  Diffusion of impurities in a granular gas undergoing homogeneous cooling
state is studied. The results are obtained by solving the Boltzmann--Lorentz
equation by means of the Chapman--Enskog method. In the first order in the
density gradient of impurities, the diffusion coefficient $D$ is determined as
the solution of a linear integral equation which is approximately solved by
making an expansion in Sonine polynomials. In this paper, we evaluate $D$ up to
the second order in the Sonine expansion and get explicit expressions for $D$
in terms of the restitution coefficients for the impurity--gas and gas--gas
collisions as well as the ratios of mass and particle sizes. To check the
reliability of the Sonine polynomial solution, analytical results are compared
with those obtained from numerical solutions of the Boltzmann equation by means
of the direct simulation Monte Carlo (DSMC) method. In the simulations, the
diffusion coefficient is measured via the mean square displacement of
impurities. The comparison between theory and simulation shows in general an
excellent agreement, except for the cases in which the gas particles are much
heavier and/or much larger than impurities. In theses cases, the second Sonine
approximation to $D$ improves significantly the qualitative predictions made
from the first Sonine approximation. A discussion on the convergence of the
Sonine polynomial expansion is also carried out.
