Let $X$ be a smooth complex algebraic variety (for simplicity). Recall that the de Rham space (or stack) of $X$ is the quotient of $X$ by the groupoid that is the formal neighborhood of the diagonally embedded copy of $X$ in $X\times X$. This groupoid degenerates naturally to the formal completion of $X$ in its normal bundle in $X\times X$, i.e. in $T^*X$. [This is the formal completion of the degeneration of $X\hookrightarrow X\times X$ to the normal cone.] There is a corresponding degeneration of the sheaf $\mathcal{D}_X$ of differential operators on $X$ to functions on $T^*X$. [More precisely, if $\pi: T^*X\rightarrow X$ is the projection, $\mathcal{D}_X$ degenerates to $\pi_*\mathcal{O}_{T^*X}$.]

**Question.** What is the nicest way (is there one?) to say a corresponding degeneration of the crystalline site on $X$, so that degeneration of $\mathcal{D}$-modules corresponds to quasicoherent sheaves on the degenerating site?

If this is completely standard, can you suggest a reference?