial lifetime.

(c) Calculate the $E0$ transition strength $\rho^2(E0)$ if the $E0$ branch is $\sim 6 \times 10^{-6}$ of the total width.