Chapter 16 — Radiation Interactions with Matter: How We Detect What Nuclei Do

"We have no sense that detects radioactivity. Everything we know about nuclei comes from radiation interacting with matter — in a photographic emulsion, a gas, a crystal, or a semiconductor. The detector IS the experiment." — Adapted from remarks by Glenn T. Seaborg

Chapter Overview

In the preceding four chapters, we studied the four modes by which unstable nuclei transform: the spontaneous decay law and its consequences (Chapter 12), alpha decay by quantum tunneling (Chapter 13), beta decay through the weak interaction (Chapter 14), and electromagnetic de-excitation via gamma emission (Chapter 15). In each case, the nucleus emits radiation — alpha particles, beta particles, neutrinos, gamma rays.

But how do we know any of this? No human sense organ detects a 1 MeV gamma ray. No one has ever seen an alpha particle. The only way we learn what nuclei do is by observing the consequences of radiation interacting with matter: ionization trails in gas, scintillation flashes in crystals, electron-hole pairs in semiconductors, tracks in emulsions, clicks in Geiger counters. Every number in every nuclear data table — every half-life, every branching ratio, every cross section — ultimately rests on the physics of this chapter.

This chapter is therefore not an aside or an appendix. It is the foundation of experimental nuclear physics. We will study:

• Charged particles (alpha, beta, protons, heavy ions): how they lose energy primarily through electromagnetic interactions with atomic electrons, described by the Bethe-Bloch formula. The remarkable Bragg peak — a sharp maximum in energy deposition at the end of the particle's range — is the physical basis of proton therapy for cancer.
• Photons (gamma rays, X-rays): the three dominant interaction mechanisms — photoelectric absorption, Compton scattering, and pair production — each dominant in a different regime of photon energy and material atomic number.
• Neutrons: electrically neutral particles that interact only through the strong and weak nuclear forces, requiring entirely different detection strategies.
• Detectors: the instruments that convert these interactions into measurable electrical signals — gas detectors, scintillators, semiconductor detectors, and neutron detectors.
• Radiation dosimetry: the units and concepts that quantify the biological impact of radiation exposure.

🏃 Fast Track: If you are primarily interested in nuclear structure or reactions, focus on Sections 16.1 (Bethe-Bloch), 16.3 (photon interactions), and 16.6 (semiconductor detectors). These are essential for understanding every experimental result in the rest of the book.

🔬 Deep Dive: The Bethe-Bloch derivation (Section 16.1) and the Klein-Nishina formula (Section 16.3) reward careful study. Both appear repeatedly in nuclear, particle, and medical physics.

📊 Spaced Review (Chapter 12): Recall the decay law $N(t) = N_0 e^{-\lambda t}$ and the concept of activity $A = \lambda N$. Every measurement of a decay constant $\lambda$ requires detecting individual decays — which requires the physics of this chapter.

📊 Spaced Review (Chapter 5): Fermi's golden rule, $W = \frac{2\pi}{\hbar}|\langle f|H'|i\rangle|^2 \rho(E_f)$, is the starting point for deriving both the Bethe-Bloch formula and the Klein-Nishina cross section.

16.1 Charged-Particle Interactions: The Bethe-Bloch Formula

16.1.1 The Physical Picture

When a fast charged particle — say a 5 MeV proton — enters a material, it is surrounded by a sea of atomic electrons (and nuclei). The proton's Coulomb field reaches out and interacts with these electrons. In each encounter, the proton transfers a small amount of energy to an electron, either exciting it to a higher atomic level or ionizing the atom entirely. The cumulative effect of billions of such small transfers is a continuous, steady energy loss as the proton traverses the medium.

This process has several key features:

1. Electromagnetic in nature. The dominant energy loss mechanism is Coulomb scattering off atomic electrons. Nuclear interactions (strong force) are rare because the nuclear cross-sectional area is $\sim 10^{-24}\,\text{cm}^2$ compared to the atomic cross section of $\sim 10^{-16}\,\text{cm}^2$.

2. Many small transfers. A 5 MeV proton in water undergoes roughly $10^5$ ionizing collisions before stopping. The average energy transfer per collision is on the order of tens of eV — tiny compared to the proton's kinetic energy — so the energy loss is effectively continuous.

3. Deterministic path. Heavy charged particles (protons, alphas, heavy ions) travel in nearly straight lines because their mass is much larger than the electron mass. The maximum energy transfer in a single collision is $T_{\max} = 2m_e c^2 \beta^2 \gamma^2$ (for $M \gg m_e$), so each collision deflects the projectile by a negligible angle.

4. Sharp range. Because the energy loss is continuous and deterministic, all particles of the same species and initial energy stop at approximately the same depth — the range. This is qualitatively different from photon attenuation (exponential, probabilistic).

💡 Intuition: A bowling ball rolling through a field of ping-pong balls. Each collision slows the bowling ball slightly but barely deflects it. After passing through enough ping-pong balls, the bowling ball stops. All bowling balls with the same initial speed stop at about the same distance.

16.1.2 Derivation of the Bethe-Bloch Formula

We derive the energy loss formula in three stages: (1) energy transfer in a single collision, (2) integration over all impact parameters, (3) quantum and relativistic corrections.

Stage 1: Energy transfer from a single collision.

Consider a heavy particle of charge $ze$, velocity $v = \beta c$, and mass $M$ passing an atomic electron at rest with impact parameter $b$ (the perpendicular distance from the particle's trajectory to the electron). The particle's trajectory is essentially a straight line (justified because $M \gg m_e$).

The momentum impulse delivered to the electron is:

$$\Delta p = \int_{-\infty}^{\infty} F_\perp \, dt$$

where $F_\perp$ is the perpendicular component of the Coulomb force. The parallel component averages to zero by symmetry. Using $F_\perp = \frac{ze^2}{4\pi\epsilon_0} \frac{b}{(b^2 + v^2 t^2)^{3/2}}$ and substituting $u = vt/b$:

$$\Delta p = \frac{ze^2}{4\pi\epsilon_0} \frac{1}{bv} \int_{-\infty}^{\infty} \frac{du}{(1+u^2)^{3/2}} = \frac{2ze^2}{4\pi\epsilon_0 bv}$$

The integral evaluates to 2. The energy transferred to the electron is:

$$T(b) = \frac{(\Delta p)^2}{2m_e} = \frac{2z^2 e^4}{(4\pi\epsilon_0)^2 m_e v^2 b^2}$$

This is a central result: the energy transfer falls as $1/b^2$. Distant collisions transfer less energy, but there are more of them (the number of electrons in an annular ring $2\pi b\,db$ grows linearly with $b$).

Stage 2: Integration over impact parameters.

