Applications — Dating, Analysis, and Nuclear Medicine

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:

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.¹ 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.

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:

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

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.

Radiation dosage

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

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 ${}_Z^A\mathrm{M}$ into a radioactive ${}_Z^{A+1}\mathrm{M}$ by ${}_Z^A\mathrm{M}(n,\gamma )$, identified afterward by its half-life and gamma energies. The activity builds toward saturation:

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.

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.

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 ${}^{14}\mathrm{C}{/}^{12}\mathrm{C}\approx 1.35\times 10^{-12}$. At death the exchange stops and the ratio decays. The living decay rate is

and a measured lower rate dates the sample.

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 $N_D=N_0-N_P$,

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 $3.7\times 10^9\mathrm{y}$; Moon rocks and all meteorites cluster at $4.5\times 10^9\mathrm{y}$, 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 $Z=20$.