††thanks: Present Address: Sumitomo Rubber Industries Ltd., Kobe, Hyogo
651-0072 Japan††thanks: Present Address: Department of Chemistry and
Biotechnology, Graduate School of Engineering, The University of Tokyo,
Bunkyo-ku, Tokyo 113-8656, Japan††thanks: email<EMAIL_ADDRESS>
# Slow polymer dynamics in poly(3-hexylthiophene) probed by muon spin
relaxation

S. Takeshita  Muon Science Laboratory, Institute of Materials Structure
Science, High Energy Accelerator Research Organization (KEK-IMSS), Tsukuba,
Ibaraki 305-0801, Japan  Graduate Institute for Advanced Studies, SOKENDAI  K.
Hori  Muon Science Laboratory, Institute of Materials Structure Science, High
Energy Accelerator Research Organization (KEK-IMSS), Tsukuba, Ibaraki
305-0801, Japan  M. Hiraishi  Muon Science Laboratory, Institute of Materials
Structure Science, High Energy Accelerator Research Organization (KEK-IMSS),
Tsukuba, Ibaraki 305-0801, Japan  Graduate School of Science and Engineering,
Ibaraki University, Bunkyo, Mito, Ibaraki 310-8512, Japan.  H. Okabe  Muon
Science Laboratory, Institute of Materials Structure Science, High Energy
Accelerator Research Organization (KEK-IMSS), Tsukuba, Ibaraki 305-0801, Japan
Institute for Materials Research, Tohoku University (IMR), Katahira, Aoba-ku,
Sendai 980-8577, Japan  A. Koda  Muon Science Laboratory, Institute of
Materials Structure Science, High Energy Accelerator Research Organization
(KEK-IMSS), Tsukuba, Ibaraki 305-0801, Japan  Graduate Institute for Advanced
Studies, SOKENDAI  D. Kawaguchi  Department of Applied Chemistry and Center
for Polymer Interface and Molecular Adhesion Science, Kyushu University,
Fukuoka 819-0395, Japan  K. Tanaka  Department of Applied Chemistry and Center
for Polymer Interface and Molecular Adhesion Science, Kyushu University,
Fukuoka 819-0395, Japan  R. Kadono  Muon Science Laboratory, Institute of
Materials Structure Science, High Energy Accelerator Research Organization
(KEK-IMSS), Tsukuba, Ibaraki 305-0801, Japan

###### Abstract

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, $\chi(\omega)\propto
1/[1-i(\omega/\tilde{\nu})^{\delta}]^{\gamma}$ (where $\tilde{\nu}$ is the
mean fluctuation rate, and $0<\gamma,\delta\leq 1$). 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 13C-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.

##  I Introduction

Conjugated polymers have been widely studied as promising semiconductor
materials because of their potential in electronic devices such as solar cells
Günes, Neugebauer, and Sariciftci (2007); Thompson and Fréchet (2008); Kim
_et al._ (2006); Reynolds, Thompson, and Skotheim (2019), field-effect
transistors Dimitrakopoulos and Malenfant (2002); Braga and Horowitz (2009);
Yan  _et al._ (2009), light-emitting diodes Burroughes  _et al._ (1990);
Krummacher  _et al._ (2006), and those for spintronics Naber, Faez, and van
der Wiel (2007); Ed. (2010); Geng  _et al._ (2016). In particular, their light
weight, flexibility, and elasticity have attracted attention as materials for
wearable devices unsuitable for inorganic semiconductors Kuribara  _et al._
(2012); Kaltenbrunner  _et al._ (2013). Among them, regio-regular
poly(3-hexylthiophene) (P3HT), in which the hexyl side chain is regularly
bound from head to tail at the 3-position of the thiophene ring, has attracted
much attention as an optoelectronic material due to its excellent properties
and solubility in organic solvents Chen, Wu, and Rieke (1995); Bao,
Dodabalapur, and Lovinger (1996); Sirringhaus  _et al._ (1999). Of particular
importance for device applications are thin films with thicknesses on the
order of 10–100 nm. However, the relationship between the thin film morphology
of organic semiconductors and their electrical and optical properties is still
largely unresolved due to the lack of suitable experimental methods for proper
investigation.

The molecular organization of conducting polymers in the condensed state and
their dynamical structure have a marked influence on their optoelectronic and
electrical properties. For example, thermally induced disorder in the alkyl
side groups of conjugated oligomers and polymers correlates with optical
absorption of bithiazole oligomers Curtis  _et al._ (2007), which may be
relevant with poly(3-alkylthiophene) (P3AT) with longer alkyl side-chains
Bolognesi  _et al._ (1993); Zhao  _et al._ (1997). While the melting behavior
of the hexyl side group and related polymorphism has not been reported for
P3HT, thermochromism phenomena suggests that thermally activated thiophene
twists from a quasi-ordered structure causes disorder in the $\pi$ conjugation
Inganäs  _et al._ (1988). Despite the importance of information on the local
molecular dynamics, such information on P3HT is so far limited.

In this respect, recent Fourier transform infrared spectroscopy (FT-IR) and
13C nuclear magnetic resonance (NMR) studies report interesting results Yazawa
_et al._ (2010). They suggested that alkyl side chain motions are activated in
P3HT crystal above $\sim$200 K, and that twisting motions of the thiophene
ring sets in above $\sim$300 K to exhibit a plastic crystal state. Similar
indications are obtained from dynamic mechanical analysis which allows direct
access to the thermo-molecular motion of the polymer in the film, where three
relaxation peaks at around 200, 300, and 390 K (corresponding to the so-called
$\beta$, $\alpha_{1}$, and $\alpha_{2}$ processes) are observed in the dynamic
loss modulus Ogata, Kawaguchi, and Tanaka (2015). From the magnitude of the
apparent activation energy and from the complementary structural analysis
using FT-IR, these processes are assigned to side chain motion, twisting
motion, and deformation of inter-lamellar crystal, respectively.

Previous theoretical studies predict that the vibrational slow twisting and
fast coupling stretching motions of $\pi$-conjugated molecules are strongly
coupled to the electronic structure and are therefore related to photoelectric
processes Tretiak  _et al._ (2002). In particular, twisting motion is also
important in that it is thought to govern the efficiency of the generation of
isolated polarons from photoinduced electron-hole polaron pairs that we have
proposed Ogata, Kawaguchi, and Tanaka (2015). Therefore, the details of the
dynamical structure of P3HT and the relationship between its structure and
physical properties such as optical absorption remain one of the most
important issues.

Here, we report on local molecular dynamics inferred from muon spin relaxation
($\mu$SR) experiments in the crystalline state P3HT. In general, the positive
muon ($\mu^{+}$) implanted into a material behaves as a light isotope of
hydrogen (denoted hereafter by the element symbol Mu). We show from the
response of $\mu$SR spectra to the longitudinal magnetic field (LF, whose
magnitude given by $B_{\rm LF}$) that about a half of implanted muons form a
muonated radical state and that the unpaired electron in the radical exerts
hyperfine fields to both $\mu^{+}$ and surrounding protons which are described
by the hyperfine (HF) and nuclear hyperfine (NHF) parameters. The unpaired
electron can also be viewed as a state in which charge carriers doped by muon
addition reactions (equivalent to hydrogenation) are localized in the vicinity
of muons. In terms of defects in solids, such a state is approximately
equivalent to the polaron often referred to in the field of conducting
polymers Mizes and Conwell (1993); Furukawa and Shimokawa (2023). $\mu$SR has
the advantage of providing detailed information on the local electronic
structure and dynamic properties of such polaron states produced by muons
themselves. We also show that the remaining diamagnetic muons show negligible
spin relaxation.

