Exercises — Chapter 15

Gamma-Ray Emission and Selection Rules

Problem 15.1 ⭐ For each of the following transitions, determine the allowed multipolarities and identify the dominant (lowest-order) multipole. State whether it is electric or magnetic.

(a) $2^+ \to 0^+$

(b) $3^- \to 1^-$

(c) $5/2^+ \to 1/2^-$

(d) $4^+ \to 4^-$

(e) $1^+ \to 0^+$

(f) $0^+ \to 0^+$

(g) $7/2^- \to 1/2^+$

Problem 15.2 ⭐ The first excited state of $^{12}$C is a $2^+$ state at 4.439 MeV. It de-excites to the $0^+$ ground state.

(a) What is the multipole character of this transition?

(b) Calculate the Weisskopf estimate for the transition rate $T_{\text{W}}$ and the corresponding half-life.

(c) The measured half-life is $t_{1/2} = 6.1 \times 10^{-14}$ s. Express the transition strength in Weisskopf units. Is this transition enhanced or hindered relative to the single-particle estimate?

Problem 15.3 ⭐ Show that for a pure E1 transition from a state with $J_i = 1$ to a state with $J_f = 0$, the angular distribution of the emitted gamma ray is:

$$W(\theta) \propto 1 + \cos^2\theta$$

when the initial state is unpolarized (equally populated $m$-substates). Here $\theta$ is the angle between the gamma-ray direction and the nuclear spin quantization axis.

Hint: The radiation pattern for each magnetic substate $m \to 0$ transition involves spherical harmonics. Sum the intensities (not amplitudes) over the three initial substates $m = -1, 0, +1$ with equal weights.

Problem 15.3b ⭐⭐ The first excited state of $^{7}$Li is a $1/2^-$ state at 0.478 MeV. The ground state is $3/2^-$.

(a) What multipole(s) are allowed for this transition? What is the dominant multipole?

(b) Calculate the Weisskopf estimate for the transition rate and half-life.

(c) The measured half-life is $t_{1/2} = 73 \pm 4$ fs. Express $B(\text{M1})$ in Weisskopf units. Is this transition enhanced, hindered, or single-particle-like?

(d) The 0.478 MeV gamma ray from $^{7}$Li is frequently used as a calibration source. What nuclear reaction is commonly used to populate this state?

Problem 15.4 ⭐⭐ The nucleus $^{60}$Ni has a $4^+$ state at 2.505 MeV, a $2^+$ state at 1.333 MeV, and a $0^+$ ground state.

(a) Determine the multipole character of the $4^+ \to 2^+$ and $2^+ \to 0^+$ transitions.

(b) Using the Weisskopf estimates, calculate the transition rates and half-lives for both gamma rays.

(c) The experimental $B(\text{E2}; 2^+ \to 0^+) = 10.5 \pm 0.3$ W.u. Is this consistent with a single-particle transition or is there collectivity? Compare to the values for $^{152}$Sm (194 W.u.) and $^{208}$Pb (0.31 W.u.) quoted in the text.

Problem 15.5 ⭐⭐ A transition between two nuclear states has $J_i = 3^+$ and $J_f = 2^+$, so both M1 and E2 are allowed.

(a) Write the total transition rate in terms of the mixing ratio $\delta$.

(b) If the mixing ratio is $\delta = +0.15$, what fraction of the total transition intensity is E2?

(c) How would you experimentally determine the mixing ratio? Name at least two methods.

Problem 15.6 ⭐⭐ The first excited state of $^{208}$Pb is a $3^-$ state at 2.615 MeV.

(a) What are the allowed multipole transitions to the $0^+$ ground state?

(b) Calculate the Weisskopf estimate for the dominant transition rate.

(c) The measured half-life is $t_{1/2} = 16.7 \pm 0.4$ ps. Express $B(\text{E3})$ in Weisskopf units. This is one of the best-measured E3 transitions — what does its strength tell you about the nuclear structure?

Problem 15.6b ⭐⭐ The giant dipole resonance (GDR) in $^{208}$Pb is centered at $E_{\text{GDR}} \approx 13.5$ MeV with a width of approximately 4 MeV and exhausts essentially the full Thomas-Reiche-Kuhn (TRK) E1 sum rule.

