  We report a novel behavior of the surface plasmon linewidth in sodium
nanospheres Na$_{1760}$ changing with the electron temperature, which
monotonically decreases and bears a discontinuous sudden drop at high electron
temperatures. Our calculation is based on the model constructed by splitting
the total Hamiltonian of all valence electrons of a metallic nanoparticle into
two sub-Hamiltonians and their coupling, and obtained results can be verified
by the pump-probe femtosecond spectroscopy experiments. In addition, we propose
that it is the size uncertainty of small nanoparticles that yields the
intrinsic linewidth of the surface plasmon resonance, which is supported by the
available data of clusters Na$_{8}$ and Na$_{20}$.
