Radioactivity — Revision Notes

⚡ 30-Second Revision

Radioactive Decay Law — $N = N_0 e^{-\lambda t}$ (Number of nuclei)
Activity — $A = \lambda N = A_0 e^{-\lambda t}$ (Rate of decay)
Half-life ($T_{1/2}$) — Time for half nuclei to decay. $T_{1/2} = \frac{\ln(2)}{lambda} = \frac{0.693}{lambda}$
Mean Life ($\tau$) — Average lifetime. $\tau = \frac{1}{lambda}$
Relationship — $\tau = T_{1/2} / 0.693 \approx 1.44 T_{1/2}$
Alpha ($\alpha$) Decay — $_Z^A\text{X} \rightarrow _{Z-2}^{A-4}\text{Y} + _2^4\text{He}$. $\Delta Z=-2, \Delta A=-4$. High ionizing, low penetrating.
Beta-minus ($\beta^-$) Decay — $_Z^A\text{X} \rightarrow _{Z+1}^A\text{Y} + _{-1}^0\text{e} + \bar{\nu}$. $\Delta Z=+1, \Delta A=0$. Moderate ionizing/penetrating.
Beta-plus ($\beta^+$) Decay — $_Z^A\text{X} \rightarrow _{Z-1}^A\text{Y} + _{+1}^0\text{e} + \nu$. $\Delta Z=-1, \Delta A=0$. Moderate ionizing/penetrating.
Gamma ($\gamma$) Decay — $_Z^A\text{X}^* \rightarrow _Z^A\text{X} + \gamma$. $\Delta Z=0, \Delta A=0$. Low ionizing, high penetrating. Not deflected by E/B fields.
Q-value — $Q = (\Delta m)c^2$. Energy released from mass defect.

2-Minute Revision

Radioactivity is the spontaneous disintegration of unstable atomic nuclei. The core principle is the Radioactive Decay Law, $N = N_0 e^{-\lambda t}$, stating that the number of undecayed nuclei decreases exponentially over time.

The **decay constant ($\lambda$)** is the probability of decay per unit time. Key parameters derived from $\lambda$ are **half-life ($T_{1/2}$)**, the time for half the nuclei to decay ($T_{1/2} = 0.693/\lambda$), and **mean life ($\tau$)**, the average lifetime ($\tau = 1/\lambda$).

The **activity ($A$)** of a sample, its rate of disintegration, also decays exponentially: $A = A_0 e^{-\lambda t}$.

There are three main types of decay: **Alpha ($\alpha$) decay** emits a helium nucleus ($_2^4\text{He}$), decreasing atomic number ($Z$) by 2 and mass number ($A$) by 4. **Beta ($\beta$) decay** involves nucleon transformation: $\beta^-$ (neutron to proton) increases $Z$ by 1, $\beta^+$ (proton to neutron) decreases $Z$ by 1, with $A$ unchanged in both.

**Gamma ($\gamma$) decay** emits high-energy photons from an excited nucleus, changing neither $Z$ nor $A$. Alpha particles are heavy, charged, highly ionizing but poorly penetrating. Beta particles are lighter, charged, with moderate ionizing and penetrating power.

Gamma rays are uncharged, massless, highly penetrating but poorly ionizing, and are not deflected by electric or magnetic fields. The energy released in decay, the Q-value, is calculated from the mass defect using $E=mc^2$.

Radioactivity is a nuclear phenomenon, unaffected by external conditions like temperature or pressure.

5-Minute Revision

Radioactivity is the process of spontaneous nuclear transformation, where unstable nuclei emit radiation to achieve stability. This process is governed by the Radioactive Decay Law, $N = N_0 e^{-\lambda t}$, where $N$ is the number of undecayed nuclei at time $t$, $N_0$ is the initial number, and $\lambda$ is the decay constant.

The **half-life ($T_{1/2}$)** is the time required for half of the nuclei to decay, related by $T_{1/2} = \frac{0.693}{lambda}$. The **mean life ($\tau$)** is the average lifetime of a nucleus, $\tau = \frac{1}{lambda}$.

The **activity ($A$)**, or rate of decay, is $A = \lambda N = A_0 e^{-\lambda t}$.

Types of Decay and their Effects:

1
Alpha ($\alpha$) Decay — Emission of a helium nucleus ($_2^4\text{He}$). The parent nucleus $_Z^A\text{X}$ transforms to $_{Z-2}^{A-4}\text{Y}$. Alpha particles are heavy, positively charged (+2e), have very high ionizing power, but very low penetrating power (stopped by paper). Example: $_{92}^{238}\text{U} \rightarrow _{90}^{234}\text{Th} + _2^4\text{He}$.
2
Beta ($\beta$) Decay — Involves nucleon transformation.

* **Beta-minus ($\beta^-$) Decay**: Neutron converts to proton ($_0^1\text{n} \rightarrow _1^1\text{p} + _{-1}^0\text{e} + \bar{\nu}$). Parent $_Z^A\text{X}$ transforms to $_{Z+1}^A\text{Y}$. $Z$ increases by 1, $A$ unchanged.

Example: $_{6}^{14}\text{C} \rightarrow _{7}^{14}\text{N} + _{-1}^0\text{e} + \bar{\nu}$. * **Beta-plus ($\beta^+$) Decay**: Proton converts to neutron ($_1^1\text{p} \rightarrow _0^1\text{n} + _{+1}^0\text{e} + \nu$).

