Exercises — Chapter 16

Charged-Particle Energy Loss (Bethe-Bloch)

Problem 16.1 ⭐ A 5.486 MeV alpha particle from ${}^{241}$Am enters a gas detector filled with argon ($Z = 18$, $A = 40$, $I = 188\,\text{eV}$, $\rho = 1.66 \times 10^{-3}\,\text{g/cm}^3$).

(a) Calculate $\beta$ and $\gamma$ for the alpha particle. (Recall $m_\alpha c^2 = 3727.4\,\text{MeV}$.)

(b) Calculate $T_{\max}$, the maximum energy transfer to a single electron.

(c) Using the Bethe-Bloch formula (neglecting shell and density corrections), compute $-\frac{1}{\rho}\frac{dE}{dx}$ in MeV cm$^2$/g.

(d) If the $W$-value for argon is 26.4 eV, how many ion pairs does the alpha particle produce per centimeter of path?

Problem 16.2 ⭐ Show that the Bethe-Bloch formula predicts $-dE/dx \propto z^2/\beta^2$ at non-relativistic velocities (keeping only the dominant terms). Use this to explain why:

(a) An alpha particle ($z = 2$) at the same velocity as a proton has 4 times the stopping power.

(b) An alpha particle at the same kinetic energy as a proton has approximately 16 times the stopping power. (Hint: at the same $T$, how does $\beta_\alpha$ compare to $\beta_p$?)

Problem 16.3 ⭐⭐ The range of a proton in water is approximately $R_p = 0.0022 \times T^{1.77}$ cm, where $T$ is in MeV (valid for 10–200 MeV).

(a) Calculate the range of a 150 MeV proton in water. Compare to the NIST PSTAR value of 15.78 cm.

(b) A proton therapy facility needs to treat a tumor at 20 cm depth in tissue (approximately water-equivalent). What initial proton energy is required?

(c) If the beam energy can be varied from 70 to 230 MeV, what range of depths can be treated?

Problem 16.4 ⭐⭐ Bragg peak width. Range straggling for protons in water can be approximated as $\sigma_R \approx 0.012 R$ (i.e., $\sigma_R/R \approx 1.2\%$).

(a) For a 150 MeV proton beam (range 15.78 cm), estimate $\sigma_R$.

(b) The Bragg peak FWHM is approximately $2.355\sigma_R$ for a Gaussian approximation. Calculate the FWHM.

(c) A tumor of 3 cm extent requires a spread-out Bragg peak (SOBP). How many individual Bragg peaks (at different energies) are needed if the peaks are spaced by one FWHM apart?

Problem 16.5 ⭐⭐ Minimum ionizing particles. Cosmic-ray muons at sea level have typical energies of $\sim 3\,\text{GeV}$ ($m_\mu c^2 = 105.7\,\text{MeV}$).

(a) Calculate $\beta\gamma$ for a 3 GeV muon. Verify that it is near the minimum of the Bethe-Bloch curve.

(b) The mass stopping power for a MIP in most materials is $\sim 2\,\text{MeV\,cm}^2/\text{g}$. How much energy does a cosmic-ray muon lose traversing 1 m of water? Is this a significant fraction of its energy?

(c) What thickness of iron ($\rho = 7.87\,\text{g/cm}^3$) would stop a 3 GeV muon, assuming constant $-\frac{1}{\rho}\frac{dE}{dx} \approx 2\,\text{MeV\,cm}^2/\text{g}$? (This is a rough estimate; discuss why the actual range is shorter.)

Problem 16.6 ⭐⭐⭐ Derivation: Energy transfer to an electron.

Starting from the impulse approximation (a fast charged particle with charge $ze$, velocity $v$, passing an electron at impact parameter $b$), derive:

(a) The transverse momentum impulse $\Delta p = \frac{2ze^2}{4\pi\epsilon_0 bv}$ by evaluating $\int_{-\infty}^{\infty} F_\perp \, dt$.

(b) The energy transfer $T(b) = \frac{2z^2 e^4}{(4\pi\epsilon_0)^2 m_e v^2 b^2}$.

(c) Show that integrating $T(b) \cdot n_e \cdot 2\pi b\,db$ over impact parameter gives a logarithmic dependence on $b_{\max}/b_{\min}$.

