Unit 3: Decay Kinetics of Unstable Nuclei: Half-Life, Successive Decay & Bateman Equilibria

Radioactive decay is a first-order unimolecular kinetic transformation governed by the fundamental quantum decay probability $\lambda$. Consider an ensemble of $N(t)$ identical, non-interacting unstable radioactive nuclei. In any infinitesimal time interval $dt$, the probability $dp$ that an individual nucleus disintegrates is:

where $\lambda$ is the decay constant ($\text{s}^{-1}$), an intrinsic nuclear property independent of chemical bonding, physical state, temperature, or pressure.

The Differential Rate Law

The expected number of nuclear disintegrations $dN$ occurring in the sample within time interval $dt$ is:

The negative sign signifies that the population of parent nuclei decreases monotonically over time.

Integral Form of the Decay Law

Separating variables and integrating from initial boundary condition $N(0) = N_0$ at $t = 0$:

The number of disintegrated daughter nuclei $N_D(t)$ formed (assuming a stable daughter and $N_D(0) = 0$) is:

``` Number of Nuclei N₀ ▲ │\ Parent N(t) = N₀ e^(-λt) Daughter N_D(t) = N₀ (1 - e^(-λt)) │ \ / N₀/2│--\------------------------------/-- (Half-Life T₁/₂) │ \ / │ \ / │ \________________________/ └──────┴────────────────────────► Time t T₁/₂ ```

Radioactive Activity Formulation

The absolute rate of disintegration is defined as activity $A(t)$:

Activity decays with the identical exponential rate constant $\lambda$ as the number of atoms.

Logarithmic Linearization

Taking the natural logarithm of both sides:

Plotting $\ln A(t)$ versus time $t$ yields a straight line with:

• Vertical intercept: $\ln A_0$
• Slope: $-\lambda$

In base-10 logarithms:

The kinetics of radionuclide disappearance can be characterized using three mathematically related parameters:

1. Decay Constant $\lambda$ (dimensions: $[\text{T}]^{-1}$, typically $\text{s}^{-1}$, $\text{h}^{-1}$, or $\text{yr}^{-1}$)

2. Half-Life $T_{1/2}$ (dimensions: $[\text{T}]$)

3. Mean Life $\tau$ (dimensions: $[\text{T}]$)

Detailed Derivation of Half-Life ($T_{1/2}$)

The half-life is the duration required for the active population (or activity) to decline to exactly half its initial value:

Taking the natural logarithm:

Using $\ln 2 \approx 0.69314718056$:

The surviving fraction after an arbitrary number of half-lives $n = t / T_{1/2}$ is:

| Half-Lives Elapsed ($n$) | Surviving Fraction ($N/N_0$) | Percentage Surviving | Decayed Fraction ($1 - N/N_0$) | | :--- | :--- | :--- | :--- | | 0 | $1$ | $100.00\%$ | $0.00\%$ | | 1 | $1/2$ | $50.00\%$ | $50.00\%$ | | 2 | $1/4$ | $25.00\%$ | $75.00\%$ | | 3 | $1/8$ | $12.50\%$ | $87.50\%$ | | 5 | $1/32$ | $3.125\%$ | $96.875\%$ | | 7 | $1/128$ | $0.781\%$ | $99.219\%$ | | 10 | $1/1024$ | $0.0977\%$ | $99.9023\%$ |

After 10 half-lives, less than $0.1\%$ ($1/1024$) of the initial radioactivity remains. In radioprotection and waste management, $10 T_{1/2}$ is the standard operational threshold for total decay of short-lived isotopes.

Detailed Derivation of Mean Life ($\tau$)

The mean lifetime $\tau$ is the arithmetic average lifespan of all nuclei in an ensemble. The probability distribution function $P(t) dt$ representing the probability that a nucleus decays between time $t$ and $t + dt$ is:

Note that this distribution is properly normalized:

The expectation value $\langle t \rangle = \tau$ is:

Using the standard definite integral $\int_0^\infty t^n e^{-\lambda t} dt = \frac{n!}{\lambda^{n+1}}$ with $n = 1$:

Converting between mean life and half-life:

Total Number of Decays Over All Time

The total cumulative disintegrations $N_{\text{tot}}$ that occur between $t = 0$ and $t = \infty$ is:

The total number of atoms initially present is equal to the initial activity multiplied by the mean life:

Because radioactive decay is a completely random stochastic process, repeated counts recorded over identical time intervals fluctuate. Understanding these statistical fluctuations is essential for determining experimental uncertainties, detection limits, and confidence intervals in radiation measurements.

The Binomial Distribution Formulation

Consider a sample containing $N_0$ radioactive atoms. Over an observation interval $\Delta t$, the probability that any specific nucleus decays is:

The probability that it survives is $q = 1 - p = e^{-\lambda \Delta t}$. The probability $P(n)$ of observing exactly $n$ disintegrations out of $N_0$ atoms is governed by the Binomial distribution:

The Poisson Approximation

In real radiochemical counting experiments:

1. The total number of atoms is extraordinarily large ($N_0 \sim 10^{12} - 10^{20}$).
2. The decay probability over typical counting intervals is tiny ($p \ll 1$).
3. The expected mean count $\mu = N_0 p$ remains finite.

Taking the mathematical limit $N_0 \to \infty$ and $p \to 0$ while holding $\mu = N_0 p$ constant, the Binomial distribution converges rigorously to the Poisson distribution:

where $\mu$ is the true mean count.