The time-dependent fluctuation of the HF and/or NHF fields, which causes
longitudinal muon spin relaxation, can serve as a measure of local molecular
dynamics at the muon site(s). We show that the LF dependence of the spin
relaxation rate, $1/T_{1\mu}$, is reproduced by the spectral density function
$J(\omega)$ derived from the local susceptibility that incorporates the
Havriliak-Negami (H-N) function, $\chi(\omega)\propto
1/[1-i(\omega/\tilde{\nu})^{\delta}]^{\gamma}$ (where $\tilde{\nu}$ is the
mean fluctuation rate, $0<\gamma,\delta\leq 1$), used in the analysis of
dielectric relaxation Havriliak and Negami (1967); Alvarez, Alegra, and
Colmenero (1991). The qualitative change in the LF dependence of $1/T_{1\mu}$
observed around 200 K is typically represented by the behavior of
$\tilde{\nu}$, and its temperature dependence are found to be consistent with
those due to the molecular motions suggested by previous study using 13C-NMR,
where the characteristic time scale of the fluctuation ($\tilde{\nu}^{-1}$) is
deduced to be $10^{-9}$–$10^{-10}$ s above $\sim$200 K.

We also find that the amplitude of the internal field fluctuations is
significantly smaller than the linewidth $\Delta_{\rm n}$ of the NHF (or HF)
field at lower temperatures. This corresponds to the expected situation that
the muonated radicals are stationary and only a part of the host molecules are
in motion, suggesting that the strong collision model (random phase
approximation) commonly assumed in dynamical models does not hold. We show
that the recently proposed model to distinguish ion dynamics from muon self-
diffusion under similar circumstances Ito and Kadono (2024) can be extended to
paramagnetic Mu to provide reasonable accounts of the experimental results on
P3HT.

The present result also suggests a need to reconsider the earlier
interpretation that $1/T_{1\mu}$ in P3HT is due to the fluctuation of HF
fields induced by rapid quasi-one-dimensional jumping of carriers (unpaired
electrons) created on the thiophene chain upon muon implantation Risdiana  _et
al._ (2010a, b).

##  II EXPERIMENTAL

The regioregular P3HT samples were prepared using the compound purchased from
Sigma-Alidrich Inc. as received. Number-average molecular weight ($M_{\rm n}$)
and polydispersity index (PDI) were 26 kg mol-1 and 2.4, respectively, which
were determined by gel permeation chromatography using polystyrene standards.
The melting temperature measured by differential scanning calorimetry was 488
K. Films of P3HT were prepared by spin-coating from chloroform solutions onto
quartz, barium fluoride (BaF2) and commercially available polyimide (PI)
substrates for absorption spectroscopy, FT-IR measurements and dynamic
mechanical analysis, whose results were reported elsewhere Ogata, Kawaguchi,
and Tanaka (2015). The films were dried under vacuum at 373 K for 12 h. The
film thickness was approximately 300 nm.

Conventional $\mu$SR measurements were performed of the P3HT films (25
mm$\phi\times 0.7$ mm) using the S1 instrument (ARTEMIS) at the Materials and
Life-science Experiment Facility, J-PARC Kojima  _et al._ (2014), where high-
precision measurements over a long time range of 20 $\mu$s can be routinely
performed using a high-flux pulsed beam of positive muons ($\approx
10^{3}$–$10^{4}$ $\mu^{+}$s per pulse, with a repetition rate 25 Hz and an
incident energy $E_{\mu}\approx 4$ MeV). The $\mu$SR spectra [the time-
dependent decay-positron asymmetry, $A(t)$] which reflects the magnetic field
distribution at the Mu site, was measured from 100 K to 350 K under zero field
(ZF), weak LF (parallel to the initial Mu polarization ${\bm{P}}_{\mu}$), and
weak transverse field (TF, perpendicular to ${\bm{P}}_{\mu}$), and were
analyzed by least-squares curve fitting Suter and Wojek (2012). The correction
factor for the instrumental asymmetry under a given LF was calibrated by a
separate set of LF-$\mu$SR measurements on a blank sample holder (made of
silver), combined with the total asymmetry corresponding to 100% muon spin
polarization determined by a weak TF-$\mu$SR measurement.

##  III RESULT

In conjugated polymers, implanted Mu can cleave double bonds to form a bound
state with cation atom: e.g., for polyacetylene,

[HC=CH]n \+ Mu $\rightarrow$

[HC=CH]k-[CHMu-$\dot{\rm{C}}$H]-[HC=CH]n-k-1,

where “$\cdot$” on C denotes the unpaired electron, whose polaron motion has
been studied by $\mu$SR in a variety of conducting polymers Nagamine  _et al._
(1984); Ishida  _et al._ (1985); Pratt  _et al._ (1997); Blundell  _et al._
(2002); Pratt (2004). Here, the Mu forms a covalent bond with carbon to form a
closed electron shell (i.e., the “diamagnetic Mu state”). However, when the
unpaired electron is localized at adjacent atoms as shown above, Mu undergoes
relatively large HF interaction with the unpaired electron, comprising a
paramagnetic defect called a “muonated radical” (denoted Mu$\dot{R}$).

In general, the spin Hamiltonian for Mu$\dot{R}$ under an external magnetic
field $B$ (parallel with $z$) is written

$\mathcal{H}/\hbar=\frac{1}{4}\omega_{0}{\bm{\sigma}}\cdot{\bf\tau}+\frac{1}{4}\omega_{*}({\bm{\sigma}}\cdot{\bm{n}})({\bm{\tau}}\cdot{\bm{n}})-\frac{1}{2}\omega_{\mu}\sigma_{z}+\frac{1}{2}\omega_{e}\tau_{z}$
$+\sum_{m}[\Omega_{\perp_{m}}{\bm{S}}_{e}\cdot{\bm{I}}_{m}+(\Omega_{\parallel_{m}}-\Omega_{\perp_{m}})S_{e}^{z}I_{m}^{z}]-\sum_{m}\omega_{n_{m}}I_{m}^{z},$
(1)

where $\omega_{0}$ and $\omega_{*}$ respectively denote the transverse
($\omega_{\perp}$) and anisotropic ($\omega_{\parallel}-\omega_{\perp}$) parts
of the HF interaction, $\omega_{\mu}=\gamma_{\mu}B$ with
$\gamma_{\mu}=2\pi\times 135.53$ MHz/T being the muon gyromagnetic ratio,
$\omega_{e}=\gamma_{e}B$ with $\gamma_{e}=2\pi\times 28024.21$ MHz/T being the
electron gyromagnetic ratio, ${\bm{\sigma}}$ and ${\bm{\tau}}$ are the Pauli
spin operators for muon and electron, and ${\bm{n}}$ is a unit vector along
the symmetry axis of the Mu$\dot{R}$ state Percival (1979); Patterson (1988).
$\Omega_{\perp_{m}}$ and $\Omega_{\parallel_{m}}-\Omega_{\perp_{m}}$ are the
transverse and anisotropic parts of the NHF parameter for the $m$th nuclear
spin $I_{m}$ ($=\frac{1}{2}$ for 1H), $\omega_{n_{m}}=\gamma_{n_{m}}B$ with
$\gamma_{n_{m}}$ being the gyromagnetic ratio for the corresponding nuclei;
$\Omega_{\parallel_{m}}$ and $\Omega_{\perp_{m}}$ are usually dominated by the
Fermi contact interaction with an order of magnitude smaller contributions
from magnetic dipolar interaction.

Figure 1(a) shows typical $\mu$SR time spectra under ZF, LF ($B_{\rm
LF}=5$–400 mT), and TF ($B_{\rm TF}=2$ mT) [the portion for $A(t)>0$] observed
at 100 K. These spectra consist of two components: the one that shows the
Larmor precession under $B_{\rm TF}$ with a small fraction of exponential
damping and another component with gradual recovery of $A(0)$ with increasing
$B_{\rm LF}$. The former includes background due to muons stopped in the
materials around the sample, and its partial asymmetry ($A_{\rm b}$) is
estimated to be $\approx 0.067$ from calibration measurements. The initial
asymmetry corresponding to 100% spin polarization was $A_{0}\simeq 0.23$ on
the blank sample holder.

