Premed · Premed · Physics 2
Lecture 24: Radioactive Decay and Medical Applications
Physics II — Electromagnetism, Optics & Modern Physics
Learning Objectives
By the end of this lecture, students will be able to:
- Describe the three main types of radioactive decay (alpha, beta, gamma) and their properties
- Apply the radioactive decay law and calculate half-life, activity, and remaining nuclei
- Explain the biological effects of radiation and the concept of radiation dose
- Describe medical applications of radioactivity including imaging and therapy
- Apply radioactive dating methods to determine the age of materials
Lecture Content
I. Types of Radioactive Decay
Radioactive decay is the spontaneous transformation of an unstable nucleus, accompanied by the emission of radiation. While the decay of any individual nucleus is random and unpredictable, the behavior of a large collection of nuclei is statistically precise.
Alpha decay involves the emission of an alpha particle, which is a ^4_2 He nucleus consisting of 2 protons and 2 neutrons. The general reaction is ^A_Z X yielding ^(A-4)_(Z-2) Y + ^4_2 He, so the mass number decreases by 4 and the atomic number decreases by 2. For example, ^226_88 Ra decays to ^222_86 Rn + ^4_2 He. Alpha particles are heavy and doubly charged, giving them low penetrating power (stopped by a sheet of paper or a few centimeters of air) but very high ionizing ability. They are particularly dangerous if the emitting material is ingested or inhaled.
Beta decay comes in two forms. In beta-minus decay, a neutron converts to a proton, emitting an electron and an antineutrino: ^A_Z X yields ^A_(Z+1) Y + e^- + anti-nu_e. The atomic number increases by 1 while the mass number is unchanged. Carbon-14 decaying to nitrogen-14 is a classic example. In beta-plus decay, a proton converts to a neutron, emitting a positron and a neutrino: ^A_Z X yields ^A_(Z-1) Y + e^+ + nu_e. Fluorine-18 decaying to oxygen-18 is the reaction exploited in PET scans. Beta particles have intermediate penetrating power (stopped by a few millimeters of aluminum). The neutrino carries away a variable fraction of the available energy, which is why beta particles exhibit a continuous energy spectrum rather than discrete energies.
Gamma decay involves the emission of a high-energy photon (gamma ray) from an excited nuclear state: ^A_Z X* yields ^A_Z X + gamma. Neither Z nor A changes; the nucleus simply sheds excess energy. Gamma emission often follows alpha or beta decay when the daughter nucleus is left in an excited state. Gamma rays are highly penetrating (requiring thick lead or concrete for shielding) but are the least ionizing of the three types per interaction.
<image>A comparison diagram of the three types of radioactive decay and their penetrating power. On the left, a radioactive source emits three types of radiation. Alpha particles (labeled with a He-4 symbol) are stopped by a sheet of paper. Beta particles (electron symbol) pass through paper but are stopped by a few millimeters of aluminum. Gamma rays (wavy line, labeled with gamma symbol) pass through both paper and aluminum but are attenuated by thick lead or concrete. A table below summarizes each type: particle emitted, charge, mass, relative ionizing power, and relative penetrating power.</image>
II. Radioactive Decay Law
The number of undecayed nuclei decreases exponentially with time: N(t) = N_0 e^(-lambda t), where N_0 is the initial number of nuclei and lambda is the decay constant, representing the probability of decay per unit time (in s^-1).
The half-life (t_1/2) is the time for half the nuclei to decay: t_1/2 = ln(2) / lambda = 0.693 / lambda. After n half-lives, N = N_0 / 2^n. Half-lives vary enormously, from fractions of a second to billions of years.
The activity (A) is the rate of decay, the number of decays per second: A(t) = lambda N(t) = A_0 e^(-lambda t), where A_0 = lambda N_0 is the initial activity. The SI unit of activity is the becquerel (Bq) = 1 decay/s. The older unit, the curie (Ci), equals 3.7 x 10^10 Bq. The mean lifetime is tau = 1/lambda = t_1/2 / ln(2), approximately 1.443 t_1/2.
III. Radioactive Dating
Carbon-14 dating determines the age of organic materials up to about 50,000 years old. Carbon-14 is produced in the atmosphere by cosmic rays and is incorporated into living organisms through the carbon cycle. While an organism is alive, its ^14C/^12C ratio matches that of the atmosphere. After death, ^14C decays with a half-life of 5730 years, and the ratio decreases. Measuring the remaining ^14C/^12C ratio gives the time since death: t = -(1/lambda) ln(A/A_0).
Other radiometric dating methods extend to much longer timescales. Potassium-argon dating (^40K yielding ^40Ar, t_1/2 = 1.25 billion years) is used for rocks on geological timescales. Uranium-lead dating (^238U yielding ^206Pb, t_1/2 = 4.47 billion years) dates the oldest rocks and has been used to determine the age of Earth. Rubidium-strontium dating is used for minerals and meteorites.