``` P(n) Probability ▲ │ Poisson Distribution (μ = 10) │ ___ │ / \ │ / \ │ / \ │ / \ │_______/ \________ └───────┴──────┴───────┴───────► Recorded Counts n 0 μ=10 20 ```

Variance and Standard Deviation of Poisson Counting

For a Poisson distribution:

• Expectation Value: $\langle n \rangle = \mu$
• Variance: $\sigma^2 = \mu$
• Standard Deviation: $\sigma = \sqrt{\mu}$

In an experimental determination, the true mean $\mu$ is unknown and is estimated by the observed single count $N$:

The result of any radiation measurement is expressed with its one-sigma standard uncertainty ($68.3\%$ confidence interval) as:

Relative Standard Deviation (RSD) and Counting Precision

The relative standard uncertainty (coefficient of variation) is:

| Observed Counts ($N$) | Standard Deviation ($\sigma = \sqrt{N}$) | Relative Standard Deviation (% RSD) | Metrological Level | | :--- | :--- | :--- | :--- | | 100 | $\pm 10$ | $10.0\%$ | Screening survey | | 1,000 | $\pm 31.6$ | $3.16\%$ | Routine monitoring | | 10,000 | $\pm 100$ | $1.00\%$ | Analytical standard | | 100,000 | $\pm 316.2$ | $0.316\%$ | High-precision metrology | | 1,000,000 | $\pm 1000$ | $0.100\%$ | Absolute activity calibration |

To improve counting precision by a factor of 10, the counting duration (or accumulated count total) must be increased by a factor of $10^2 = 100$.

Net Count Rate and Background Subtraction Uncertainty

Every radiation detector records ambient background radiation counts $N_b$ during background counting time $t_b$, in addition to gross sample counts $N_g$ accumulated over sample counting time $t_g$. The net count rate $R_n$ is:

By propagation of independent random errors:

Many unstable nuclei can decay through multiple competing pathways. For example, potassium-40 ($^{40}\text{K}$) undergoes both negative beta decay ($\beta^-$) into calcium-40 and electron capture (EC) into argon-40: ``` ┌───► ⁴⁰Ca + e⁻ + ν̄_e (β⁻ decay: 89.28%) ⁴⁰K ───────┤ └───► ⁴⁰Ar + ν_e + γ (EC decay: 10.72%) ``` Similarly, copper-64 ($^{64}\text{Cu}$) decays via three competing modes: $\beta^-$ ($39.0\%$), $\beta^+$ ($17.6\%$), and electron capture ($43.4\%$).