In order to use Mu as a probe, it is often crucial to know the electronic
state of Mu in the target material. To this end, $\mu$SR measurements under a
weak transverse field ($B_{\rm TF}$) is useful to evaluate the yield of the
diamagnetic Mu state (Mu+ or Mu- for the isolated Mu) by the initial asymmetry
distribution in the $\mu$SR time spectra:

$A(t)\simeq A_{\rm p}\sum_{i,j,k}f_{ijk}\cos\omega^{\prime}_{ijk}t+A_{\rm
d0}e^{-\lambda t}\cos\omega_{\mu}t,$ (2)

where $A_{\rm p}$ [$\equiv A(0)-A_{\rm d0}$] is the partial asymmetry of
paramagnetic component (= Mu$\dot{R}$), $A_{\rm d0}$ is that of diamagnetic
component with the damping rate $\lambda$, $\omega^{\prime}_{ijk}$ and
$f_{ijk}$ are the transition frequencies between the relevant Mu energy levels
and their relative amplitudes ($i,j=1$–4 for the HF levels, $k$ is the number
of sub-levels due to the NHF interaction which depends on ${\bm{I}}_{m}$)
Patterson (1988). In the present measurements, the depolarization induced by
the broad distribution of $\omega^{\prime}_{ijk}$ due to the NHF interaction
far exceeds the time resolution determined by the muon pulse width
($\tau\approx$80 ns at J-PARC, yielding the corresponding Nyquist frequency
$1/2\tau\approx 6.3$ MHz), and the first term is averaged out to yield

$A(t)\simeq A_{\rm d0}e^{-\lambda t}\cos\omega_{\mu}t.$ (3)

As shown below, it was inferred from the analysis of ZF spectra at 100–150 K
that $\lambda$ is attributed to the background signal from muons stopped in
the thermal shield (made of copper) around the sample with a small partial
asymmetry $A_{s}\approx 0.03$ (which comprises a part of $A_{\rm b}$).
Therefore, we rewrite Eq. (3) as follows:

$A(t)\simeq[A_{\rm d}+(A_{\rm h}+A_{\rm s}e^{-\lambda_{\rm
TF}t})]\cos\omega_{\mu}t,$ (4)

where $A_{\rm d}=A_{\rm d0}-A_{\rm b}$ refers to the contribution of the P3HT
sample, and $A_{\rm h}=A_{\rm b}-A_{\rm s}$ to that of silver sample holder.
The curve fitting using Eq. (4) showed excellent agreement with $A_{\rm s}$
fixed to 0.03, suggesting that the diamagnetic component coming from the
sample shows negligible relaxation. $A_{\rm d}$ increased slightly from
$\sim$0.08 at 100 K to $\sim$0.10 as temperature increased [see Figs. 2(a)].
In case $\lambda$ could be entirely attributed to the sample, the upper bound
of the relaxation rate for the $A_{\rm d}$ component is estimated by
$\lambda_{\rm d}=\lambda\cdot A_{\rm d0}/A_{\rm d}$, which is also shown in
Fig. 2(b). From these evaluations, we conclude that the spin relaxation in the
diamagnetic Mu contribute little to the ZF/LF-$\mu$SR spectra at least above
200 K.

Figure 1:  (a) TF- and ZF/LF-$\mu$SR time spectra observed at typical
temperatures, which consists of paramagnetic ($A_{\rm p0}$) and diamagnetic
($A_{\rm d0}$) components, where the latter involves contribution from copper
thermal shield showing relaxation ($A_{\rm s}$). The solid curves represent
the least-square fits by Eqs. (3) and (5). The horizontal dashed/dot-dashed
lines show $A_{\rm d}$ and $A_{\rm s}$ determined by TF/ZF-$\mu$SR
measurements.  Figure 2: (a)–(b) Temperature dependence of the parameters for
the diamagnetic Mu in P3HT: (a) partial asymmetry ($A_{\rm d}$), (b)
transverse relaxation rate ($\lambda_{\rm TF}$) for the $A_{\rm s}$ component,
and the upper bound of relaxation rate ($\lambda_{\rm d}$). The relaxation
rate under a zero field ($\lambda_{\rm ZF}=1/T_{\rm 1\mu}$ in Eq. (5) with
$B_{\rm LF}=0$) is also shown in (b) for comparison. (c) An example for the
initial asymmetry versus $B_{\rm LF}$ for the paramagnetic component obtained
at 100 K, where solid line shows the result of fitting using Eq. (6), dashed
line shows the case without NHF interaction, and dot-dashed line is for the
case of fast fluctuations ($\nu\gg\omega_{0}$). (d) Temperature dependence of
$\Delta_{\rm n}$ obtained by curve fitting. (e) Crystal structure of P3HT with
yellow, brown, and white balls representing S, C, and H (after Ref.Yao  _et
al._ (2021)), and (f) a schematic illustration of the local electronic
structure for Mu$\dot{R}$ and associated NHF interactions with nearby protons
(arrows indicate their nuclear spins).

In the ZF spectra at 100–150 K, the slowly relaxing $A_{\rm s}$ component is
well reproduced by the dynamical Kubo-Toyabe (KT) function, $G_{z}^{\rm
KT}(t)=G_{z}^{\mathrm{KT}}(t;\sigma,\nu_{\rm d})$ (with $\nu_{\rm d}$ being
the fluctuation rate of $\sigma$), and its linewidth ($\sigma\simeq 0.37$ MHz)
identifies it as coming from muons stopped in a copper thermal shield Kadono
_et al._ (1989): the remaining non-relaxing part ($A_{\rm d}+A_{\rm h}$) is
attributed to the diamagnetic Mu in P3HT and silver sample holder. We found
that the KT function can be approximated by an exponential relaxation due to
relatively fast fluctuations ($\nu_{\rm d}\gtrsim\sigma$).

Here, let us summarize the separation/identification of $\mu$SR signal
components: There are four components (described by the partial asymmetry),
two from Mu$\dot{R}$ [$A_{\rm p}(x_{\rm p})$] and diamagnetic Mu ($A_{\rm d}$)
in the P3HT sample, and two from diamagnetic Mu (as background) in the silver
sample holder ($A_{\rm h}$) and copper thermal shield ($A_{\rm s}$). Among
these, $A_{\rm p}(x_{\rm p})$ and $A_{\rm s}$ respectively exhibit a
characteristic field-dependence of $A_{\rm p}$ and ZF spin relaxation
described by the Kubo-Toyabe function, and are thereby readily distinguished.
In addition, the asymmetry of the background ($A_{\rm b}=A_{\rm h}+A_{\rm s}$)
and that of the entire diamagnetic component ($A_{\rm d0}=A_{\rm d}+A_{\rm
b}$) are determined by experiments, allowing the identification of the P3HT-
derived diamagnetic Mu as $A_{\rm d}=A_{\rm d0}-A_{\rm b}$. Since the
relaxation rate for $A_{\rm h}$ (high-purity silver) is below the lower limit
of the present experiment, the only uncertainty is the distinction between the
spin relaxation rate exhibited by $A_{\rm s}$ and that of $A_{\rm d}$
(particularly above $\sim$200 K), and to account for this the spin relaxation
rate of the latter ($\lambda_{\rm d}$) is given as an upper bound.