<image>Panel A: An exponential decay curve showing N/N_0 versus time in units of half-lives. At t = 0, N/N_0 = 1. At t = t_1/2, N/N_0 = 0.5. At t = 2t_1/2, N/N_0 = 0.25. At t = 3t_1/2, N/N_0 = 0.125. The curve is labeled N(t) = N_0 e^(-lambda t) and key points are marked. Panel B: A diagram of carbon-14 dating. A living tree absorbs ^14C from the atmosphere (constant ratio). After the tree dies, ^14C decays. A comparison bar chart shows the ^14C activity in a living sample, a 5,730-year-old sample (half the activity), and an 11,460-year-old sample (one-quarter the activity).</image>
IV. Biological Effects of Radiation
Radiation damages biological tissue primarily by ionizing atoms and molecules. Direct damage occurs when radiation breaks chemical bonds, especially in DNA. Indirect damage occurs when radiation ionizes water molecules, producing free radicals that go on to damage biomolecules.
Several dose quantities are used to characterize radiation exposure. The absorbed dose is the energy absorbed per unit mass, measured in grays (Gy) = 1 J/kg (the older unit is the rad = 0.01 Gy). The equivalent dose accounts for the differing biological effectiveness of different radiation types: H = D x Q, where Q is the quality factor (Q = 1 for beta and gamma, Q = 20 for alpha particles). The unit is the sievert (Sv), with the older unit being the rem = 0.01 Sv. The effective dose further accounts for the varying sensitivity of different organs.
Typical doses provide useful context. Background radiation delivers about 2-3 mSv per year from cosmic rays, radon, food, and soil. A chest X-ray delivers roughly 0.02 mSv, while a CT scan of the abdomen delivers about 8 mSv. The lethal dose for whole-body acute exposure is approximately 4-5 Sv (LD50/30). Acute radiation syndrome at high doses causes nausea, immune suppression, hemorrhage, and death. Long-term effects include an increased cancer risk of approximately 5% per Sv of effective dose. The ALARA principle guides radiation safety: exposure should be As Low As Reasonably Achievable.
V. Medical Imaging with Radiation
X-ray imaging works by passing X-rays through the body. Denser tissues such as bone absorb more radiation, creating contrast in a two-dimensional projection image. Applications include diagnosing fractures, dental imaging, and chest X-rays.
CT (computed tomography) acquires multiple X-ray projections from different angles and uses computer reconstruction to produce cross-sectional images. It provides detailed three-dimensional anatomical information but delivers a higher radiation dose than plain X-rays.
Nuclear medicine uses radioactive tracers to produce functional images that show how organs work, not just their structure. In gamma camera and SPECT (Single Photon Emission Computed Tomography) imaging, a tracer such as ^99m Tc (t_1/2 = 6 hours) is injected into the patient and accumulates in specific organs. The gamma rays emitted by the tracer are detected by a camera to create images. SPECT adds three-dimensional capability.
PET (Positron Emission Tomography) uses positron-emitting tracers such as ^18F-FDG (t_1/2 = 110 min). When the emitted positron annihilates with an electron, two 511 keV gamma rays are produced, traveling in opposite directions (180 degrees apart). Coincidence detection of these two gamma rays allows precise localization of the annihilation event. PET measures metabolic activity and is critical for cancer detection, cardiology, and neurology. It is often combined with CT (PET/CT) to correlate functional and anatomical information.
VI. Radiation Therapy
External beam radiation therapy directs high-energy X-rays or gamma rays (from ^60Co sources or linear accelerators) at tumors. Multiple beams from different angles concentrate the dose on the tumor while sparing surrounding tissue. Treatment is typically fractionated, with the total dose delivered over many sessions to allow normal tissue to recover. Advanced techniques include intensity-modulated radiation therapy (IMRT), stereotactic radiosurgery, and proton therapy.
Brachytherapy places radioactive sources directly inside or adjacent to the tumor, delivering a high local dose with rapid fall-off that spares nearby tissues. It is commonly used for cervical, prostate, and breast cancers.
Radiopharmaceutical therapy uses radioactive drugs that target specific tissues. Iodine-131 treats thyroid cancer because iodine naturally concentrates in the thyroid gland. Radium-223 targets bone metastases because radium mimics calcium. Lutetium-177 PSMA treats prostate cancer by targeting prostate-specific membrane antigen.
The fundamental principles of radiation therapy rest on the fact that rapidly dividing cells, including cancer cells, are more sensitive to radiation damage. The goal is to deliver a tumoricidal dose while minimizing damage to normal tissue, a balance quantified by the therapeutic ratio comparing tumor control probability to normal tissue complication probability.
<image>A diagram of PET imaging. Panel A: ^18F-FDG is injected into a patient and accumulates in metabolically active tissue (e.g., a tumor). A fluorine-18 nucleus emits a positron, which travels a short distance (~1 mm) before annihilating with an electron. Two gamma rays (each 511 keV) are emitted in opposite directions (180 degrees apart). Panel B: The patient lies inside a ring of gamma ray detectors. Coincidence circuits identify pairs of simultaneously detected gamma rays to reconstruct the line of response. Panel C: A resulting PET image showing a whole-body scan with bright spots indicating areas of high metabolic activity (brain, bladder, and a tumor highlighted).</image>
