**Figure 2.** Difference between the reflectivity |*R*|_{n} calculated numerically and the reflectivity |*R*|_{h} calculated by the high-order formula (22).

**Abstract**

We investigate the quantum reflection governed by the Casimir–Polder potential which scales as −1/*r*^{3} at short separation and as −1/*r*^{4} at long separation. Using the available analytical solutions for the −1/*r*^{4} potential and quantum defect theory, we develop a three-parameter model for quantum reflection. The three parameters characterizing the potential are essentially constant near the threshold. The application scope of the model is examined. For a potential dominated by the −1/*r*^{4} behaviour, which is the case in many realistic atom–surface systems, our model gives accurate results over the energy range in which the quantum reflection is significant. Moreover, this parameterized model allows one to fit the experimental data efficiently.