  Transitions of nuclear compositions in the crust of a neutron star induced by
stellar spin-down are evaluated at zero temperature. We construct a
compressible liquid-drop model for the energy of nuclei immersed in a neutron
gas, including pairing and shell correction terms, in reference to the known
properties of the ground state of matter above neutron drip density, $4.3
\times 10^{11} g cm^{-3}$. Recent experimental values and extrapolations of
nuclear masses are used for a description of matter at densities below neutron
drip. Changes in the pressure of matter in the crust due to the stellar
spin-down are calculated by taking into account the structure of the crust of a
slowly and uniformly rotating relativistic neutron star. If the initial
rotation period is $\sim$ ms, these changes cause nuclei, initially being in
the ground-state matter above a mass density of about $3 \times 10^{13} g
cm^{-3}$, to absorb neutrons in the equatorial region where the matter
undergoes compression, and to emit them in the vicinity of the rotation axis
where the matter undergoes decompression. Heat generation by these processes is
found to have significant effects on the thermal evolution of old neutron stars
with low magnetic fields; the surface emission predicted from this heating is
compared with the $ROSAT$ observations of X-ray emission from millisecond
pulsars and is shown to be insufficient to explain the observed X-ray
luminosities.