Partial Decay Constants

Let an unstable nuclide decay through $m$ independent competing branches, each characterized by its own partial decay constant $\lambda_i$:

The total rate of disappearance of the parent nucleus is the sum of the disappearance rates across all individual decay modes:

Defining the total decay constant $\lambda_{\text{tot}}$:

The population of the parent decays exponentially according to the total decay constant:

Effective Total Half-Life

Because $T_{1/2} = \ln 2 / \lambda$, the total half-life $T_{1/2,\text{tot}}$ satisfies:

Dividing through by $\ln 2$:

Notice that this is formally identical to the formula for resistors connected in parallel. For two competing branches:

The total half-life is always strictly shorter than the shortest partial half-life.

Branching Ratio (Branching Fraction)

The branching ratio $BR_i$ (or branching fraction $f_i$) of the $i$-th decay channel is defined as the fraction of all decays that proceed through that channel:

By definition, the sum of all branching ratios equals unity:

Daughter Growth Kinetics in Branching Decay

The production rate of daughter nucleus $D_i$ formed via the $i$-th branch is:

Assuming $N_{D,i}(0) = 0$ and that the daughter is stable:

At any time $t$, the ratio of the numbers of atoms of two alternative stable daughters produced is:

This constant ratio forms the physical foundation of K-Ar (potassium-argon) geological dating, where the accumulated $^{40}\text{Ar}$ relative to $^{40}\text{Ca}$ determines rock crystallization age over billions of years.

In many radioactive decay chains, an unstable parent radionuclide decays into a daughter which is itself radioactive, giving rise to a cascade of successive decays:

The System of Coupled Differential Rate Equations

The time evolution of the population of each nuclide in the chain is governed by a balance between production from its immediate parent and disappearance via its own decay:

Derivation for a Two-Step Chain ($N_1 \to N_2 \to N_3$)

Let the initial conditions at $t = 0$ be $N_1(0) = N_1^0$ and $N_2(0) = 0, N_3(0) = 0$.

1. Parent Solution:

2. Daughter Solution:

Substitute $N_1(t)$ into the daughter differential equation:

This is a first-order linear ordinary differential equation. Multiplying by the integrating factor $e^{\lambda_2 t}$:

Integrating both sides from $0$ to $t$:

Assuming $\lambda_1 \ne \lambda_2$:

Multiplying by $e^{-\lambda_2 t}$:

The daughter activity $A_2(t) = \lambda_2 N_2(t)$ is:

General Harry Bateman Solution (1910)

For an arbitrary linear chain of $n$ nuclides with initial condition $N_1(0) = N_1^0$ and $N_2(0) = N_3(0) = \dots = N_n(0) = 0$, Harry Bateman proved via Laplace transforms that the population of the $n$-th nuclide is:

Expanded explicitly:

where each coefficient $C_j$ is given by:

This elegant theorem solves the isotopic inventory of any nuclear decay chain, from uranium decay series to complex reactor core transmutations.

In a successive decay chain $N_1 \xrightarrow{\lambda_1} N_2 \xrightarrow{\lambda_2} N_3$, the temporal relationship between parent and daughter activities depends fundamentally on the relative magnitudes of the decay constants $\lambda_1$ and $\lambda_2$. Three distinct kinetic regimes emerge:

1. Secular Equilibrium ($\lambda_1 \ll \lambda_2$, i.e., $T_{1/2,1} \gg T_{1/2,2}$)

When the parent half-life is thousands or millions of times longer than the daughter half-life, the parent activity remains essentially constant over experimental timescales:

