We study the reflectivity of Casimir–van der Waals potentials, which behave as $-C_4{/r}^4$ at large distances and as $-C_3{/r}^3$ at small distances. The overall behavior of the reflection amplitude R depends crucially on the parameter $\rho =\sqrt{2M}{C}_3/\left(ħ\sqrt{C_4}\right)$ which determines the relative importance of the $-{1/r}^3$ and the $-{1/r}^4$ parts of the potential. Near threshold, $E={ħ}^2{k}^2/\left(2M\right)\mathrm{}\to 0,$ the reflectivity is given by $|R|\sim \exp \left(-2bk\right),$ with b depending on $\rho$ and the shape of the potential at intermediate distances. In the limit of large energies, $\ln |R|$ is proportional to $-k^{1/3}$ with a known constant of proportionality depending only on $C_3.$ For small values of $\rho ,$ the reflectivity behaves as for a homogeneous $-{1/r}^3$ potential in the whole range of energies and does not depend on $C_4$ or the shape of the potential beyond the $-{1/r}^3$ region. For moderate and large values of $\rho ,$ the reflectivity depends on $C_4$ and on the potential shape. For sufficiently large values of $\rho ,$ which are ubiquitous in realistic systems, there is a range of energies beyond the near-threshold region, where the reflectivity shows the high-energy behavior appropriate for a homogeneous $-{1/r}^4$ potential, i.e., $\ln |R|$ is proportional to $-\sqrt{k}$ with a proportionality constant depending only on $C_4.$ This conspicuous and model-independent signature of the Casimir effect is illustrated for the reflectivities of neon atoms scattered off a silicon surface, which were recently measured by Shimizu [Phys. Rev. Lett. 86, 987 (2001)].

• Received 6 November 2001

DOI:https://doi.org/10.1103/PhysRevA.65.032902