Using spherical harmonica method for non-multiplying 1-plane geometry, write the exact infinite coupled equations. Hence for a large non-absorbing system, obtain equations under Py approximation Discuss the Imitation of P approximation.

In reactor physics, the spherical harmonics method is a mathematical technique used to approximate the solution of the neutron transport equation, especially in cases of non-multiplying media, which is common in nuclear reactors. The spherical harmonics expansion allows the neutron flux to be expressed as a series of spherical harmonics functions, which simplifies the complex geometry of neutron transport problems.

For a non-multiplying, one-plane geometry system, the neutron transport equation in steady-state can be written as:

1v∂ϕ(r⃗,Ω)∂t+Ω⃗⋅∇ϕ(r⃗,Ω)+σtϕ(r⃗,Ω)=0\frac{1}{v} \frac{\partial \phi(\vec{r}, \Omega)}{\partial t} + \vec{\Omega} \cdot \nabla \phi(\vec{r}, \Omega) + \sigma_t \phi(\vec{r}, \Omega) = 0

where ϕ(r⃗,Ω)\phi(\vec{r}, \Omega) is the angular flux, Ω⃗\vec{\Omega} represents the direction of neutron travel, σt\sigma_t is the total cross-section, and vv is the neutron velocity. In spherical harmonics expansion, the angular flux is expanded in terms of the spherical harmonics Yl(Ω)Y_l(\Omega), as follows:

ϕ(r⃗,Ω)=∑l=0∞(2l+1)ϕ^l(r⃗)Yl(Ω)\phi(\vec{r}, \Omega) = \sum_{l=0}^{\infty} (2l+1) ____ ________ ________ _________ ________ __________ __________ ___ __________ _____ _______.
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