The rainbow effect in ion channeling (in very thin crystals) is considered. In the analysis the results of catastrophe theory are used. The calculations were performed for 10-MeV H + ions and the (100) channel of a 1000-A-thick Au crystal.PACS numbers: 61.80. Mk, 02.40.+m It has been shown recently 1 that in ion channeling in very thin crystals the differential transmission cross section is singular, i.e., the rainbow effect occurs. This was explained by the fact that contributions of the atomic strings of the crystal to the differential transmission cross section interfere. Shortly after the prediction the effect was observed. 2 In this Letter we shall consider the rainbow effect in ion channeling in the case of 10-MeV H + ions and the (100) channel of a 1000-A-thick Au crystal. The results obtained will be compared to the corresponding ones given by catastrophe theory. 3 It should be noted that a similar effect, occurring in particle scattering from surfaces, was analyzed by Berry. 4 The z axis is taken to coincide with the channel axis and the origin to lie in the median plane of the crystal. The incident ion velocity is parallel to the z axis. The crystal is assumed to be sufficiently thin for the ion trajectory to be approximated by a straight line. In this case the components of the scattering angle and the differential transmision cross section as functions of the impact parameter can be easily obtained with use of the momentum approximation. l We assume that the ionatom interaction potential is of the Thomas-Fermi type; for the integral of that potential along the trajectory, needed in applying the momentum approximation, we use Lindhard's expression, 5Vi=ZiZ 2 e 2 \n Ca's (x-Xi) 2 +(y-yi) :where Z\ and Z 2 are the atomic numbers of the ion and the crystal, respectively, x and y are the components of the impact parameter, x, and y t are the transverse coordinates of the atoms of the /th atomic string of the crystal, a s =(97r 2 /\28Z2) 1/3 tfo is the screening radius, ao is the Bohr radius, and C = 3 is a fitting parameter. Variable Vj/Z\ed, where d is the distance between the atoms of the atomic strings, is the continuum potential of the /th string. 5 As has been said above, we consider here 10-MeV H + ions in the (100) channel of a 1000-A-thick Au crystal. Since in this case the distance between the atoms of the atomic strings of the crystal equals the unit-cell parameter, which is 4.078 97 A, 6 the number of atoms in one atomic string is 246. The number of strings is 24, i.e., we take into account the strings lying on the four nearest (relative to the channel center) coordination circles. * Figure 1 gives the rainbow line, i.e., the line along which the differential transmission cross section is singular, in the impact-parameter plane in this H + -• Au case. The coordinates of the atomic strings lying on the first coordination circle, in a.u., are (2.73,0), (0,2.73), (-2.73,0), and (0,-2.73). The coordinates of the points defining this line are determined only by the arrangement of the atomic string...