  The molecular dynamics of regioregulated poly(3-hexylthiophene) P3HT is
investigated using muon spin relaxation ($\mu$SR). The response of the $\mu$SR
spectra to a longitudinal magnetic field ($B_{\rm LF}$, parallel to the initial
muon spin direction) indicates that the implanted muons form both muonated
radicals localized on the thiophene ring and diamagnetic states with comparable
yields. Moreover, the unpaired electron in the radical undergoes hyperfine
interactions with muon bound to thiophene and with neighboring protons, whose
fluctuations can serve as a measure for the molecular dynamics. The $B_{\rm
LF}$ dependence of the longitudinal muon spin relaxation rate ($1/T_{1\mu}$)
measured in detail at several temperatures is found to be well reproduced by
the spectral density function $J(\omega)$ derived from the local susceptibility
that incorporates the Havriliak-Negami (H-N) function used in the analysis of
dielectric relaxation,
$\chi(\omega)\propto1/[1-i(\omega/\tilde{\nu})^\delta]^\gamma$ (where
$\tilde{\nu}$ is the mean fluctuation rate, and $0<\gamma, \delta\le1$). The
magnitude of $\tilde{\nu}$ and its temperature dependence deduced from the
analysis of $1/T_{1\mu}$ are found to be consistent with the motion of hexyl
chains and thiophene rings suggested by $^{13}$C-NMR. The present result marks
a methodological milestone in the application of $\mu$SR to the dynamics of
complex systems with coexisting fluctuations over a wide range of time scales,
such as polymers.
