  In \cite{Nac2,Nac1} Nachtergaele obtained explicit lower bounds for the
spectral gap above many frustration free quantum spin chains by using the
``martingale method''. We present simple improvements to his main bounds which
allow one to obtain a sharp lower bound for the spectral gap above the spin-1/2
ferromagnetic XXZ chain. As an illustration of the method, we also calculate a
lower bound for the spectral gap of the AKLT model, which is about 1/3 the size
of the expected gap.
