Exercises — Chapter 30

Electrostatic Accelerators

Problem 30.1 ⭐ A tandem Van de Graaff accelerator has a terminal voltage of $V = 15$ MV.

(a) Calculate the kinetic energy of a proton beam (starting as H$^{-}$, stripped to H$^{+}$ at the terminal).

(b) Calculate the kinetic energy of a ${}^{32}\text{S}$ beam, assuming the sulfur ion is stripped to charge state $q = 12e$ at the terminal.

(c) Express the sulfur beam energy in MeV per nucleon. Is this above the Coulomb barrier for ${}^{32}\text{S} + {}^{208}\text{Pb}$? (Use $V_C = 1.44 z_1 z_2 / (R_1 + R_2)$ MeV with $R_i = 1.2 A_i^{1/3}$ fm.)

Problem 30.2 ⭐ Why can't helium be accelerated in a tandem Van de Graaff using the standard method? Explain what HeH$^{-}$ molecular ions are and how they solve the problem. What maximum energy per nucleon can a ${}^{4}\text{He}^{2+}$ beam achieve in a 20 MV tandem?

Problem 30.3 ⭐⭐ A Van de Graaff accelerator with terminal voltage $V = 10.0 \pm 0.5$ kV (energy stability $\Delta V/V = 5 \times 10^{-5}$) is used to study the ${}^{27}\text{Al}(p,\gamma){}^{28}\text{Si}$ reaction near the $E_p = 992$ keV resonance, which has a width $\Gamma = 0.10$ keV.

(a) What is the energy spread $\Delta T$ of the proton beam at 992 keV?

(b) Is the beam energy resolution adequate to resolve this resonance? What if the resonance width were $\Gamma = 0.02$ keV?

(c) If a cyclotron with $\Delta T / T = 10^{-3}$ were used instead, what would the beam energy spread be at 992 keV? Comment on the relative merit of electrostatic accelerators for resonance studies.

Cyclotrons

Problem 30.4 ⭐ A classical cyclotron has a magnetic field $B = 1.2$ T and a dee radius of $R = 0.75$ m. Calculate:

(a) The cyclotron frequency for protons (in MHz).

(b) The maximum proton kinetic energy (in MeV).

(c) The speed of the protons at extraction as a fraction of $c$. Is the non-relativistic approximation justified?

Problem 30.5 ⭐ The RIKEN SRC superconducting cyclotron accelerates ${}^{238}\text{U}^{86+}$ to 345 MeV/u. The extraction radius is $R = 5.36$ m.

(a) Calculate the total kinetic energy of the uranium beam in MeV and in GeV.

(b) Calculate $\beta$ and $\gamma$ for the extracted beam.

(c) Calculate the magnetic field at the extraction radius. (Use the relativistic cyclotron formula.)

(d) What is the magnetic rigidity $B\rho$ of this beam (in T$\cdot$m)?

Problem 30.6 ⭐⭐ Show that for a non-relativistic cyclotron, the kinetic energy per nucleon can be written as:

$$\frac{T}{A} = K \left(\frac{Z}{A}\right)^2$$

where $K = e^2 B^2 R^2 / (2u)$ is the "K-value" of the cyclotron (in MeV). Calculate $K$ for a cyclotron with $B = 3.0$ T and $R = 2.0$ m, and determine the maximum energy per nucleon for (i) protons, (ii) ${}^{12}\text{C}^{6+}$, (iii) ${}^{48}\text{Ca}^{20+}$, and (iv) ${}^{238}\text{U}^{92+}$.

Problem 30.7 ⭐⭐ A compact superconducting cyclotron for medical isotope production has $B = 4.0$ T and $R = 0.25$ m.

(a) Calculate the maximum proton energy.

(b) The ${}^{18}\text{O}(p,n){}^{18}\text{F}$ reaction threshold is $Q = -2.44$ MeV (endothermic). What is the threshold proton kinetic energy in the lab frame? (Use the formula from Chapter 17: $T_{\text{th}} = -Q(1 + m_p/m_T)$ for $|Q| \ll m_T c^2$.)

