Atomic Structure

The atom is the smallest unit of an element. At its centre lies the nucleus — tiny (diameter ~$10^{-15}\ \text{m}$) yet extremely dense — containing:

• Protons (charge: $+1$): the number of protons defines the element and is called the **atomic number ($Z$)**.
• Neutrons (charge: 0): contribute to nuclear mass and help stabilise the nucleus via the strong nuclear force.

Surrounding the nucleus are electrons (charge: $-1$, mass $\approx 1/1836$ of a proton), which occupy discrete energy levels (shells). In a neutral atom the number of electrons equals $Z$.

| Particle | Charge | Relative mass |<br>|---|---|---|<br>| Proton | $+1$ | 1 |<br>| Neutron | $0$ | 1 |<br>| Electron | $-1$ | $\approx 1/1836$ |

The **mass number ($A$)** counts all nucleons (protons + neutrons): $A = Z + N$, where $N$ is the neutron count.

Isotopes and Nuclide Notation

Isotopes are atoms of the same element (same $Z$, identical chemical behaviour) that differ in neutron number $N$ and therefore in mass number $A$. The standard nuclide notation is:

$${}^{A}_{Z}X$$

where $X$ is the chemical symbol of the element.

Carbon isotopes:

• ${}^{12}_{6}\text{C}$: 6 protons, 6 neutrons — stable; ~98.9 % of natural carbon.
• ${}^{13}_{6}\text{C}$: 6 protons, 7 neutrons — stable; ~1.1 %.
• ${}^{14}_{6}\text{C}$: 6 protons, 8 neutrons — radioactive ($t_{1/2} = 5{,}730$ years); used in radiocarbon dating of organic material.

Types of Nuclear Radiation

Nuclei with an unstable proton-to-neutron ratio spontaneously emit radiation to reach a more stable configuration. Three principal types exist:

Alpha Radiation ($\alpha$)

Nature: A helium-4 nucleus — two protons and two neutrons (${}^{4}_{2}\text{He}$).<br>Penetration: Stopped by a sheet of paper or a few centimetres of air; even unprotected skin provides effective shielding.<br>Ionising power: Very high — the large charge and mass cause dense ionisation over a very short path, depositing energy rapidly.<br>Example source: Americium-241, present in most domestic smoke detectors.

Beta Radiation ($\beta$)

Nature: A high-speed electron ($\beta^-$ decay) emitted when a neutron converts to a proton inside the nucleus, or a positron ($\beta^+$ decay) when a proton converts to a neutron.<br>Penetration: Stopped by a few millimetres of aluminium or several millimetres of thick plastic.<br>Ionising power: Moderate — lighter and singly charged, so it interacts less frequently with matter than alpha.<br>Example source: Strontium-90, used in medical tracers and industrial thickness gauges.

Gamma Radiation ($\gamma$)

Nature: High-energy electromagnetic radiation — the nucleus sheds excess energy as photons without emitting a particle.<br>Penetration: Requires several centimetres of lead or metres of concrete for significant attenuation; it is never completely blocked.<br>Ionising power: Lowest per unit path length, yet most dangerous externally due to its penetrating power and ability to reach internal organs.<br>Example source: Cobalt-60, used in cancer radiotherapy (gamma knife treatment).

| Type | Symbol | Nature | Stopped by | Relative ionisation |<br>|---|---|---|---|---|<br>| Alpha | $\alpha$ | ${}^{4}_{2}\text{He}$ nucleus | Paper / skin | Very high |<br>| Beta | $\beta$ | Electron or positron | Aluminium (mm) | Moderate |<br>| Gamma | $\gamma$ | EM photons | Lead (cm) | Low |

Half-Life

Radioactive decay is random and spontaneous at the level of individual nuclei — no external factor can determine exactly when a given nucleus will decay. Yet in a large sample the collective decay rate follows a precise statistical pattern: the activity decreases exponentially over time.

The **half-life ($t_{1/2}$) is the time required for half** of the radioactive nuclei in a sample to decay (equivalently, for the activity to fall to half its current value).

After each successive half-life, the remaining quantity is halved again:

$$N(t) = N_0 \times \left(\frac{1}{2}\right)^{n} \qquad \text{where } n = \frac{t}{t_{1/2}}$$

| Half-lives elapsed ($n$) | Fraction remaining | Percentage remaining |<br>|---|---|---|<br>| 0 | $1$ | 100 % |<br>| 1 | $1/2$ | 50 % |<br>| 2 | $1/4$ | 25 % |<br>| 3 | $1/8$ | 12.5 % |<br>| 4 | $1/16$ | 6.25 % |

Worked Example: Iodine-131 ($t_{1/2} = 8$ days) is used in thyroid cancer treatment. If a patient receives an initial activity of $A_0 = 800\ \text{MBq}$, what activity remains after 32 days?

$$n = \frac{t}{t_{1/2}} = \frac{32}{8} = 4 \text{ half-lives}$$

$$A = A_0 \times \left(\frac{1}{2}\right)^4 = 800 \times \frac{1}{16} = 50\ \text{MBq}$$

Half-lives span an enormous range — from microseconds (highly unstable synthetic nuclei) to $4.5 \times 10^9$ years for Uranium-238, comparable to Earth's age and useful for uranium–lead radiometric dating of ancient rocks.