The neutron generation time ${\mathit{t}}_{\mathbf{g}\mathbf{e}\mathbf{n}}$in a reactor is the average time needed for a fast neutron emitted in one fission event to be slowed to thermal energies by the moderator and then initiate another fission event. Suppose the power output of a reactor at timeis $\mathit{t}\mathbf{=}\mathbf{0}$is ${\mathit{P}}_{\mathbf{0}}$. Show that the power output a time tlater is $\mathit{P}\left(t\right)$, whererole="math" localid="1661757074768" $\mathit{P}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathit{P}}_{\mathbf{0}}{\mathit{t}}^{\mathbf{t}\mathbf{/}{\mathbf{t}}_{\mathbf{g}\mathbf{e}\mathbf{n}}}$ and kis the multiplication factor. For constant power output,$\mathit{k}\mathbf{=}\mathbf{1}$.

1. In a fission event, a neutron is emitted to be slowed to thermal energies by the moderator and then another fission event is initiated.
2. For constant power output,$k=1$

The moderator of a nuclear reactor is a substance that slows neutrons down. Two fission events of occur one after the other. Again, in a chain reaction, the number of nuclides multiples by a certain factor that is related to the amount of neutron generation with time. Again, it is know that the power output is directly proportional to the number of nuclides produced or released, thus using this concept, we get the required power value.

Formula:

The output power of a reactor is as follows:

$P=\frac{NQ}{t}$ …… (i)

After each time interval$t_{gen}$the number of nuclides in the chain reaction gets multiplied byk. The number of such time intervals that has gone by at timetis$\frac{t}{t_{gen}}$.

For example, if the multiplication factor is 5 and there were 12 nuclei involved in the reaction to start with, then after one interval 60 nuclei are involved. And after another interval 300 nuclei are involved.

Thus, the number of nuclides engaged in the chain reaction at time t is given as follows:

role="math" localid="1661757300846" $N=N_0{k}^{t/t_{gen}}$

Now, from equation (i), we know that$P\alpha N$

Thus, the above equation becomes as follows:

role="math" localid="1661757394030" $P\left(t\right)=P_0{k}^{t/t_{gen}}$

Here, k is the multiplication factor and at $k=1$, the power value becomes $P=P_0$that is constant throughout the reaction.

Hence, the output power is $P\left(t\right)=P_0{k}^{t/t_{gen}}$.