(a) Calculate the TRK sum rule value $\sum B(\text{E1})\cdot E_\gamma = \frac{9}{4\pi}\frac{\hbar^2}{2m_N}\frac{NZ}{A}$ for $^{208}$Pb in units of e$^2$fm$^2$MeV.

(b) If the GDR carries 100% of the E1 strength and has $B(\text{E1})_{\text{GDR}} = \text{TRK sum} / E_{\text{GDR}}$, calculate $B(\text{E1})$ in W.u. for the GDR.

(c) Compare to the low-energy $3^- \to 0^+$ E3 transition from Problem 15.6. Why is the low-lying E1 strength so much weaker than the GDR despite both being electric multipole transitions?

Transition Rates and Weisskopf Estimates

Problem 15.7 ⭐ Calculate the Weisskopf estimates for the following transition rates (in s$^{-1}$) and the corresponding half-lives:

(a) E1 transition, $E_\gamma = 0.5$ MeV, $A = 50$

(b) M1 transition, $E_\gamma = 1.0$ MeV, $A = 100$

(c) E2 transition, $E_\gamma = 0.2$ MeV, $A = 150$

(d) E3 transition, $E_\gamma = 1.5$ MeV, $A = 200$

(e) M4 transition, $E_\gamma = 0.1$ MeV, $A = 100$

Problem 15.8 ⭐⭐ A nucleus with $A = 120$ has an excited state that de-excites by an E2 transition with $E_\gamma = 0.800$ MeV.

(a) Calculate the nuclear recoil energy $E_R$.

(b) Calculate the Weisskopf estimate for the transition rate and half-life.

(c) Calculate the natural linewidth $\Gamma$ of the gamma-ray line.

(d) Compare $2E_R$ to $\Gamma$. Could this transition exhibit the Mossbauer effect? Explain.

Problem 15.9 ⭐⭐ Consider a rotational band in a deformed nucleus ($A = 168$) with $E(I) = \frac{\hbar^2}{2\mathcal{J}} I(I+1)$ and $\mathcal{J} = 40\;\hbar^2/\text{MeV}$.

(a) Calculate the energies of the $2^+ \to 0^+$, $4^+ \to 2^+$, $6^+ \to 4^+$, and $8^+ \to 6^+$ gamma-ray transitions.

(b) All are E2 transitions. Using the Weisskopf estimates, calculate the transition rate for each.

(c) In a real rotational nucleus, the $B(\text{E2})$ values within the band are related by Clebsch-Gordan coefficients: $B(\text{E2}; I \to I-2) \propto \langle I\,2\,0\,0 | I-2\,0 \rangle^2 \cdot Q_0^2$. Look up or calculate these CG coefficients and determine the ratios of transition rates.

Problem 15.10 ⭐⭐⭐ The E1 hindrance factor. In many nuclei, observed B(E1) values are $10^{-3}$ to $10^{-6}$ W.u.

(a) Using a simple argument based on the center-of-mass constraint (the E1 operator, after removing center-of-mass motion, becomes $\propto \sum_p \mathbf{r}_p - (Z/A)\sum_k \mathbf{r}_k$), explain why E1 transitions between low-lying single-particle states are suppressed relative to the Weisskopf estimate.

(b) The giant dipole resonance (GDR) at excitation energy $E_{\text{GDR}} \approx 78 A^{-1/3}$ MeV exhausts most of the E1 strength (Thomas-Reiche-Kuhn sum rule). If the total E1 strength is $\sum B(\text{E1}) = \frac{9}{4\pi}\frac{NZ}{A}\frac{\hbar^2}{2m_N}$ (energy-weighted sum rule), estimate the fraction of the E1 strength available below 5 MeV excitation for a nucleus with $A = 100$, $Z = 44$.

Problem 15.10b ⭐⭐ A deformed nucleus with $A = 170$ has a ground-state rotational band with the moment of inertia $\mathcal{J} = 35\;\hbar^2/\text{MeV}$.

(a) Calculate the Weisskopf estimates for the E2 transition rates for the $2^+ \to 0^+$, $4^+ \to 2^+$, $6^+ \to 4^+$, and $8^+ \to 6^+$ transitions.

