  We analyze the low-energy properties of superconductors near the onset of
accidental nodes, i.e. zeroes of the gap function not enforced by symmetry. The
existence of such nodes has been motivated by recent experiments suggesting a
transition between nodeless and nodal superconductivity in iron-based
compounds. We find that the low-temperature behavior of the penetration depth,
the specific heat, and the NMR spin-lattice relaxation rate are determined by
the scaling properties of a quantum critical point associated with the nascent
nodes. Although the power-law exponents are insensitive to weak short-range
electronic interactions, they can be significantly altered by the curvature of
the Fermi surface or by the three-dimensional character of the gap.
Consequently, the behavior of macroscopic quantities near the onset of nodes
can be used as a criterion to determine the nodal structure of the gap
function.