The lineshape of the LF spectra suggests that the $A_{\rm p}$ component
consists of multiple components with different exponential relaxation rates
Risdiana  _et al._ (2010a), similar to those observed in NMR Yazawa  _et al._
(2010). Although the stretched exponential decay is often used in such
situation Mashita  _et al._ (2016), the relaxation was approximated by a
single exponential component to simplify the analysis. A closer look at the LF
spectrum shows that $A(0)$ in the spectrum at 0–10 mT is much more reduced
than $A_{\rm d0}+A_{\rm p0}/2$ expected for the spin triplet Mu$\dot{R}$,
where $A_{\rm p0}=A_{0}-A_{\rm d0}$. This indicates that the unpaired electron
is subjected to NHF interactions with the surrounding protons. Based on these
observations, the ZF/LF spectra were analyzed by least-squares fitting with
the following function:

$A(t)\simeq[A_{\rm p0}g_{z}(x_{\rm p})+A_{\rm s}]\cdot\exp[-t/T_{1\mu}]+A_{\rm
h},$ (5) $\displaystyle g_{z}(x_{\rm p})$ $\displaystyle=$
$\displaystyle\frac{\frac{1}{2}h_{z}(x_{\rm n})+x_{\rm p}^{2}}{1+x_{\rm
p}^{2}},\>\>x_{\rm p}=\gamma_{\rm av}B_{\rm LF}/\omega_{0}$ (6) $\displaystyle
h_{z}(x_{\rm n})$ $\displaystyle\simeq$ $\displaystyle\frac{\frac{1}{3}+x_{\rm
n}^{2}}{1+x_{\rm n}^{2}},\>\>x_{\rm n}\simeq\gamma_{\rm av}B_{\rm
LF}/\Delta_{\rm n},$ (7)

where $g_{z}(x_{\rm p})$ is the initial polarization of Mu$\dot{R}$ as a
function of the normalized field $x_{\rm p}$ Patterson (1988); we presume
$\omega_{0}\simeq 2\pi\times 220$ MHz and that $\omega_{*}$ is relatively
small ($\sim 2\pi\times 10$ MHz) based on unpublished data Pratt .
$1/T_{1\mu}$ is the relaxation rate which depends on the magnitude of $B_{\rm
LF}$. $h_{z}(x_{\rm n})$ is the approximated field dependence of initial
polarization under the NHF interaction characterized by the second moment
$\Delta_{\rm n}$, where the $\frac{1}{3}$ term corresponds to the possibility
that the direction of the NHF field is parallel with the initial spin
polarization of the triplet Mu$\dot{R}$ state Beck, Meier, and Schenck (1975),
$\gamma_{\rm av}=(\gamma_{e}+\gamma_{\mu})/2=2\pi\times 14.08$ MHz/mT is the
gyromagnetic ratio of the unpaired electron in the triplet state. This
approximation is also supported by the fact that the spin relaxation of the
triplet muonium (Mu0) in solid Kr is well represented by the quasi-static KT
relaxation function $G_{z}^{\rm KT}(t;\Delta_{\rm n},0)$ with
$\gamma_{\mu}B_{\rm LF}$ replaced by $\gamma_{\rm av}B_{\rm LF}$, where a
relatively small $\Delta_{\rm n}$ ($\approx 0.7$ MHz) due to dilute 83Kr
nuclei allows us to observe the detailed time evolution Storchak  _et al._
(1996). In the present experimental condition with relatively large
$\Delta_{\rm n}$, only the asymptotic behavior, $h_{z}(x_{\rm n})\approx
G_{z}^{\rm KT}(\Delta_{\rm n}t\gg 1)$, can be observed. (The exact form of
$h_{z}(x_{\rm n})$ under LF is found elsewhere Yaouanc and Dalmas de Réotier
(2010).) The NHF parameter in the static limit can be written more explicitly
using the parameters in Eq. (1) Kadono  _et al._ (1990),

$\Delta_{\rm
n}^{2}=\sum_{m}\frac{1}{3}(\Omega_{\perp_{m}}^{2}+2\Omega_{\parallel_{m}}^{2})\frac{I_{m}(I_{m}+1)}{3}.$
(8)

In the actual curve fits, we included the $A_{\rm s}$ component as a part of
relaxing term as shown in Eq. (5), considering the difficulty to distinguish
the relaxation between that exhibited by $A_{\rm s}$ and by the diamagnetic Mu
in the sample. Note that $\lambda_{\rm ZF}$ in Fig. 2(b) corresponds to the
decay of $A_{\rm p0}g_{z}(0)+A_{\rm s}$. The fact that $\lambda_{\rm ZF}$ is
much greater than $\lambda_{\rm TF}$ above 200 K indicates that $\lambda_{\rm
ZF}$ primarily reflects spin relaxation of the Mu$\dot{R}$ state.

As seen in Fig. 1, the results of the curve fitting (with $\omega_{0}$ fixed
to $2\pi\times 220$ MHz) reasonably reproduce the observed behavior of time
spectra: the global fits for the each set of data at a given temperature,
where common values of parameters ($A_{\rm p0}$, $\Delta_{\rm n}$, etc.) are
assumed for the time spectra with different LFs, allowed to deduce these
parameters without much uncertainty. The LF dependence of $A(0)$ and the
temperature dependence of $\Delta_{\rm n}$ obtained from these fits are shown
in Figs. 2(c) and 2(d). $\Delta_{\rm n}$ is least dependent on temperature,
indicating that the radical state is stable over the entire temperature range
of measurements. The mean value $\Delta_{\rm n}=2\pi\times 51.9(1.4)$ MHz is
consistent with the NHF parameters estimated by DFT calculations (50–55 MHz)
which is dominated by the contribution of the proton situated next to Mu Pratt
. The model of local electronic structure based on these results is shown in
Figs. 2(e) and 2(f), where Mu is bonded to the 4-position of the thiophene
ring with the neighboring proton making primary contribution to $\Delta_{\rm
n}$ Pratt .

The temperature and magnetic field dependence of $1/T_{1\mu}$ obtained
simultaneously in the above analysis is shown in Fig. 3(a) (where the ZF data
were excluded for the following analysis to minimize the influence of the
copper thermal shield). The dashed lines deduced from the curve fit (see
below) indicate that $1/T_{1\mu}$ at 200 K and below is strongly suppressed
for $B_{\rm LF}\gtrsim 10^{1}$ mT. Meanwhile, there is a marked increase in
$1/T_{1\mu}$ on the higher magnetic field side above 200 K, which accompanies
a change in the $B_{\rm LF}$ dependence from concave to convex behavior with
elevated temperature. These features seem to be difficult to describe with the
conventional model for the spin dynamics. Therefore, here we first return to
the basics of spin relaxation theory and introduce a model that can describe
the relaxation due to more general fluctuations, and then discuss the behavior
of $1/T_{1\mu}$ based on this model.

Since the spin relaxation is found to be dominated by that of Mu$\dot{R}$, it
is attributed to the fluctuation of HF and/or NHF fields $H(t)$. The
relaxation rate is derived from the autocorrelation function $C(t)=\langle
H(t)H(0)\rangle/\langle H(0)^{2}\rangle$ via the dynamical susceptibility

$\chi(\omega)=\frac{1}{2}\coth\left(\frac{\hbar\omega}{2k_{B}T}\right)\frac{\chi_{\rm
s}}{2\pi}\int C(t)e^{i\omega t}dt,$ (9) $\chi_{\rm s}=\Delta^{2}/k_{B}T,$ (10)

where the factor $\frac{1}{2}\coth(\frac{\hbar\omega}{2k_{B}T})$ comes from
the thermal average over the canonical ensemble (denoted by
$\langle...\rangle$) See  _et al._ (2020), $\chi_{\rm s}$ is the static
susceptibility obeying the Curie law, and $\Delta^{2}$ ($=\gamma_{\rm
av}^{2}\langle H(0)^{2}\rangle$) is the second moment of the fluctuating local
field. The general expression for the spin relaxation exhibited by Mu$\dot{R}$
is given in terms of the transitions between four energy levels,

$1/T_{1\mu}=\sum_{i,j}\frac{a_{ij}{\rm
Im}\>\chi(\omega_{ij})}{\frac{1}{2}\coth(\frac{\hbar\omega_{ij}}{2k_{B}T})}\approx\Delta^{2}\sum_{i,j}\frac{a_{ij}J(\omega_{ij})}{\omega_{ij}},$
(11)

where $J(\omega)$ is the spectral density corresponding to the imaginary part
of $\chi(\omega)$, $\omega_{ij}$ are the relevant Mu$\dot{R}$ Zeeman
frequencies with their respective amplitude $a_{ij}$.