(d) Argue physically why $b_{\max}$ is set by adiabatic cutoff and $b_{\min}$ by the uncertainty principle.

Problem 16.7 ⭐⭐⭐ Delta rays. A 200 MeV proton in water produces delta rays (knocked-on electrons) with a spectrum $\frac{dN}{dT_e} = \frac{C}{T_e^2}$ for $T_{\min} \leq T_e \leq T_{\max}$.

(a) Determine $T_{\max}$ for a 200 MeV proton.

(b) A delta ray is "visible" (produces a detectable secondary track) if $T_e > 1\,\text{keV}$. Calculate the number of visible delta rays per cm of proton track. (Use $C = \frac{2\pi r_e^2 m_e c^2 z^2 n_e}{\beta^2}$ with $n_e$ for water.)

(c) What fraction of the proton's total energy loss goes into delta rays with $T_e > 1\,\text{keV}$?

Photon Interactions

Problem 16.8 ⭐ For each of the following photon energies, identify the dominant interaction mechanism in (i) water and (ii) lead. Use the approximate crossover energies from Section 16.3.

(a) 10 keV (b) 100 keV (c) 1 MeV (d) 10 MeV (e) 100 MeV

Problem 16.9 ⭐ A 1.0 MeV photon Compton-scatters off an electron at rest.

(a) Calculate the scattered photon energy $E_\gamma'$ for scattering angles $\theta = 30°$, $60°$, $90°$, $120°$, $180°$.

(b) Calculate the electron recoil kinetic energy $T_e$ for each angle.

(c) What is the Compton edge energy? At what scattering angle does the electron receive half the photon's energy?

Problem 16.10 ⭐⭐ Compton edge and backscatter peak. A gamma-ray spectrum from an unknown source shows: - A photopeak at 1274 keV - A Compton edge at 1061 keV - A backscatter peak at 213 keV

(a) Verify that the Compton edge and backscatter peak positions are consistent with a 1274 keV gamma ray.

(b) Identify the source. (Hint: 1274 keV is a common calibration energy.)

(c) This source also emits a 511 keV gamma ray. Explain why, and predict the Compton edge energy for the 511 keV line.

Problem 16.11 ⭐⭐ Beer's law and shielding. The mass attenuation coefficients $\mu/\rho$ (cm$^2$/g) for 662 keV photons (${}^{137}$Cs) are:

(a) Calculate the half-value layer (HVL) in cm for each material.

(b) How many HVLs of lead are needed to reduce the intensity by a factor of 1000?

(c) A radiation worker needs to reduce the dose rate from a ${}^{137}$Cs source by a factor of 100. Calculate the required thickness of (i) lead, (ii) concrete.

(d) Which material requires the least mass per unit area ($\rho x$ in g/cm$^2$) for the same attenuation? What does this tell you about the optimal shielding material?

Problem 16.12 ⭐⭐ Pair production threshold. Show that:

(a) Pair production by a photon in free space ($\gamma \to e^+ + e^-$) is forbidden by simultaneous conservation of energy and momentum.

(b) In the Coulomb field of a nucleus of mass $M$, the threshold photon energy is $E_{\text{th}} = 2m_e c^2 (1 + m_e/M)$, which approaches $2m_e c^2 = 1.022\,\text{MeV}$ for $M \gg m_e$.

(c) In the field of an atomic electron (triplet production: $\gamma + e^- \to e^+ + e^- + e^-$), the threshold is $4m_e c^2 = 2.044\,\text{MeV}$.

Problem 16.13 ⭐⭐⭐ Klein-Nishina integration. The total Compton cross section per electron is obtained by integrating the Klein-Nishina formula over all solid angles.

(a) Define $\alpha = E_\gamma / (m_e c^2)$. Show that the total cross section is:

$$\sigma_{\text{KN}} = 2\pi r_e^2 \left\{\frac{1+\alpha}{\alpha^2}\left[\frac{2(1+\alpha)}{1+2\alpha} - \frac{1}{\alpha}\ln(1+2\alpha)\right] + \frac{1}{2\alpha}\ln(1+2\alpha) - \frac{1+3\alpha}{(1+2\alpha)^2}\right\}$$

(b) Verify that for $\alpha \ll 1$, this reduces to $\sigma_T = (8\pi/3)r_e^2 = 0.665\,\text{barn}$.

