The Nuclear Reactor — Question 5

Question 5

Delayed-neutron precursors continue emitting after production stops

One hypothetical precursor group obeys $dC/dt=S_C-\lambda C$. Before shutdown it is in equilibrium at constant $S_C=10^8\,\mathrm{s^{-1}}$; its half-life is 10.0 s. At t=0 production stops instantly. Each precursor decay emits one neutron.

1. Find the equilibrium precursor inventory and the subsequent neutron-emission rate.

2. Calculate the number of delayed neutrons emitted in the first 20.0 s.

Question 5 - Solution

Delayed-neutron precursors continue emitting after production stops

(a) Method and calculation

$$\dot C=0\Rightarrow\boxed{C_0=S_C/\lambda=\frac{10^8(10)}{\ln2}=1.44270\times10^9}.$$$$C(t)=C_0e^{-\lambda t}\Rightarrow\boxed{S_n(t)=\lambda C(t)=10^8\,2^{-t/(10\,\mathrm s)}\,\mathrm{s^{-1}}}.$$

(b) Interpretation and further calculation

$$N_n(0,T)=\int_0^TS_n(t)dt=C_0(1-e^{-\lambda T}).$$$$\boxed{N_n(0,20)=\frac34C_0=1.08202\times10^9}.$$

Physical check

The delayed source has a decay timescale even after new fissions cease. A real reactor has several precursor groups and coupled neutron dynamics.