 Solution

ID: past-exam-of-the-mphil-in-nuclear-energy-course-of-the-university-of-cambridge/2021/paper-ne1/4/c/solution

Φ

as the independent variable and abbreviate

a = σ_{a Th}

b = σ_{a U}

and

c = σ_{c Th}

. Substituting the parent solution into the second linear differential equation gives

\frac{d N _{U}}{d Φ} + b N_{U} = c N_{Th0} e^{- a Φ} .

e^{b Φ}

. Integrating from the initial state yields

N_{U} (Φ) = N_{U0} e^{- b Φ} + \frac{c N _{Th0}}{b - a} (e^{- a Φ} - e^{- b Φ}), b \neq = a .

The first term is survival of the initial population; the second accumulates production at earlier exposures, weighted by subsequent survival. For

b = a

, the continuous limit is

(N_{U0} + c N_{Th0} Φ) e^{- a Φ}

. The same formula applies to a time-varying prescribed flux through

Φ (t)

; intervals of zero flux leave both populations unchanged.

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