Substituting into the daughter activity equation:

``` Activity A(t) A₁ ▲──────────────────────────────────── Parent A₁(t) = A₁⁰ (Constant) │ /──────────────────────── Daughter A₂(t) in Secular Eq: A₂ = A₁ │ / │ / │ / │_______/ └───────┴────────────────────────────► Time t ~ 7 T₁/₂,₂ ```

As $t \gg T_{1/2,2}$ ($t \ge 7 T_{1/2,2}$), $e^{-\lambda_2 t} \to 0$:

In secular equilibrium, the daughter activity becomes exactly equal to the parent activity, and the abundance ratio of parent to daughter is directly proportional to their half-lives! Classic natural examples:

• $^{226}\text{Ra}$ ($T_{1/2} = 1600\text{ yr}$) $\xrightarrow{\alpha} \, ^{222}\text{Rn}$ ($T_{1/2} = 3.82\text{ days}$)
• $^{238}\text{U}$ ($T_{1/2} = 4.47 \times 10^9\text{ yr}$) $\xrightarrow{\alpha} \, ^{234}\text{Th}$ ($T_{1/2} = 24.1\text{ days}$)

2. Transient Equilibrium ($\lambda_1 < \lambda_2$, but parent decays noticeably, $T_{1/2,1} \approx 10 T_{1/2,2}$)

When the parent half-life is moderately longer than the daughter, the daughter activity rises, reaches a maximum, and then decays with the apparent half-life of the parent. For large $t$ such that $e^{-\lambda_2 t} \ll e^{-\lambda_1 t}$:

Notice that in transient equilibrium, the daughter activity exceeds the parent activity by the constant factor:

Derivation of Time to Maximum Daughter Activity ($t_{\max}$)

To determine when the daughter activity reaches its peak, differentiate $N_2(t)$ with respect to $t$ and set the derivative to zero:

Taking natural logarithms:

Yielding the Time-to-Peak Equation:

At exactly $t = t_{\max}$, substituting $\lambda_1 e^{-\lambda_1 t} = \lambda_2 e^{-\lambda_2 t}$ into $A_2(t)$:

At the peak, the parent and daughter activity curves cross each other! The premier radiopharmaceutical example is the $^{99}\text{Mo} \to ^{99m}\text{Tc}$ generator:

• $^{99}\text{Mo}$: $T_{1/2} = 66.0\text{ hours}$
• $^{99m}\text{Tc}$: $T_{1/2} = 6.01\text{ hours}$
• $t_{\max} \approx 22.9\text{ hours}$ post-elution, determining the optimal clinical elution schedule.

The third kinetic regime occurs when the parent decays faster than the daughter.

No Equilibrium ($\lambda_1 > \lambda_2$, i.e., $T_{1/2,1} < T_{1/2,2}$)

When the parent has a shorter half-life than the daughter, no equilibrium is ever established. As time progresses:

The exponential term of the parent vanishes rapidly, leaving only the daughter's decay term:

All initial parent nuclei disintegrate into daughter nuclei, which then decay away at their own leisurely characteristic rate $\lambda_2$. Examples:

• $^{140}\text{Ba}$ ($T_{1/2} = 12.75\text{ days}$) $\xrightarrow{\beta^-} \, ^{140}\text{La}$ ($T_{1/2} = 1.678\text{ days}$)
• $^{218}\text{Po}$ ($T_{1/2} = 3.10\text{ min}$) $\xrightarrow{\alpha} \, ^{214}\text{Pb}$ ($T_{1/2} = 26.8\text{ min}$)

``` Activity A(t) ▲ │ Parent A₁(t) (Rapid Fall) │ \ │ \ Daughter Growth and Slow Decay A₂(t) │ \ /──────────────\ │ \____/ \_________________ └────────────────────────────────────────────────► Time t ```

Comprehensive Comparison of Equilibrium Regimes

| Equilibrium Type | Half-Life Relationship | Ratio of Decay Constants | Activity Ratio at Late Times | Cross-Over at $t_{\max}$? | Representative Real-World Example | | :--- | :--- | :--- | :--- | :--- | :--- | | Secular | $T_{1/2,1} \gg T_{1/2,2}$ ($>10^4\times$) | $\lambda_1 \ll \lambda_2$ | $\frac{A_2(t)}{A_1(t)} \to 1.000$ | No ($A_2$ approaches $A_1$ asymptotically) | $^{226}\text{Ra} \xrightarrow{\alpha} ^{222}\text{Rn}$ | | Transient | $T_{1/2,1} > T_{1/2,2}$ ($\sim 10\times$) | $\lambda_1 < \lambda_2$ | $\frac{A_2(t)}{A_1(t)} \to \frac{\lambda_2}{\lambda_2 - \lambda_1} > 1$ | Yes ($A_2 = A_1$ at $t = t_{\max}$) | $^{99}\text{Mo} \xrightarrow{\beta^-} ^{99m}\text{Tc}$ | | No Equilibrium | $T_{1/2,1} < T_{1/2,2}$ | $\lambda_1 > \lambda_2$ | $\frac{A_2(t)}{A_1(t)} \to \infty$ ($A_1 \to 0$) | Yes ($A_2$ overtakes $A_1$, parent vanishes) | $^{218}\text{Po} \xrightarrow{\alpha} ^{214}\text{Pb}$ |

