Law of successive disintegration
The Law of successive disintegration describes the decay of radioactive nuclei that transform into a daughter product, which itself is radioactive and decays further. This process continues until a stable nucleus is formed.
Consider a parent nucleus N1 decaying into a daughter nucleus N2, which then decays into a stable nucleus N3:
N1 → N2 → N3 (stable)
The rate of change of the number of daughter nuclei (N2) at any time t is given by:
dN2/dt = λ1N1 - λ2N2
Where λ1 and λ2 are the decay constants of the parent and daughter nuclei, respectively.
The solution for N2(t) is complex but describes how the amount of the daughter product initially increases, reaches a maximum, and then decreases as the parent nuclei deplete.
Secular and transient equilibrium
In a successive disintegration chain, equilibrium refers to a state where the ratio of the number of parent and daughter nuclei, or their activities, becomes constant.
Secular Equilibrium
Secular equilibrium occurs when the half-life of the parent nucleus (T1/2,1) is much longer than the half-life of the daughter nucleus (T1/2,2), meaning λ1 « λ2.
In this state, after a sufficiently long time, the activity of the daughter nucleus becomes equal to the activity of the parent nucleus:
A1 = A2 or λ1N1 = λ2N2
This implies N1/N2 = λ2/λ1 (a constant ratio).
The amount of the daughter product effectively decays with the half-life of the parent, as it is continuously replenished by the long-lived parent.
Transient Equilibrium
Transient equilibrium occurs when the half-life of the parent nucleus (T1/2,1) is longer than the half-life of the daughter nucleus (T1/2,2), but not significantly longer (λ1 < λ2).
In this state, after an initial period, the ratio of the activities of the parent and daughter nuclei becomes constant:
A1 / A2 = constant ≠ 1
More specifically, the ratio is A2/A1 = λ2 / (λ2 - λ1).
Both the parent and daughter activities decay with the effective decay constant of the parent (λ1).
The daughter activity tracks the parent activity after reaching equilibrium, but the daughter activity is always greater than the parent activity.
Alpha ray spectrum
Alpha decay involves the emission of an alpha particle (a helium nucleus, 4He2). The alpha ray spectrum refers to the distribution of kinetic energies of these emitted alpha particles.
Alpha decay is a two-body disintegration process (parent nucleus → daughter nucleus + alpha particle).
Due to the conservation of energy and momentum in a two-body decay, the emitted alpha particles have discrete kinetic energies. This means alpha particles from a given decay transition will all have the same energy, or a few specific discrete energies if the daughter nucleus can be left in different excited states.
The energy of the alpha particle is determined by the mass difference (Q-value) between the parent nucleus and the combined masses of the daughter nucleus and alpha particle. Any excess energy is shared between the alpha particle and the recoiling daughter nucleus.
Geiger-Nuttal law
The Geiger-Nuttal law describes an empirical relationship between the decay constant (λ) of an alpha-emitting nucleus and the range (R) of the alpha particles it emits in air.
The law states:
log λ = A + B log R
Where:
- λ is the decay constant of the alpha emitter.
- R is the range of the alpha particles (related to their energy).
- A and B are constants, with B being approximately the same for various decay series.
This law implies that alpha emitters with shorter half-lives (larger λ) emit alpha particles with higher energies (larger R). It was historically significant in understanding the quantum mechanical tunneling effect in alpha decay.
Beta ray spectrum and Pauli’s neutrino hypothesis
Beta ray spectrum
Beta decay involves the transformation of a neutron into a proton (β- decay) or a proton into a neutron (β+ decay), accompanied by the emission of an electron or positron, respectively.
The beta ray spectrum refers to the distribution of kinetic energies of the emitted beta particles (electrons or positrons).
Unlike alpha decay, the beta ray spectrum is continuous, ranging from zero up to a maximum kinetic energy (Emax) specific to the decay. This observation initially posed a challenge to the laws of conservation of energy and momentum.
If beta decay were a two-body process (nucleus → daughter nucleus + beta particle), the beta particles should have discrete energies, similar to alpha particles. The continuous spectrum suggested that some energy, momentum, and angular momentum were unaccounted for.
Pauli’s neutrino hypothesis
To resolve the