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Roughness exponents and grain shapes

In surfaces with grainy features, the local roughness $w$ shows a crossover at a characteristic length $r_c$, with roughness exponent changing from $α_1\approx 1$ to a smaller $α_2$. The grain shape, the choice of $w$ or height-height correlation function (HHCF) $C$, and the procedure to calculate root mean-square averages are shown to have remarkable effects on $α_1$. With grains of pyramidal shape, $α_1$ can be as low as 0.71, which is much lower than the previous prediction 0.85 for rounded grains. The same crossover is observed in the HHCF, but with initial exponent $χ_1\approx 0.5$ for flat grains, while for some conical grains it may increase to $χ_1\approx 0.7$. The universality class of the growth process determines the exponents $α_2=χ_2$ after the crossover, but has no effect on the initial exponents $α_1$ and $χ_1$, supporting the geometric interpretation of their values. For all grain shapes and different definitions of surface roughness or HHCF, we still observe that the crossover length $r_c$ is an accurate estimate of the grain size. The exponents obtained in several recent experimental works on different materials are explained by those models, with some surface images qualitatively similar to our model films.