  We studied the short proton free induction decay signals and the broad ^1H
NMR spectra of Mg(OH)2 and Ca(OH)2 powders at 77-355 K and 42 MHz using pulsed
NMR techniques. Using a Gaussian-type back extrapolation procedure for the
obscured data of the proton free induction decay signals, we obtained more
precise values of the second moments of the Fourier-transformed broad NMR
spectra than those in a previous report [Y. Itoh and M. Isobe, J. Phys. Soc.
Jpn. Vol. 84, 113601 (2015)] and compared with the theoretical second moments.
The decrease in the second moment could not account for the large decrease in
the magnitude of the intrinsic proton spin-lattice relaxation rate 1/T_1 from
Mg(OH)2 to Ca(OH)2. The analysis of 1/T_1 $propto$ exp(-E_g/k_BT) with E_g~0.01
eV points to a local hopping mechanism, and that of 1/T_1 $propto$ T^n with
n~0.5 points to an anharmonic rattling mechanism.

Materials Science;Other Condensed Matter;Strongly Correlated Electrons