(b) In a well-deformed rotor, the intrinsic quadrupole moment $Q_0$ relates to the $B(\text{E2})$ within the band by: $$B(\text{E2}; I \to I-2) = \frac{5}{16\pi}e^2 Q_0^2 |\langle I\,2\,0\,0|I-2\,0\rangle|^2$$ If $Q_0 = 7.0$ e$\cdot$b for this nucleus, calculate $B(\text{E2}; 2^+ \to 0^+)$ in W.u. and the actual transition rate. By what factor is it enhanced over the Weisskopf estimate?

(c) Explain physically why rotational E2 transitions are so strongly enhanced. What does the $Q_0^2$ factor represent microscopically?

Internal Conversion

Problem 15.11 ⭐ The $7/2^+ \to 5/2^+$ transition in $^{57}$Fe at 14.413 keV has a total internal conversion coefficient $\alpha_{\text{total}} = 8.56$.

(a) For every 100 nuclear transitions, how many result in gamma-ray emission and how many in conversion electron emission?

(b) Calculate the total transition rate given that the half-life of the excited state is 98.3 ns.

(c) What are the separate gamma-ray and internal conversion partial rates?

Problem 15.12 ⭐ The binding energies of the K, $L_I$, $L_{II}$, and $L_{III}$ shells in iron ($Z = 26$) are 7.112, 0.846, 0.721, and 0.708 keV, respectively.

(a) Calculate the kinetic energies of the K, $L_I$, $L_{II}$, and $L_{III}$ conversion electrons for the 14.413 keV transition in $^{57}$Fe.

(b) Would these conversion electrons be distinguishable with a magnetic spectrometer of resolution $\Delta T/T = 0.1\%$? Calculate the required resolution to separate the $L_{II}$ and $L_{III}$ lines.

Problem 15.13 ⭐⭐ For the 661.7 keV $M4$ transition in $^{137}$Ba (following $^{137}$Cs beta decay), the internal conversion coefficients are $\alpha_K = 0.0916$, $\alpha_L = 0.0142$, $\alpha_M = 0.00325$.

(a) Calculate $\alpha_{\text{total}}$ and the fraction of transitions producing gamma rays vs. conversion electrons.

(b) The $K/L$ ratio is $\alpha_K/\alpha_L = 6.45$. Compare this to the values quoted in the text for different multipolarities. Is the $K/L$ ratio consistent with an M4 assignment?

(c) If the beta decay populates the 661.7 keV state at a rate of $10^6$ disintegrations per second, how many 661.7 keV gamma rays are emitted per second? How many K-shell conversion electrons?

Problem 15.14 ⭐⭐ Internal conversion coefficient systematics. Using the approximate formulas given in Section 15.4.4, estimate $\alpha_K$ for the following cases and compare to the tabulated values:

(a) E2 transition, $E_\text{tr} = 100$ keV, $Z = 50$ (tabulated: 1.84)

(b) M1 transition, $E_\text{tr} = 200$ keV, $Z = 50$ (tabulated: 0.062)

(c) E1 transition, $E_\text{tr} = 500$ keV, $Z = 50$ (tabulated: 0.0013)

Discuss the limitations of the nonrelativistic approximation.

Problem 15.15 ⭐⭐⭐ After internal conversion removes a K-shell electron from $^{57}$Fe, the resulting atomic vacancy is filled by an L-shell electron.

(a) Calculate the energy of the K$\alpha$ X-ray emitted in this process.

(b) Alternatively, an Auger electron can be emitted. If a K vacancy is filled by an $L_I$ electron and the energy is transferred to an $L_{III}$ electron, calculate the kinetic energy of the Auger electron.

(c) The fluorescence yield $\omega_K$ (probability of X-ray emission vs. Auger emission) for iron is 0.340. Out of 100 K-shell conversion events, how many K X-rays and how many Auger electrons are produced?

Problem 15.15b ⭐⭐ The 393.5 keV isomeric transition in $^{113}$In (from the $1/2^-$ isomeric state to the $9/2^+$ ground state) has $\alpha_K = 0.0487$ and $\alpha_L = 0.0073$.

(a) This is an M4 transition. Calculate $\alpha_K/\alpha_L$ and compare to the expected ratio for M4 multipolarity.

