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Analytical theory for the crossover from retarded to non-retarded interactions between metal plates

The van der Waals force established between two surfaces plays a central role in many phenomena, such as adhesion or friction. However, the dependence of this forces on the distance of separation between plates is very complex. Two widely different non-retarded and retarded regimes are well known, but these have been traditionally studied separately. Much less is known about the important experimentally accesible cross-over regime. In this study, we provide analytical approximations for the van der Waals forces between two plates that interpolates exactly between the short distance and long distance behavior, and provides new insight into the crossover from London to Casimir forces at finite temperature. At short distance, where the behavior is dominated by non-retarded interactions, we work out a very accurate simplified approximation for the Hamaker constant which adopts analytical form for both the Drude and Lorentz models of dielectric response. We apply our analytical expressions for the study of forces between metallic plates, and observe very good agreement with exact results from numerical calculations. Our results show that contributions of interband transitions remain important in the experimentally accessible regime of decades nm for several metals, including gold.