IGNOU MPHE 26 2026 - What do you understand by constant cross-section approximation? Derive

Question

What do you understand by constant cross-section approximation? Derive the steady state equation for a plane geometry. Also write the transport equation for a spherical geometry.

19 Jan 2026

Answer :

Word Count : 315

In neutron transport theory, the constant cross-section approximation assumes that the macroscopic cross-section ( \Sigma ) of the medium is independent of position, direction, and energy. This simplifies the transport equations, as interactions such as absorption, scattering, and fission are treated as uniform throughout the medium, allowing analytical solutions for flux distributions in simple geometries. It is particularly useful in steady-state, one-group analyses. For steady-state neutron transport in a plane (slab) geometry, consider a slab of thickness ( L ) along the ( x )-axis. Let ( \phi(x) ) denote the neutron flux and ( J(x) ) the neutron current. Under the constant cross-section approximation, the one-speed neutron transport equation is: [ \mu \frac{d \psi(x, \mu)}{dx} + \Sigma_t \psi(x, \mu) = \frac{\Sigma_s}{2} \int_{-1}^{1} \psi(x, \mu') ,

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