(b) The K-shell binding energy of indium ($Z = 49$) is $B_K = 27.94$ keV. Calculate the K conversion electron energy.

(c) For each K-shell conversion event, an In K$\alpha$ X-ray (energy $\approx 24.0$ keV) or an Auger electron may be produced. The fluorescence yield for indium is $\omega_K = 0.85$. Out of 1000 isomeric transitions, how many produce: (i) a 393.5 keV gamma ray, (ii) a K conversion electron, (iii) a K X-ray, (iv) a K Auger electron?

E0 Transitions and Nuclear Isomers

Problem 15.16 ⭐ Explain why the $0^+_2 \to 0^+_1$ transition in $^{72}$Se cannot proceed by:

(a) Single photon emission

(b) M1 or E2 radiation

(c) What de-excitation mechanisms are available?

Problem 15.17 ⭐⭐ $^{40}$Ca has a $0^+$ excited state at 3.353 MeV.

(a) What de-excitation pathway(s) are available for this state to reach the $0^+$ ground state?

(b) Given that $E_\text{tr} = 3.353$ MeV $> 2m_e c^2 = 1.022$ MeV, internal pair production (E0 pair emission) is energetically allowed. The kinetic energy shared by the electron-positron pair is $T_{e^+ e^-} = E_\text{tr} - 2m_e c^2 = 2.331$ MeV. Sketch the expected energy spectrum of the electron (or positron) from this process.

(c) Compare the E0 pair emission rate to the expected internal conversion rate for this transition energy.

Problem 15.18 ⭐⭐ $^{99\text{m}}$Tc has $J^\pi = 1/2^-$ at 142.6 keV above the $9/2^+$ ground state.

(a) What is the multipolarity of the direct isomeric transition to the ground state?

(b) Calculate the Weisskopf estimate for this transition rate and the corresponding half-life.

(c) The actual de-excitation proceeds dominantly through the $7/2^+$ state at 140.5 keV, with a 2.17 keV E2 gamma ray followed by a 140.5 keV transition. What multipolarities are allowed for the 140.5 keV $7/2^+ \to 9/2^+$ transition?

(d) Why does the indirect cascade dominate over the direct $1/2^- \to 9/2^+$ M4 transition?

Problem 15.19 ⭐⭐⭐ $^{178\text{m2}}$Hf has $K^\pi = 16^+$ at 2.446 MeV with $t_{1/2} = 31$ years.

(a) The ground-state band has $K = 0$. What is $\Delta K$ for a direct transition to any member of the ground-state band?

(b) Using the $K$-selection rule $\Delta K \le \lambda$, what is the minimum multipolarity required for a direct transition to the ground-state band?

(c) Estimate the energy stored per mole of $^{178\text{m2}}$Hf in the isomeric state and compare to the energy released per mole in typical chemical reactions ($\sim 500$ kJ/mol).

(d) Explain why the idea of "triggering" the release of this stored energy with X-rays was both tantalizing and ultimately unsuccessful.

Problem 15.20 ⭐⭐⭐ $^{180\text{m}}$Ta has $J^\pi = 9^-$ at 77.1 keV. The ground state has $J^\pi = 1^+$.

(a) Determine the multipole character of the direct isomeric transition.

(b) Calculate the Weisskopf estimate for the transition rate and half-life, assuming the dominant multipolarity you found in (a).

(c) The experimental lower limit on the half-life is $> 4.5 \times 10^{16}$ years. How does this compare to your Weisskopf estimate? What does the comparison tell you about additional hindrance mechanisms?

(d) If $^{180\text{m}}$Ta is produced in the neutrino process during a core-collapse supernova, what temperature would be needed to thermally populate intermediate states that could mediate decay? (Hint: consider the energy gap to the nearest intermediate state.)

The Mossbauer Effect

Problem 15.21 ⭐ Calculate the following for the 14.413 keV transition in $^{57}$Fe:

(a) The nuclear recoil energy $E_R$ for a free $^{57}$Fe atom.

(b) The natural linewidth $\Gamma$ from the 98.3 ns half-life.

(c) The ratio $2E_R/\Gamma$ — by how many linewidths are the emission and absorption lines separated?

(d) The velocity (in mm/s) corresponding to the natural linewidth $\Gamma$.

