Decay Chains, Branching, and Constant Production
Population balance is production minus removal. For constant production alpha, dN/dt = alpha - lambda N and N = (alpha/lambda)(1 - e^(-lambda t)) when N(0)=0. Successive decay uses coupled Bateman equations; branching decay uses total lambda equal to the sum of branch constants.
Why this shows up in the exam
Radioactive daughter growth · Reactor production of nuclides · Branching-ratio and decay-series problems
Learn the idea
When nuclides are produced as well as removed, write a separate rate equation for every population. A daughter can grow because its parent decays and shrink because it also decays. Likewise, a reactor can create a nuclide continuously while decay removes it, eventually reaching a steady population and activity.
🧠 Memory hook: Every population gets an inflow minus an outflow.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- dN/dt = alpha - lambda N — constant production with radioactive removal
- N(t) = (alpha/lambda)(1 - e^(-lambda t)) — population for N(0)=0 under constant production
- lambda_total = sum lambda_i — total decay constant for independent competing branches
How to approach it
- 1Draw arrows between nuclides
- 2Write one differential equation per species
- 3Check the t = 0 and t -> infinity limits
Common slip-ups that cost marks
- •Applying simple decay when production continues
- •Adding half-lives instead of branch decay constants
- •Ignoring daughter decay in a chain
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
In hydrogen, an electron transitions from n = 2 to n = 1. Using E_n = -13.6/n^2 eV, find the emitted photon energy.
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