Figure 3: (a) The longitudinal relaxation rate versus $B_{\rm LF}$ exhibited
by the Mu$\dot{R}$ state at various temperatures (semilog plot). The dashed
lines are the results of curve fits by the Havriliak-Negami type spectral
function against the magnetic field dependence (see text). (b) The angular
frequencies of Zeeman transitions ($\omega_{i,j}$) for Mu$\dot{R}$ in the
absence of NHF interaction ($\omega_{0}=2\pi\times$220 MHz).

In the conventional models for spin dynamics, the Lorentz-type dynamic
susceptibility, $\chi(\omega)=\chi_{\rm s}/(1-i\omega/\nu)$ derived from
$C(t)=\exp(-\nu t)$, is often used to yield
$J(\omega)=(\omega/\nu)/[1+(\omega/\nu)^{2}]$, which corresponds to the Debye
model in dielectric relaxation Debye (1929) (or that introduced by
Bloembergen-Purcell-Pound (BPP) or Redfield in NMR Bloembergen, Purcell, and
Pound (1948)). However, in complex systems such as polymers, the
characteristic fluctuation frequencies are expected to be widely distributed.
In this case, it is useful to resort to one of the most general forms that
introduced by Havriliak and Negami Havriliak and Negami (1967):

$\chi(\omega)=\chi_{\rm HN}(\omega)=\chi_{\rm
s}\frac{1}{[1-i(\omega/\tilde{\nu})^{\delta}]^{\gamma}}$ (12)

which leads to

$\frac{J(\omega)}{\omega}=\frac{\sin\gamma\theta}{\omega|z|^{\gamma}}\simeq\left\\{\begin{array}[]{ll}\frac{\omega^{\delta-1}}{\tilde{\nu}^{\delta}}&(\omega\ll\tilde{\nu})\\\
\frac{\tilde{\nu}^{\gamma\delta}}{\omega^{\gamma\delta+1}}&(\omega\gg\tilde{\nu})\end{array}\right.$
(13)

where $\gamma$ and $\delta$ are the generalized power indices
($0<\gamma,\delta\leq 1$), $\tilde{\nu}$ is the mean fluctuation frequency,
$|z|=(x^{2}+y^{2})^{1/2}$ with
$x=1+(\omega/\tilde{\nu})^{\delta}\cos(\delta\pi/2)$,
$y=(\omega/\tilde{\nu})^{\delta}\sin(\delta\pi/2)$, and
$\theta=\tan^{-1}(y/x)$. An important feature of Eq. (13) is that
$J(\omega)/\omega$ is maximized when $\omega=\tilde{\nu}$ (i.e., the “$T_{1}$
minimum” in NMR).

Introducing $\chi_{\rm HN}(\omega)$ is also useful in considering the
complexity that Mu$\dot{R}$ is subject to fluctuations of local fields at
multiple Zeeman frequencies for a single magnetic field [see Fig. 3(b)]. For
example, the minimum intra-triplet transition frequency $\omega_{12}$ is given
by the following equation:

$\omega_{12}=\frac{1}{2}\omega_{0}\left[1-(1+x_{\rm
p}^{2})^{1/2}\right]+\omega_{-},$ (14)

where $\omega_{-}=\gamma_{\rm av}B_{\rm LF}$. Therefore,
$\omega_{12}\simeq\omega_{-}$ for $x_{\rm p}\lesssim 1$ ($B_{\rm LF}\lesssim
18$ mT) and $\omega_{12}\simeq\omega_{0}/2-\omega_{\mu}$ for $x_{\rm p}\gg 1$
($B_{\rm LF}\gg 18$ mT), indicating that $\omega_{12}$ changes nonlinearly
with respect to $B_{\rm LF}$. Considering the foreseeable complexity for the
behavior of $\tilde{\nu}$ in polymers, it is not realistic to compare
$1/T_{1\mu}$ calculated using Eq. (11) with experimental values. Therefore, we
treat the obtained $1/T_{1\mu}$ as a function of the magnetic field, namely

$1/T_{1\mu}\simeq\Delta^{2}\frac{J(\omega)}{\gamma_{\rm av}B_{\rm
LF}},\>\>\>\>\omega/\tilde{\nu}=B_{\rm LF}/B_{\tilde{\nu}},$ (15)

and attempt to characterize it in terms of the parameters in the Havriliak-
Negami function; here, $B_{\tilde{\nu}}$ is the characteristic field defined
by $\tilde{\nu}=\gamma_{\rm av}B_{\tilde{\nu}}$. We then discuss the nature of
the fluctuations while semi-quantitatively considering the relationship with
the Zeeman level of Mu$\dot{R}$.

The dashed lines in Fig. 3 show the results of curve fitting by the least-
squares method assuming Eqs. (13) and (15). These curves reasonably reproduce
the magnetic field dependence of the data, although the corresponding
parameters exhibit considerable uncertainty due to the relatively strong
mutual correlations. The temperature dependence of $\gamma$, $\delta$,
$B_{\tilde{\nu}}$, and $\Delta$ obtained from this analysis is shown in Fig.
4.

Figure 4:  The parameters in the Havriliak-Negami function obtained by curve
fits shown in Fig. 3: (a) power indices, (b) mean field (with corresponding
mean frequency in the right axis), and (c) hyperfine parameter. (d) Dynamic
loss modulus ($E^{\prime\prime}$) for a P3HT film at a frequency of 20 Hz,
where three relaxation peaks corresponding to $\beta$, $\alpha_{1}$, and
$\alpha_{2}$ processes are observed at 200, 300 and 390 K Ogata, Kawaguchi,
and Tanaka (2015). The colored bands (200–300 K and above $\sim$300 K)
indicate the temperature regions discerned by the difference in the mode of
molecular motion inferred from 13C-NMR Yazawa  _et al._ (2010) and
$E^{\prime\prime}$.

According to previous 13C-NMR and FT-IR studies, the behavior of the P3HT
chains can be roughly divided into three regions in the temperature range
observed here. At low temperatures below 200 K, both the thiophene ring and
the hexyl side chain are in a frozen state, while at 200–300 K, only the hexyl
side chain is in motion (crystalline state). It is further suggested that
above 300 K, in addition to the motion of the hexyl side chain, the twisting
motion of the thiophene ring is also excited (plastic crystal state). It is
noteworthy that most of the parameters in Fig. 4 appear to exhibit
correlations with such a change in the motional mode: $B_{\tilde{\nu}}$ and
$\Delta$ gradually increases above 200 K; $\delta$ and $\gamma$ respectively
exhibit a slight increase above 200 K and 300 K (although the behavior is
blurred by large error bars reflecting uncertainty in the curve fits). These
correlations suggest that the spin relaxation seen in $\mu$SR originates from
molecular motion. Note that the values of $\gamma$ and $\delta$ are similar to
those of other polymers obtained from dielectric relaxation analysis at the
same ambient temperature Havriliak and Negami (1967).

Let us now examine these changes more quantitatively: in Eq. (13), the
relatively small $\gamma$ and $\delta$ below $\sim$200 K indicate that
$J(\omega)$ spreads over the wide region of $\omega$ centered at
$\tilde{\nu}$. Thus, they imply a slightly broader spectral density
distribution at low temperatures. However, above 200 K, $\delta\simeq 1$ and
$B_{\tilde{\nu}}$ shows gradual increase from $\sim$101 to $10^{2}$ mT. This
suggests that fluctuations with frequencies around $\tilde{\nu}$ (see right
vertical axis of Fig. 4(b)) become dominant in $J(\omega)$ in this temperature
region.