(c) Is this cyclotron adequate for ${}^{18}\text{F}$ production? What is the excess energy above threshold?

Problem 30.8 ⭐⭐⭐ Relativistic cyclotron corrections. A proton cyclotron has $B_0 = 1.5$ T and $R = 0.50$ m.

(a) Calculate the maximum proton kinetic energy using the non-relativistic formula.

(b) Calculate the exact (relativistic) maximum kinetic energy using $T = (\gamma - 1)m_p c^2$ with $\gamma = \sqrt{1 + (eBR/(m_p c))^2}$.

(c) What is the percentage error from using the non-relativistic formula?

(d) At what fraction of the extraction radius does the cyclotron frequency deviate from the non-relativistic value by more than 1%?

Magnetic Rigidity and Fragment Separators

Problem 30.9 ⭐ Calculate the magnetic rigidity $B\rho$ (in T$\cdot$m) for:

(a) A 200 MeV proton.

(b) A ${}^{48}\text{Ca}^{20+}$ ion at 140 MeV/u.

(c) A ${}^{132}\text{Sn}^{50+}$ fragment at 150 MeV/u.

Use the relativistic formula in each case.

Problem 30.10 ⭐⭐ A fragment separator has a maximum magnetic rigidity of $B\rho_{\max} = 8.0$ T$\cdot$m (FRIB's ARIS).

(a) What is the maximum energy per nucleon for fully stripped ${}^{238}\text{U}^{92+}$ that can pass through the separator?

(b) What about ${}^{78}\text{Ni}^{28+}$?

(c) A ${}^{132}\text{Sn}^{50+}$ fragment is produced at 170 MeV/u. Can it be transmitted through ARIS?

Problem 30.11 ⭐⭐ The wedge degrader. A fragment separator uses a wedge-shaped aluminum degrader to introduce a $Z$-dependent energy loss for isotope separation. Two fully stripped fragments, ${}^{78}\text{Ni}^{28+}$ and ${}^{78}\text{Zn}^{30+}$, have the same mass number $A = 78$ but different atomic numbers.

(a) Both enter the degrader at 150 MeV/u. Using the Bethe-Bloch formula approximation $\Delta E \propto Z^2 / \beta^2$ (Chapter 16), estimate the ratio of their energy losses.

(b) After the degrader, will the two species have the same or different magnetic rigidities? Explain qualitatively how this enables $Z$-separation.

(c) Why is a wedge (varying thickness across the beam) used rather than a uniform degrader?

Problem 30.12 ⭐⭐⭐ Designing a particle identification scheme. At FRIB, you are setting up an experiment to study the neutron-rich isotope ${}^{54}\text{Ca}$ ($Z = 20$, $N = 34$) produced by fragmentation of a ${}^{76}\text{Ge}$ beam at 150 MeV/u.

(a) Estimate $B\rho$ for the ${}^{54}\text{Ca}^{20+}$ fragment, assuming it emerges at 100 MeV/u after the target and degrader. Use the relativistic formula.

(b) What other isotopes might have a similar $B\rho$? (Hint: Ions with the same $A/Z$ ratio have the same $B\rho$ at the same velocity.)

(c) If the flight path through the separator is $L = 40$ m and the TOF resolution is $\Delta t = 50$ ps, what mass resolving power $A/\Delta A$ is achieved? (Use $\Delta A/A = 2\Delta t / \text{TOF}$ for fixed $B\rho$.)

Penning Traps and Mass Measurements

Problem 30.13 ⭐ In a Penning trap with magnetic field $B = 7.0$ T:

(a) Calculate the cyclotron frequency $\nu_c$ for ${}^{39}\text{K}^{+}$ ($m = 38.9637 \; \text{u}$).

(b) If the reference ion is ${}^{39}\text{K}^{+}$ and the unknown ion has $\nu_c = 1,746,231.5 \pm 0.3$ Hz, determine the mass of the unknown ion in atomic mass units. Compare to ${}^{41}\text{K}$ ($m = 40.9618 \; \text{u}$) — is this consistent?

