  In recent experiments on nanoscale Al particles, whose electron number was
fixed by charging effects, a ``negative gap'' was observed in particles with an
odd number of electrons. This observation has called into question the use of a
grand canonical ensemble in describing superconductivity in such ultrasmall
particles.
  We have studied the effects of fixed electron number and finite size in
nanoscale superconductors, by applying the canonical BCS theory for the
attractive Hubbard model. The ground state energy and the energy gap are
compared with the conventional and parity-projected grand canonical BCS
results, and in one dimension also with the exact solutions by the Bethe
ansatz. The crossover from the bulk to quantum limit is studied for various
regimes of electron density and coupling strength. The effects of boundary
conditions and different lattice structures are also examined.
  A ``negative gap'' for odd electron number emerges most naturally in the
canonical scheme. For even electron number, the gap is particularly large for
``magic numbers'' of electrons for a given system size or of atoms for a fixed
electron density. These features are in accordance with the exact solutions,
but are essentially missed in the grand canonical results.
