We present details of the crystal structure and magnetic behavior of Fe/Rh (001) sputter deposited multilayer thin films. We find that the films undergo a structural phase transformation from a body-centered-tetragonal (bct) to a face-centered-tetragonal (fct) lattice which is concomitant with a magnetic transformation from a ferromagnetic to a nonferromagnetic state. We identify the Rh spacer thickness where both bet and fct phases first coexist as the critical thickness (t(crit)). In the region prior to the critical thickness, both Fe and Rh are ferromagnetic with Rh moments of at least 1/mu(B) and Fe moments enhanced as much as 30% relative to their bulk value. In the postcritical region, we find the films to be nonferromagnetic. The experimental results are compared with theoretical calculations on similar structures.