Radioactive Decay

Unstable nuclei emit ionising radiation as they decay. Three types dominate the specification, and you must know the nature, charge, mass, ionising power and penetrating power of each.

Radiation	Nature	Charge	Mass	Ionising power	Stopped by
Alpha (α)	helium-4 nucleus $^{4}_{2}\text{He}$ (2 protons + 2 neutrons)	$+2e$	$4u$	strong	a sheet of paper / few cm of air
Beta-minus (β⁻)	fast electron $^{\;\;0}_{-1}\text{e}$	$-e$	$\approx \frac{1}{1840}u$	moderate	a few mm of aluminium
Gamma (γ)	high-energy EM photon	$0$	$0$	weak	thick lead / concrete (only reduced)

There is an inverse relationship between ionising power and penetration. Alpha is a large, slow, doubly-charged particle, so it collides with and strips electrons from many atoms — it ionises strongly but loses its energy quickly and travels only a few centimetres in air. Gamma has no charge and interacts weakly, so it ionises little but penetrates deeply.

Beta-plus (β⁺) is a positron ($^{0}_{+1}\text{e}$) — the antiparticle of the electron, charge $+e$ — emitted in some decays. Electron capture is a related process in which a nucleus absorbs an inner-shell electron.

Behaviour in fields. Because α and β⁻ carry opposite charges, they are deflected in opposite directions in an electric or magnetic field; γ (uncharged) is undeflected. Beta deflects far more than alpha for the same field because it has a much smaller mass (and hence a much larger charge-to-mass ratio) despite carrying less charge.

A nucleus is written $^{A}_{Z}\text{X}$, where $A$ is the nucleon number (protons + neutrons) and $Z$ is the proton number (charge). In every decay equation the totals of $A$ and of $Z$ must be the same on both sides.

Alpha decay — the nucleus emits a helium-4 nucleus, losing 4 nucleons and 2 protons: $^{A}_{Z}\text{X} \;\rightarrow\; ^{A-4}_{Z-2}\text{Y} + ^{4}_{2}\text{He}$ For example: $^{238}_{\;92}\text{U} \rightarrow ^{234}_{\;90}\text{Th} + ^{4}_{2}\text{He}$ (check: $238=234+4$ and $92=90+2$ ✔).

Beta-minus decay — inside the nucleus a neutron turns into a proton, emitting an electron and an (electron) antineutrino: $^{1}_{0}n \;\rightarrow\; ^{1}_{1}p + ^{\;\;0}_{-1}\text{e} + \bar{\nu}_e$ so for the whole nucleus $A$ is unchanged and $Z$ increases by 1: $^{A}_{Z}\text{X} \;\rightarrow\; ^{\;\;A}_{Z+1}\text{Y} + ^{\;\;0}_{-1}\text{e} + \bar{\nu}_e$ For example: $^{14}_{\;6}\text{C} \rightarrow ^{14}_{\;7}\text{N} + ^{\;\;0}_{-1}\text{e} + \bar{\nu}_e$.

Beta-plus decay — a proton turns into a neutron, emitting a positron and a neutrino; $Z$ decreases by 1: $^{A}_{Z}\text{X} \;\rightarrow\; ^{\;\;A}_{Z-1}\text{Y} + ^{0}_{+1}\text{e} + \nu_e$

Electron capture — the nucleus captures an inner electron, so a proton becomes a neutron; $Z$ decreases by 1 (like β⁺): $^{A}_{Z}\text{X} + ^{\;\;0}_{-1}\text{e} \;\rightarrow\; ^{\;\;A}_{Z-1}\text{Y} + \nu_e$

Gamma emission — a nucleus in an excited state loses energy as a γ photon. Neither $A$ nor $Z$ changes; only the energy of the nucleus falls: $^{A}_{Z}\text{X}^{*} \;\rightarrow\; ^{A}_{Z}\text{X} + \gamma$

The antineutrino/neutrino carries no charge and (essentially) no mass, so it does not affect the $A$ and $Z$ balance — but it must be included to conserve energy and (lepton) number.

Radioactive decay is random and spontaneous:

• Random — it is impossible to predict which nucleus will decay next or when a particular nucleus will decay. Each undecayed nucleus has the same fixed probability of decaying per second.
• Spontaneous — decay is a property of the nucleus alone and is unaffected by external conditions such as temperature, pressure, or the chemical compound the atom is part of.