Radionuclide Inventory in Nuclear Reactors and Spent Fuel

Understanding these three regimes is the core principle of nuclear fuel inventory codes (e.g., ORIGEN). In freshly discharged spent nuclear fuel:

1. Short-lived fission products ($T_{1/2} \sim \text{seconds to days}$) undergo rapid non-equilibrium cascades into longer-lived daughters.
2. In intermediate storage ponds ($1 - 5\text{ years}$), isotopes like $^{137}\text{Cs} \to ^{137m}\text{Ba}$ ($T_{1/2} = 2.55\text{ min}$) and $^{90}\text{Sr} \to ^{90}\text{Y}$ ($T_{1/2} = 64.1\text{ hours}$) exist in strict secular equilibrium, where the daughter activity matches the 30-year parent activity.
3. In deep geological repositories ($>10^4\text{ years}$), actinide decay chains ($^{241}\text{Pu} \to ^{241}\text{Am} \to ^{237}\text{Np}$) govern long-term radiotoxicity.

While the classical Bateman equations provide elegant solutions for simple linear chains ($N_1 \to N_2 \to N_3$), real-world nuclear fuel cycles and reactor transmutation networks involve branched, cyclic, and cross-feeding pathways (e.g., $(n, \gamma)$ activation competing with simultaneous beta and alpha decays). For such complex networks, modern nuclear inventory codes (e.g., ORIGEN, CINDER) formulate the problem via matrix differential equations.

Matrix Formulation of Transmutation Chains

Let the column vector $\vec{N}(t) = [N_1(t), N_2(t), \dots, N_k(t)]^T$ denote the time-dependent population inventory of $k$ distinct nuclides. The governing system of coupled first-order differential equations is:

where $\mathbf{A}$ is the $k \times k$ transition rate matrix (or burnup matrix):

• Diagonal elements represent the total destruction rate (decay constant plus neutron absorption):

• Off-diagonal elements represent the production rate of nuclide $i$ from precursor $j$:

Formal Solution via the Matrix Exponential

The exact analytical solution of this autonomous linear system with initial inventory $\vec{N}(0)$ is:

where the matrix exponential is defined by the convergent Taylor series:

Numerical Solution via the Chebyshev Rational Approximation Method (CRAM)

Direct Taylor summation of $\exp(\mathbf{A} t)$ suffers from numerical instability when the spectrum of eigenvalues $\lambda_i$ spans dozens of orders of magnitude (from fractions of a second to billions of years). Modern reactor burnup solvers utilize the Chebyshev Rational Approximation Method (CRAM), approximating $\exp(\mathbf{A} t)$ by a rational function $\hat{r}_k(z) = p_k(z) / q_k(z)$ on the negative real axis:

where $\alpha_j$ and $\theta_j$ are pre-computed complex poles and residues. CRAM evaluates the complete isotopic inventory of thousands of isotopes across an entire nuclear reactor core in milliseconds with precision exceeding $10^{-14}$!

Mathematical Taxonomy of Radioactive Decay Kinetics & Equilibrium Regimes

``` SUCCESSIVE DECAY KINETICS N₁ ──(λ₁)──► N₂ ──(λ₂)──► N₃

REGIME CONDITION ACTIVITY RATIO (Late Times) ───────────── ─────────────────────── ─────────────────────────────── Secular Eq λ₁ ≪ λ₂ (T₁/₂ ≫ 10⁴×) A₂(t) / A₁(t) ──► 1.000 Transient Eq λ₁ < λ₂ (T₁/₂ ≈ 10×) A₂(t) / A₁(t) ──► λ₂ / (λ₂ - λ₁) > 1 No Equilibrium λ₁ > λ₂ (T₁/₂ < T₁/₂,₂) A₂(t) / A₁(t) ──► ∞ (A₁ vanishes) ```

In the secular regime, the abundance of all intermediate daughters in an undisturbed rock is directly proportional to their half-lives:

This simple relation allows geochemists to determine the age of continents, meteorites, and lunar samples.