---
title: Applications — Dating, Analysis, and Nuclear Medicine
module: Radiation and Applications
moduleNumber: 11
lessonNumber: 5
order: 1105
summary: >
  Charged particles lose energy continuously and stop at a well-defined range
  with a Bragg peak, while gamma rays are attenuated exponentially. These
  interactions define radiation detectors and dosimetry (gray and sievert) and
  drive the applications: neutron activation analysis, magnetic resonance
  imaging, PET, and radiometric dating with carbon-14 and long-lived rock clocks.
topics: [Radiation and Applications]
sources:
  - book: Tipler & Llewellyn
    ref: "Ch. 11 — Nuclear Physics; §11-9 Applications, Interaction of Particles and Matter"
  - book: Tipler & Llewellyn
    ref: "§11-9 Radiation Dosage; Neutron Activation Analysis; Radioactive Dating"
draft: false
---

Every use of nuclear radiation — imaging, dating, cancer therapy, dosimetry —
rests on how radiation deposits energy in matter. Charged particles and photons
lose energy by different laws, and those laws set the design of detectors and the
definition of a radiation dose.

## Energy loss of charged particles

A fast charged particle traversing matter loses energy mainly by ionizing atoms
along its path, in many small Coulomb collisions with electrons. The stopping
power (energy lost per unit length) grows as the particle slows:

$$
-\frac{\d E}{\d x} \propto \frac{z^2}{v^2}\,n_e\,\ln\!\frac{2m_e v^2}{I},
$$

where $z$ and $v$ are the particle's charge and speed, $n_e$ the electron density
of the medium, and $I$ its mean ionization energy.[^tl-interaction] The $1/v^2$
factor is decisive: as the particle slows, it loses energy faster, so most of the
energy is dumped near the very end of the track. The result is the **Bragg peak** —
a sharp maximum in energy deposition just before the particle stops at a
well-defined **range**.

$$
% caption: The Bragg curve: a charged particle deposits energy slowly at first
% and dumps most of it in a sharp peak just before stopping at its range, because
% stopping power grows as the particle slows.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.4,0) node[right, black!70] {depth $x$};
  \draw[->, black] (0,0) -- (0,4.2) node[above, black!70] {stopping power};
  % slow rise then sharp Bragg peak then drop to zero
  \draw[acc, very thick]
    (0.2,1.0)
    .. controls (2.0,1.15) and (3.6,1.5) .. (4.6,2.0)
    .. controls (5.3,2.5) and (5.6,3.8) .. (5.85,3.8)
    .. controls (6.05,3.8) and (6.2,0.4) .. (6.4,0.1)
    -- (6.7,0.05);
  % range marker
  \draw[black, dashed] (6.4,0) -- (6.4,0.9);
  \node[anchor=north, black!70] at (6.4,0) {range};
  \node[acc, anchor=south east, font=\scriptsize] at (5.95,3.8) {Bragg peak};
\end{tikzpicture}
$$

The Bragg peak is why proton and heavy-ion beams can target a tumor at depth:
the dose is concentrated at the end of the range, sparing tissue in front of it.
Alpha and beta particles have short ranges in tissue (millimeters or less), which
is why medical isotopes meant to be detected from outside the body are gamma
emitters instead.

## Attenuation of gamma rays

Photons are removed from a beam not gradually but in single events — each photon
travels until one interaction removes it. A beam of initial intensity $I_0$ through
thickness $x$ is attenuated exponentially:

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

where $\mu$ is the **linear attenuation coefficient**. The thickness that halves
the beam is the half-value layer $x_{1/2} = (\ln 2)/\mu$.

