  We study time-harmonic scattering in $\mathbb{R}^n$ ($n=2,3$) by a planar
screen (a "crack" in the context of linear elasticity), assumed to be a
non-empty bounded relatively open subset $\Gamma$ of the hyperplane
$\mathbb{R}^{n-1}\times \{0\}$, on which impedance (Robin) boundary conditions
are imposed. In contrast to previous studies, $\Gamma$ can have arbitrarily
rough (possibly fractal) boundary. To obtain well-posedness for such $\Gamma$
we show how the standard impedance boundary value problem and its associated
system of boundary integral equations must be supplemented with additional
solution regularity conditions, which hold automatically when $\partial\Gamma$
is smooth. We show that the associated system of boundary integral operators is
compactly perturbed coercive in an appropriate function space setting,
strengthening previous results. This permits the use of Mosco convergence to
prove convergence of boundary element approximations on smoother "prefractal"
screens to the limiting solution on a fractal screen. We present accompanying
numerical results, validating our theoretical convergence results, for
three-dimensional scattering by a Koch snowflake and a square snowflake.
