Atomic diffraction under oblique incidence: An analytical expression

The semiclassical perturbation method developed by Henkel et al. [J. Phys. II 4, 1955 (1994)] to model cold-atom diffraction by optical standing waves, is applied to the diffraction of fast atoms on crystal surfaces at grazing incidence (GIFAD or FAD). We first show that the interaction time and interaction length embedded in the obliquity factor is well suited to explain the transition from three-dimensional to two-dimensional (2D) diffraction. The situation of a slightly misaligned primary beam, corresponding to oblique incidence in the effective 2D system, is addressed pointing out discrepancies such as the absence of net deflection of the atomic beam. Guided by time-reversal considerations, we propose an arbitrarily symmetrized form significantly improving the agreement with experimental data recorded in oblique incidence.

Figure 1

Schematic of the scattering geometry. A fast He atom is incident with a total wave vector ${\vec{k}}_i$ close to the $\langle 110\rangle$ direction of LiF(100) surface with polar incidence angle ${\varphi }_{\text{lab}}$ and misalignment angle ${\Gamma }_{\text{lab}}$. This 3D scattering problem can be reduced to a 2D problem of the He atom probing, with a wave vector ${\vec{k}}_{i,\text{2D}}=(k_{iy},k_{iz})$, the corrugation function $\xi \left(y\right)$ under oblique incidence ${\theta }_{\text{2D}}=arctan(k_{iy}/k_{iz})$. The diffracted beams are depicted with dashed vectors with the specular one in red.

Figure 2

Diffraction charts for ${}^4\mathrm{He}$ with $E_i=460$ eV and ${\varphi }_{\text{lab}}=0.{93}^{\circ }$ incident on a LiF surface close to the $\langle 110\rangle$ direction, i.e., $k_i=942$ Å${}^{-1}$ and $k_{iz}=15.3$ Å${}^{-1}$, $G_y$ is 2.2 Å${}^{-1}$ corresponding to a deflection angle of 0.134 deg = ${\sin }^{-1}$($G_y/k_y$). The interaction parameters are $R_c=1.18$ Å and $h_c=0.325$ Å. (a) Experimental data [14]. (b) Semiclassical calculations using Eq. (2). (c) “Exact” quantum calculation [14]. (d) Semiclassical calculations using Eq. (1). Note that both horizontal and vertical scale have the same units ($G_y$) relevant for diffraction. In addition, the misalignment angle ${\Gamma }_{\text{lab}}$ is reported also in degrees in the laboratory frame (red) as well as the corresponding obliquity angle ${\theta }_{\text{2D}}=tan\left({\Gamma }_{\text{lab}}\right)/tan\left({\varphi }_{\text{lab}}\right)$ (blue). (a) and (c) have been adapted from [14]. The dashed line in (b) correspond to the maximum transfer of momentum obtained from classical calculations. The rainbow angle of $k_{iy}\approx 4G_y$ used for quantitative comparison in Fig. 3 is marked by the arrows labeled “rb.”

Figure 3

Diffraction probabilities extracted from Fig. 2 for $k_{iy}=4G_y$ (rainbow position) for experiment (black pattern), quantum theory (dashed pattern), and semiclassical theory from Eq. (2) (blue pattern).