Difficulty: Introductory to Intermediate | Prerequisites: Nuclear structure and binding energy (Ch. 30.1-30.3). You should understand what protons, neutrons, and alpha particles are before starting this material.
This topic covers what happens when unstable nuclei transform: the types of radiation they emit, the mathematics of how quickly they decay, and how we measure and use radiation in medicine. It follows directly from nuclear structure and stability. If you understand why some nuclei are unstable, this section explains what they do about it and how we put that behaviour to practical use. You should already know what alpha particles are and be comfortable with exponential functions.
Unstable nuclei decay by emitting alpha particles, beta particles (electrons or positrons), or gamma rays. The rate of decay follows an exponential curve described by the half-life. In medicine, radiation dose is measured in grays (energy absorbed) and converted to sieverts (biological damage) using a weighting factor called RBE.
Alpha decay (α decay)
A nucleus emits an alpha particle (⁴₂He), losing 2 protons and 4 nucleons total. Think of it as the nucleus spitting out a chunk of helium.
Beta-minus decay (β⁻ decay)
A neutron in the nucleus converts into a proton, emitting an electron (e⁻) and energy. The mass number stays the same; the atomic number increases by 1.
Beta-plus decay (β⁺ decay)
A proton converts into a neutron, emitting a positron (e⁺) and energy. The mass number stays the same; the atomic number decreases by 1.
Gamma decay (γ decay)
A nucleus in an excited state drops to a lower energy level by emitting a high-energy photon (gamma ray). No change in Z or A. Think of it as the nucleus releasing leftover energy after an alpha or beta decay.
Half-life (t₁/₂)
The time it takes for half of a radioactive sample to decay. After one half-life, half remains; after two, a quarter remains; and so on.
Time constant (τ)
Related to the half-life by τ = 1.44 × t₁/₂. It is the average lifetime of a single radioactive nucleus before it decays.
Activity (R)
The number of decays per second in a radioactive sample. Measured in becquerels (Bq). In simple terms, it tells you how "hot" a sample is right now.
Becquerel (Bq)
The SI unit of activity. 1 Bq = 1 decay per second.
Curie (Ci)
An older unit of activity. 1 Ci = 3.7 × 10¹⁰ Bq. Named after Marie Curie.
Radiation dose / Gray (Gy)
The energy absorbed from ionising radiation per kilogram of tissue. 1 Gy = 1.00 J/kg of absorbed energy.
Relative Biological Effectiveness (RBE)
A weighting factor that accounts for the fact that different types of radiation cause different amounts of biological damage per unit of absorbed energy. Alpha particles (RBE = 20) do far more damage per gray than gamma rays (RBE = 1).
Dose equivalent / Sievert (Sv)
The product of dose in grays and the RBE. One sievert of any type of radiation produces the same biological damage. In simple terms, sieverts are the "biologically fair" unit of radiation exposure.
Rem
An older unit of dose equivalent. 1 rem = 0.01 Sv.
There are three principal modes of radioactive decay: alpha, beta, and gamma.
ᴬ_Z X → ᴬ⁻⁴_(Z−2) Y + α + energy
The parent nucleus emits an alpha particle (2 protons, 2 neutrons). The daughter has a different element identity.
Most of the released energy ends up as kinetic energy of the alpha particle.
ᴬ_Z X → ᴬ_(Z+1) Y + e⁻ + energy
A neutron converts to a proton inside the nucleus. An electron is emitted.
The mass number A is unchanged. The atomic number increases by 1.
This is how neutron-rich nuclei move toward stability.
ᴬ_Z X → ᴬ_(Z−1) Y + e⁺ + energy
A proton converts to a neutron. A positron (the electron's antiparticle) is emitted.
The mass number A is unchanged. The atomic number decreases by 1.
This is how proton-rich nuclei move toward stability.
A nucleus in an excited energy state transitions to a lower energy state by emitting a gamma-ray photon.
