Semiquantum approach for fast atom diffraction: Solving the rainbow divergence

In this work we introduce a distorted-wave method, based on the initial value representation approach of the quantum evolution operator, in order to improve the description of rainbow effects given by the surface eikonal (SE) model for diffraction from crystal surfaces produced by grazing scattering of fast atoms. The proposed theory, named the surface initial value representation (SIVR) approximation, is applied to He atoms colliding with a LiF(001) surface along low-indexed crystallographic channels. For this collision system the SIVR approach provides a very good representation of the quantum interference structures of experimental projectile distributions, even in the angular region around classical rainbow angles where the semiclassical SE method diverges.

Figure 1

Sketch of the angular coordinates for the FAD process.

Figure 2

Angular projectile distribution, as a function of the deflection angle $\Theta$, for ${}^4\mathrm{He}$ atoms scattered from LiF(001) along the $\left[100\right]$ direction with the glancing incidence angle ${\theta }_i=0.71$ deg. Two different impact energies are considered: (a) $E=7.3$ keV and (b) $E$ $=8.6$ keV. Solid red line, SIVR results for the supernumerary rainbow mechanism, derived by integrating the starting position ${\vec{R}}_{os}$ over a reduced unit cell; shadow gray line, experimental data from Ref. [1]; vertical arrows, positions of the classical rainbow angles $\pm {\Theta }_{\text{rb}}$.

Figure 3

Similar to Fig. 2 for ${}^4\mathrm{He}$ atoms scattered from LiF(001) along the $\left[110\right]$ channel. The incidence energy and angle are $E$ $=7.5$ keV and ${\theta }_i=0.67$ deg, respectively. The experimental data were extracted from Ref. [19].

Figure 4

Similar to Fig. 2 considering different theoretical descriptions of the supernumerary rainbow mechanism. Solid red line, SIVR approximation; dashed green line, SE approach; dash-dotted blue line, SE approach without including the Maslov correction term, as explained in the text. All the theories were evaluated integrating the starting position ${\vec{R}}_{os}$ over a reduced unit cell.

Figure 5

Similar to Fig. 2 comparing the contributions of the different mechanisms. Dashed red line, SIVR results derived from Eq. (8) by integrating the starting position ${\vec{R}}_{os}$ over a reduced unit cell (supernumerary rainbow contribution); solid blue line, similar by using an extended integration area, as explained in the text; dotted vertical lines, theoretical peak positions based on the Bragg condition [Eq. (15)].