It is reported in the previous 13C-NMR study that the temperature dependence
of the longitudinal relaxation rate ($1/T_{1}$) for the 13C signals in the
hexyl side chain exhibit broad peaks centered around 300 K (i.e., the $T_{1}$
minimum), which indicates that the fluctuation rate around this temperature is
equal to the Zeeman angular frequency $\omega_{\rm NMR}\approx 10^{3}$ MHz
Yazawa  _et al._ (2010). This is in reasonable agreement with
$\tilde{\nu}\simeq 10^{3}$–$10^{4}$ MHz at 200–300 K, and suggesting that
$1/T_{1\mu}$ is also dominated by hexyl chain motion. Above $\sim$300 K,
$B_{\tilde{\nu}}$ reaches 3–$4\times 10^{2}$ mT ($\tilde{\nu}\gtrsim 10^{4}$
MHz) with $\Delta$ exceeding $\Delta_{\rm n}$, suggesting that the
contribution of thiophene ring motion sets in. The 13C-NMR results also
suggest that $1/T_{1}$ of 13C signals in the thiophene ring continues to
increase with increasing temperature over the range from 170 K to 370 K, where
the fluctuation rate of the thiophene ring increases on a scale 1–2 orders of
magnitude smaller than that of the hexyl side chain. This indicates that the
contribution of the thiophene ring motion to $1/T_{1\mu}$ is expected to be
observed only at higher temperatures above $\sim$300 K, which is also
consistent with the present experimental results and their interpretation.

##  IV DISCUSSION

By introducing the Havriliak-Negami function into the spectral density
function $J(\omega)$ describing $1/T_{1\mu}$, we were able to identify
molecular motion as the microscopic origin of $1/T_{1\mu}$ on the semi-
quantitative basis. In the following, we will examine the consistency of
parameters in the dynamical model with respect to the mean fluctuation
frequency $\tilde{\nu}$.

As shown in Fig 4(c), $\Delta$ is considerably smaller than $\Delta_{\rm n}$
below 200 K ($\Delta\approx 2\pi\times 20$ MHz), and gradually increases with
increasing temperature. This is in contrast to the naive expectation that
$\Delta$ is independent of temperature and equal to $\Delta_{\rm n}$. However,
as can be inferred from Figs. 2(e) and (f), the motion of molecules in the
crystal is spatially restricted, so that the fluctuations in $H(t)$ induced by
the associated relative motion of protons and unpaired electrons are also
partially suppressed. Under such circumstances, a part of the autocorrelation
of $H(t)$ is non-vanishing for $t\rightarrow\infty$, and the observed $\Delta$
may correspond to the magnitude of the remaining fluctuating component. More
specifically, in the case of Lorentz-type $\chi(\omega)$, this situation can
be described by the Edwards–Anderson order parameter $Q$ in the
autocorrelation function Edwards and Anderson (1975, 1976),

$\frac{\langle H(t)H(0)\rangle}{\langle H(0)^{2}\rangle}=\frac{\langle
H(t)H(0)\rangle}{\Delta_{\rm n}^{2}/\gamma_{\rm av}^{2}}=Qe^{-\nu t}+(1-Q).$
(16)

The fact that $\Delta$ is smaller than $\Delta_{\rm n}$ is consistent with
this situation, $\Delta=\sqrt{Q}\Delta_{\rm n}$, where
$\sqrt{Q}=\Delta/\Delta_{\rm n}\approx 0.4$ (i.e., $Q\approx 0.16$) below 200
K. It is then interpreted that $\Delta$ below $\sim$200 K mainly reflects
relatively restricted fluctuations of protons on the hexyl chain which becomes
progressively more intense as the temperature rises above $\sim$200 K.

On the other hand, it is also possible that $\Delta$ is dominated by the
fluctuation of $\omega_{0}$ (and/or $\omega_{*}$), as suggested for isolated
molecules in the gas phase or in solution Fleming  _et al._ (1996); McKenzie
(2011). In this case, $\Delta$ is expected to be as large as $\omega_{0}$ for
free rotation/vibration of molecules which is far greater than the observed
$\Delta$. However, provided that only a tiny part of it fluctuates for subtle
twist motions, etc., we can use the same scenario as for the NHF field above,
where $\Delta\approx\sqrt{Q}\omega_{0}$. Assuming an even smaller value of $Q$
than in the NHF case [$\approx 0.01$ for the Lorentz-type $\chi$] is
sufficient to explain the observed magnitude of $\Delta$. The trend seen in
Fig. 4(c) of $\Delta$ increasing over $\Delta_{\rm n}$ above 300 K may suggest
an additional contribution from the fluctuating HF field.

Now, the above interpretation gives rise to another issue regarding the
behavior of the initial asymmetry $A_{\rm p}$ vs. $B_{\rm LF}$. As can be seen
from Figs. 4(b) and (c), the mean frequency $\tilde{\nu}$ is larger than
$\Delta$ at all temperatures, indicating that spin relaxation is subject to
the motional narrowing effect. This appears at first glance to contradict the
behavior of $A_{\rm p}$ shown in Fig. 2(c) (which is common over the entire
range of temperature, as is clear from Fig. 1). Namely, while $A_{\rm p}$ at
zero field would decrease to $A_{\rm p0}g_{z}(0)\approx A_{\rm p0}/6$ for
$\tilde{\nu}\lesssim\Delta$ due to relaxation by the static distribution of
$H(t)$, the effective $\Delta$ is expected to decrease due to so-called
motional narrowing for $\tilde{\nu}>\Delta$, and $A_{\rm p}$ at zero field is
expected to recover to $A_{\rm p0}/2$.

However, such a quasistatic behavior of $A_{\rm p}$ is expected when the
dynamical and static components coexist in the fluctuations of $H(t)$,
corresponding to $Q<1$ in Eq. (16). We extended the recently proposed
dynamical model for diamagnetic Mu based on Eq. (16) Ito and Kadono (2024) to
the present case (where $\tilde{\nu}$ is replaced with $\nu$) and the
simulation results are shown in Fig. 5: when the full amplitude $\Delta_{\rm
n}$ fluctuates ($Q=1$), the initial polarization $g_{z}(x_{\rm p})$ converges
to $\frac{1}{2}$ at $x_{\rm p}=0$ for $\nu>\Delta_{\rm n}$, indicating that
the static depolarization due to $\Delta_{\rm n}$ is quenched by the motional
narrowing. On the other hand, for $Q=0.2$, the field dependence of
$g_{z}(x_{\rm p})$ is close to that for the quasistatic muonium and almost
independent of $\nu$. Similarly, when $\omega_{0}$ in Mu$\dot{R}$ fluctuates
in its full magnitude (i.e., $Q=1$), it is predicted that $g_{z}(x_{\rm
p})\rightarrow 1$ for $\nu\gg\omega_{0}$, i.e., Mu$\dot{R}$ becomes
effectively equivalent with the diamagnetic Mu in terms of magnetic response.
(This situation cannot be discerned by the behavior of $1/T_{1\mu}$ alone, so
that special care must be taken for that of the initial asymmetry.)

Figure 5:  Simulation of the initial polarization $g_{z}(x_{\rm p})$ for
Mu$\dot{R}$, where $\omega_{0}=2\pi\times 220$ MHz and $\Delta_{\rm
n}=2\pi\times 50$ MHz. The longitudinal field dependence of $g_{z}(x_{\rm p})$
corresponding to fluctuations $\nu=0$ and 5 GHz is shown for $Q=1$ (a) and 0.2
(b).

The same situation has already been demonstrated in the simulation for the
diamagnetic Mu which exhibits depolarization due to random local fields from
surrounding nuclear magnetic moments. The behavior of relaxation function with
respect to $\nu$ deviates from that of the dynamical KT function for $Q<1$,
and it becomes less dependent on $\nu$ and converges to a quasi-static KT
function as $Q$ approaches zero. An important point to note in this simulation
is that $Q=1$ is expected when the fluctuations are due to self-diffusion of
the diamagnetic Mu, while $Q<1$ when they are governed by the dynamics of the
surrounding ions Ito and Kadono (2024). Since the similar situation can be
presumed for the paramagnetic Mu, the small $Q$ suggested in P3HT can be
regarded as evidence that the cause of the fluctuations is not the hopping
motion of Mu$\dot{R}$ itself.

