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We calculate the rate of hydrogen burning for neutron stars (NSs) withhydrogen atmospheres and an underlying reservoir of nuclei capable of protoncapture. This burning occurs in the exponentially suppressed diffusive tail ofH that extends to the hotter depths of the envelope where protons are rapidlycaptured. This process, which we call diffusive nuclear burning (DNB), canchange the H abundance at the NS photosphere on timescales as short as$10^{2-4}$ years. In the absence of diffusion, the hydrogen at the photosphere(where $T\approx 10^6 {\rm K}$ and $\rho\sim 0.1 {\rm g cm^{-2}}$) would lastfor far longer than a Hubble time. Our work impacts the understanding of theevolution of surface abundances of isolated NSs, which is important to theirthermal spectrum and their effective temperature-core temperature relation. Inthis paper, we calculate the rate of H burning when the overall consumptionrate is controlled by the nuclear timescales, rather than diffusion timescales.The immediate application is for H burning on millisecond radio pulsars and inquiescence for the accreting NS Cen X-4. We will apply this work to young radiopulsars and magnetars once we have incorporated the effects of strong$B>10^{12} {\rm G}$ magnetic fields.Comment: 18 pages, 8 figures, accepted for publication by Ap

Diffusive Nuclear Burning in Neutron Star Envelopes