Decay Chain Calculator
Free Decay chain Calculator for nuclear physics. Enter variables to compute results with formulas and detailed steps.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Decay Chain Calculator
Calculator
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Formula: N2(t) = N0 * (lambda1/(lambda2-lambda1)) * (exp(-lambda1*t) - exp(-lambda2*t))
Worked example — Ra-226: ~1.00e20 atoms | Rn-222: ~6.51e14 atoms | Secular equilibrium nearly reached
Formula
N2(t) = N0 * (lambda1/(lambda2-lambda1)) * (exp(-lambda1*t) - exp(-lambda2*t))
Where N2 is the daughter population at time t, N0 is the initial parent atoms, lambda1 and lambda2 are the parent and daughter decay constants (ln2/half-life), and t is the elapsed time. This is the Bateman equation for a two-member chain.
Worked Examples
Example 1: Radium-226 to Radon-222 Chain
Problem:Starting with 1e20 atoms of Ra-226 (t1/2 = 1600 years), calculate the Rn-222 (t1/2 = 3.82 days) population after 30 days.
Solution:lambda1 = ln2 / 1600 yr = 4.332e-4 yr^-1 lambda2 = ln2 / (3.82/365.25) yr = 66.25 yr^-1 t = 30/365.25 = 0.0821 yr N_Ra = 1e20 * exp(-4.332e-4 * 0.0821) = 9.9996e19 (essentially unchanged) N_Rn = 1e20 * (4.332e-4 / (66.25 - 4.332e-4)) * (exp(-4.332e-4*0.0821) - exp(-66.25*0.0821)) = 1e20 * 6.539e-6 * (0.9999 - 0.00432) = 6.51e14 atoms
Result:Ra-226: ~1.00e20 atoms | Rn-222: ~6.51e14 atoms | Secular equilibrium nearly reached
Example 2: Mo-99 to Tc-99m Medical Generator
Problem:A Mo-99 (t1/2 = 66 hours) generator starts with 1e18 atoms. Find Tc-99m (t1/2 = 6.01 hours) population after 24 hours.
Solution:lambda1 = ln2 / 66h = 0.01050 h^-1 lambda2 = ln2 / 6.01h = 0.11534 h^-1 t = 24 h N_Mo = 1e18 * exp(-0.01050 * 24) = 7.77e17 N_Tc = 1e18 * (0.01050/(0.11534-0.01050)) * (exp(-0.01050*24) - exp(-0.11534*24)) = 1e18 * 0.1001 * (0.7772 - 0.0633) = 7.15e16 atoms
Result:Mo-99: 7.77e17 atoms | Tc-99m: 7.15e16 atoms | Transient equilibrium
Frequently Asked Questions
What is a radioactive decay chain and how does it work?
A radioactive decay chain (also called a decay series) is a sequence of successive radioactive decays where an unstable parent nucleus transforms into a daughter nucleus, which may itself be radioactive and decay further into a granddaughter, and so on, until a stable isotope is reached. Each step in the chain involves the emission of alpha particles, beta particles, or gamma rays, with each decay governed by its own characteristic half-life. The most well-known natural decay chains start from uranium-238, uranium-235, and thorium-232, each passing through about 10 to 15 intermediate radioactive isotopes before reaching stable lead isotopes. Understanding decay chains is essential for nuclear waste management, radiometric dating, and radiation protection.
What is the Bateman equation and how does it describe decay chain kinetics?
The Bateman equations are a set of coupled first-order differential equations that describe the number of atoms of each member in a radioactive decay chain as a function of time. For a simple two-member chain, the daughter population equals N0 times lambda1 divided by (lambda2 minus lambda1) times the difference of two exponentials. For longer chains, the general Bateman solution involves sums of exponential terms with coefficients determined by all the decay constants in the chain. These equations assume that each decay produces exactly one daughter atom and that branching ratios are unity. The Bateman equations are the mathematical foundation for Decay Chain Calculator and enable prediction of the complete time evolution of every isotope in the chain.
What is secular equilibrium in a decay chain and when does it occur?
Secular equilibrium occurs when the half-life of the parent isotope is much longer (at least 100 times) than the half-life of any daughter in the chain. Under these conditions, the parent activity remains essentially constant over many daughter half-lives, and after a transient buildup period, each daughter reaches a steady-state population where its activity equals the parent activity. In secular equilibrium, the number of daughter atoms equals N_parent times lambda_parent divided by lambda_daughter. A classic example is radium-226 (half-life 1,600 years) in secular equilibrium with radon-222 (half-life 3.82 days). This principle is widely used in radiometric dating and environmental radiation monitoring.
