Generally, balance neutrons in - neutrons out. Σ 𝐸 𝜑 𝐸 = (Σ𝑠 int(𝐸′ → 𝐸) φ(𝐸′)𝑑𝐸′) + χ 𝐸 𝑠𝑓′′′
collision rate Σ(𝐸)𝜑(𝐸) includes absorption and scattering, and is how energy is lost, hence on left.
(Σ𝑠 int(𝐸′ → 𝐸) φ(𝐸′)𝑑𝐸′) is probability of scattering down to energy E from E′, and χ(𝐸)𝑠𝑓′′′ is rate of fission neutrons produced at energy E (χ prob of production at energy E, sf''' is rate of fission neutron production)
Fast neutrons, if scattered, are removed from fast regime, hence simplifies to Σ(𝐸)𝜑(𝐸) ≈ χ(𝐸)𝑠𝑓′′′, or 𝜑(𝐸)≈χ(𝐸)𝑠𝑓′′′Σ(𝐸)
For the intermediate range, no fission neutron production and no absorption, just scattering
Therefore, Σ𝑠(𝐸)𝜑(𝐸) =int(Σ𝑠(𝐸′→ 𝐸) φ(𝐸))𝑑𝐸′