The number of electrons in a cylindrical shell of radius $b$, thickness $db$, and length $dx$ along the particle's path is:

$$dN_e = n_e \cdot 2\pi b \, db \, dx$$

where $n_e = \rho N_A Z_{\text{mat}} / A_{\text{mat}}$ is the electron number density ($\rho$ = material density, $Z_{\text{mat}}$ and $A_{\text{mat}}$ = atomic number and mass number of the material). The total energy loss per unit path length is:

$$-\frac{dE}{dx} = \int_{b_{\min}}^{b_{\max}} T(b) \cdot n_e \cdot 2\pi b \, db = \frac{4\pi z^2 e^4 n_e}{(4\pi\epsilon_0)^2 m_e v^2} \ln\frac{b_{\max}}{b_{\min}}$$

The physics is in the limits. The integral is logarithmic — it depends only weakly on the exact cutoffs — but we need to specify them:

• $b_{\max}$: The maximum impact parameter is set by the requirement that the collision time $\tau \sim b/(\gamma v)$ must be shorter than the orbital period of the electron ($\sim \hbar / \bar{I}$, where $\bar{I}$ is a mean excitation energy characteristic of the material). Adiabatic collisions with $\tau$ longer than the orbital period transfer no energy. This gives $b_{\max} \sim \gamma \hbar v / \bar{I}$.

• $b_{\min}$: The minimum impact parameter is set by either the maximum classically allowed energy transfer or, in the quantum regime, by the de Broglie wavelength of the electron in the collision. The result is $b_{\min} \sim \hbar / (m_e \gamma v)$ in the quantum limit (which applies for fast projectiles).

Stage 3: The Bethe-Bloch formula.

Substituting the limits into the logarithm and applying relativistic corrections (including the density effect $\delta$ and shell corrections $C/Z$), we arrive at the Bethe-Bloch formula for the mean energy loss per unit path length:

$$\boxed{-\left\langle\frac{dE}{dx}\right\rangle = \frac{4\pi N_A r_e^2 m_e c^2}{\beta^2} \frac{Z_{\text{mat}} \rho}{A_{\text{mat}}} z^2 \left[\ln\frac{2m_e c^2 \beta^2 \gamma^2 T_{\max}}{I^2} - 2\beta^2 - \delta(\beta\gamma) - \frac{2C}{Z_{\text{mat}}}\right]}$$

where: - $r_e = e^2/(4\pi\epsilon_0 m_e c^2) = 2.818\,\text{fm}$ is the classical electron radius - $N_A = 6.022 \times 10^{23}\,\text{mol}^{-1}$ is Avogadro's number - $I$ is the mean excitation energy of the absorber material (e.g., $I \approx 75\,\text{eV}$ for water, $\approx 173\,\text{eV}$ for aluminum, $\approx 790\,\text{eV}$ for lead) - $T_{\max} = \frac{2m_e c^2 \beta^2 \gamma^2}{1 + 2\gamma m_e/M + (m_e/M)^2}$ is the maximum energy transfer in a single collision - $\delta(\beta\gamma)$ is the density-effect correction (important at high energies where the electric field is screened by the polarization of the medium) - $C/Z_{\text{mat}}$ is the shell correction (important at low energies when $v$ is comparable to orbital electron velocities)

The prefactor evaluates to:

$$K \equiv 4\pi N_A r_e^2 m_e c^2 = 0.3071\,\text{MeV}\,\text{cm}^2/\text{g}$$

when $\rho$ and $A_{\text{mat}}$ are factored out (i.e., using the mass stopping power $-\frac{1}{\rho}\frac{dE}{dx}$ in units of $\text{MeV}\,\text{cm}^2/\text{g}$).

16.1.3 Key Features of the Bethe-Bloch Curve

The dependence of $-dE/dx$ on the projectile's kinetic energy (or equivalently $\beta\gamma$) has a characteristic shape:

1. $1/\beta^2$ rise at low energies. For slow particles, the energy loss is dominated by the $1/\beta^2 = 1/v^2$ prefactor. Slow particles spend more time near each electron, transferring more energy. This explains why charged particles deposit more energy near the end of their range, producing the Bragg peak.

2. Minimum at $\beta\gamma \approx 3$–$4$. The $1/\beta^2$ decrease and the logarithmic increase balance at a minimum near $\beta\gamma \approx 3.5$ (corresponding to $T/M \approx 3$ for any particle). At the minimum: $$\left(-\frac{1}{\rho}\frac{dE}{dx}\right)_{\min} \approx 1\text{–}2\,\text{MeV}\,\text{cm}^2/\text{g}$$ A particle at this minimum is called a minimum ionizing particle (MIP). Cosmic-ray muons at Earth's surface are approximately MIPs.

3. Relativistic rise. Beyond the minimum, the logarithm grows slowly with $\beta\gamma$ (the relativistic rise). Physically, the transverse electric field of the particle is Lorentz-contracted, extending the effective range of interaction. The density effect $\delta$ limits this rise at very high energies.

4. $z^2$ dependence on projectile charge. The energy loss scales as the square of the projectile charge. An alpha particle ($z = 2$) loses four times as much energy per unit path length as a proton at the same velocity. This is why alpha particles have much shorter ranges than protons of the same kinetic energy.

5. Depends on material via $Z_{\text{mat}}/A_{\text{mat}}$ and $I$. For most materials except hydrogen, $Z/A \approx 0.4$–$0.5$, so the mass stopping power is roughly material-independent. The mean excitation energy $I$ varies logarithmically.

⚠️ Common Misconception: The Bethe-Bloch formula describes the mean energy loss. Individual particles fluctuate around this mean — these fluctuations are called energy straggling and follow a Landau distribution (asymmetric, with a long high-energy tail from rare close collisions).

16.1.4 Numerical Example: Protons in Water

For a 150 MeV proton (typical for proton therapy) in water ($Z = 7.42_{\text{eff}}$, $A = 14.3_{\text{eff}}$, $I = 75\,\text{eV}$, $\rho = 1.0\,\text{g/cm}^3$):

• $\gamma = (T + m_p c^2)/(m_p c^2) = (150 + 938.3)/938.3 = 1.160$
• $\beta = \sqrt{1 - 1/\gamma^2} = 0.508$
• $\beta\gamma = 0.589$

Substituting into the Bethe-Bloch formula (ignoring $\delta$ and $C/Z$, which are small at this energy):

$$-\frac{1}{\rho}\frac{dE}{dx} \approx \frac{0.3071 \times 0.555}{0.258} \times 1 \times \left[\ln\frac{2 \times 0.511 \times 0.258 \times 1.345 \times 282}{(75 \times 10^{-6})^2} - 0.516\right]$$

$$\approx 0.661 \times [12.05 - 0.52] \approx 7.6\,\text{MeV\,cm}^2/\text{g}$$

In water ($\rho = 1$), this gives $-dE/dx \approx 7.6\,\text{MeV/cm}$. The NIST PSTAR database gives $7.43\,\text{MeV\,cm}^2/\text{g}$ at 150 MeV, in excellent agreement.

At 10 MeV (near end of range): $\beta = 0.145$, $\beta\gamma = 0.146$, and $-dE/dx \approx 47\,\text{MeV/cm}$ — more than six times higher. This dramatic increase at low energy creates the Bragg peak.

16.2 The Bragg Peak and Charged-Particle Range

16.2.1 The Bragg Curve

As a charged particle slows down in matter, it loses energy at an increasing rate (because $-dE/dx \propto 1/v^2$ at low velocities). The result is that the energy deposited per unit depth — the dose — increases dramatically just before the particle stops. A plot of dose versus depth is called the Bragg curve, and the sharp maximum is the Bragg peak.