Problem 15.22 ⭐ In a Mossbauer experiment with $^{57}$Fe, the source is moved at velocity $v$ relative to the absorber.

(a) Calculate the Doppler energy shift $\Delta E$ for $v = 1.0$ mm/s.

(b) What velocity range (in mm/s) is needed to scan through a magnetically split $^{57}$Fe Mossbauer spectrum with hyperfine field $B_{\text{hf}} = 33.0$ T? (The splitting of the ground state is $\Delta E_g = g_g \mu_N B_{\text{hf}} / I_g$ and the excited state splitting is $\Delta E_e = g_e \mu_N B_{\text{hf}} / I_e$, with $g_g = 0.0906$, $g_e = -0.1549$, $I_g = 1/2$, $I_e = 3/2$.)

(c) How many absorption lines appear for magnetically split $^{57}$Fe, and what selection rule governs the allowed transitions?

Problem 15.23 ⭐⭐ The recoil-free fraction for $^{57}$Fe in metallic iron ($\Theta_D = 470$ K) at $T = 0$ K:

(a) Calculate $E_R / (k_B \Theta_D)$ and hence $f(T=0)$.

(b) Calculate $f$ at room temperature ($T = 300$ K) using the Debye model. You may use the approximation $\langle x^2 \rangle (300\;\text{K}) \approx 1.28 \times \langle x^2 \rangle (0)$ for $\Theta_D = 470$ K.

(c) Explain why the Mossbauer effect is not observed for the 1.33 MeV gamma ray of $^{60}$Co ($A = 60$), even if the source and absorber are cooled to 4 K.

Problem 15.24 ⭐⭐ The peak resonance cross section for $^{57}$Fe Mossbauer absorption.

(a) Derive the expression $\sigma_0 = \frac{2\pi \lambda^2}{1 + \alpha_{\text{total}}} \cdot \frac{2J_e + 1}{2J_g + 1}$ where $\lambda = \hbar c / E_\gamma$ is the reduced photon wavelength.

(b) Calculate $\sigma_0$ numerically for $^{57}$Fe ($J_g = 1/2$, $J_e = 3/2$, $\alpha_{\text{total}} = 8.56$, $E_\gamma = 14.413$ keV).

(c) Compare $\sigma_0$ to the geometric nuclear cross section $\pi R^2$ (with $R = 1.21 \times 57^{1/3}$ fm). By what factor does the resonance cross section exceed the geometric cross section?

Problem 15.25 ⭐⭐ In the Pound-Rebka experiment, $^{57}$Fe gamma rays travel a vertical distance $h = 22.5$ m.

(a) Calculate the fractional energy shift $\Delta E/E = gh/c^2$.

(b) Express this shift in units of the natural linewidth $\Gamma$.

(c) The experiment achieved a precision of $\sim 1\%$ on the measured shift. Given that the shift is only $\sim 0.5\Gamma$, explain qualitatively how the center of a Lorentzian line can be determined to a fraction of the linewidth (hint: think about counting statistics and the shape of the absorption dip).

(d) What source velocity (in mm/s) corresponds to the gravitational redshift over 22.5 m?

Problem 15.26 ⭐⭐⭐ Mossbauer spectroscopy of an iron compound shows a spectrum with two absorption lines (a quadrupole doublet) at velocities $v_1 = +0.72$ mm/s and $v_2 = +1.58$ mm/s relative to metallic iron.

(a) Calculate the isomer shift $\delta = (v_1 + v_2)/2$ in mm/s. What oxidation state of iron is suggested?

(b) Calculate the quadrupole splitting $\Delta E_Q = (v_2 - v_1) \times E_0/c$ in mm/s and in eV.

(c) If the compound is now cooled below its magnetic ordering temperature, predict qualitatively what happens to the Mossbauer spectrum.

Problem 15.20b ⭐⭐ The $^{99}$Mo/$^{99\text{m}}$Tc generator.

(a) Write the Bateman equation for the activity of $^{99\text{m}}$Tc as a function of time after the generator is eluted (milked) at $t = 0$. Assume the generator contains $A_0(^{99}\text{Mo})$ activity of $^{99}$Mo at $t = 0$ and the elution removes all $^{99\text{m}}$Tc.