What is transient equilibrium and how does it differ from secular equilibrium?
Transient equilibrium occurs when the parent half-life is longer than the daughter half-life but not by an overwhelmingly large factor (typically less than 100 times longer). In this case, after an initial buildup, the daughter activity actually exceeds the parent activity and then both decay at the rate determined by the parent half-life. The daughter-to-parent activity ratio stabilizes at lambda_daughter divided by (lambda_daughter minus lambda_parent), which is always greater than unity. An example is barium-140 (12.75 days) decaying to lanthanum-140 (1.68 days). Unlike secular equilibrium where parent activity is effectively constant, in transient equilibrium both parent and daughter activities decrease measurably over the observation period.
How do you calculate the time of maximum daughter activity in a decay chain?
The time at which the daughter population reaches its maximum value can be calculated analytically from the Bateman equations. Taking the derivative of the daughter population equation with respect to time and setting it equal to zero gives t_max equals ln(lambda2/lambda1) divided by (lambda2 minus lambda1). At this time, the daughter decay rate exactly equals the daughter production rate from parent decay. Before t_max, the daughter is building up faster than it decays; after t_max, it decays faster than it is produced. This maximum time depends only on the two decay constants, not on the initial parent population. Knowledge of t_max is important for medical isotope production and nuclear waste management planning.
What are the four natural radioactive decay series found in nature?
There are four natural decay series, named after their longest-lived members. The uranium series starts with U-238 (half-life 4.47 billion years) and ends at stable Pb-206 through 14 decay steps. The actinium series starts with U-235 (704 million years) and ends at Pb-207 through 11 steps. The thorium series starts with Th-232 (14 billion years) and ends at Pb-208 through 10 steps. The fourth series, the neptunium series starting with Np-237 (2.14 million years) ending at Bi-209, has no surviving natural members because Np-237 is too short-lived relative to the age of the Earth. These decay series are fundamental to understanding natural radioactivity, geological dating, and environmental radiation.
How does branching decay affect the calculations in a decay chain?
Branching decay occurs when a radioactive isotope can decay through more than one mode, such as both alpha and beta decay, producing different daughter products. When branching occurs, the decay constant for each branch equals the total decay constant multiplied by the branching ratio (the fraction decaying through that branch). This means the total half-life remains unchanged, but the effective production rate of each daughter is reduced by the branching fraction. In complex decay chains like the uranium series, several isotopes exhibit branching, creating parallel paths that eventually converge. Decay Chain Calculator assumes a single decay path for simplicity, but the input decay constants can be adjusted to represent the effective constants for a specific branch.
What practical applications rely on decay chain calculations?
Decay chain calculations have numerous practical applications across science and industry. In nuclear medicine, production of technetium-99m from molybdenum-99 generators requires precise knowledge of the Mo-99 to Tc-99m decay chain dynamics to optimize elution schedules. In geological dating, the uranium-lead method exploits the complete U-238 decay chain to date rocks billions of years old. Nuclear power plants must track the buildup of fission product decay chains to predict decay heat after reactor shutdown. Environmental monitoring uses radon-222 measurements (a member of the U-238 chain) to assess radioactive contamination. Nuclear forensics analyzes isotope ratios along decay chains to determine the age and origin of nuclear materials.
How does the initial condition affect decay chain evolution?
The initial conditions profoundly affect how a decay chain evolves over time. If only the parent isotope is present initially (a freshly purified sample), all daughters must build up from zero, and the system takes several daughter half-lives to approach equilibrium. If the system starts in secular equilibrium (as in an undisturbed ore body), all activities are equal and remain so as long as the parent activity does not change significantly. If daughters are selectively removed (as in a molybdenum-technetium generator), the daughter repopulation follows the Bateman equations with modified initial conditions. Understanding these different starting scenarios is crucial for correctly interpreting measurements and planning experiments in nuclear science.
What units are used to express radioactive activity and how do they relate?
Radioactive activity is expressed in two main unit systems. The SI unit is the becquerel (Bq), defined as one disintegration per second. The older unit is the curie (Ci), originally defined as the activity of one gram of radium-226, which equals 3.7 times 10 to the 10th disintegrations per second. Common prefixes are used: megabecquerel (MBq), gigabecquerel (GBq), millicurie (mCi), and microcurie (uCi). Activity is related to the number of atoms by A equals lambda times N, where lambda is the decay constant (ln2 divided by the half-life). This relationship is central to decay chain calculations because it connects the measurable quantity (activity) to the fundamental atomic populations governed by the Bateman equations.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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