The Bragg peak is one of the most distinctive features of charged-particle interactions. Its properties:

1. Sharp localization of dose. A 150 MeV proton in water deposits relatively little energy for the first ~15 cm, then deposits most of its energy in a narrow peak about 5 mm wide at the end of its range (~15.8 cm).

2. Distal falloff. Beyond the Bragg peak, the dose drops to essentially zero over a distance comparable to the peak width. No energy is deposited beyond the range.

3. Peak-to-entrance ratio. The ratio of dose at the Bragg peak to dose at the entrance is typically 3:1 to 5:1 for protons, and even higher for heavier ions (carbon: ~8:1 or higher due to nuclear fragmentation effects).

4. Nuclear fragmentation tail. For heavier ions (carbon, oxygen), nuclear reactions between the projectile and target nuclei produce lighter fragments that travel beyond the Bragg peak, creating a small dose tail. This tail is absent for protons (which cannot fragment further).

16.2.2 Range and Range-Energy Relations

The range $R$ of a charged particle with initial kinetic energy $T_0$ is defined as:

$$R(T_0) = \int_0^{T_0} \frac{dT}{-dE/dx(T)} = \int_0^{T_0} \left(-\frac{dE}{dx}\right)^{-1} dT$$

This integral can be evaluated numerically using tabulated stopping powers. Approximate range-energy relations for protons in various materials:

These values are from the NIST PSTAR database. Notice that ranges scale roughly as $T^{1.75}$ for protons in the therapeutic energy range (50–250 MeV), though the exact power depends on the energy regime and material.

Scaling relations. For particles with the same velocity ($\beta$), the range scales as:

$$R \propto \frac{M}{z^2}$$

Therefore, an alpha particle ($M = 4m_p$, $z = 2$) at the same velocity as a proton has a range $R_\alpha = R_p \cdot 4/4 = R_p$ — the same range. But at the same kinetic energy, the alpha particle has a velocity $v_\alpha = v_p/2$, and its range is much shorter: $R_\alpha \approx R_p/4$ at the same $T$.

16.2.3 Straggling

Although the Bethe-Bloch formula gives the mean energy loss, individual particles experience statistical fluctuations because each collision transfers a random amount of energy. As a result:

• Range straggling: Particles of identical initial energy stop at slightly different depths. The range distribution has a standard deviation $\sigma_R / R \approx 1$–$3\%$ for protons in the therapeutic range — small enough that the Bragg peak remains sharp.

• Energy straggling: After traversing a fixed thickness of material, particles have a distribution of residual energies. The Landau distribution describes this for thin absorbers (few collisions); the Vavilov distribution generalizes to intermediate thicknesses; for thick absorbers, the central limit theorem applies and the distribution approaches Gaussian.

16.2.4 Delta Rays

Occasionally, a collision transfers enough energy to an atomic electron that the electron itself becomes a fast charged particle — a delta ray (or $\delta$-ray). Delta rays are knocked-on electrons with enough kinetic energy to produce their own ionization tracks branching off from the primary track. They are visible in cloud chamber and emulsion photographs as short secondary tracks.

The maximum energy transfer to a single electron is:

$$T_{\max} = \frac{2 m_e c^2 \beta^2 \gamma^2}{1 + 2\gamma m_e / M + (m_e/M)^2}$$

For a proton with $T = 200\,\text{MeV}$ ($\gamma = 1.213$, $\beta\gamma = 0.664$): $T_{\max} \approx 0.45\,\text{MeV}$. An electron with this energy has a range of about 1.5 mm in water — long enough to be detected as a separate track.

📜 Historical Context: Delta rays were crucial in the early days of nuclear physics. Observing their density along a track in a cloud chamber allowed physicists to measure the charge of the primary particle (since $-dE/dx \propto z^2$). This technique was used to identify new particles in cosmic-ray experiments.

16.3 Photon Interactions with Matter

Photons (gamma rays, X-rays) interact with matter through three principal mechanisms, each dominant in a different energy and atomic number regime. Unlike charged particles, photons travel without continuous energy loss — each photon either passes through unaffected or undergoes a single catastrophic interaction. This gives photon attenuation its characteristic exponential (Beer's law) behavior.

16.3.1 Photoelectric Absorption

In the photoelectric effect, a photon is completely absorbed by a bound atomic electron, which is ejected with kinetic energy:

$$T_e = E_\gamma - B_e$$

where $B_e$ is the binding energy of the electron. The photoelectric effect cannot occur with a free electron (momentum-energy conservation forbids it — the atom must absorb the recoil momentum).

Cross section. The photoelectric cross section per atom is approximately:

$$\sigma_{\text{pe}} \propto \frac{Z^n}{E_\gamma^{7/2}}$$

where $n \approx 4$–$5$ (the exact power depends on the energy range and which atomic shell dominates). A commonly used approximation for $E_\gamma$ well above the K-edge but below $\sim 200\,\text{keV}$:

$$\sigma_{\text{pe}} \approx \sigma_0 \times Z^5 \left(\frac{m_e c^2}{E_\gamma}\right)^{7/2}$$

The strong $Z^5$ dependence explains why high-$Z$ materials are effective radiation shields (and why lead, $Z = 82$, is the standard shielding material for X-rays and low-energy gamma rays).

Characteristic features: - Absorption edges. The cross section shows sharp discontinuities (edges) at photon energies equal to the binding energies of atomic shells. The K-edge of lead is at 88 keV; just above the K-edge, $\sigma_{\text{pe}}$ jumps by a factor of $\sim 6$ because K-shell electrons suddenly become available for ejection. - Subsequent emission. The atom is left in an excited state with a vacancy in an inner shell. It relaxes by emitting characteristic X-rays or Auger electrons. In high-$Z$ materials, X-ray fluorescence dominates; in low-$Z$ materials, Auger emission dominates. - Full energy deposit. If the characteristic X-rays and Auger electrons are absorbed within the detector, the photoelectric effect deposits the full photon energy. This is why photoelectric absorption produces the full-energy peak (photopeak) in gamma-ray spectra.

16.3.2 Compton Scattering

In Compton scattering, a photon scatters off a quasi-free atomic electron. The photon transfers part of its energy to the electron and continues as a scattered photon with reduced energy. The kinematics are derived from relativistic energy-momentum conservation.

Compton kinematics. The scattered photon energy as a function of scattering angle $\theta$:

$$E_\gamma' = \frac{E_\gamma}{1 + \frac{E_\gamma}{m_e c^2}(1 - \cos\theta)}$$

The electron recoil kinetic energy is:

$$T_e = E_\gamma - E_\gamma' = E_\gamma \cdot \frac{\frac{E_\gamma}{m_e c^2}(1-\cos\theta)}{1 + \frac{E_\gamma}{m_e c^2}(1-\cos\theta)}$$

Special cases: - Forward scattering ($\theta = 0$): $E_\gamma' = E_\gamma$, $T_e = 0$ — the photon passes through unchanged. - 90-degree scattering ($\theta = \pi/2$): $E_\gamma' = E_\gamma / (1 + E_\gamma/m_e c^2)$. - Backscattering ($\theta = \pi$): $E_\gamma' = E_\gamma / (1 + 2E_\gamma/m_e c^2)$, $T_e = E_\gamma \cdot \frac{2E_\gamma/m_e c^2}{1 + 2E_\gamma/m_e c^2}$.