Problem 30.14 ⭐⭐ The ISOLTRAP Penning trap measures the mass of ${}^{82}\text{Zn}$ (a neutron-rich nucleus with $N = 52$, just above the $N = 50$ shell closure). The cyclotron frequency ratio to ${}^{82}\text{Kr}^{+}$ (mass known to 0.1 keV) is measured as $r = \nu_c({}^{82}\text{Kr}^{+})/\nu_c({}^{82}\text{Zn}^{+}) = 1.000\,000\,423(15)$.

(a) From this ratio, calculate the mass of ${}^{82}\text{Zn}$ in atomic mass units, given $m({}^{82}\text{Kr}) = 81.913\,483\,6(3)$ u.

(b) Calculate the two-neutron separation energy $S_{2n}({}^{82}\text{Zn}) = B(82,30) - B(80,30)$. (You will need the mass of ${}^{80}\text{Zn}$; use $m({}^{80}\text{Zn}) = 79.944\,29$ u.)

(c) A sharp drop in $S_{2n}$ at $N = 50$ is a signature of the shell closure. If $S_{2n}({}^{80}\text{Zn}) = 13.4$ MeV and your calculated $S_{2n}({}^{82}\text{Zn})$ is significantly lower, what does this tell you about the $N = 50$ shell gap in zinc?

Problem 30.15 ⭐⭐ MR-TOF vs. Penning trap. An MR-TOF device achieves mass resolving power $R = m/\Delta m = 2 \times 10^5$ after a flight time of 10 ms. A Penning trap achieves $\delta m/m = 5 \times 10^{-8}$ with a measurement time of 500 ms.

(a) For $A = 100$, what mass uncertainty (in keV) does each device achieve?

(b) The isotope of interest has $t_{1/2} = 30$ ms. What fraction of produced ions survive to complete the measurement in each device? (Assume exponential decay.)

(c) Which device is better suited for this isotope? Explain the trade-off between precision and speed.

Laser Spectroscopy

Problem 30.16 ⭐ The isotope shift of the $D_1$ transition ($5s \to 5p_{1/2}$) in rubidium between ${}^{85}\text{Rb}$ and ${}^{87}\text{Rb}$ is $\delta\nu^{85,87} = 78.1$ MHz. The total isotope shift has two contributions:

$$\delta\nu^{A,A'} = \delta\nu_{\text{mass}}^{A,A'} + F \cdot \delta\langle r^2 \rangle^{A,A'}$$

where $F$ is the field shift constant. For this transition, the mass shift is calculated to be $\delta\nu_{\text{mass}}^{85,87} = 56.2$ MHz.

(a) Extract the product $F \cdot \delta\langle r^2 \rangle^{85,87}$.

(b) If $F = -700$ MHz/fm$^2$, determine $\delta\langle r^2\rangle^{85,87}$ in fm$^2$.

(c) Given that $\langle r^2 \rangle^{1/2}({}^{85}\text{Rb}) = 4.20$ fm, estimate $\langle r^2 \rangle^{1/2}({}^{87}\text{Rb})$.

Problem 30.17 ⭐⭐ Hyperfine structure. The magnetic hyperfine interaction splits an atomic level with total electronic angular momentum $J$ into $2\min(I,J)+1$ components, where $I$ is the nuclear spin. The splitting is proportional to the magnetic hyperfine constant $A_{hf}$, which depends on the nuclear magnetic moment.

(a) For an atomic transition between a level with $J = 1/2$ and one with $J = 3/2$, how many hyperfine components are expected for a nucleus with $I = 3/2$? (List the allowed $F$ values for each level, where $\mathbf{F} = \mathbf{I} + \mathbf{J}$.)

(b) If an unknown nucleus is measured and 6 hyperfine components are observed (3 in each level), what constraint does this place on the nuclear spin $I$?

Doppler Shift and Gamma-Ray Detection

Problem 30.18 ⭐ A ${}^{54}\text{Ca}$ nucleus moving at $\beta = v/c = 0.40$ emits a 2.04 MeV gamma ray (the expected $2^+ \to 0^+$ transition).

