Radioactivity — Alpha, Beta, Gamma and Half-Life

Radioactivity is the spontaneous disintegration of an unstable atomic nucleus, emitting radiation in the process. The key word is spontaneous — it happens on its own, driven by the nucleus’s desire to reach a more stable configuration. No chemical reaction, no external trigger.

Henri Becquerel discovered it accidentally in 1896 when uranium fogged a photographic plate through wrapping paper. Marie Curie named the phenomenon and identified two radioactive elements: polonium and radium. Their work opened the door to nuclear physics.

Understanding radioactivity isn’t just exam content — it’s the basis of nuclear medicine, carbon dating, and nuclear power.

Key Terms and Definitions

Radioactive decay: The process by which an unstable nucleus spontaneously emits radiation to become more stable.

Activity (A): The number of decays per second. Measured in Becquerel (Bq): 1 Bq = 1 decay/second. Also measured in Curie (Ci): 1 Ci = $3.7 \times 10^{10}$ Bq.

Half-life ( $t_{1/2}$ ): The time taken for half the nuclei in a sample to decay. It’s a fixed property of each isotope — completely independent of temperature, pressure, or chemical state.

Decay constant ( $\lambda$ ): The probability per unit time that a given nucleus will decay. Higher $\lambda$ = faster decay = shorter half-life.

Parent nucleus: The original unstable nucleus before decay.

Daughter nucleus: The nucleus formed after decay.

The Three Types of Radiation

Alpha ( $\alpha$ ) Radiation

An alpha particle is a helium nucleus: 2 protons + 2 neutrons, written as ${}^4_2\text{He}$ .

When a nucleus emits an alpha particle, it loses 2 protons and 2 neutrons:

{}^A_Z X \rightarrow {}^{A-4}_{Z-2} Y + {}^4_2 \text{He}

Properties:

Positively charged (+2)
Heavy and slow — stopped by a sheet of paper or a few centimetres of air
Highly ionising (deposits energy densely along its path)
Most damaging if inhaled/ingested (inside the body)

Example: Uranium-238 emits alpha to become Thorium-234.

Beta ( $\beta$ ) Radiation

Beta-minus ( $\beta^-$ ) decay: a neutron in the nucleus converts to a proton, emitting an electron and an antineutrino.

{}^A_Z X \rightarrow {}^A_{Z+1} Y + e^- + \bar{\nu}_e

The mass number stays the same; the atomic number increases by 1.

Beta-plus ( $\beta^+$ ) decay: a proton converts to a neutron, emitting a positron and a neutrino.

Properties:

$\beta^-$ : negatively charged; $\beta^+$ : positively charged
Penetrates a few metres of air; stopped by a few mm of aluminium
Less ionising than alpha
Can penetrate skin

Gamma ( $\gamma$ ) Radiation

Gamma rays are high-energy electromagnetic radiation emitted when a nucleus drops from an excited energy state to a lower one — no change in mass number or atomic number.

{}^A_Z X^* \rightarrow {}^A_Z X + \gamma

Properties:

No charge, no mass
Very high penetration — requires several cm of lead or metres of concrete to stop
Least ionising per unit path length, but deeply penetrating
Used in radiotherapy and sterilisation

Summary Table

Property	Alpha ( $\alpha$ )	Beta ( $\beta^-$ )	Gamma ( $\gamma$ )
Nature	${}^4_2$ He nucleus	Electron	Electromagnetic wave
Charge	+2	–1	0
Stopped by	Paper	Aluminium (few mm)	Lead (several cm)
Ionising power	Highest	Medium	Lowest
Range in air	Few cm	Few metres	Very large

Radioactive Decay Law

The number of undecayed nuclei $N$ at time $t$ follows an exponential decay:

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

where $N_0$ is the initial number of nuclei, $\lambda$ is the decay constant, and $t$ is time.

Activity: $A(t) = \lambda N(t) = A_0 \, e^{-\lambda t}$

t_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}

After $n$ half-lives: $N = N_0 \times \left(\frac{1}{2}\right)^n$

Also: $n = \frac{t}{t_{1/2}}$

The exponential nature of decay means the activity is always proportional to the amount of substance remaining. A large sample decays fast; as material depletes, the rate of decay slows proportionally.