(c) Evaluate $\sigma_{\text{KN}}$ at $E_\gamma = 0.662\,\text{MeV}$, $1.33\,\text{MeV}$, and $10\,\text{MeV}$. Plot $\sigma_{\text{KN}}/\sigma_T$ versus $\alpha$.

Neutron Interactions

Problem 16.14 ⭐ A 2 MeV neutron undergoes elastic collisions in various moderators.

(a) Calculate $\alpha = [(A-1)/(A+1)]^2$ for H ($A = 1$), D ($A = 2$), C ($A = 12$), and Fe ($A = 56$).

(b) After a single head-on collision ($\theta_{\text{cm}} = \pi$), what is the neutron energy in each case?

(c) Calculate $\xi$ (average logarithmic energy decrement) for each moderator.

(d) How many collisions are needed to thermalize the 2 MeV neutron ($E_{\text{th}} = 0.025\,\text{eV}$) in each moderator?

Problem 16.15 ⭐⭐ The 1/v law. The thermal neutron capture cross section for ${}^{10}$B is $\sigma_0 = 3837\,\text{barn}$ at $E_0 = 0.0253\,\text{eV}$.

(a) Calculate $\sigma(E)$ at $E = 1\,\text{eV}$, $0.1\,\text{eV}$, and $0.001\,\text{eV}$.

(b) A neutron beam with a Maxwell-Boltzmann energy distribution at temperature $T$ has a mean cross section: $\bar{\sigma} = \sigma_0 \sqrt{\pi/4} \cdot (T_0/T)^{1/2}$, where $T_0 = 293\,\text{K}$. Calculate $\bar{\sigma}$ for ${}^{10}$B at $T = 293\,\text{K}$ and $T = 600\,\text{K}$.

(c) Explain physically why the Maxwell-averaged cross section differs from $\sigma_0$.

Problem 16.16 ⭐⭐ A BF$_3$ proportional counter (enriched in ${}^{10}$B) is used to detect thermal neutrons via ${}^{10}$B(n,$\alpha$)${}^{7}$Li. The ${}^{10}$B number density in BF$_3$ gas at 1 atm, 300 K is $n_{10} = 1.06 \times 10^{19}\,\text{cm}^{-3}$.

(a) Calculate the macroscopic cross section $\Sigma = n_{10} \sigma$ for thermal neutrons ($\sigma = 3837\,\text{barn}$).

(b) What fraction of thermal neutrons are captured in a 30 cm long counter? (Use $f = 1 - e^{-\Sigma L}$.)

(c) For the 94% branch ($Q = 2.31\,\text{MeV}$), calculate the kinetic energies of the $\alpha$ and ${}^{7}$Li$^*$ using conservation of momentum (the thermal neutron momentum is negligible).

Detectors

Problem 16.17 ⭐ A parallel-plate ionization chamber with a 2 cm gap is filled with air ($W = 33.97\,\text{eV}$) at STP. A 5.486 MeV alpha particle from ${}^{241}$Am stops completely in the gas.

(a) How many ion pairs are produced?

(b) What is the total charge collected (in coulombs) at each electrode?

(c) If the collection time is 10 $\mu$s and the measurement is repeated at a rate of 100 events/s, what is the average current?

Problem 16.18 ⭐⭐ Gas multiplication. In a proportional counter, the gas multiplication factor $M$ depends on the voltage. At $V = 800\,\text{V}$, $M = 500$.

(a) If a 5.486 MeV alpha particle produces 161,000 primary ion pairs (in P-10 gas), what is the total collected charge with gas multiplication?

(b) Compare to the charge from a 662 keV gamma ray that deposits its full energy via photoelectric absorption (producing $\sim 19{,}500$ primary ion pairs in P-10). Can the proportional counter distinguish alphas from gammas?

(c) In a Geiger-Müller counter, $M \approx 10^8$–$10^{10}$. Explain why the GM counter cannot distinguish alphas from gammas.

Problem 16.19 ⭐⭐ Scintillator comparison. Three detectors measure the 662 keV gamma ray from ${}^{137}$Cs:

(a) Two gamma rays are emitted at 661 keV and 663 keV. Can each detector resolve them as separate peaks? (A rule of thumb: two peaks are resolved if their separation exceeds 1 FWHM.)