Gamma-ray energies are typically around 1 MeV.
No change in A or Z. The nucleus is the same element before and after; it just loses excess energy.
Gamma emission often follows an alpha or beta decay, as the daughter nucleus is frequently left in an excited state.
The number of undecayed atoms remaining after time t is: N = N₀ × (1/2)^(t/t₁/₂)
N₀ is the initial number of atoms. t and t₁/₂ must be in the same units.
Each half-life reduces the remaining sample by half. After 3 half-lives, 1/8 of the original remains.
Nucleus A decays to Nucleus B with a half-life of 10 minutes. A pure sample starts with 24,000 particles of A.
After 30 minutes (3 half-lives): N = 24,000 × (1/2)^(30/10) = 24,000 × (1/8) = 3,000 particles of A remaining.
That means 21,000 particles of B have been produced.
τ = 1.44 × t₁/₂
τ is the mean lifetime of a single nucleus. It is not the same as the half-life, but they are proportional.
Activity R = N/τ = 0.693N / t₁/₂
Activity is inversely proportional to the half-life: a short half-life means a high activity (many decays per second).
Activity also decays exponentially: R = R₀ × (1/2)^(t/t₁/₂) = R₀ × e^(−t/τ)
The SI unit is the becquerel: 1 Bq = 1 decay per second.
The older unit is the curie: 1 Ci = 3.7 × 10¹⁰ Bq.
Radiation dose measures the energy deposited by ionising radiation per kilogram of tissue.
SI unit: the gray (Gy). 1 Gy = 1.00 J/kg of absorbed energy.
Different radiation types cause different levels of biological damage for the same absorbed dose.
RBE values: X-rays = 1, gamma rays = 1, beta particles = 1, protons = 5, neutrons = 5 to 20, alpha particles = 20.
Alpha particles have the highest RBE because they are large, heavy, and deposit all their energy over a very short distance, causing concentrated damage to cells.
Dose equivalent (in sieverts) = dose (in Gy) × RBE.
One sievert of any radiation type produces the same biological effect, regardless of the type of radiation.
The older unit is the rem: 1 rem = 0.01 Sv.
This is the unit that matters for health and safety assessments. Grays tell you the physics; sieverts tell you the biology.
Exponential decay
N = N₀ × (1/2)^(t / t₁/₂)
Time constant
τ = 1.44 × t₁/₂
Activity
R = N / τ = 0.693 × N / t₁/₂
Activity over time
R = R₀ × (1/2)^(t / t₁/₂) = R₀ × e^(−t/τ)
Dose equivalent
Dose equivalent (Sv) = Dose (Gy) × RBE
Unit conversions
1 Gy = 1.00 J/kg, 1 Bq = 1 decay/s, 1 Ci = 3.7 × 10¹⁰ Bq, 1 rem = 0.01 Sv
Carbon-14 dating relies on the known half-life of ¹⁴C (about 5,730 years) to determine the age of organic material. By measuring how much ¹⁴C remains in a sample compared to the expected initial amount, scientists can work backwards using the decay equation.
In medical imaging, tracers such as technetium-99m (t₁/₂ ≈ 6 hours) are injected into patients. The short half-life means the tracer is active long enough to produce images but decays quickly, limiting the patient's radiation exposure. PET scans use positron-emitting isotopes (beta-plus decay), detecting the gamma rays produced when the positron annihilates with an electron.
Radiation therapy for cancer uses the biological damage caused by radiation intentionally, targeting tumour cells. The choice of radiation type and dosimetry (grays, sieverts) is central to treatment planning.
Students often confuse half-life with the time for a sample to fully decay. A radioactive sample never fully reaches zero; it halves repeatedly, approaching zero asymptotically.
A common mistake is mixing up grays and sieverts. Grays measure raw energy absorbed. Sieverts account for biological damage. 1 Gy of alpha radiation is far more dangerous than 1 Gy of gamma radiation, but 1 Sv of each produces the same biological effect.