Parent $_Z^A\text{X}$ transforms to $_{Z-1}^A\text{Y}$. $Z$ decreases by 1, $A$ unchanged. Example: $_{11}^{22}\text{Na} \rightarrow _{10}^{22}\text{Ne} + _{+1}^0\text{e} + \nu$. Beta particles (electrons/positrons) are light, charged ($\pm$e), have moderate ionizing and penetrating power (stopped by aluminum).

1
Gamma ($\gamma$) Decay — Emission of high-energy photons from an excited nucleus ($_Z^A\text{X}^* \rightarrow _Z^A\text{X} + \gamma$). No change in $Z$ or $A$. Gamma rays are massless, uncharged, have very low ionizing power, but very high penetrating power (stopped by thick lead/concrete). They are not deflected by electric or magnetic fields.

Q-value: The energy released in a decay is $Q = (\Delta m)c^2$, where $\Delta m$ is the mass defect. This energy is shared as kinetic energy among the products. Radioactivity is a nuclear process and is unaffected by external factors like temperature, pressure, or chemical state. Applications include carbon dating, medical diagnostics (e.g., Technetium-99m), and cancer therapy (e.g., Cobalt-60).

Example: A sample has a half-life of $10\,\text{s}$. What fraction remains after $30\,\text{s}$? Number of half-lives $n = 30\,\text{s} / 10\,\text{s} = 3$. Fraction remaining $= (1/2)^n = (1/2)^3 = 1/8$.

Prelims Revision Notes

Radioactivity is the spontaneous disintegration of unstable atomic nuclei. It's a nuclear phenomenon, unaffected by external factors like temperature, pressure, or chemical state. The fundamental law is the Radioactive Decay Law: $N = N_0 e^{-\lambda t}$, where $N$ is the number of undecayed nuclei at time $t$, $N_0$ is the initial number, and $\lambda$ is the decay constant.

The **decay constant ($\lambda$)** is the probability of decay per unit time, measured in $\text{s}^{-1}$ or $\text{min}^{-1}$.

**Half-life ($T_{1/2}$)** is the time for half of the radioactive nuclei to decay. It's related to $\lambda$ by $T_{1/2} = \frac{\ln(2)}{lambda} = \frac{0.693}{lambda}$. For $n$ half-lives, the remaining fraction is $(1/2)^n$. **Mean life ($\tau$)** is the average lifetime of a nucleus, $\tau = \frac{1}{lambda}$. Note that $\tau \approx 1.44 T_{1/2}$.

**Activity ($A$)** is the rate of disintegration ($-\frac{dN}{dt}$). $A = \lambda N$. It also decays exponentially: $A = A_0 e^{-\lambda t}$. Units: Becquerel (Bq) = 1 decay/s; Curie (Ci) = $3.7 \times 10^{10}$ Bq.

Types of Radioactive Decay:

1
Alpha ($\alpha$) Decay — Emission of a helium nucleus ($_2^4\text{He}$). Atomic number ($Z$) decreases by 2, Mass number ($A$) decreases by 4. Example: $_{92}^{238}\text{U} \rightarrow _{90}^{234}\text{Th} + _2^4\text{He}$. Alpha particles are heavy, +2e charge, very high ionizing power, very low penetrating power (stopped by paper/skin). Deflected by E/B fields.
2
Beta ($\beta$) Decay — Involves transformation of a nucleon.

* **Beta-minus ($\beta^-$) Decay**: Neutron $\rightarrow$ Proton + Electron + Antineutrino. $Z$ increases by 1, $A$ unchanged. Example: $_{6}^{14}\text{C} \rightarrow _{7}^{14}\text{N} + _{-1}^0\text{e} + \bar{\nu}$.

* **Beta-plus ($\beta^+$) Decay**: Proton $\rightarrow$ Neutron + Positron + Neutrino. $Z$ decreases by 1, $A$ unchanged. Example: $_{11}^{22}\text{Na} \rightarrow _{10}^{22}\text{Ne} + _{+1}^0\text{e} + \nu$.

Beta particles are light, $\pm$e charge, moderate ionizing power, moderate penetrating power (stopped by aluminum). Deflected by E/B fields.

1
Gamma ($\gamma$) Decay — Emission of high-energy photons (electromagnetic waves) from an excited nucleus. No change in $Z$ or $A$. Example: $_Z^A\text{X}^* \rightarrow _Z^A\text{X} + \gamma$. Gamma rays are massless, uncharged, very low ionizing power, very high penetrating power (stopped by thick lead/concrete). NOT deflected by E/B fields.

Q-value: Energy released in decay, $Q = (\Delta m)c^2$. $\Delta m$ is the mass defect (mass of reactants - mass of products). $1\,\text{u} = 931.5\,\text{MeV/c}^2$. For $\beta^-$ decay, when using atomic masses, the electron masses effectively cancel out. For $\beta^+$ decay, two electron masses must be subtracted from the parent atomic mass to get the nuclear mass difference for Q-value calculation.

Vyyuha Quick Recall

Alpha Beta Gamma: Charge, Mass, Penetration, Ionization, Deflection.

Alpha: Charge +2, Mass 4, Penetration Low, Ionization High, Deflected (less).
Beta: Charge $\pm$1, Mass 1/1836, Penetration Medium, Ionization Medium, Deflected (more).
Gamma: Charge 0, Mass 0, Penetration High, Ionization Low, Deflected (None).