Spontaneous and Random Nuclear Decay: MDCAT Physics notes
Spontaneous and Random Nuclear Decay MDCAT notes: alpha, beta and gamma emission, decay equations, ionizing power, deflection, decay constant and activity.
Radioactivity
Unstable nuclei emit radiation spontaneously (not affected by temperature, pressure or chemical state) and randomly (we cannot predict which nucleus will decay next, only the probability). Natural radioactive elements include uranium, radium and thorium. Artificial radioisotopes are made by bombarding stable nuclei with high-energy particles (e.g. neutrons).
Three radiations are emitted: alpha, beta and gamma. There is no "delta" radiation.
Alpha, beta and gamma
| Property | Alpha ($\alpha$) | Beta ($\beta^-$) | Gamma ($\gamma$) |
|---|---|---|---|
| Nature | Helium nucleus $^4_2$He | Fast electron | High-energy photon |
| Charge | $+2e$ | $-e$ | 0 |
| Ionizing power | Highest | Medium | Lowest |
| Penetrating power | Lowest (paper stops it) | Medium (few mm aluminium) | Highest (thick lead) |
| Deflection in fields | Small (heavy) | Large (light) | None |
Beta particles are deflected more than alpha particles in a magnetic field because their mass is thousands of times smaller, even though their charge is only half.
Decay equations
- Alpha decay: $^{A}_{Z}X\to{}^{A-4}_{Z-2}Y+{}^4_2\mathrm{He}$. Mass number falls by 4, atomic number by 2.
- Beta decay: a neutron changes into a proton and an electron: $^{A}_{Z}X\to{}^{A}_{Z+1}Y+{}^{\;\,0}_{-1}e$. Mass number unchanged, atomic number rises by 1.
- Gamma emission: $^{A}_{Z}X^*\to{}^{A}_{Z}X+\gamma$. Neither $A$ nor $Z$ changes; the nucleus simply loses energy, going from an excited to a lower state. Conversely, a nucleus that absorbs a gamma photon is excited.
Worked examples: $^{234}_{90}$Th $\to$ $^{234}_{91}$Pa $+\beta$; $^{14}_{6}$C $\to$ $^{14}_{7}$N $+\beta$; $^{238}_{92}$U $\to$ $^{234}_{90}$Th $+\alpha$.
Decay law, decay constant and activity
The number of nuclei decaying per second (the activity) is proportional to the number present:
$$A=\frac{\Delta N}{\Delta t}=-\lambda N,\qquad N=N_0e^{-\lambda t}$$$\lambda$ is the decay constant: the probability that a nucleus decays per unit time. Its unit is s$^{-1}$. A smaller $\lambda$ means a more stable nuclide, so stability is judged by the decay constant. Because $N$ falls exponentially, the activity also decreases exponentially with time. The SI unit of activity is the becquerel (1 Bq = 1 decay per second).
Example: $N=6.7\times10^{21}$, $\lambda=8.3\times10^{-10}$ s$^{-1}$ gives $A=\lambda N\approx5.6\times10^{12}$ Bq.
Key formulas
- $A=\lambda N$; $N=N_0e^{-\lambda t}$
- $\lambda=\dfrac{0.693}{T_{1/2}}$
- Alpha: $A-4,\ Z-2$; Beta: $A,\ Z+1$; Gamma: no change
Common MDCAT traps
- Beta decay keeps the mass number the same; do not subtract 4.
- Alpha has the greatest ionizing power but the least penetration.
- Gamma emission does not create a new element.
- Unit of $\lambda$ is s$^{-1}$, not s.
- Convert hours to seconds before computing $\lambda$ in s$^{-1}$.
Quick revision
- Decay is spontaneous and random.
- Alpha = helium nucleus; beta = electron; gamma = photon.
- Beta is bent more than alpha in a magnetic field.
- Activity decreases exponentially.
- Artificial radioisotopes: bombard stable nuclei with fast particles.