The maximum electron recoil energy (backscatter, $\theta = \pi$) defines the Compton edge in a gamma-ray spectrum:

$$T_e^{\max} = \frac{2E_\gamma^2 / (m_e c^2)}{1 + 2E_\gamma/(m_e c^2)}$$

For $E_\gamma = 662\,\text{keV}$ (${}^{137}$Cs): $T_e^{\max} = 478\,\text{keV}$. For $E_\gamma = 1332\,\text{keV}$ (${}^{60}$Co): $T_e^{\max} = 1118\,\text{keV}$.

The Klein-Nishina formula. The differential cross section for Compton scattering per electron was derived by Klein and Nishina in 1929 using quantum electrodynamics (one of the first successful applications of QED):

$$\frac{d\sigma}{d\Omega} = \frac{r_e^2}{2}\left(\frac{E_\gamma'}{E_\gamma}\right)^2 \left(\frac{E_\gamma'}{E_\gamma} + \frac{E_\gamma}{E_\gamma'} - \sin^2\theta\right)$$

where $r_e = 2.818\,\text{fm}$ is the classical electron radius and $E_\gamma'/E_\gamma$ is given by the Compton formula above.

Limiting cases: - Low energy ($E_\gamma \ll m_e c^2$): The Klein-Nishina formula reduces to the Thomson cross section $d\sigma/d\Omega = (r_e^2/2)(1 + \cos^2\theta)$, and the total cross section approaches $\sigma_T = (8\pi/3)r_e^2 = 0.665\,\text{barn}$ — the classical limit where the photon wavelength is much larger than the Compton wavelength. - High energy ($E_\gamma \gg m_e c^2$): The cross section decreases approximately as $\sigma_{\text{KN}} \sim \sigma_T \cdot (m_e c^2 / E_\gamma) \cdot [\ln(2E_\gamma/m_e c^2) + 1/2]$. The scattering becomes increasingly forward-peaked.

The total Compton cross section per atom scales as $Z$ (one free electron per atomic electron), while the photoelectric cross section scales as $Z^{4-5}$. This is why Compton scattering dominates at intermediate energies where the photoelectric cross section has fallen but pair production has not yet turned on.

16.3.3 Pair Production

At photon energies above the threshold $E_\gamma > 2m_e c^2 = 1.022\,\text{MeV}$, a photon can convert into an electron-positron pair in the Coulomb field of a nucleus. The nucleus is required to conserve momentum (a free photon cannot produce a pair).

$$\gamma \to e^- + e^+$$

The threshold is exactly $2m_e c^2 = 1.022\,\text{MeV}$ in the field of a heavy nucleus (which absorbs negligible recoil). In the field of an atomic electron (triplet production), the threshold is $4m_e c^2 = 2.044\,\text{MeV}$ because the electron must also recoil.

Cross section. The pair production cross section per atom scales as:

$$\sigma_{\text{pair}} \propto Z^2 \left[\ln\left(\frac{2E_\gamma}{m_e c^2}\right) - \frac{7}{9} - f(Z)\right]$$

where $f(Z)$ is a correction for the nuclear Coulomb field. The $Z^2$ dependence (compared to $Z^{4-5}$ for photoelectric and $Z^1$ for Compton) means pair production is relatively more important in low-$Z$ materials at very high energies.

The cross section rises from zero at threshold and approaches a constant value at very high energies (limited by screening of the nuclear field by atomic electrons).

Subsequent annihilation. The positron slows down, thermalizes, and annihilates with an electron, producing two 511 keV gamma rays emitted in opposite directions (in the center-of-mass frame, which is nearly the lab frame for slow positrons):

$$e^+ + e^- \to \gamma + \gamma \quad (E_\gamma = 511\,\text{keV each})$$

These 511 keV annihilation photons are the basis of positron emission tomography (PET) in nuclear medicine.

📊 Spaced Review (Chapter 14): Recall that $\beta^+$ decay also produces positrons. The 511 keV annihilation radiation from ${}^{18}$F (half-life 110 min) is the signal detected in PET. Here we see the detection physics.

16.3.4 Summary: Three Regimes

The three mechanisms dominate in different regimes:

The crossover energies depend on $Z$: - Low $Z$ (water, $Z_{\text{eff}} \approx 7.4$): Photoelectric dominates below $\sim 25\,\text{keV}$; Compton dominates from $\sim 25\,\text{keV}$ to $\sim 25\,\text{MeV}$; pair production above $\sim 25\,\text{MeV}$. - High $Z$ (lead, $Z = 82$): Photoelectric dominates below $\sim 500\,\text{keV}$; Compton dominates from $\sim 500\,\text{keV}$ to $\sim 5\,\text{MeV}$; pair production above $\sim 5\,\text{MeV}$.

16.3.5 Beer's Law: Photon Attenuation

Because each photon either passes through or is removed from the beam in a single interaction, the intensity of a narrow beam of monoenergetic photons decreases exponentially with depth:

$$I(x) = I_0 \, e^{-\mu x}$$

where $\mu$ is the linear attenuation coefficient (units: cm$^{-1}$), equal to the sum of the attenuation coefficients for each mechanism:

$$\mu = \mu_{\text{pe}} + \mu_{\text{Compton}} + \mu_{\text{pair}}$$

The mass attenuation coefficient $\mu/\rho$ (units: cm$^2$/g) is more useful because it is nearly independent of density:

$$I(x) = I_0 \, e^{-(\mu/\rho) \rho x}$$

Useful related quantities:

• Half-value layer (HVL): The thickness that attenuates the beam by 50%: $x_{1/2} = \ln 2 / \mu$. For 662 keV photons (${}^{137}$Cs) in lead: HVL $\approx 0.65\,\text{cm}$. For the same photons in water: HVL $\approx 8.9\,\text{cm}$.

• Mean free path: $\lambda = 1/\mu$, the average distance a photon travels before interacting.

• Tenth-value layer (TVL): The thickness that attenuates by 90%: $x_{1/10} = \ln 10 / \mu = 3.32 \times x_{1/2}$.

⚠️ Important Caveat: Beer's law applies strictly only to narrow-beam (good geometry) conditions where every scattered photon is removed from the beam. In practice, scattered photons (especially Compton-scattered) can be re-directed into the detector, increasing the apparent intensity. This is accounted for by a buildup factor $B(x)$: $I(x) = B(x) I_0 e^{-\mu x}$ where $B > 1$ for broad-beam geometry.

Numerical example. How much lead shielding is needed to reduce 1 MeV gamma-ray intensity by a factor of 1000?

From NIST XCOM tables: $\mu/\rho = 0.0710\,\text{cm}^2/\text{g}$ for 1 MeV photons in lead ($\rho = 11.35\,\text{g/cm}^3$). Thus $\mu = 0.806\,\text{cm}^{-1}$.

$$x = \frac{\ln 1000}{\mu} = \frac{6.91}{0.806} = 8.6\,\text{cm} \approx 3.4\,\text{inches}$$

This is for narrow-beam geometry; practical shielding calculations must include the buildup factor.

16.4 Neutron Interactions with Matter

Neutrons, being electrically neutral, do not interact with atomic electrons via the Coulomb force. They pass through the electron cloud without noticing it and interact only with nuclei via the strong (and weak) nuclear force. This makes neutrons both penetrating and difficult to detect.