Students sometimes think beta-minus decay involves an electron being stripped from the atom's electron cloud. It does not. The electron is created inside the nucleus when a neutron converts to a proton.
Another error is assuming gamma decay changes the element. It does not. Gamma decay only releases energy; Z and A remain the same.
⚠️ Half-life calculations are almost guaranteed on the exam. Practise plugging into N = N₀ × (1/2)^(t/t₁/₂) until it is automatic.
⚠️ Be able to write the decay equations for all three types (alpha, beta-minus, beta-plus) and identify what changes in Z and A for each.
⚠️ Know the difference between activity and the number of remaining atoms. Activity tells you the decay rate; N tells you how much is left.
⚠️ The gray-to-sievert conversion (dose equivalent = Gy × RBE) is a common short calculation. Know the RBE values for at least alpha particles (20), beta/gamma/X-rays (1), and protons (5).
⚠️ Gamma decay is the one that does not change the element. This is a frequent multiple-choice distractor.
True or False: In beta-minus decay, the mass number A changes.
False. A stays the same. Only Z increases by 1.
Fill in the blank: After 4 half-lives, the fraction of the original sample remaining is ___.
1/16 (i.e. (1/2)⁴).
True or False: 1 Gy of alpha radiation and 1 Gy of gamma radiation produce the same biological damage.
False. Alpha particles have an RBE of 20, so 1 Gy of alpha radiation produces 20 times the biological damage of 1 Gy of gamma radiation.
Fill in the blank: The SI unit for activity is the ___, equal to 1 decay per second.
Becquerel (Bq).
True or False: Gamma decay changes the element of the nucleus.
False. Neither Z nor A changes in gamma decay.
Q: A sample contains 16,000 radioactive atoms with a half-life of 5 hours. How many atoms remain after 20 hours?
A: 20 hours = 4 half-lives. N = 16,000 × (1/2)⁴ = 16,000 / 16 = 1,000 atoms.
Q: Write the beta-minus decay equation for a generic nucleus ᴬ_Z X.
A: ᴬ_Z X → ᴬ_(Z+1) Y + e⁻ + energy. A stays the same; Z increases by 1.
Q: A patient receives a radiation dose of 0.3 Gy from alpha particles (RBE = 20). What is the dose equivalent in sieverts?
A: Dose equivalent = 0.3 Gy × 20 = 6.0 Sv.
Q: Why is the activity of a sample with a short half-life higher than a sample with a long half-life, assuming the same number of atoms?
A: Activity R = 0.693N / t₁/₂. A shorter t₁/₂ in the denominator gives a larger R. The atoms decay more quickly, so more decays happen per second.
Q: A radioactive sample has an initial activity of 800 Bq and a half-life of 2 hours. What is the activity after 6 hours?
A: 6 hours = 3 half-lives. R = 800 × (1/2)³ = 800 / 8 = 100 Bq.
This material builds directly on nuclear structure and stability (Ch. 30.1-30.3). The reason nuclei decay in specific ways is determined by the balance of forces and binding energy covered there.
Exponential decay appears in many other areas of physics: RC circuits (charge decay), thermal cooling (Newton's law of cooling), and damped oscillations all follow the same mathematical form. If you are comfortable with the half-life equation, you already understand the pattern for all of these.
The dose and dosimetry concepts connect to biology and health physics. If you continue into biophysics or medical physics, the gray/sievert framework scales directly into clinical radiation protection.
radioactive decay, alpha decay, beta decay, beta-minus, beta-plus, positron emission, gamma decay, gamma ray, half-life, exponential decay, time constant, activity, becquerel, curie, radiation dose, gray, sievert, rem, RBE, relative biological effectiveness, dose equivalent, nuclear medicine, PET scan, carbon-14 dating, radiocarbon dating, radiation therapy, ionising radiation, PHYS 116, general physics, University of Washington