$$
% caption: Gamma intensity falls exponentially with absorber thickness; each
% half-value layer removes half the remaining beam.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.4,0) node[right, black!70] {thickness $x$};
  \draw[->, black] (0,0) -- (0,4.2) node[above, black!70] {$I(x)$};
  \draw[acc, very thick] (0,4.0)
    .. controls (0.9,2.7) and (1.6,2.1) .. (2.3,1.9)
    .. controls (3.3,1.55) and (3.9,1.2) .. (4.3,1.0)
    .. controls (5.5,0.65) and (6.1,0.55) .. (6.9,0.45);
  \node[anchor=east, black!70] at (0,4.0) {$I_0$};
  \draw[black, dashed] (0,2.0) -- (2.3,2.0) -- (2.3,0);
  \node[anchor=east, black, font=\scriptsize] at (0,2.0) {$\tfrac{1}{2}I_0$};
  \node[anchor=north, black!70] at (2.3,0) {$x_{\frac{1}{2}}$};
  \draw[black, dashed] (0,1.0) -- (4.3,1.0) -- (4.3,0);
  \node[anchor=east, black, font=\scriptsize] at (0,1.0) {$\tfrac{1}{4}I_0$};
  \node[anchor=north, black!70] at (4.3,0) {$2\,x_{\frac{1}{2}}$};
\end{tikzpicture}
$$

Three mechanisms contribute to $\mu$, dominating in different energy ranges:

- **Photoelectric effect** (low energy): the photon is fully absorbed by a bound
  electron; cross section rises steeply with atomic number, roughly $Z^n/E^3$,
  which is why high-$Z$ lead shields gamma rays well.
- **Compton scattering** (intermediate energy): the photon scatters off an
  electron, losing part of its energy.
- **Pair production** (above $2m_e c^2 = 1.022\,\mathrm{MeV}$): the photon
  converts to an electron-positron pair in a nucleus's field.

## Detectors

Detectors convert the deposited energy into a measurable signal.

- **Scintillation counter**: radiation excites a crystal (such as NaI(Tl)), which
  emits a flash of light collected by a photomultiplier. A lead collimator gives
  the direction, forming a **gamma camera** for imaging the distribution of a
  source isotope in the body.
- **Geiger counter**: an ionizing particle triggers an avalanche in a gas-filled
  tube, giving a large, easily counted pulse (but no energy resolution).
- **Semiconductor detector**: ionization creates electron-hole pairs in a
  reverse-biased junction; the collected charge measures energy with high
  resolution.

$$
% caption: A collimated scintillation detector (gamma camera): a lead collimator
% selects a direction, gamma rays flash the crystal, and a photomultiplier turns
% the flash into a pulse a computer maps to source position.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % source
  \fill[black] (0.4,1.4) circle (2.4pt);
  \node[black, anchor=east, font=\scriptsize] at (0.3,1.4) {source};
  % gamma ray
  \draw[->, black] (0.6,1.45) -- (2.35,1.7);
  \node[black, font=\scriptsize, anchor=south] at (1.4,1.55) {gamma};
  % lead collimator (two blocks)
  \draw[black, thick, fill=black!10] (2.4,0.6) rectangle (2.9,1.4);
  \draw[black, thick, fill=black!10] (2.4,2.0) rectangle (2.9,2.8);
  \node[black, font=\scriptsize, anchor=north] at (2.65,0.55) {Pb};
  % crystal
  \draw[thick] (3.1,1.2) rectangle (4.3,2.2);
  \node[black, font=\scriptsize, align=center] at (3.7,1.7) {NaI\\crystal};
  % photomultiplier
  \draw[black, thick, fill=black!6] (4.4,1.35) -- (5.9,1.1) -- (5.9,2.3) -- (4.4,2.05) -- cycle;
  \node[black, font=\scriptsize, align=center] at (5.2,1.7) {photo\\mult.};
  % output
  \draw[->, black] (5.95,1.7) -- (6.9,1.7);
  \node[black, font=\scriptsize, anchor=west] at (6.95,1.7) {pulse};
\end{tikzpicture}
$$

## Radiation dosage

The biological effect of radiation depends on how much energy it deposits and on
what kind of radiation it is.

> **Definition (Absorbed and equivalent dose).** The **absorbed dose** is energy
> deposited per unit mass, measured in **gray** ($1\,\mathrm{Gy} = 1\,\mathrm{J/kg}$;
> the older **rad** is $0.01\,\mathrm{Gy}$). The **equivalent dose**
> $H = Q\,D$ weights it by a quality factor $Q$ (relative biological
> effectiveness) and is measured in **sievert** ($\mathrm{Sv}$; the older **rem**
> is $0.01\,\mathrm{Sv}$).