16.4.1 Neutron Moderation (Elastic Scattering)

A fast neutron ($E \sim \text{MeV}$) can be slowed down to thermal energies ($E \sim 0.025\,\text{eV}$ at room temperature) through successive elastic collisions with nuclei — a process called moderation.

Kinematics of elastic scattering. In a head-on elastic collision between a neutron (mass $m_n$) and a nucleus of mass $M = A \cdot m_n$ (approximating $m_n \approx 1\,\text{u}$), the minimum neutron energy after the collision is:

$$E_{\min} = E_0 \left(\frac{A-1}{A+1}\right)^2 \equiv \alpha E_0$$

where $\alpha = \left(\frac{A-1}{A+1}\right)^2$ is the maximum fractional energy retained. For isotropic scattering in the center-of-mass frame (valid for s-wave scattering at low energies), the average logarithmic energy loss per collision is:

$$\xi = \langle \ln(E_{\text{before}}/E_{\text{after}}) \rangle = 1 + \frac{\alpha}{1-\alpha}\ln\alpha$$

For hydrogen ($A = 1$): $\alpha = 0$, $\xi = 1$. A neutron can lose all its energy in a single collision with hydrogen (maximum energy transfer to a nucleus of equal mass).

For carbon ($A = 12$): $\alpha = 0.716$, $\xi = 0.158$. It takes $\sim 110/0.158 \approx 115$ collisions to thermalize a 2 MeV neutron ($\ln(2 \times 10^6 / 0.025) / \xi \approx 115$).

For lead ($A = 208$): $\alpha = 0.981$, $\xi = 0.0096$. It takes $\sim 1810$ collisions — lead is a terrible moderator but an excellent neutron reflector.

💡 Intuition: A billiard ball hitting another billiard ball of equal mass can stop dead (hydrogen moderator). A billiard ball hitting a bowling ball bounces back with nearly the same speed (lead). Equal masses maximize energy transfer.

16.4.2 Neutron Capture

Once thermalized, neutrons are efficiently captured by nuclei through radiative capture: ${}^{A}Z(n,\gamma){}^{A+1}Z$. The capture cross section for thermal neutrons follows the $1/v$ law over a wide energy range:

$$\sigma(E) = \sigma_0 \sqrt{\frac{E_0}{E}} = \sigma_0 \frac{v_0}{v}$$

where $\sigma_0$ is the cross section at a reference thermal energy $E_0 = 0.0253\,\text{eV}$ ($v_0 = 2200\,\text{m/s}$, the most probable velocity at $T = 293\,\text{K}$).

The $1/v$ law has a simple physical explanation: the probability of capture is proportional to the time the neutron spends in the vicinity of the nucleus, which is $\sim 1/v$. More precisely, it follows from the s-wave ($l = 0$) capture cross section in the limit where the neutron wavelength is much larger than the nuclear radius.

Notable thermal neutron capture cross sections:

The enormous variation — from 0.332 barn for hydrogen to $2.65 \times 10^6$ barn for ${}^{135}$Xe — reflects the detailed nuclear structure of the compound nucleus (Chapter 18). When the neutron energy matches a resonance in the compound nucleus, the cross section can be enormously enhanced.

16.4.3 Neutron-Induced Fission

For certain heavy nuclei (${}^{233}$U, ${}^{235}$U, ${}^{239}$Pu), thermal neutron capture leads to fission rather than gamma emission. The fission cross section for ${}^{235}$U is 585 barn at thermal energies — enough that a single kilogram of enriched uranium contains enough fissile atoms to sustain a chain reaction (Chapter 26).

Other nuclides (${}^{238}$U, ${}^{232}$Th) undergo fission only with fast neutrons ($E > 1\,\text{MeV}$). The distinction between fissile (thermal-fission) and fissionable (fast-fission-only) nuclides is central to reactor physics.

16.5 Gas-Filled Detectors

All radiation detectors exploit the same basic principle: radiation interacts with matter (the detector medium), producing a detectable signal — usually ionization that is collected as an electrical current or pulse. We begin with the oldest and simplest class: gas-filled detectors.

16.5.1 Basic Principle

A gas-filled detector consists of a gas volume (typically argon, or a noble gas mixture) between two electrodes with a voltage $V$ applied between them. When radiation ionizes the gas, the resulting ion pairs (positive ion + free electron) drift toward the electrodes under the electric field. The collected charge produces an electrical signal.

The average energy required to produce one ion pair in gas is the $W$-value: about 26 eV for argon, 35 eV for air, 42 eV for helium. For a 1 MeV particle stopping in argon, approximately $10^6 / 26 \approx 38,000$ ion pairs are produced.

16.5.2 Operating Regions

The behavior of the detector depends critically on the applied voltage, defining five distinct operating regions:

1. Recombination region (low voltage): The electric field is too weak to collect all ion pairs before some recombine. Signal depends on voltage. Not useful for detection.

2. Ionization chamber region ($\sim 100$–$300\,\text{V}$ for typical geometries): All primary ion pairs are collected, but no gas multiplication occurs. The collected charge is proportional to the energy deposited. Ionization chambers operate here. They produce small signals ($\sim 10^{-14}$ C for a 5 MeV alpha particle in air) requiring sensitive electrometers, but they are the most accurate devices for measuring absorbed dose. Used in survey meters, area monitors, and as the gold standard for therapy beam calibration.

3. Proportional region ($\sim 500$–$1000\,\text{V}$): The electric field near the thin anode wire is strong enough that drifting electrons gain enough energy between collisions to ionize additional gas molecules — gas multiplication (Townsend avalanche). The multiplication factor $M$ is typically $10^3$–$10^4$ and is the same for each primary electron, so the total charge is proportional to the initial ionization. Proportional counters operate here. They provide energy information and can distinguish alpha particles from beta particles and gamma rays by pulse height.

4. Limited proportional region ($\sim 1000$–$1200\,\text{V}$): The avalanche becomes large enough that space-charge effects (UV photons from de-excitation) trigger secondary avalanches. Proportionality is degraded. Rarely used.

5. Geiger-Müller region ($\sim 1200$–$1500\,\text{V}$): A single ion pair triggers a complete discharge throughout the entire gas volume. The output pulse is always the same size regardless of the type or energy of the radiation — the detector "counts" events but provides no energy information. Geiger-Müller (GM) counters operate here. They are simple, rugged, inexpensive, and produce large signals ($\sim 1\,\text{V}$) that require no amplification. The characteristic "click" of a GM counter is one of the most recognizable sounds in science. However, after each discharge, the counter is "dead" for $\sim 100$–$300\,\mu\text{s}$ (dead time), limiting the maximum count rate to $\sim 10^3$–$10^4$ counts/s.

📜 Historical Context: Hans Geiger and Walther Müller developed the Geiger-Müller counter in 1928 at the University of Kiel. It was the first electronic radiation detector and revolutionized nuclear physics by replacing the tedious visual counting of scintillations on zinc sulfide screens (the method used in the Geiger-Marsden experiment of 1909).

