The neutron transport equation is solved in plane geometry for a moderator with a periodic temperature distribution using the synthetic scattering kernel of Williams. A simple correspondence between the new model and the heavy-gas model is found for physical quantities dependent on the first two eigenvalues of the kernel. A recursion procedure for solving the energy moments of the flux is also presented. The flux is determined by a method using singular eigenfunctions. Some numerical results for the mean energy of the flux as a function of the lattice length are presented for A = 8 or for A = 10 employing the heavy-gas model. In order to consider the effect of the periodicity of the temperature distribution on the mean energy of the neutron spectrum obtained, the results are compared to the mean energy of the neutron spectrum in Kottwitz geometry. There is a considerable deviation for lattices with lengths of the order of the rethermalization length. In this respect, the lattices with lengths of the order of ten rethermalization lengths describe Kottwitz geometry fairly well.