In view of the above discussion, it is necessary to reconsider the
interpretation claimed in previous studies that the muon $1/T_{1\mu}$ observed
in P3HT is due to one-dimensional (1D) motion of unpaired electron on the
conjugated molecular chain Risdiana  _et al._ (2010a, b). The main rationale
in these studies is that the $B_{\rm LF}$ dependence of $1/T_{1\mu}$ at low
temperatures follows $J(\omega)/\omega\propto\omega^{-1/2}$ as theoretically
predicted for 1D jumping motion of electrons Risch and Kehr (1992). However,
the linewidth $\Delta$ for the corresponding fluctuation is expected to be
comparable to $\omega_{0}$ with large $Q$ ($\simeq 1$). Moreover, the 1D jump
frequency is reported to be $\nu\simeq 5.3\times 10^{14}$ s-1 from NMR Mabboux
_et al._ (1995), which is much greater than $\omega_{0}$ ($\simeq 1.38\times
10^{9}$ s-1). Therefore, if the unpaired electron associated with Mu$\dot{R}$
are moving at such high jumping frequencies, it is expected that $A_{\rm
p}\simeq A_{\rm p0}$ [i.e., $g_{z}(x_{\rm p})\simeq 1$] regardless of $B_{\rm
LF}$; see the dashed line in Fig. 2(c). A similar behavior of $g_{z}(x_{\rm
p})$ has also been shown theoretically Nosov and Yakovleva (1963); Risch and
Kehr (1992); Patterson (1988), which is inconsistent with the observations
that $A_{\rm p}$ strongly depends on $B_{\rm LF}$. Another issue is that they
report the large deviation of $1/T_{1\mu}$ from this $\omega^{-1/2}$ behavior
around room temperature, which they attribute to the onset of three-
dimensional (3D) diffusion of carriers Risdiana  _et al._ (2010a, b). However,
this is not consistent with the implication from NMR that the 1D-3D anisotropy
remains large ($D_{\parallel}/D_{\perp}\sim 10^{6}$) in the relevant
temperature range Mabboux  _et al._ (1995). Needless to mention, there is no
such contradiction in our interpretation that $T_{1\mu}$ is due to molecular
motion rather than unpaired electron motion.

##  V Summary & Conclusion

We have shown that muons implanted to crystalline P3HT fall into a muoniated
radical state and a diamagnetic state, and that the unpaired electron in the
former undergoes HF interaction with muons bound to the thiophene ring and NHF
interaction with surrounding protons. Furthermore, we have succeeded in
observing the local molecular motions of P3HT from longitudinal spin
relaxation due to fluctuations of the HF and/or NHF fields. By introducing the
Havriliak-Negami function for the interpretation of the spectral density
$J(\omega)$ inferred from the LF-$\mu$SR measurements, we have been able to
make a semi-quantitative correspondence between the phenomenology of the
spectral density in dielectric relaxation and the microscopic origin of muon
spin relaxation due to molecular motion. This also served to establish a basis
for comparison of $\mu$SR result with those of other microscopic probes such
as NMR. The detailed magnetic field dependence measurements of LF-$\mu$SR have
revealed a qualitative change in $J(\omega)$ with temperature, which is
consistent with the 13C-NMR result that revealed the hexyl chain motion at
200-300 K and the twisting motion of the thiophene ring above 300 K. Similar
studies are expected to be widely applicable to other macromolecular systems.

###### Acknowledgements.

This work was supported by the Photon and Quantum Basic Research Coordinated
Development (QBRCD) Program from the Ministry of Education, Culture, Sports,
Science, and Technology (MEXT), and partly by the JST-Mirai Program
(JPMJMI18A2), Japan. The authors would like to thank F. Pratt for providing
information on $\mu$SR experiment and DFT calculations for muonated radicals
in P3HT prior to publication. Thanks are also to H. Seto for acquisition of
the financial support from the QBRCD Program, and to the MUSE staff for the
help during experiment at MLF, J-PARC. The muon experiment was conducted under
the support of Inter-University-Research Programs by Institute of Materials
Structure Science, KEK (Proposal No. 2014MS03).