16.5.3 Multi-Wire Proportional Chambers

Georges Charpak (Nobel Prize 1992) revolutionized particle detection by inventing the multi-wire proportional chamber (MWPC) in 1968. By arranging many parallel anode wires in a plane, the MWPC provides position information — the wire(s) that fire indicate where the radiation passed through. MWPCs can handle rates of $\sim 10^6\,\text{counts/s/wire}$ and achieve spatial resolution of $\sim 1\,\text{mm}$, making them essential for tracking in high-energy and nuclear physics experiments.

16.6 Scintillation Detectors

16.6.1 Operating Principle

Certain materials emit visible or ultraviolet light (scintillation) when ionizing radiation deposits energy in them. A scintillation detector couples such a material to a photosensor — historically a photomultiplier tube (PMT), increasingly a silicon photomultiplier (SiPM) — that converts the light into an electrical signal.

The light output is approximately proportional to the deposited energy, so scintillation detectors provide energy information. The key performance metrics are:

• Light yield (photons per MeV): determines the statistical precision of the energy measurement
• Decay time (ns to $\mu$s): determines the count-rate capability
• Emission wavelength: must match the photosensor sensitivity
• Density and $Z_{\text{eff}}$: determine the stopping power for gamma rays

16.6.2 Inorganic Scintillators

Inorganic crystals produce scintillation through the band-structure mechanism: radiation creates electron-hole pairs, which migrate to luminescence centers (activator impurities) and recombine with emission of visible photons.

NaI(Tl) (sodium iodide doped with thallium) has been the workhorse of gamma-ray spectroscopy since the 1940s: - Light yield: $\sim 38{,}000$ photons/MeV (one of the highest) - Decay time: 250 ns - Density: 3.67 g/cm$^3$, $Z_{\text{eff}} = 51$ - Energy resolution: $\sim 6$–$7\%$ FWHM at 662 keV (${}^{137}$Cs) - Hygroscopic (must be hermetically sealed) - Available in large sizes (up to 16" $\times$ 16") at moderate cost

LaBr$_3$(Ce) (lanthanum bromide doped with cerium) is a newer scintillator with significantly better energy resolution: - Light yield: $\sim 63{,}000$ photons/MeV - Decay time: 16 ns (16$\times$ faster than NaI) - Energy resolution: $\sim 2.8$–$3.2\%$ FWHM at 662 keV - Contains intrinsic radioactivity (${}^{138}$La, ${}^{227}$Ac contamination from raw materials) - Much more expensive than NaI

CsI(Tl) (cesium iodide with thallium): - Light yield: $\sim 54{,}000$ photons/MeV - Decay time: 1000 ns (slow) - Not hygroscopic — mechanically rugged - Used in calorimeters (Belle II, BaBar)

16.6.3 Organic Scintillators (Plastic and Liquid)

Organic scintillators produce light through molecular fluorescence. The scintillation mechanism does not require a crystal lattice, so organic scintillators can be made as plastics, liquids, or dissolved in solvents.

Key properties: - Very fast: decay time $\sim 2$–$5\,\text{ns}$ — excellent for timing applications - Low $Z$ ($Z_{\text{eff}} \approx 5$–$6$): poor for gamma-ray detection (low photoelectric efficiency), but good for beta particles and fast neutrons - Pulse shape discrimination (PSD): organic scintillators have different time profiles for scintillation induced by different particles. Heavily ionizing particles (protons, alphas) produce a larger slow component than lightly ionizing particles (electrons). This allows discrimination between neutrons (which produce recoil protons) and gamma rays (which produce recoil electrons) — essential for neutron detection.

16.7 Semiconductor Detectors

16.7.1 Operating Principle

Semiconductor detectors are essentially solid-state ionization chambers. Radiation creates electron-hole pairs in a semiconductor (silicon or germanium), and these charge carriers are collected by an applied electric field. The key advantage is the small energy per electron-hole pair:

Because $\epsilon$ is $\sim 10\times$ smaller in semiconductors than in gas, a given energy deposit produces $\sim 10\times$ more charge carriers. The statistical fluctuation in the number of carriers determines the energy resolution (Poisson statistics: $\sigma_N = \sqrt{FN}$, where $F \approx 0.1$ is the Fano factor), so semiconductor detectors have fundamentally superior energy resolution.

16.7.2 Silicon Detectors

Silicon detectors are fabricated as p-n junction diodes operated under reverse bias. The depleted region (free of mobile carriers) serves as the sensitive volume.

• Charged-particle spectroscopy: Si surface-barrier detectors and ion-implanted detectors provide energy resolution of $\sim 10$–$15\,\text{keV}$ FWHM for 5 MeV alpha particles — sufficient to resolve individual nuclear levels.
• Particle tracking: Silicon strip detectors and pixel detectors provide spatial resolution of $\sim 10$–$50\,\mu\text{m}$. The inner tracking layers of every major particle physics detector (ATLAS, CMS, ALICE) use silicon.
• Limitations for gamma rays: Si has low $Z$ ($Z = 14$) and low density ($\rho = 2.33\,\text{g/cm}^3$), giving poor photoelectric efficiency above $\sim 30\,\text{keV}$.

16.7.3 High-Purity Germanium (HPGe) Detectors: The Gold Standard

Germanium ($Z = 32$, $\rho = 5.32\,\text{g/cm}^3$) has much higher stopping power for gamma rays than silicon. High-purity germanium (HPGe) detectors are the gold standard for gamma-ray spectroscopy, with energy resolution of $\sim 1.8$–$2.0\,\text{keV}$ FWHM at 1332 keV (${}^{60}$Co). Compare this to NaI at the same energy: $\sim 60\,\text{keV}$ FWHM — a factor of 30 worse.

Why HPGe has the best resolution:

1. Small $\epsilon$. Ge has $\epsilon = 2.96\,\text{eV/pair}$, producing $\sim 450{,}000$ electron-hole pairs for a 1332 keV gamma ray. The Poisson fluctuation: $\sigma_N = \sqrt{FN} = \sqrt{0.13 \times 450{,}000} \approx 242$ carriers, giving $\sigma_E = 242 \times 2.96\,\text{eV} \approx 0.72\,\text{keV}$ and $\text{FWHM} = 2.355 \sigma_E \approx 1.69\,\text{keV}$. The measured resolution is slightly worse due to electronic noise and charge collection effects.

2. Excellent charge transport. High-purity Ge ($\sim 10^{10}$ net impurities per cm$^3$) can be depleted over distances of several centimeters, creating large sensitive volumes.

3. Must be cooled. Ge has a small band gap ($E_g = 0.67\,\text{eV}$ vs. 1.12 eV for Si), so thermal excitation at room temperature creates too many carriers ($\sim 10^{13}\,\text{cm}^{-3}$), swamping the signal. HPGe detectors must be operated at liquid nitrogen temperature (77 K) or cooled by electromechanical coolers.

HPGe detector types: - Coaxial: Large volume (up to $\sim 600\,\text{cm}^3$), good efficiency over a wide energy range. Standard for laboratory gamma spectroscopy. - Broad-energy Ge (BEGe): Smaller electrode, lower capacitance, better resolution at low energies. Used in low-background experiments. - Segmented: Electrodes divided into segments for position sensitivity. Used in gamma-ray tracking arrays (GRETINA, AGATA).

🔗 Connection (Chapter 15): The gamma-ray spectra we analyzed in Chapter 15 — the precise energies of nuclear transitions, the multipolarities determined from angular correlations — all rely on HPGe detectors. Without germanium crystal growing technology, modern nuclear spectroscopy would be impossible.