Solved Examples

Easy (CBSE Class 10): Complete the decay equation

Q: Complete: ${}^{226}_{88}\text{Ra} \rightarrow {}^{?}_{?}\text{Rn} + {}^4_2\text{He}$

Solution: Alpha decay: A decreases by 4, Z decreases by 2.

{}^{226}_{88}\text{Ra} \rightarrow {}^{222}_{86}\text{Rn} + {}^4_2\text{He}

This is the radium-to-radon decay that produces the radon gas found in poorly ventilated basements.

Medium (JEE Main): Half-life calculation

Q: A radioactive element has a half-life of 5 years. What fraction remains after 20 years?

Solution: Number of half-lives = $n = \frac{20}{5} = 4$ .

\frac{N}{N_0} = \left(\frac{1}{2}\right)^4 = \frac{1}{16}

One-sixteenth of the original sample remains.

Hard (JEE Advanced): Finding initial activity

Q: A sample has an activity of 2000 Bq at $t = 0$ . After 6 hours, the activity is 250 Bq. Find the half-life and the decay constant.

Solution:

$A(t) = A_0 \cdot 2^{-t/t_{1/2}}$

$250 = 2000 \cdot 2^{-6/t_{1/2}}$

$\frac{250}{2000} = \frac{1}{8} = 2^{-3}$

So $\frac{6}{t_{1/2}} = 3$ → $t_{1/2} = 2$ hours.

$\lambda = \frac{0.693}{t_{1/2}} = \frac{0.693}{2} = 0.3465 \text{ hr}^{-1}$

Exam-Specific Tips

JEE Main 2024 (Paper 1): Asked about the activity remaining after 3 half-lives. Standard formula application — $A = A_0/8$ . Worth 4 marks.

CBSE Class 12 Board: Questions on: (1) balancing decay equations (identifying daughter nucleus), (2) half-life calculations, (3) properties of alpha/beta/gamma. A 3-mark question on comparing penetrating powers appears almost every year.

NEET: Radioactivity appears in the nuclear physics section. Focus on the decay law, the relation between $\lambda$ and $t_{1/2}$ , and distinguishing properties of the three radiation types. 2–3 questions expected.

Common Mistakes to Avoid

Mistake 1: Confusing mass number and atomic number in alpha decay. Alpha decay removes 4 from mass number and 2 from atomic number. Students often subtract 2 from both.

Mistake 2: Forgetting that gamma decay does NOT change the nucleus. After alpha or beta decay, the daughter nucleus may be in an excited state and emits gamma. The gamma alone doesn’t change $A$ or $Z$ .

Mistake 3: Using half-life as the time until all nuclei decay. Half-life means half of the current sample decays — not half of the original. After each half-life, you’re halving what’s left, not the original. Theoretically, some nuclei always remain.

Mistake 4: Getting the beta-minus daughter wrong. In $\beta^-$ decay, a neutron becomes a proton — so $Z$ increases by 1 while $A$ stays the same. Students sometimes increase $A$ by 1 instead.

Mistake 5: Confusing activity and amount. Activity decreases over time; so does the amount of radioactive material. But the ratio $A/N = \lambda$ is constant. If you double the mass of a sample, you double its activity.

Practice Questions

Q1: Write the complete alpha decay equation for ${}^{210}_{84}\text{Po}$ .

{}^{210}_{84}\text{Po} \rightarrow {}^{206}_{82}\text{Pb} + {}^4_2\text{He}

Polonium-210 decays to Lead-206 with alpha emission. This is the decay Marie Curie studied.

Q2: A sample of a radioactive element has 6.4 × 10¹⁶ atoms at $t = 0$ . Its half-life is 3 days. How many atoms remain after 9 days?

$n = 9/3 = 3$ half-lives. $N = 6.4 \times 10^{16} \times (1/2)^3 = 6.4 \times 10^{16}/8 = 8 \times 10^{15}$ atoms.

Q3: What is the penetrating power order of the three types of radiation?

$\gamma > \beta > \alpha$ . Gamma has the highest penetrating power, alpha the lowest.

Q4: A radioactive substance has a decay constant of $0.231$ per day. Find the half-life.

$t_{1/2} = 0.693/\lambda = 0.693/0.231 = 3$ days.

Q5: After 3 half-lives, what percentage of the original sample has decayed?

Remaining = $(1/2)^3 = 1/8 = 12.5\%$ . Decayed = $100 - 12.5 = \mathbf{87.5\%}$ .

