  The Casimir interaction in a stack of equally spaced infinitely thin layers
is investigated within the zero-frequency mode summation method. The response
properties are considered to be described by a constant conductivity or by a
Drude-Lorentz model with a finite set of oscillators consistent with the
optical characteristics for graphite. It is found that the asymptotic distance
dependence is affected significantly by the specific response. While the energy
is $\sim 1/d^3$ for the constant conductivity model, the energy exhibits
fractional dependence $\sim 1/d^{5/2}$ for the Drude-Lorentz description. The
Casimir force on a plane is also strongly dependent upon the particular plane
location in the stack. Furthermore, the calculated Casimir energy within the
Drude-Lorentz model yields results in good agreement with measured cohesion
energy in graphite.
