Radioactive decay series Knowledge-Based Study

Study notes

1. What is a Radioactive Decay Series?

   Many naturally occurring and synthetic radionuclides do not become stable after a single nuclear transformation. Instead, an unstable heavy nucleus undergoes a succession of spontaneous nuclear decays until a permanent, non-radioactive nuclide is reached. This sequential pathway of disintegrations is termed a radioactive decay series or radioactive cascade.

   In any decay chain, the initial unstable radionuclide is referred to as the parent nuclide. The intermediate radionuclides produced along the sequence are called daughter nuclides, and the final non-radioactive nuclide that terminates the cascade is known as the stable end product.

   Alpha and Beta Emissions

   Radioactive decay series among heavy elements proceed primarily via two types of radioactive decay:

   • Alpha ($\alpha$) Decay: The nucleus ejects an alpha particle, which is a helium-4 nucleus ($_{2}^{4}\text{He}$). This process decreases the mass number ($A$) by 4 and decreases the atomic number ($Z$) by 2:$$_{Z}^{A}\text{X} ightarrow \,_{Z-2}^{A-4}\text{Y} + \,_{2}^{4}\text{He}$$
   • Beta-Minus ($\beta^{-}$) Decay: A neutron in an unstable nucleus converts into a proton, emitting an electron ($_{-1}^{0}\beta$ or $_{-1}^{0}\text{e}$) and an antineutrino. This process leaves the mass number ($A$) unchanged while increasing the atomic number ($Z$) by 1:$$_{Z}^{A}\text{X} ightarrow \,_{Z+1}^{A}\text{Y} + \,_{-1}^{0}\beta$$

   Gamma ($\gamma$) emission frequently accompanies these transitions to release excess nuclear excitation energy, but it alters neither $A$ nor $Z$.

   Classification by Mass Number ($4n$ Rules)

   Because alpha emission changes the mass number by exactly 4 and beta decay causes no change in mass number, the mass number of any member in a series can be expressed in terms of the mathematical formula $4n + k$, where $n$ is an integer and $k \in \{0, 1, 2, 3\}$. This defines four distinct decay series:

   • Thorium Series ($4n$): Initiates with $_{90}^{232}\text{Th}$ and terminates at stable lead-208 ($_{82}^{208}\text{Pb}$).
   • Neptunium Series ($4n + 1$): Initiates with synthetic $_{93}^{237}\text{Np}$ and terminates at stable bismuth-209 ($_{83}^{209}\text{Bi}$) or thallium-205 ($_{81}^{205}\text{Tl}$). Unlike the other three, this series is not prominent in nature because the half-lives of all intermediate members are short relative to the age of the Earth.
   • Uranium (Radium) Series ($4n + 2$): Initiates with $_{92}^{238}\text{U}$ and terminates at stable lead-206 ($_{82}^{206}\text{Pb}$).
   • Actinium Series ($4n + 3$): Initiates with $_{92}^{235}\text{U}$ and terminates at stable lead-207 ($_{82}^{207}\text{Pb}$).

   Determining Numbers of $\alpha$ and $\beta$ Decays

   When given a parent nuclide $_{Z_{1}}^{A_{1}}\text{X}$ and a stable terminal nuclide $_{Z_{2}}^{A_{2}}\text{Y}$, the total number of alpha particles ($N_{\alpha}$) and beta particles ($N_{\beta}$) emitted can be determined systematically using conservation laws:

   1. Conservation of Mass Number: Since only alpha particles affect mass number, calculate $N_{\alpha}$ first:$$N_{\alpha} = \frac{A_{1} - A_{2}}{4}$$
   2. Conservation of Atomic Number (Charge): Account for the proton loss from alpha decays and proton gain from beta decays:$$Z_{2} = Z_{1} - 2(N_{\alpha}) + 1(N_{\beta})$$$$N_{\beta} = Z_{2} - Z_{1} + 2(N_{\alpha})$$

   Example: In the decay of $_{92}^{238}\text{U}$ to $_{82}^{206}\text{Pb}$, the mass decrease is $238 - 206 = 32$. Thus, $N_{\alpha} = \frac{32}{4} = 8$. The resulting expected atomic number before beta decay would be $92 - 2(8) = 76$. To reach $Z = 82$, the nucleus must undergo $N_{\beta} = 82 - 76 = 6$ beta decays.

   Secular vs. Transient Equilibrium

   In a closed geological or chemical system, parent and daughter nuclides eventually establish a dynamic condition termed radioactive equilibrium:

   • Secular Equilibrium: Occurs when the half-life of the parent nuclide is exceptionally longer than those of all subsequent daughter nuclides ($t_{1/2,\text{ parent}} \gg t_{1/2,\text{ daughter}}$). Under this condition, the decay rate (activity, $A = \lambda N$) of each intermediate daughter becomes virtually equal to the activity of the parent:$$A_{\text{parent}} = A_{\text{daughter 1}} = A_{\text{daughter 2}} = \dots$$
   • Transient Equilibrium: Occurs when the half-life of the parent is longer than that of the daughter, but not infinitely so (typically by a factor of 10 to 100). The daughter activity reaches a maximum and subsequently decreases at a rate governed by the parent's half-life.

   Secular equilibrium provides the theoretical foundation for radiometric dating methods, such as uranium-lead ($\text{U}$–$\text{Pb}$) geochronology.