(a) Calculate the gamma-ray energy observed at $\theta_{\text{lab}} = 0°$ (forward), $90°$, and $180°$ (backward).

(b) If a germanium detector has intrinsic energy resolution $\Delta E / E = 0.2\%$ at 1 MeV, what angular opening $\Delta\theta$ can the detector subtend at $\theta = 90°$ before the Doppler broadening exceeds the intrinsic resolution?

(c) How does gamma-ray tracking (position resolution $\sim 2$ mm at a distance of 20 cm from the target) improve the situation?

Problem 30.19 ⭐⭐ Compton scattering in a tracking array. A 1.33 MeV gamma ray (from the decay of ${}^{60}\text{Co}$, used for calibration) undergoes Compton scattering in a HPGe detector.

(a) If the gamma ray scatters at angle $\theta_1 = 30°$ in the first interaction, depositing energy $E_1$, and is then fully absorbed at a second interaction point, calculate $E_1$ and the remaining energy $E_2 = 1.33 - E_1$ using the Compton formula:

$$E_1 = E_\gamma - \frac{E_\gamma}{1 + (E_\gamma / m_e c^2)(1 - \cos\theta_1)}$$

(b) In the tracking algorithm, the two interaction points are identified, and $\theta_1$ is reconstructed from their positions. If the position resolution is $\sigma_x = 2$ mm and the two interaction points are separated by $d = 30$ mm, estimate the angular resolution $\delta\theta_1$.

(c) From $\delta\theta_1$, estimate the reconstructed energy resolution for this event.

Active Targets and Reaction Studies

Problem 30.20 ⭐⭐ The AT-TPC is filled with deuterium gas at a pressure of $P = 200$ Torr and temperature $T_{\text{gas}} = 293$ K. A ${}^{54}\text{Ca}$ beam at 70 MeV/u enters the 1-meter-long chamber.

(a) Calculate the number density of deuterium molecules and the areal density (atoms/cm$^2$) of deuterium atoms along the beam path. (Use ideal gas: $n = P/(k_B T)$.)

(b) Express the areal density in mg/cm$^2$ of deuterium. Compare to a typical solid deuterated polyethylene (CD$_2$) target thickness of 10 mg/cm$^2$.

(c) If the $(d,p)$ reaction cross section at this energy is $\sigma \sim 10$ mb and the beam rate is 500 ions/second, estimate the reaction rate per second.

Problem 30.20b ⭐⭐⭐ Luminosity comparison. Compare the luminosity $\mathcal{L} = R_{\text{beam}} \times n_t$ (in cm$^{-2}$s$^{-1}$) for three experimental configurations:

(a) A stable ${}^{12}\text{C}$ beam at $10^{10}$ ions/s on a solid ${}^{9}\text{Be}$ target of 100 mg/cm$^2$.

(b) A radioactive ${}^{54}\text{Ca}$ beam at 500 ions/s on a solid ${}^{9}\text{Be}$ target of 100 mg/cm$^2$.

(c) A radioactive ${}^{54}\text{Ca}$ beam at 500 ions/s in the AT-TPC filled with deuterium gas at 200 Torr (use your result from Problem 30.20 for $n_t$).

(d) For a reaction cross section $\sigma = 100$ mb, calculate the reaction rate (events/s) in each case. Why does the AT-TPC partially compensate for the low beam rate?

(e) If the AT-TPC detects reaction products with 80% efficiency versus 5% for a conventional setup (due to the $4\pi$ solid angle coverage), what is the ratio of detected events per second between configurations (b) and (c)?

Synchrotrons and Energy Ramping

Problem 30.20c ⭐⭐ The GSI SIS-18 synchrotron has a bending radius $\rho = 10$ m and a maximum magnetic field of $B_{\max} = 1.8$ T.

(a) Calculate the maximum momentum and kinetic energy per nucleon for ${}^{238}\text{U}^{73+}$ (a partially stripped uranium beam — the charge state typically available from the SIS-18 injection chain).

