Repulsive Casimir interaction: Boyer oscillators at nanoscale

We study the effect of temperature on the repulsive Casimir interaction between an ideally permeable and an ideally polarizable plate {\it in vacuo}. At small separations or for low temperatures the quantum fluctuations of the electromagnetic field give the main contribution to the interaction, while at large separations or for high temperatures the interaction is dominated by the classical thermal fluctuations of the field. At intermediate separations or finite temperatures both the quantum and thermal fluctuations contribute. For a system composed of one infinitely permeable plate between two ideal conductors at a finite temperature, we identify a {\it stable mechanical equilibrium} state, if the infinitely permeable plate is located in the middle of the cavity. For small displacements the restoring force of this {\it Boyer oscillator} is linear in the deviation from the equilibrium position, with a spring constant that depends inversely on the separation between the two conducting plates and linearly on temperature. Furthermore, an array of such oscillators presents an ideal Einsteinian crystal that displays a fluctuation force between its outer boundaries stemming from the displacement fluctuations of the Boyer oscillators.