Radioactivity

Radioactivity: A Nuclear Phenomenon

Radioactivity was discovered in 1896 by Henri Becquerel, quite by accident. While studying how certain compounds glow after being exposed to visible light, Becquerel placed some uranium-potassium sulphate on a photographic plate wrapped in black paper, separated by a silver sheet. When he developed the plate hours later, it was blackened — something had passed through both the paper and the silver. That "something" was radiation from the uranium compound.

Further experiments revealed that radioactivity is a nuclear phenomenon: an unstable nucleus spontaneously decays, emitting particles or energy. This process is called radioactive decay. Three types occur in nature:

1. $\alpha$-decay — the nucleus emits a helium nucleus, ${}_{2}^{4}\text{He}$.
2. $\beta$-decay — the nucleus emits an electron ($\beta^-$) or a positron ($\beta^+$). A positron has the same mass as an electron but opposite charge.
3. $\gamma$-decay — the nucleus emits high-energy photons (hundreds of keV or more).

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The Three Decay Modes in Detail

$\alpha$-Decay

In $\alpha$-decay, the parent nucleus loses two protons and two neutrons — exactly the composition of a helium-4 nucleus. The general equation is:

The emitted $\alpha$-particle carries a positive charge and is relatively heavy. Because it interacts strongly with matter, it has a short range in air (a few centimetres) and can be stopped by a sheet of paper.

$\beta$-Decay

In $\beta$-decay, a neutron inside the nucleus transforms into a proton (or vice versa), and an electron or positron is emitted. There are two types:

• $\beta^-$ decay: A neutron converts into a proton, an electron, and an antineutrino ($\bar{\nu}_e$). The electron is emitted.

• $\beta^+$ decay: A proton converts into a neutron, a positron, and a neutrino ($\nu_e$). The positron is emitted.

The neutrino and antineutrino are nearly massless, neutral particles that interact very weakly with matter. They were first postulated by Wolfgang Pauli to conserve energy and momentum in $\beta$-decay, and were later detected experimentally.

$\gamma$-Decay

$\gamma$-decay occurs when a nucleus in an excited state (often left after $\alpha$ or $\beta$ decay) drops to a lower energy state by emitting a high-energy photon. The nucleus itself does not change its composition:

The asterisk denotes an excited nuclear state. $\gamma$-rays are electromagnetic radiation with wavelengths shorter than X-rays and are extremely penetrating — they can pass through several centimetres of lead.

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Key Properties of Radioactive Decay

The textbook lists several fundamental properties of radioactive decay. Each is stated and proved below.

Property 1: Radioactive decay is a statistical process

You cannot predict exactly when a given unstable nucleus will decay. However, for a large sample of identical nuclei, the decay follows a well-defined statistical law. This is the law of radioactive decay.

›Proof

Consider a sample containing $N(t)$ identical radioactive nuclei at time $t$. The number of decays $dN$ occurring in a short time interval $dt$ is proportional to $N(t)$ and to $dt$:

$dN = -\lambda N(t)\, dt$

where $\lambda$ is the decay constant (a positive constant characteristic of the isotope). The negative sign indicates that $N$ decreases with time.

Rearranging:

$\frac{dN}{N} = -\lambda\, dt$

Integrate both sides:

$\int_{N_0}^{N(t)} \frac{dN}{N} = -\lambda \int_0^t dt$

$\ln N(t) - \ln N_0 = -\lambda t$

$\ln \frac{N(t)}{N_0} = -\lambda t$

Exponentiating:

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

This is the exponential decay law. $N_0$ is the number of nuclei at $t=0$.

Property 2: The decay constant $\lambda$ is related to the half-life $T_{1/2}$

The half-life is the time required for half of the original nuclei to decay. Set $N(T_{1/2}) = N_0/2$:

Take natural logarithms:

Property 3: The mean life $\tau$ is the average lifetime of a nucleus

The mean life (or average life) $\tau$ is the reciprocal of the decay constant:

›Proof

The number of nuclei that decay between $t$ and $t+dt$ is $|dN| = \lambda N_0 e^{-\lambda t} dt$. Each of these lived for time $t$. The total "lifetime" of all nuclei is:

$\int_0^\infty t\,|dN| = \int_0^\infty t \lambda N_0 e^{-\lambda t} dt$

The average lifetime is this total divided by $N_0$:

$\tau = \frac{1}{N_0} \int_0^\infty t \lambda N_0 e^{-\lambda t} dt = \lambda \int_0^\infty t e^{-\lambda t} dt$

Using the standard integral $\int_0^\infty t e^{-at} dt = 1/a^2$:

$\tau = \lambda \cdot \frac{1}{\lambda^2} = \frac{1}{\lambda}$

The relationship between half-life and mean life is:

Property 4: Activity $R$ is the rate of decay …