(b) If the injection energy is 11.4 MeV/u (from the UNILAC linac), calculate the magnetic field at injection. What is the ratio $B_{\max}/B_{\min}$?

(c) The revolution frequency at injection is approximately $f_{\text{rev}} = \beta c / (2\pi \rho_{\text{avg}})$ where $\rho_{\text{avg}} \approx 34$ m is the average orbit circumference divided by $2\pi$. Calculate $f_{\text{rev}}$ at injection and at maximum energy. The RF must track this frequency — what is the required frequency sweep range?

Problem 30.20d ⭐ K-value exercise. The "K-value" of a cyclotron is defined as $K = q^2 B^2 R^2 / (2u)$ in units where $q$ and $B$ are in SI and the result is in MeV. A cyclotron's maximum energy per nucleon for any ion is:

$$\frac{T}{A} = K \left(\frac{Z}{A}\right)^2$$

(a) The TRIUMF main cyclotron has $K = 520$ MeV. What maximum energy can it give to protons ($Z/A = 1$)? To ${}^{12}\text{C}^{6+}$ ($Z/A = 0.5$)?

(b) The RIKEN SRC has $K = 2600$ MeV. What maximum energy can it give to ${}^{238}\text{U}^{92+}$ ($Z/A = 0.387$)? Compare to the actual beam energy of 345 MeV/u — why might the actual energy differ from this simple estimate?

(c) Two cyclotrons have the same K-value but different radii. Which has the stronger field? If Cyclotron A has $R_A = 2R_B$, what is the ratio $B_A / B_B$?

Experimental Design

Problem 30.21 ⭐⭐ You are designing an experiment to measure the mass of ${}^{130}\text{Cd}$ ($Z = 48$, $N = 82$, a key r-process waiting-point nucleus) at FRIB using a Penning trap. The expected FRIB production rate after the gas stopper is ~10 ions/second.

(a) The Penning trap measurement requires a minimum of 100 ions for adequate statistics. If the trapping/measurement cycle is 200 ms per ion, how long will data collection take?

(b) The half-life of ${}^{130}\text{Cd}$ is 162 ms. What fraction of ions survive the 200 ms measurement cycle?

(c) To compensate for decay losses, how many ions must be delivered to the trap? How long does data collection now take?

(d) Is this a feasible experiment in a typical 7-day beam time? What fraction of the beam time is available for physics data (assuming 2 days for setup and calibration)?

Problem 30.22 ⭐⭐⭐ Proposal calculation: Coulomb excitation of ${}^{132}\text{Sn}$. You plan to Coulomb-excite the first $2^+$ state of the doubly magic ${}^{132}\text{Sn}$ using a ${}^{132}\text{Sn}$ beam at 5 MeV/u on a ${}^{208}\text{Pb}$ target. The Coulomb excitation cross section is approximately:

$$\sigma_{CE} \approx \frac{1}{4} \pi \left(\frac{a}{2}\right)^2 \left(\frac{Z_1 Z_2 e^2}{\hbar c}\right)^2 \frac{B(E2; 0^+ \to 2^+)}{e^2 \text{fm}^4} \cdot f(E, a_{\text{half}})$$

where $a = d_0 / 2$ is half the distance of closest approach and $f \sim 10^{-2}$ is a kinematic factor. For a rough estimate, use $\sigma_{CE} \sim 100$ mb (this is typical for Coulomb excitation of doubly magic nuclei).

(a) If the beam rate is $10^4$ ions/s and the target is 1 mg/cm$^2$ of ${}^{208}\text{Pb}$, calculate the Coulomb excitation rate per second.

(b) To measure the $B(E2)$ to 10% statistical precision, you need $\sim 100$ Coulomb excitation events detected. If the gamma-ray detection efficiency (GRETINA at forward angles) is 5%, how many Coulomb excitation events must occur? How many hours of beam time does this require?

(c) Write a one-paragraph PAC motivation for this experiment.