The quality factor is about $1$ for x-rays, gamma rays, and beta particles, but
about $20$ for alpha particles and fast neutrons, which deposit their energy
densely along short tracks and do more damage per unit energy.

## Neutron activation analysis

Exposing a sample to a slow-neutron flux $I$ transmutes a stable isotope
$^{A}_{Z}\mathrm{M}$ into a radioactive $^{A+1}_{Z}\mathrm{M}$ by
$^{A}_{Z}\mathrm{M}(n,\gamma)$, identified afterward by its half-life and gamma
energies. The activity builds toward saturation:

$$
R(t) = R_0\bigl(1 - e^{-\lambda t}\bigr),
\qquad
R_0 = N_0\,\sigma\,I,
$$

where $N_0$ is the number of target nuclei and $\sigma$ the capture cross section.
As $t \to \infty$, $R \to R_0$, and measuring the saturation activity gives the
mass of the element present. The technique is isotope-specific and sensitive to
trace amounts, used from environmental pollutant analysis to authenticating
paintings by their pigment elements.

$$
% caption: Neutron-activation activity rises toward saturation R0 = N0 sigma I as
% production balances decay; a short-lived product saturates quickly.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.4,0) node[right, black!70] {irradiation time $t$};
  \draw[->, black] (0,0) -- (0,4.0) node[above, black!70] {activity $R(t)$};
  % saturating curve toward R0
  \draw[acc, very thick] (0,0)
    .. controls (1.0,2.4) and (2.2,3.2) .. (3.6,3.45)
    .. controls (4.8,3.6) and (6.0,3.62) .. (7.0,3.63);
  \draw[black, dashed] (0,3.65) -- (7.0,3.65);
  \node[anchor=east, black!70] at (0,3.65) {$R_0$};
  \node[acc, font=\scriptsize, anchor=west] at (3.6,2.9) {saturation};
\end{tikzpicture}
$$

## Magnetic resonance and imaging

A proton in a field $B$ has two spin orientations split by
$\Delta E = 2\mu_z B$. Irradiating with photons of matching energy
$hf = \Delta E$ drives resonant transitions; in a $1\,\mathrm{T}$ field the
frequency is about $42.5\,\mathrm{MHz}$, in the radio band. Because the field at a
proton is the applied field plus the local field of its molecular surroundings,
the resonance frequency probes molecular structure — **nuclear magnetic
resonance**. Placing a patient in a field that varies with position makes the
resonance frequency map to location, the basis of **magnetic resonance imaging
(MRI)**. The radio-frequency photons carry far less energy than molecular bonds, so
MRI produces almost no biological damage, unlike x-rays.

## Emission tomography

A gamma camera records a two-dimensional projection, discarding depth. **Computer-assisted
tomography (CT)** restores depth by rotating a fan beam around the patient and
reconstructing a transverse slice from the transmitted intensities. **Positron
emission tomography (PET)** uses a positron-emitting tracer: the positron
annihilates within millimeters into two back-to-back $0.511\,\mathrm{MeV}$
photons, and a ring of detectors registering the pair in coincidence fixes the
decay to the line between them. Common PET emitters ($^{11}\mathrm{C}$,
$^{13}\mathrm{N}$, $^{15}\mathrm{O}$, $^{18}\mathrm{F}$) are short-lived and must
be made near a cyclotron.

| Modality | Probe | Signal | Distinctive strength |
| --- | --- | --- | --- |
| MRI | RF photons in a field gradient | proton resonance frequency | soft-tissue contrast, no ionizing dose |
| CT | external x-ray fan beam | transmitted intensity | fast three-dimensional structure |
| PET | positron-emitting tracer | back-to-back annihilation photons | functional/metabolic imaging |

## Radiometric dating

A radioactive clock compares present isotope abundances against their known
initial ratio.