(b) ${}^{134}$Cs emits gamma rays at 605 keV and 796 keV. Can each detector resolve these from the 662 keV line of ${}^{137}$Cs?

(c) In the decay of ${}^{152}$Eu, gamma rays are emitted at 344.3, 411.1, 444.0, 778.9, 867.4, 964.1, 1085.8, 1112.1, and 1408.0 keV. Which pairs of peaks would be unresolvable in NaI but resolved in HPGe?

Problem 16.20 ⭐⭐ HPGe energy resolution. The Fano factor for germanium is $F = 0.13$ and $\epsilon = 2.96\,\text{eV}$.

(a) Calculate the statistical (Fano) contribution to the FWHM at 122 keV (${}^{57}$Co), 662 keV (${}^{137}$Cs), and 1332 keV (${}^{60}$Co).

(b) A manufacturer specifies the FWHM at 1332 keV as 1.85 keV. What is the contribution from electronic noise and charge collection, assuming it adds in quadrature with the statistical contribution?

(c) Show that the statistical contribution to the resolution $R = \text{FWHM}/E_0$ scales as $1/\sqrt{E_0}$.

Problem 16.21 ⭐⭐⭐ Detector efficiency. A coaxial HPGe detector with 50% relative efficiency (relative to a 3" $\times$ 3" NaI at 25 cm for the 1332 keV line of ${}^{60}$Co, where the NaI absolute efficiency is $\epsilon_{\text{NaI}} = 1.2 \times 10^{-3}$) is used to measure a ${}^{60}$Co source.

(a) What is the absolute full-energy peak efficiency of the HPGe detector at 1332 keV and 25 cm?

(b) The source has an activity of 10 kBq. How many counts per second appear in the 1332 keV photopeak? (Remember ${}^{60}$Co emits one 1332 keV photon per decay, but also a 1173 keV photon — consider only the 1332 keV line.)

(c) How long must you count to accumulate 10,000 counts in the 1332 keV peak (for $\sim 1\%$ statistical uncertainty)?

Dosimetry

Problem 16.22 ⭐ Convert the following doses between old and SI units:

(a) 500 mrad to Gray (b) 100 mrem to Sievert (c) 2 Gy to rad (d) 0.05 Sv to rem

Problem 16.23 ⭐ A radiation worker receives the following exposures in one year: - Whole-body gamma irradiation: 5 mGy - Hand exposure (from handling sources): 50 mGy from beta particles

(a) Calculate the equivalent dose (in mSv) to the whole body and to the hand.

(b) Using tissue weighting factors ($w_T = 0.12$ for remainder tissues including hands), calculate the contribution of each exposure to the effective dose.

(c) Is the worker within the ICRP annual effective dose limit of 20 mSv?

Problem 16.24 ⭐⭐ Dose from a radioactive source. A point source of ${}^{137}$Cs ($E_\gamma = 662\,\text{keV}$, $\Gamma_\delta = 0.33\,\text{R\cdot m}^2/(\text{hr}\cdot\text{Ci})$) has an activity of 100 mCi.

(a) Calculate the exposure rate (in R/hr) at a distance of 1 m.

(b) The dose rate in tissue is approximately $D \approx 0.87 \times X$ (where $X$ is in R/hr and $D$ is in rad/hr) for medium-energy gamma rays. Convert to mGy/hr.

(c) How long can a worker remain at 1 m before receiving the weekly dose limit of 0.4 mSv?

(d) At what distance does the dose rate drop below 2.5 $\mu$Sv/hr (the boundary of a radiation area under US NRC regulations)?

Problem 16.25 ⭐⭐ Comparing radiation types. A person receives the following exposures: - 1 mGy whole body from gamma rays - 0.05 mGy to the lungs from inhaled alpha-emitting radon daughters - 0.1 mGy to the skin from beta particles

(a) Calculate the equivalent dose to each tissue.

(b) Calculate the effective dose from each exposure.

(c) Which exposure contributes the most to the effective dose? This result explains why radon is the dominant source of natural radiation exposure.

