Diffraction Theory of Sinusoidal Gratings and Application to In Situ Surface Self-Diffusion Measurements

Abstract

Computer calculations of the diffracted intensities from a sinusoidal surface grating have been carried out using analytical methods developed independently by Brekhovskikh, Senior, Stroke, and Petit from the electromagnetic theory of diffraction. These calculations provide a theoretical foundation for the newly developed laser-diffraction technique for continuously monitoring surface self-diffusion of solids under ultrahigh vacuum conditions. The calculated patterns were characterized by an absolute maximum in intensity, which occurs at increasingly higher orders as the amplitude of the profile increases; the order of the maximum is designated nmax. Calculations were carried out at normal incidence for both Ē and H̄ parallel to the groove direction, but no sensible polarization effect was found in the range of amplitude and periods investigated. The effect of angle of incidence θi was also determined, and it was shown theoretically that, as θi increases, nmax on the +n-side of the pattern shifts toward the zero order, but on the −n-side nmax passes through a maximum before slowly decreasing. This effect also has been verified by experiment. In addition to these analyses, the phase modulation treatment used in communication theory has been examined and it is shown that this approach is the limiting case of the rigorous electromagnetic theory when the period of the grating becomes very large compared to the amplitude. Excellent agreement has been obtained between theoretical and experimental intensity distributions. This agreement provides a fundamental basis for the linear relationship between the amplitude A of the profile and the diffraction order of maximum intensity, nmax. which has been previously established experimentally and is used to follow the decay in A during a diffusion experiment. While nmax (A) is linear in the range of amplitudes and periods most frequently used in surface-diffusion measurements, the theory predicts nonlinearities in nmax (A) when the amplitude-to-period ratio is greater than about 0.08; these deviations are important in very low temperature diffusion studies. As an example of the application of the diffraction theory of sinusoidal gratings, the experimental determination of the surface self-diffusivity of a (110) nickel surface is presented. The laser-diffraction method has been used in conjunction with a low-energy electron-diffraction system, permitting the surface structure and cleanliness to be determined at any time during a diffusion experiment. An activation energy of 19.0 kcal/mole and a D0=0.01 cm2/sec have been measured for a (110) Ni C2 structure in the temperature range 800° to 1080°C.