Because it is random, we work with probabilities and large numbers. The decay constant $\lambda$ is the probability per unit time that a given nucleus decays (unit $\text{s}^{-1}$). If there are $N$ undecayed nuclei, the number decaying per second — the activity $A$ — is

$A = \lambda N$

Activity is measured in becquerel (Bq), where $1\ \text{Bq} = 1$ decay per second. A large decay constant means a nucleus is very unstable (high activity, short half-life); a small $\lambda$ means a long-lived, less active source.

Because a detector measures counts, the recorded count rate is proportional to the true activity (a fixed fraction of the decays reaches the detector). So a count rate $C$ falls in exactly the same exponential way as the activity: $C=C_0 e^{-\lambda t}$ — provided you first subtract the background.

Because the rate of decay is proportional to the number of nuclei present, the number of undecayed nuclei falls exponentially:

$N = N_0\,e^{-\lambda t}$

where $N_0$ is the number at $t=0$. Since $A=\lambda N$, the activity (and the corrected count rate) obey the same law:

$A = A_0\,e^{-\lambda t}, \qquad C = C_0\,e^{-\lambda t}$

Half-life. The half-life $t_{1/2}$ is the time for the number of undecayed nuclei (or the activity) to fall to half its initial value. Setting $A = \tfrac12 A_0$: $\tfrac12 A_0 = A_0\,e^{-\lambda t_{1/2}} \;\Rightarrow\; \tfrac12 = e^{-\lambda t_{1/2}}$ Taking natural logs, $\ln\tfrac12 = -\lambda t_{1/2}$, i.e. $-\ln 2 = -\lambda t_{1/2}$, so

$\boxed{\;t_{1/2} = \dfrac{\ln 2}{\lambda}\;}\qquad\text{equivalently}\qquad \lambda\, t_{1/2} = \ln 2 \approx 0.693.$

The half-life is constant — it takes the same time to halve again and again, which is why an activity–time curve halves in equal steps ($A_0\to A_0/2\to A_0/4\to\dots$).

Linearising the exponential. To find $\lambda$ accurately from data, take natural logs of $A = A_0 e^{-\lambda t}$: $\ln A = \ln A_0 - \lambda t$ Comparing with $y = c + mx$ (with $y=\ln A$, $x=t$): the gradient is $-\lambda$ and the intercept is $\ln A_0$. So a graph of $\ln A$ (or $\ln C$) against $t$ is a straight line, and

$\lambda = -\,\text{gradient}, \qquad t_{1/2} = \frac{\ln 2}{\lambda}.$

This is the same log-linearisation trick used for capacitor discharge in Unit 4, where $\ln V$ against $t$ has gradient $-1/RC$. The physics differs but the mathematics is identical: an exponential becomes a straight line whose gradient is the (negative of the) decay constant.

Background radiation. A detector always records some counts even with no source present — from cosmic rays, rocks (radon), building materials, food and medical sources. This background count rate must be measured (source removed) and subtracted from every reading to give the corrected count rate before any decay analysis or half-life is found. Because decay is random, counts fluctuate statistically, so you should count for a longer time (or repeat and average) to reduce the percentage uncertainty, which for $N$ counts is of order $\dfrac{\sqrt{N}}{N}=\dfrac{1}{\sqrt{N}}$ — bigger counts are proportionally more precise.

CP15 — absorption of gamma radiation by lead. Gamma is attenuated exponentially as it passes through an absorber of thickness $x$:

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

where $I_0$ is the incident intensity (corrected count rate with no absorber), $I$ is the transmitted intensity, and $\mu$ is the linear absorption (attenuation) coefficient of the material ($\text{m}^{-1}$ or $\text{mm}^{-1}$).

Method and analysis.

1. With the source removed, measure the background count over a long time; find the background count rate.
2. Place a gamma source a fixed distance from a Geiger–Müller tube. Insert lead sheets of increasing thickness $x$ between them; for each thickness record the count over a fixed time and subtract the background to get the corrected count rate $I$.
3. Take natural logs of $I = I_0 e^{-\mu x}$: $\ln I = \ln I_0 - \mu x$
4. Plot $\ln I$ (corrected count rate) against thickness $x$. This is a straight line of gradient $-\mu$ and intercept $\ln I_0$. So $\mu = -\,\text{gradient}$.

Safety precautions (be specific). Handle sealed sources with tongs/forceps, never bare hands, to keep them away from the body; hold the source at arm's length and point it away from people; keep it in a lead-lined container when not in use; minimise the time of exposure; and maximise the distance between source and people (intensity falls as $1/r^2$). "Be careful" earns nothing — the marks are for named, specific precautions with a reason.