= Solution
Substitute the separated solution into the <one-group neutron diffusion equation>. The radial and axial <Laplacian> contributions are $-\alpha^2\phi$ and $-\beta^2\phi$. A nonzero solution therefore requires
$$
\boxed{\alpha^2+\beta^2
=\frac{(\eta-1)\Sigma_a}{D}
=\frac{\nu\Sigma_f-\Sigma_a}{D}.}
$$
The left side is the <geometric buckling>; the right side is fixed by the stipulated homogeneous material. Their equality is the model's <nuclear criticality> condition, expressing the balance of production, absorption and leakage. Both sides have inverse-length-squared units.
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