**Carbon-14 dating.** Cosmic-ray neutrons make $^{14}\mathrm{C}$ in the atmosphere
by $^{14}\mathrm{N}(n,p)^{14}\mathrm{C}$; it beta-decays back with
$t_{1/2} = 5730\,\mathrm{y}$. Living organisms exchange carbon with the atmosphere
and hold the equilibrium ratio $^{14}\mathrm{C}/^{12}\mathrm{C} \approx 1.35\times10^{-12}$.
At death the exchange stops and the ratio decays. The living decay rate is

$$
R = \lambda N = \frac{0.693}{t_{1/2}}\,N \approx 15.6\ \frac{\mathrm{decays}}{\mathrm{min{\cdot}g\ of\ carbon}},
$$

and a measured lower rate dates the sample.

> **Example (Age of a bone).** A fragment with $200\,\mathrm{g}$ of carbon decays
> at $400\,\mathrm{decays/min}$. A living sample would give
> $200 \times 15.6 = 3120\,\mathrm{decays/min}$. The ratio $3120/400 = 7.8 \approx 2^n$
> gives $n = \ln 7.8/\ln 2 = 2.96$ half-lives, so the age is
> $2.96 \times 5730 \approx 17{,}000\,\mathrm{years}$ — far older than any
> historical attribution to the 1500s.

$$
% caption: Carbon-14 dating: a sample's C-14/C-12 ratio decays from its living
% value, and the measured fraction gives the age in half-lives of 5730 years.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.6,0) node[right, black!70] {age (half-lives)};
  \draw[->, black] (0,0) -- (0,4.2) node[above, black!70] {$^{14}$C fraction};
  \draw[acc, very thick] (0,4.0)
    .. controls (0.8,2.9) and (1.4,2.4) .. (2.0,2.0)
    .. controls (3.0,1.5) and (3.6,1.15) .. (4.0,1.0)
    .. controls (5.2,0.7) and (5.8,0.55) .. (6.8,0.42);
  \node[anchor=east, black!70, font=\scriptsize] at (0,4.0) {living};
  \foreach \x/\lab/\y in {2.0/1/2.0, 4.0/2/1.0, 6.0/3/0.55} {
    \draw[black, dashed] (\x,0) -- (\x,\y);
    \fill[black] (\x,\y) circle (1.6pt);
    \node[anchor=north, black, font=\scriptsize] at (\x,-0.03) {\lab};
  }
  \node[black, font=\scriptsize, anchor=west] at (2.6,2.9) {5730 y each};
\end{tikzpicture}
$$

**Rock clocks.** For a parent decaying to a stable daughter with no initial
daughter present, the age follows from the parent-to-daughter ratio. Since
$N_P = N_0\,e^{-\lambda t}$ and $N_D = N_0 - N_P$,

$$
t = \frac{t_{1/2}}{\ln 2}\,\ln\!\Bigl(1 + \frac{N_D}{N_P}\Bigr).
$$

Long-lived ratios ($^{238}\mathrm{U}/^{206}\mathrm{Pb}$,
$^{87}\mathrm{Rb}/^{87}\mathrm{Sr}$, $^{40}\mathrm{K}/^{40}\mathrm{Ar}$) date old
rocks. The oldest Earth rocks are about $3.7\times10^9\,\mathrm{y}$; Moon rocks and
all meteorites cluster at $4.5\times10^9\,\mathrm{y}$, the accepted age of the
solar system.

**Accelerator mass spectrometry (AMS)** counts the rare atoms directly rather than
waiting for their decays, extending $^{14}\mathrm{C}$ dating to $50{,}000\,\mathrm{y}$
with milligram samples; **particle-induced x-ray emission (PIXE)** identifies
elements by the characteristic x-rays a proton beam knocks out, complementing
neutron activation analysis for elements above $Z = 20$.

[^tl-interaction]: **Tipler & Llewellyn**, _Modern Physics_, §11-9 — Interaction of Particles and Matter: the $1/v^2$ stopping power and Bragg peak for charged particles, exponential $e^{-\mu x}$ attenuation of gamma rays, and the gray/sievert dosimetry units. Detector and dosimetry details follow §11-9's applications and radiation-dosage material.