FAQs

Q: Why is half-life constant even as the sample depletes? Because radioactive decay is a probabilistic process. Each nucleus has a fixed probability $\lambda$ of decaying per unit time, regardless of how many other nuclei surround it. The rate drops proportionally as $N$ decreases, keeping the half-life constant.

Q: Can alpha decay be useful? Yes — smoke detectors use Americium-241, which emits alpha particles to ionise air and complete a circuit. When smoke enters, it absorbs the alpha particles, the circuit breaks, and the alarm triggers.

Q: What is carbon dating? Carbon-14 ( ${}^{14}$ C) is a radioactive isotope produced in the atmosphere. Living organisms continuously absorb it. After death, the C-14 decays with a half-life of 5730 years. By measuring the remaining C-14 ratio, we can estimate how long ago an organism died.

Q: Are gamma rays dangerous? Yes — they penetrate deeply and can damage DNA, increasing cancer risk with prolonged exposure. But controlled doses are used in radiotherapy to kill cancer cells.

Advanced Concepts

Successive disintegration and radioactive series

When a parent nucleus decays, the daughter may also be radioactive. This creates a decay chain. Four natural decay series exist: uranium series ( $^{238}$ U), actinium series ( $^{235}$ U), thorium series ( $^{232}$ Th), and neptunium series ( $^{237}$ Np, artificial).

Quick calculation: Given a decay chain from $^A_Z X$ to $^{A'}_Z' Y$ , the number of $\alpha$ decays = $\frac{A - A'}{4}$ and the number of $\beta^-$ decays = $2 \times \text{(number of } \alpha\text{)} - (Z - Z')$ .

$^{238}_{92}$ U decays to $^{206}_{82}$ Pb. Find the number of $\alpha$ and $\beta^-$ decays.

$\alpha$ decays = $(238-206)/4 = 32/4 = 8$

$\beta^-$ decays = $2 \times 8 - (92-82) = 16 - 10 = 6$

Answer: 8 $\alpha$ and 6 $\beta^-$ decays.

Nuclear reactions and mass-energy equivalence

\Delta E = \Delta m \cdot c^2

The mass defect ( $\Delta m$ ) is the difference between the mass of constituents and the mass of the nucleus. This “missing mass” is converted to binding energy that holds the nucleus together.

1 \text{ u} = 931.5 \text{ MeV}/c^2

If $\Delta m = 0.02$ u, the energy released = $0.02 \times 931.5 = 18.63$ MeV.

Mean life vs half-life

The mean life ( $\tau$ ) is the average time a nucleus survives before decaying:

\tau = \frac{1}{\lambda} = \frac{t_{1/2}}{0.693} = 1.443 \times t_{1/2}

After one mean life, the fraction remaining = $e^{-1} \approx 36.8\%$ .

Additional Practice Questions

Q6. $^{238}$ U decays to $^{206}$ Pb. How many $\alpha$ and $\beta^-$ decays occur?

$\alpha$ decays = $(238-206)/4 = 8$ . $\beta^-$ decays = $2(8) - (92-82) = 6$ . Total: 8 $\alpha$ + 6 $\beta^-$ .

Q7. The mean life of a radioactive element is 10 days. Find its half-life.

$t_{1/2} = 0.693 \times \tau = 0.693 \times 10 = 6.93$ days.

Q8. In beta-minus decay, which particle is emitted along with the electron?

An antineutrino ( $\bar{\nu}_e$ ). The neutrino was postulated by Pauli to conserve energy and momentum — without it, beta decay would violate conservation laws.

Key Terms and Definitions

The Three Types of Radiation

Alpha (α\alphaα) Radiation

Beta (β\betaβ) Radiation

Gamma (γ\gammaγ) Radiation

Summary Table

Radioactive Decay Law

Solved Examples

Easy (CBSE Class 10): Complete the decay equation

Medium (JEE Main): Half-life calculation

Hard (JEE Advanced): Finding initial activity

Exam-Specific Tips

Common Mistakes to Avoid

Practice Questions

FAQs

Advanced Concepts

Successive disintegration and radioactive series

Nuclear reactions and mass-energy equivalence

Mean life vs half-life

Additional Practice Questions

Practice Questions

Alpha ( $\alpha$ ) Radiation

Beta ( $\beta$ ) Radiation

Gamma ( $\gamma$ ) Radiation