(b) Show that the maximum $^{99\text{m}}$Tc activity occurs at time $t_{\max} = \frac{\ln(\lambda_d/\lambda_p)}{\lambda_d - \lambda_p}$ where $\lambda_p$ and $\lambda_d$ are the parent and daughter decay constants. Calculate $t_{\max}$ numerically.

(c) A hospital receives a generator containing 10 GBq of $^{99}$Mo on Monday morning at 8:00 AM. The generator is first eluted at 8:00 AM and again at 8:00 AM Tuesday. Calculate the $^{99\text{m}}$Tc activity obtained at each elution. (Account for the 87.6% branching ratio and assume 95% elution efficiency.)

(d) By Friday morning (4 days later), how much $^{99}$Mo remains? How much $^{99\text{m}}$Tc can be obtained from the Friday morning elution?

Challenging and Research Problems

Problem 15.27 ⭐⭐⭐ Consider a gamma-ray cascade $4^+ \xrightarrow{\gamma_1} 2^+ \xrightarrow{\gamma_2} 0^+$ in a nucleus populated by an unpolarized reaction.

(a) Both transitions are pure E2. Show that the angular correlation function for the $\gamma_1$-$\gamma_2$ cascade is:

$$W(\theta) = 1 + A_{22} P_2(\cos\theta) + A_{44} P_4(\cos\theta)$$

where $\theta$ is the angle between the two gamma rays. (You do not need to evaluate the coefficients, but state what they depend on.)

(b) For the specific case $4^+ \to 2^+ \to 0^+$ with pure E2 transitions, the coefficients are $A_{22} = 0.1020$ and $A_{44} = 0.0091$. Plot $W(\theta)$ vs. $\theta$ and determine the angles of maximum and minimum correlation.

(c) How does the angular correlation change if the $4^+ \to 2^+$ transition is mixed M1+E2 with mixing ratio $\delta$?

Problem 15.28 ⭐⭐⭐ The nuclear recoil in gamma-ray emission.

(a) Derive the exact expression for the emitted gamma-ray energy, accounting for both recoil and the relativistic mass-energy relation:

$$E_\gamma = E_0 \left(1 - \frac{E_0}{2Mc^2}\right)$$

Show that this reduces to $E_\gamma \approx E_0 - E_R$ when $E_0 \ll Mc^2$.

(b) For $^{57}$Fe at 14.413 keV, calculate the fractional difference between the exact and approximate results.

(c) For a hypothetical 10 MeV gamma ray from a nucleus with $A = 10$, is the exact treatment necessary?

Problem 15.29 ⭐⭐⭐ (Research) The second-order Doppler shift (time dilation) in Mossbauer spectroscopy. In addition to the first-order Doppler shift from source motion, there is a temperature-dependent second-order Doppler shift (SOD):

$$\frac{\delta E}{E} = -\frac{\langle v^2 \rangle}{2c^2}$$

where $\langle v^2 \rangle$ is the mean-square velocity of the emitting nucleus.

(a) In the Debye model, $\frac{1}{2}M\langle v^2\rangle = \frac{3}{2} k_B T$ at high temperatures ($T \gg \Theta_D$). Estimate the SOD shift for $^{57}$Fe at room temperature and express it in mm/s.

(b) At low temperatures, $\langle v^2 \rangle$ depends on the zero-point motion. Show that the SOD shift approaches a constant as $T \to 0$.

(c) Discuss how the SOD shift affects the measurement of true isomer shifts. Why must temperature be controlled carefully in quantitative Mossbauer experiments?

Problem 15.30 ⭐⭐⭐ (Research) Gamma-ray lasers (grasers) — a concept that has fascinated nuclear physicists for decades.

(a) For stimulated emission of gamma rays (analogous to a laser), one needs population inversion — more nuclei in the excited state than the ground state. For the 14.413 keV state of $^{57}$Fe with a 98.3 ns lifetime, estimate the pumping rate required to maintain population inversion in a sample of $10^{18}$ $^{57}$Fe nuclei.

(b) The main obstacle is the enormous recoil-free resonance cross section $\sigma_0 \approx 256$ barns, which means that a gamma-ray photon traveling through the inverted medium would be amplified but also strongly absorbed by ground-state nuclei. Calculate the mean free path for 14.413 keV photons in metallic iron (density 7.87 g/cm$^3$, $^{57}$Fe natural abundance 2.2%).

