Using the flux synthesis method and energy -dependent boundary conditions, we have solved the three-dimensional multigroup diffusion equation, without as suming space-energy separability and without explicitly introducing the concept of buckling, to study the diffusion of neutrons inside beryllium assemblies with finite transverse dimensions. The energy -dependent neutron spectra have been reported at various distances inside the two experimental assemblies of Lake and Kallfelz (35.6 × 35.6 × 50.8 cm3 and 25.4 × 25.4 × 50.8 cm3). We have discussed in detail the problem of the existence of a true discrete or a pseudo-asymptotic mode in these assemblies. We have also defined an “equivalent buckling” and find that the equivalent buckling agrees with the conventional definition of buckling only in large assemblies and only then in the epicold energy region. We have also discussed the validity of using diffusion theory in small assemblies.