Synthesis and Advanced Problems

Problem 16.26 ⭐⭐⭐ Proton therapy vs. photon therapy. A tumor is located at 15 cm depth in water-equivalent tissue. Compare the ratio of entrance dose to tumor dose for:

(a) A 6 MV photon beam (approximate the depth-dose curve as $D(x) \propto e^{-0.05x}$ for $x > 1.5\,\text{cm}$, with buildup for $x < 1.5\,\text{cm}$ reaching a maximum at 1.5 cm).

(b) A proton beam with range adjusted to place the Bragg peak at 15 cm. (Approximate the entrance dose as $-dE/dx$ at the surface energy, and the peak dose as the value at $\sim 1\,\text{cm}$ before the range end.)

(c) Discuss qualitatively how a spread-out Bragg peak (SOBP) changes the entrance-to-tumor ratio for protons.

Problem 16.27 ⭐⭐⭐ Gamma-ray spectrum interpretation. An HPGe detector measures the following peaks from an unknown source:

• 511 keV (moderate intensity)
• 1173 keV (strong)
• 1332 keV (strong, equal to 1173)
• 2505 keV (weak)

(a) Identify the source. Explain each peak.

(b) The 2505 keV peak is a sum peak. Explain under what conditions sum peaks occur, and why this peak is weak.

(c) At what energies do you expect to find the Compton edges for the 1173 and 1332 keV lines?

(d) At what energies do you expect single and double escape peaks?

Problem 16.28 ⭐⭐⭐ Neutron detection challenge. A mixed radiation field contains both neutrons and gamma rays. A ${}^{3}$He proportional counter detects neutrons via ${}^{3}$He(n,p)${}^{3}$H ($Q = 0.764\,\text{MeV}$).

(a) What is the pulse height (in terms of deposited energy) for thermal neutron capture events? Are all events at the same pulse height?

(b) Gamma rays also produce pulses in the detector (via Compton scattering off electrons in the gas and walls). Explain why the gamma-ray pulse heights are typically much lower than the neutron pulse heights, and how pulse-height discrimination can separate them.

(c) At what gamma-ray flux does the gamma-ray count rate begin to overwhelm the neutron signal, even with discrimination? This is called the "gamma sensitivity" of the neutron detector.

Problem 16.29 ⭐⭐⭐ 🔬 Research problem: Stopping power data. Access the NIST PSTAR database (physics.nist.gov) and:

(a) Plot $-\frac{1}{\rho}\frac{dE}{dx}$ versus kinetic energy for protons in water from 1 MeV to 1000 MeV on a log-log scale. Identify the minimum and the Bethe-Bloch rise.

(b) Compare the NIST data to the Bethe-Bloch formula (without density or shell corrections). At what energies does the simple formula work well, and where does it fail?

(c) Repeat for aluminum and lead. Verify that the mass stopping powers are similar across materials (within a factor of 2) at the same $\beta\gamma$.

Problem 16.30 ⭐⭐⭐ 🔬 Research problem: Bragg peak optimization. Using the stopping_power.py code from this chapter:

(a) Calculate and plot the depth-dose (Bragg) curve for protons in water at initial energies of 70, 100, 150, 200, and 250 MeV.

(b) Construct a spread-out Bragg peak (SOBP) covering the depth range 10–15 cm by weighting multiple Bragg peaks at different energies. What weighting function produces a flat dose over the target volume?

(c) Compare the integral dose (total energy deposited in tissue) for the SOBP to that of a single 6 MeV photon beam delivering the same dose to the 10–15 cm region. Quantify the dose reduction advantage of protons.

Problem 16.31 ⭐⭐⭐ 🔬 Research problem: HPGe spectrum simulation. Write a Monte Carlo simulation that generates a gamma-ray spectrum for a ${}^{60}$Co source measured by an HPGe detector.

(a) For each photon (1173 or 1332 keV), determine probabilistically whether it undergoes photoelectric absorption, Compton scattering, or pair production (using approximate cross sections for Ge at these energies).

(b) For Compton events, sample the scattering angle from the Klein-Nishina distribution and record the deposited energy (assuming the scattered photon escapes).

(c) Include Gaussian smearing with the HPGe resolution. Plot the simulated spectrum and compare to published ${}^{60}$Co spectra.