16.8 Neutron Detectors

Neutrons, being uncharged, cannot directly ionize matter. All neutron detectors work indirectly: the neutron undergoes a nuclear reaction that produces a charged particle (or gamma ray), and the charged particle is detected by one of the methods described above.

16.8.1 Thermal Neutron Detection

The most common reactions for thermal neutron detection:

${}^{10}$B(n,$\alpha$)${}^{7}$Li (Q = 2.79 MeV for ground state, 2.31 MeV for excited state):

$${}^{10}\text{B} + n \to \begin{cases} {}^{7}\text{Li}^* + \alpha + 2.31\,\text{MeV} & (94\%) \\ {}^{7}\text{Li} + \alpha + 2.79\,\text{MeV} & (6\%) \end{cases}$$

Used in BF$_3$ proportional counters (gas filled with boron trifluoride enriched in ${}^{10}$B) and boron-lined detectors.

${}^{3}$He(n,p)${}^{3}$H (Q = 0.764 MeV):

$${}^{3}\text{He} + n \to {}^{3}\text{H} + p + 0.764\,\text{MeV}$$

${}^{3}$He proportional counters have the highest efficiency for thermal neutrons due to the enormous cross section (5333 barn). However, ${}^{3}$He is scarce and expensive (produced by tritium decay), leading to a worldwide shortage that has driven the development of alternative detectors.

${}^{6}$Li(n,$\alpha$)${}^{3}$H (Q = 4.78 MeV):

Used in ${}^{6}$Li-loaded scintillators (e.g., ${}^{6}$LiI(Eu)) and in ${}^{6}$Li glass scintillators. The large Q-value allows good discrimination between neutron events and gamma-ray background.

16.8.2 Fast Neutron Detection

Fast neutrons ($E > 100\,\text{keV}$) are detected by first moderating them to thermal energies (then detecting with the methods above), or by detecting recoil protons from elastic scattering:

• Moderation-based detectors: A polyethylene sphere (Bonner sphere) or cylindrical moderator surrounds a thermal neutron detector. The moderation efficiency depends on neutron energy, so measurements at multiple moderator thicknesses can unfold the neutron energy spectrum.

• Organic scintillators: Fast neutrons scatter off hydrogen in plastic or liquid scintillators, producing recoil protons that are detected by their scintillation. Pulse shape discrimination separates neutron events from gamma-ray background.

• Fission chambers: A thin layer of fissile material (${}^{235}$U or ${}^{238}$U) coats an electrode inside an ionization chamber. Neutron-induced fission produces heavily ionizing fission fragments that are easily detected. ${}^{238}$U fission chambers are threshold detectors ($E_n > 1\,\text{MeV}$) used for fast neutron measurements in reactor environments.

16.9 Detector Performance: Resolution and Efficiency

16.9.1 Energy Resolution

The ability to distinguish two closely spaced peaks in an energy spectrum is quantified by the energy resolution, defined as:

$$R = \frac{\text{FWHM}}{E_0}$$

where FWHM is the full width at half maximum of the peak at energy $E_0$. The FWHM typically arises from the convolution of several contributions added in quadrature:

$$\text{FWHM}^2 = \text{FWHM}_{\text{stat}}^2 + \text{FWHM}_{\text{elec}}^2 + \text{FWHM}_{\text{coll}}^2$$

Statistical contribution. The number of information carriers (ion pairs, photons, electron-hole pairs) $N$ fluctuates with variance $\sigma_N^2 = FN$, where $F$ is the Fano factor ($F = 1$ for Poisson statistics, $F \approx 0.1$–$0.15$ for Ge and Si because not all fluctuation modes are independent). The statistical FWHM is:

$$\text{FWHM}_{\text{stat}} = 2.355 \sqrt{FN} \cdot \epsilon = 2.355 \sqrt{F \epsilon E_0}$$

Comparison at 662 keV (${}^{137}$Cs):

For NaI, the measured resolution is much worse than the statistical limit because of non-uniformities in light collection and PMT gain variations. For HPGe, the measured resolution is close to the statistical limit, with a small contribution from electronic noise.

16.9.2 Detection Efficiency

The absolute efficiency $\epsilon_{\text{abs}}$ is the probability that a given radiation quantum emitted by the source produces a count in the detector:

$$\epsilon_{\text{abs}} = \frac{\text{counts detected}}{\text{quanta emitted by source}} = \epsilon_{\text{geom}} \times \epsilon_{\text{int}}$$

where $\epsilon_{\text{geom}} = \Omega / 4\pi$ is the geometric efficiency (fraction of solid angle subtended by the detector) and $\epsilon_{\text{int}}$ is the intrinsic efficiency (probability that a quantum entering the detector produces a count).

For gamma-ray detectors, the intrinsic efficiency depends on the detector material and size. The full-energy peak efficiency (counting only events in the photopeak) is always less than the total efficiency (counting all detected events):

$$\epsilon_{\text{peak}} = \epsilon_{\text{abs}} \times \frac{\text{photopeak counts}}{\text{all counts}} = \epsilon_{\text{abs}} \times \frac{P}{T}$$

where $P/T$ is the peak-to-total ratio. For a typical 3" $\times$ 3" NaI detector at 662 keV and 25 cm source-to-detector distance: $\epsilon_{\text{abs}} \approx 0.01$ and $P/T \approx 0.75$. For HPGe: similar $\epsilon_{\text{abs}}$ (depending on crystal size), but $P/T \approx 0.25$–$0.40$ because the smaller volume means more partial energy deposits (Compton scattering followed by photon escape).

⚠️ Resolution vs. Efficiency Trade-off: HPGe has far superior resolution but often lower full-energy peak efficiency compared to a large NaI crystal. The choice depends on the application: HPGe when you need to resolve closely spaced peaks, NaI when you need high counting statistics and cost is a concern.

16.10 Gamma-Ray Spectroscopy: Putting It Together

A measured gamma-ray energy spectrum encodes the three photon interaction mechanisms discussed in Section 16.3. Understanding the spectral features is essential for every nuclear physicist.

16.10.1 Spectral Features from a Monoenergetic Source

Consider a ${}^{137}$Cs source (single gamma ray at 662 keV) measured with an HPGe detector:

1. Full-energy peak (photopeak) at 662 keV: Events where the full photon energy is deposited — either by a single photoelectric absorption, or by one or more Compton scatterings followed by photoelectric absorption of the degraded photon, all within the detector volume.

2. Compton continuum: A broad distribution from zero up to the Compton edge at $T_e^{\max} = 478\,\text{keV}$. These are events where the photon Compton-scatters once and the scattered photon escapes the detector.

3. Backscatter peak at $\sim 184\,\text{keV}$: Photons that Compton-scatter at $\sim 180°$ in the shielding or surrounding materials and then enter the detector. The backscattered photon energy is $E_\gamma' = E_\gamma/(1 + 2E_\gamma/m_ec^2) = 662/(1 + 2.59) = 184\,\text{keV}$.

