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$1/T_1$ in the D-wave superconducting state with coexisting antiferromagnetism

We consider a general D-wave superconducting gap function in the presence of coexisting antiferromagnetic (AFM) long-rang order. We find that the D-wave order parameter develops additional nodes without lowering the symmetry of the state due to the interplay of the superconducting and AFM correlations. This AFM correlated D-wave gap has a small gap structure inside the generic D-wave gap. As a result of this additional structure it shows a quite different response to impurities compared to a standard D-wave gap. We calculate the nuclear spin-lattice relaxation rate $1/T_1$ of this gap in the presence of impurities and discuss the implications for the experiments of some AFM heavy- fermion superconductors and in particular for CeRhIn5.