(c) Discuss at least two additional fundamental obstacles to building a gamma-ray laser. Why has this remained an unsolved problem despite decades of effort?

Problem 15.31 ⭐⭐ (Synthesis) Connecting gamma decay to nuclear structure. The nucleus $^{154}$Sm has a ground-state rotational band ($K^\pi = 0^+$) built on the $0^+$ ground state, with the following measured $B(\text{E2})$ values:

(a) These values are strongly enhanced over the single-particle estimate. Calculate the transition rates (in s$^{-1}$) for each, given $E_\gamma(2^+ \to 0^+) = 82$ keV, $E_\gamma(4^+ \to 2^+) = 185$ keV, and $E_\gamma(6^+ \to 4^+) = 274$ keV.

(b) Calculate the half-lives. Are these "fast" or "slow" transitions on the scale of gamma-ray physics?

(c) The internal conversion coefficients for these transitions are $\alpha(82\;\text{keV, E2}) = 0.96$, $\alpha(185\;\text{keV, E2}) = 0.046$, and $\alpha(274\;\text{keV, E2}) = 0.010$. For each transition, what fraction of de-excitations produce a gamma ray? Discuss the role of transition energy in determining whether a gamma ray or conversion electron is more likely.

(d) In a rotational model, $B(\text{E2}; I \to I-2)$ is proportional to $|\langle I\,2\,0\,0|I-2\,0\rangle|^2$. Verify that the ratios of the measured $B(\text{E2})$ values are consistent with the rotational prediction. What does this tell you about the nuclear deformation?

Problem 15.32 ⭐⭐ (Conceptual) Consider the following nuclear states and their possible de-excitation pathways. For each, determine the dominant mechanism (gamma emission, internal conversion, E0 internal conversion, or internal pair production) and explain your reasoning.

(a) A $2^+$ state at 800 keV in $^{60}$Ni de-exciting to the $0^+$ ground state.

(b) A $0^+$ excited state at 600 keV in $^{72}$Ge de-exciting to the $0^+$ ground state.

(c) A $0^+$ excited state at 2.50 MeV in $^{16}$O de-exciting to the $0^+$ ground state.

(d) A $9/2^+$ state at 140 keV in a nucleus with $Z = 72$ ($A = 178$) de-exciting to a $1/2^-$ ground state.

(e) A $3^+$ state at 1.5 MeV in $^{28}$Si de-exciting to the $2^+$ first excited state at 1.779 MeV. (Note: the $3^+$ state is below the $2^+$ state — this is a trick question. What happens?)

Problem 15.33 ⭐⭐⭐ (Integrative) The complete decay scheme of $^{137}$Cs.

$^{137}$Cs ($J^\pi = 7/2^+$, $t_{1/2} = 30.17$ y) beta-decays to $^{137}$Ba. The decay populates two states in $^{137}$Ba: - $11/2^-$ isomeric state at 661.66 keV (94.7% of decays) - $3/2^+$ ground state (5.3% of decays, direct)

The $11/2^-$ state ($^{137\text{m}}$Ba, $t_{1/2} = 2.552$ min) de-excites to the $3/2^+$ ground state via the 661.66 keV gamma ray.

(a) What is the multipolarity of the 661.66 keV transition? Determine the change in spin and parity.

(b) The internal conversion coefficient for this transition is $\alpha_{\text{total}} = 0.1103$. Calculate the fraction of de-excitations producing a 661.66 keV gamma ray.

(c) Starting with $10^6$ $^{137}$Cs atoms, how many 661.66 keV gamma rays are eventually emitted? (Account for the branching ratio to the isomeric state and the conversion coefficient.)

(d) Why is the 661.66 keV gamma ray from $^{137}$Cs one of the most widely used energy calibration standards in gamma-ray spectroscopy? What properties make it ideal?

(e) The Mossbauer effect has been observed for the 661.66 keV transition (in frozen Cs/Ba compounds at low temperature). However, the recoil-free fraction is extremely small. Calculate $E_R$ and estimate why $f$ is so small for this transition compared to the 14.413 keV transition in $^{57}$Fe.