Research-Level Problems

Problem 30.23 ⭐⭐⭐ (Research) ISOL vs. fragmentation for ${}^{100}\text{Sn}$. The doubly magic nucleus ${}^{100}\text{Sn}$ ($Z = N = 50$) is the heaviest known $N = Z$ nucleus and has been produced at both ISOL and fragmentation facilities.

(a) Explain why ${}^{100}\text{Sn}$ is difficult to produce by ISOL. Consider: What is its half-life (~1 s)? What is the chemistry of tin? Is diffusion from a thick target likely to be fast enough?

(b) At RIKEN-RIBF, ${}^{100}\text{Sn}$ has been produced by fragmentation of ${}^{124}\text{Xe}$ at 345 MeV/u. Estimate the magnetic rigidity of ${}^{100}\text{Sn}^{50+}$ at 200 MeV/u (a plausible fragment energy). Can BigRIPS ($B\rho_{\max} = 9.5$ T$\cdot$m) accommodate this?

(c) At ISOLDE, ${}^{100}\text{Sn}$ could in principle be produced by proton-induced spallation of a tin or lanthanum target. What advantages would the ISOL beam quality offer for a Penning trap mass measurement?

(d) Discuss the scientific importance of a precision mass measurement of ${}^{100}\text{Sn}$ for nuclear structure and the $N = Z$ line.

Problem 30.24 ⭐⭐⭐ (Research) Charge radii across the calcium isotope chain. Laser spectroscopy measurements at ISOLDE have determined the charge radii of calcium isotopes from ${}^{36}\text{Ca}$ to ${}^{52}\text{Ca}$.

(a) The magic numbers in calcium are $Z = 20$ and $N = 20, 28, 32, 34$ (?). How would you expect the charge radius to behave at each magic number? (Hint: At shell closures, the charge radius often increases more slowly than the $A^{1/3}$ trend, producing a "kink" in $\delta\langle r^2\rangle$.)

(b) The measurement of $\langle r^2 \rangle^{1/2}$ for ${}^{52}\text{Ca}$ ($N = 32$) was a landmark result that confirmed ab initio nuclear theory predictions including three-nucleon forces. If the measured value is $3.570 \pm 0.005$ fm and the $A^{1/3}$ extrapolation from ${}^{48}\text{Ca}$ ($3.477$ fm) would give $3.509$ fm, by how many standard deviations does the measurement deviate from the naive extrapolation?

(c) Explain why this measurement required a radioactive beam facility rather than conventional spectroscopy. (Hint: What is the half-life of ${}^{52}\text{Ca}$?)

Problem 30.25 ⭐⭐⭐ (Research) Estimating FRIB's reach. FRIB's production rate for a given isotope can be roughly estimated from:

$$R_{\text{secondary}} = R_{\text{primary}} \times \sigma_{\text{prod}} \times n_t \times \epsilon_{\text{sep}} \times \epsilon_{\text{transport}}$$

where $R_{\text{primary}}$ is the primary beam rate, $\sigma_{\text{prod}}$ is the production cross section, $n_t$ is the target areal density, $\epsilon_{\text{sep}}$ is the separator transmission efficiency, and $\epsilon_{\text{transport}}$ is the transport efficiency to the experimental station.

For FRIB: $R_{\text{primary}} \approx 8 \times 10^{13}$ ${}^{238}\text{U}$/s (at 400 kW, 200 MeV/u), $n_t \approx 10^{22}$ atoms/cm$^2$ (for a Be target of $\sim 15$ mm), $\epsilon_{\text{sep}} \approx 0.5$, $\epsilon_{\text{transport}} \approx 0.8$.

(a) The fragmentation cross section for producing ${}^{78}\text{Ni}$ from ${}^{238}\text{U}$ is roughly $\sigma \approx 0.01$ $\mu$b. Estimate the ${}^{78}\text{Ni}$ rate at the experiment.

(b) How many ${}^{78}\text{Ni}$ ions are produced in a 7-day beam time?

(c) Is this enough for a Penning trap mass measurement (requires ~100 ions)? For in-beam gamma-ray spectroscopy (requires ~$10^4$ gamma-ray events, with ~5% efficiency)?