4. X-ray peaks: Characteristic lead X-rays ($\sim 75\,\text{keV}$) from photoelectric absorption in lead shielding.

For higher-energy sources ($E_\gamma > 1.022\,\text{MeV}$), additional features appear: - Single escape peak at $E_\gamma - 511\,\text{keV}$: pair production where one annihilation photon escapes. - Double escape peak at $E_\gamma - 1022\,\text{keV}$: pair production where both annihilation photons escape. - 511 keV annihilation peak: from positrons produced by pair production in surrounding materials.

16.11 Radiation Dosimetry: Quantifying Biological Effects

The biological effects of radiation depend not just on the type and energy of the radiation, but on how much energy is deposited in tissue, the spatial distribution of that energy deposit, and the sensitivity of the irradiated tissue. Radiation dosimetry provides the quantitative framework.

16.11.1 Absorbed Dose

The absorbed dose $D$ is the energy deposited per unit mass:

$$D = \frac{dE_{\text{absorbed}}}{dm}$$

SI unit: Gray (Gy) = 1 J/kg. Older unit: rad = 0.01 Gy (100 rad = 1 Gy).

Absorbed dose is a purely physical quantity — it says nothing about the biological effectiveness of the radiation. A dose of 1 Gy from gamma rays has a very different biological effect than 1 Gy from alpha particles.

16.11.2 Equivalent Dose

The equivalent dose $H$ accounts for the different biological effectiveness of different radiation types:

$$H = w_R \times D$$

where $w_R$ is the radiation weighting factor (dimensionless):

SI unit: Sievert (Sv) = 1 J/kg (dimensionally the same as the Gray, but distinguished by the weighting). Older unit: rem = 0.01 Sv.

The large weighting factor for alpha particles ($w_R = 20$) reflects their high linear energy transfer (LET): alpha particles deposit all their energy in a short track, creating a dense column of ionization that causes severe, clustered DNA damage. Gamma rays, by contrast, produce sparse ionization — the same total energy is spread over a much longer path.

💡 Intuition: Imagine the same amount of water (energy) distributed as a light drizzle over a large garden (gamma rays) versus a fire hose aimed at one flower (alpha particles). The total water is the same, but the damage to the flower is very different.

16.11.3 Effective Dose

The effective dose $E$ further accounts for the varying sensitivity of different organs and tissues:

$$E = \sum_T w_T \times H_T$$

where $w_T$ is the tissue weighting factor for tissue/organ $T$, and $H_T$ is the equivalent dose to that tissue. The tissue weighting factors sum to 1 and reflect the relative risk of stochastic effects (cancer, hereditary damage):

These values are from the International Commission on Radiological Protection (ICRP) Publication 103 (2007).

16.11.4 Dose Limits and Context

To put the numbers in context:

The factor of $\sim 10{,}000$ between a chest X-ray and a lethal dose — and the factor of $\sim 10{,}000$ between the occupational limit and a therapy dose — illustrate the enormous range over which radiation effects must be quantified.

⚠️ ALARA Principle: Radiation protection follows the principle of keeping doses "As Low As Reasonably Achievable" (ALARA). There is no known threshold below which radiation is guaranteed to be harmless — the linear no-threshold (LNT) model assumes that cancer risk is proportional to dose even at very low doses, although this assumption is debated.

16.12 Comparing Charged Particles and Photons: Implications for Therapy

The fundamental difference between charged-particle and photon dose distributions has profound implications for radiation therapy:

Photons (conventional radiotherapy): - Exponential attenuation (Beer's law): dose decreases monotonically with depth after an initial buildup region. - Dose is highest near the surface and decreases continuously — the tumor at depth always receives less dose than the tissue in front of it. - To deliver adequate dose to a deep tumor, the entrance dose must be high, and cross-firing from multiple angles is used to spread the entrance dose while concentrating dose at the tumor.

Protons (proton therapy): - Dose increases with depth (Bragg peak), with a sharp maximum at a depth determined by the initial energy. - Beyond the Bragg peak, the dose drops to essentially zero — no exit dose. - By modulating the beam energy (using a range modulator or active scanning), the Bragg peak can be spread over the tumor volume (Spread-Out Bragg Peak, SOBP).

The advantage is stark: proton therapy deposits 2–3 times less total energy in the patient than photon therapy for the same dose to the tumor. This is particularly important for pediatric tumors (reducing second cancer risk), tumors near critical structures (optic nerve, brainstem, spinal cord), and re-irradiation.

Carbon ions offer an even sharper Bragg peak and higher biological effectiveness in the peak (due to higher LET), but the nuclear fragmentation tail and higher facility costs limit their availability.

🔗 Connection (Chapter 27): The medical applications of nuclear physics — radiotherapy, imaging, radiopharmaceuticals — all build on the interaction physics of this chapter. Chapter 27 develops these applications in detail.

16.13 Chapter Summary

This chapter established the physical foundations of radiation detection:

1. Charged particles lose energy continuously through electromagnetic interactions with atomic electrons. The Bethe-Bloch formula quantifies this, predicting the $1/v^2$ rise that produces the Bragg peak. Heavy charged particles have well-defined ranges; the Bragg peak makes protons and heavy ions superior to photons for targeted energy deposition.

2. Photons interact through three mechanisms — photoelectric absorption ($\propto Z^{4-5}/E^{7/2}$), Compton scattering ($\propto Z$, Klein-Nishina), and pair production ($\propto Z^2$, threshold 1.022 MeV). Photon attenuation follows Beer's law: $I = I_0 e^{-\mu x}$.

3. Neutrons interact via the strong force with nuclei. Moderation by elastic scattering slows fast neutrons to thermal energies; capture cross sections follow the $1/v$ law. Detection requires a nuclear reaction producing a charged particle.

4. Gas detectors (ionization chambers, proportional counters, GM counters) exploit ionization in gas. Scintillation detectors (NaI, LaBr$_3$, plastic) produce light. Semiconductor detectors (Si, HPGe) produce electron-hole pairs with the best energy resolution.

5. Radiation dosimetry quantifies biological impact: absorbed dose (Gy), equivalent dose (Sv, weighted by radiation type), and effective dose (Sv, weighted by tissue sensitivity).

Looking Ahead: With this chapter's detector physics in hand, we are ready to study nuclear reactions (Chapter 17), where the interplay of kinematics, cross sections, and detection defines everything we measure. Every cross section, angular distribution, and excitation function in the rest of this book depends on the detection principles we have just established.

Further Reading Pointers

The topics of this chapter are covered in: - Krane, Introductory Nuclear Physics (1988), Chapter 7 — concise treatment of interactions and detectors - Knoll, Radiation Detection and Measurement, 4th ed. (2010) — the definitive graduate reference for detector physics - Leo, Techniques for Nuclear and Particle Physics Experiments, 2nd ed. (1994) — excellent derivations of Bethe-Bloch and interaction cross sections - Particle Data Group, "Passage of Particles Through Matter" — authoritative summary of stopping powers and ranges, updated annually

Data tables referenced in this chapter: - NIST PSTAR: Stopping powers and ranges for protons (physics.nist.gov/PhysRefData/Star/Text/PSTAR.html) - NIST XCOM: Photon cross sections (physics.nist.gov/PhysRefData/Xcom/html/xcom1.html) - NIST ASTAR: Stopping powers for alpha particles - IAEA Nuclear Data Services: Neutron cross sections (www-nds.iaea.org)