Problem 30.26 ⭐ Quick estimates — accelerator types. Match each accelerator type (A–D) with its primary advantage and the facility where it is used. No calculation required.

Advantages (each used once): (i) Highest beam power for rare-isotope production. (ii) Continuously variable energy to very high energies. (iii) Best energy resolution for resonance studies. (iv) Continuous (CW) beam with high average current.

Facilities (each used once): (i) GSI/FAIR. (ii) FRIB. (iii) University of Notre Dame. (iv) RIKEN-RIBF.

Integrated Conceptual Problems

Problem 30.27 ⭐⭐ ISOL chemistry. Explain why the ISOL method is highly efficient for producing beams of noble gases (He, Ne, Ar, Kr, Xe, Rn) and alkali metals (Na, K, Rb, Cs, Fr) but inefficient for refractory metals (Zr, Nb, Mo, W, Re).

(a) What physical process limits the speed at which isotopes are released from an ISOL target? How does this depend on the melting point and vapor pressure of the element?

(b) The Resonance Ionization Laser Ion Source (RILIS) at ISOLDE uses tuned lasers to selectively ionize specific elements. Explain how this achieves element selectivity (choosing $Z$), going beyond the mass separator (which selects $A$).

(c) ISOLDE recently developed a new target/ion-source combination for producing beams of actinium ($Z = 89$). Actinium is a refractory element with a melting point of 1050$°$C. The target is operated at 2000$°$C. Qualitatively, why is the high target temperature essential? What fraction of produced actinium isotopes with $t_{1/2} = 100$ ms might be expected to survive the ~500 ms release time?

Problem 30.28 ⭐⭐ Beam quality comparison. An ISOL beam of ${}^{132}\text{Sn}$ post-accelerated to 5 MeV/u has an energy spread $\Delta T / T = 10^{-3}$ and an emittance of $\epsilon = 3\pi$ mm$\cdot$mrad. A fragmentation beam of ${}^{132}\text{Sn}$ at 150 MeV/u from BigRIPS has $\Delta T / T = 10^{-2}$ and $\epsilon = 40\pi$ mm$\cdot$mrad.

(a) For a Coulomb excitation experiment requiring beam-energy resolution of $\Delta T / T < 5 \times 10^{-3}$ to cleanly separate elastic from inelastic scattering, which beam is suitable?

(b) For a knockout reaction study ${}^{132}\text{Sn}({}^{9}\text{Be}, X)$ where the desired observable is the inclusive cross section (no energy resolution needed, but the beam must be fast enough to use the thick-target technique), which beam is more appropriate? Why?

(c) This complementarity is often described as "ISOL beams are like a rifle bullet (precise, aimed) and fragmentation beams are like a shotgun blast (fast, broad)." Evaluate this analogy: where does it work and where does it break down?

Problem 30.29 ⭐⭐ The gas stopping cell. At FRIB, fast fragments ($\sim 100$–$200$ MeV/u) are slowed by a solid degrader and then stopped in a helium-filled gas cell at about 100 mbar.

(a) Estimate the range (in mg/cm$^2$) of a 150 MeV/u ${}^{78}\text{Ni}$ ion in aluminum using the approximation $R \approx 0.5 \times (T/A)^{1.7} \times A / Z^2$ mg/cm$^2$, where $T/A$ is in MeV/u. What thickness of aluminum degrader is needed to reduce the energy from 150 MeV/u to approximately 5 MeV/u?

(b) Once in the gas cell, the ions are thermalized by collisions with helium atoms. The thermalization time is roughly $\sim 100\;\mu$s. If the extracted ion is singly charged ($q = +e$) and is guided to the exit by a DC electric field and RF carpet, the total extraction time is approximately 50 ms. For an isotope with $t_{1/2} = 20$ ms, what fraction survives extraction?

(c) Gas stopping efficiencies at current facilities are typically 20–50%. Identify at least two physical effects that cause losses (hint: consider charge exchange, space charge, and recombination).