## References

## References

  * Günes, Neugebauer, and Sariciftci (2007) S. Günes, H. Neugebauer, and N. S. Sariciftci, “Conjugated polymer-based organic solar cells,” Chem. Rev. 107, 1324–1338 (2007).
  * Thompson and Fréchet (2008) B. C. Thompson and J. M. J. Fréchet, “Polymer–fullerene composite solar cells,” Angew. Chem. Inte. Ed. Engl. 47, 58–77 (2008).
  * Kim  _et al._ (2006) Y. Kim, S. Cook, S. M. Tuladhar, S. A. Choulis, J. Nelson, J. R. Durrant, D. D. C. Bradley, M. Giles, I. McCulloch, C.-S. Ha, and M. Ree, “A strong regioregularity effect in self-organizing conjugated polymer films and high-efficiency polythiophene:fullerene solar cells,” Nature Mater. 5, 197–203 (2006).
  * Reynolds, Thompson, and Skotheim (2019) J. R. Reynolds, B. C. Thompson, and T. A. Skotheim,  _Handbook of Conducting Polymers_ , 4th ed. (CRC Press, Boca Raton, 2019).
  * Dimitrakopoulos and Malenfant (2002) C. D. Dimitrakopoulos and P. R. L. Malenfant, “Organic thin film transistors for large area electronics,” Adv. Mater. 14, 99–117 (2002).
  * Braga and Horowitz (2009) D. Braga and G. Horowitz, “High-performance organic field-effect transistors,” Adv. Mater. 21, 1473–1486 (2009).
  * Yan  _et al._ (2009) H. Yan, Z. Chen, Y. Zheng, C. Newman, J. R. Quinn, F. Dötz, M. Kastler, and A. Facchetti, “A high-mobility electron-transporting polymer for printed transistors,” Nature 457, 679–686 (2009).
  * Burroughes  _et al._ (1990) J. H. Burroughes, D. D. C. Bradley, A. R. Brown, R. N. Marks, K. Mackay, R. H. Friend, P. L. Burns, and A. B. Holmes, “Light-emitting diodes based on conjugated polymers,” Nature 347, 539–541 (1990).
  * Krummacher  _et al._ (2006) B. C. Krummacher, V.-E. Choong, M. K. Mathai, S. A. Choulis, F. So, F. Jermann, T. Fiedler, and M. Zachau, “Highly efficient white organic light-emitting diode,” Appl. Phys. Lett. 88, 113506 (2006).
  * Naber, Faez, and van der Wiel (2007) W. J. M. Naber, S. Faez, and W. G. van der Wiel, “Organic spintronics,” J. Phys. D: Appl. Phys. 40, R205 (2007).
  * Ed. (2010) Z. V. V. Ed.,  _Organic spintronics_ (CRC Press, Boca Raton, 2010).
  * Geng  _et al._ (2016) R. Geng, H. M. Luong, T. T. Daugherty, L. Hornak, and T. D. Nguyen, “A review on organic spintronic materials and devices: II. magnetoresistance in organic spin valves and spin organic light emitting diodes,” J. Sci.: Adv. Mater. Dev. 1, 256–272 (2016).
  * Kuribara  _et al._ (2012) K. Kuribara, H. Wang, N. Uchiyama, K. Fukuda, T. Yokota, U. Zschieschang, C. Jaye, D. Fischer, H. Klauk, T. Yamamoto, K. Takimiya, M. Ikeda, H. Kuwabara, T. Sekitani, Y.-L. Loo, and T. Someya, “Organic transistors with high thermal stability for medical applications,” Nature Commun. 3, 723 (2012).
  * Kaltenbrunner  _et al._ (2013) M. Kaltenbrunner, T. Sekitani, J. Reeder, T. Yokota, K. Kuribara, T. Tokuhara, M. Drack, R. Schwödiauer, I. Graz, S. Bauer-Gogonea, S. Bauer, and T. Someya, “An ultra-lightweight design for imperceptible plastic electronics,” Nature 499, 458–463 (2013).
  * Chen, Wu, and Rieke (1995) T.-A. Chen, X. Wu, and R. D. Rieke, “Regiocontrolled synthesis of poly(3-alkylthiophenes) mediated by rieke zinc: Their characterization and solid-state properties,” J. Am. Chem. Soc. 117, 233–244 (1995).
  * Bao, Dodabalapur, and Lovinger (1996) Z. Bao, A. Dodabalapur, and A. J. Lovinger, “Soluble and processable regioregular poly(3-hexylthiophene) for thin film field-effect transistor applications with high mobility,” Appl. Phys. Lett. 69, 4108–4110 (1996).
  * Sirringhaus  _et al._ (1999) H. Sirringhaus, P. J. Brown, R. H. Friend, M. M. Nielsen, K. Bechgaard, B. M. W. Langeveld-Voss, A. J. H. Spiering, R. A. J. Janssen, E. W. Meijer, P. Herwig, and D. M. de Leeuw, “Two-dimensional charge transport in self-organized, high-mobility conjugated polymers,” Nature 401, 685–688 (1999).
  * Curtis  _et al._ (2007) M. D. Curtis, J. I. Nanos, H. Moon, and W. S. Jahng, “Side chain disorder and phase transitions in alkyl-substituted, conjugated oligomers. relation to side-chain melting in P3ATs,” J. Am. Chem. Soc. 129, 15072–15084 (2007).
  * Bolognesi  _et al._ (1993) A. Bolognesi, W. Porzio, F. Provasoli, and T. Ezquerra, “The thermal behaviour of low-molecular-weight poly(3-decylthiophene),” Makromol. Chem. 194, 817–827 (1993).
  * Zhao  _et al._ (1997) Y. Zhao, D. Keroack, G. Yuan, A. Massicotte, R. Hanna, and M. Leclerc, “Melting behavior of poly(3-alkylthiophene)s with long alkyl side-chains,” Macromol. Chem. Phys. 198, 1035–1049 (1997).
  * Inganäs  _et al._ (1988) O. Inganäs, W. R. Salaneck, J.-E. Österholm, and J. Laakso, “Thermochromic and solvatochromic effects in poly(3-hexylthiophene),” Synth. Met. 22, 395–406 (1988).
  * Yazawa  _et al._ (2010) K. Yazawa, Y. Inoue, T. Shimizu, M. Tansho, and N. Asakawa, “Molecular dynamics of regioregular poly(3-hexylthiophene) investigated by NMR relaxation and an interpretation of temperature dependent optical absorption,” J. Phys. Chem. B 114, 1241–1248 (2010).
  * Ogata, Kawaguchi, and Tanaka (2015) Y. Ogata, D. Kawaguchi, and K. Tanaka, “An effect of molecular motion on carrier formation in a poly(3-hexylthiophene) film,” Sci. Rep. 5, 8436 (2015).
  * Tretiak  _et al._ (2002) S. Tretiak, A. Saxena, R. L. Martin, and A. R. Bishop, “Conformational dynamics of photoexcited conjugated molecules,” Phys. Rev. Lett. 89, 097402 (2002).
  * Mizes and Conwell (1993) H. A. Mizes and E. M. Conwell, “Stability of polarons in conducting polymers,” Phys. Rev. Lett. 70, 1505–1508 (1993).
  * Furukawa and Shimokawa (2023) Y. Furukawa and D. Shimokawa, “Polarons, Bipolarons, and Electrical Properties of Crystalline Conducting Polymers,” Bull. Chem. Soc. Jpn. 96, 1243–1251 (2023).
  * Havriliak and Negami (1967) S. Havriliak and S. Negami, “A complex plane representation of dielectric and mechanical relaxation processes in some polymers,” Polymer 8, 161–210 (1967).
  * Alvarez, Alegra, and Colmenero (1991) F. Alvarez, A. Alegra, and J. Colmenero, “Relationship between the time-domain kohlrausch-williams-watts and frequency-domain havriliak-negami relaxation functions,” Phys. Rev. B 44, 7306–7312 (1991).
  * Ito and Kadono (2024) T. U. Ito and R. Kadono, “Distinguishing ion dynamics from muon diffusion in muon spin relaxation,” J. Phys. Soc. Jpn. 93, 044602 (2024).
  * Risdiana  _et al._ (2010a) Risdiana, Fitrilawati, R. Hidayat, R. E. Siregar, M. O. Tjia, and I. Watanabe, “$\mu$SR study of electron radical dynamics in regio-regular polythiophene,” J. Phys.: Conf. Ser. 200, 052024 (2010a).
  * Risdiana  _et al._ (2010b) Risdiana, Fitrilawati, R. Hidayat, A. Nugroho, R. Siregar, M. Tjia, and I. Watanabe, “Intra- and inter-chain polaron diffusion in regio-random polythiophene studied by muon spin relaxation,” Physica B: Condens. Matt. 405, S381–S383 (2010b).
  * Kojima  _et al._ (2014) K. M. Kojima, T. Murakami, Y. Takahashi, H. Lee, S. Y. Suzuki, A. Koda, I. Yamauchi, M. Miyazaki, M. Hiraishi, H. Okabe, S. Takeshita, R. Kadono, T. Ito, W. Higemoto, S. Kanda, Y. Fukao, N. Saito, M. Saito, M. Ikeno, T. Uchida, and M. M. Tanaka, “New $\mu\text{SR}$ spectrometer at J-PARC MUSE based on Kalliope detectors,” J. Phys.: Conf. Ser. 551, 012063 (2014).
  * Suter and Wojek (2012) A. Suter and B. Wojek, “Musrfit: A free platform-independent framework for $\mu$sr data analysis,” Phys. Proc. 30, 69–73 (2012).
  * Nagamine  _et al._ (1984) K. Nagamine, K. Ishida, T. Matsuzaki, K. Nishiyama, Y. Kuno, T. Yamazaki, and H. Shirakawa, “Solitons in polyacetylene produced and probed by positive muons,” Phys. Rev. Lett. 53, 1763–1766 (1984).
  * Ishida  _et al._ (1985) K. Ishida, K. Nagamine, T. Matsuzaki, Y. Kuno, T. Yamazaki, E. Torikai, H. Shirakawa, and J. H. Brewer, “Diffusion properties of the muon-produced soliton in trans-polyacetylene,” Phys. Rev. Lett. 55, 2009–2012 (1985).
  * Pratt  _et al._ (1997) F. L. Pratt, S. J. Blundell, W. Hayes, K. Nagamine, K. Ishida, and A. P. Monkman, “Anisotropic polaron motion in polyaniline studied by muon spin relaxation,” Phys. Rev. Lett. 79, 2855–2858 (1997).
  * Blundell  _et al._ (2002) S. J. Blundell, F. L. Pratt, I. M. Marshall, C. A. Steer, W. Hayes, A. Husmann, C. Fischmeister, R. E. Martin, and A. B. Holmes, “Muon-spin relaxation study of anisotropic charge carrier motion in polyphenylene vinylene-based polymers,” J. Phys.: Condens. Matt. 14, 9987 (2002).
  * Pratt (2004) F. L. Pratt, “Muon spin relaxation as a probe of electron motion in conducting polymers,” J. Phys.: Condens. Matt. 16, S4779 (2004).
  * Percival (1979) P. W. Percival, Radiochimica Acta 26, 1–14 (1979).
  * Patterson (1988) B. D. Patterson, “Muonium states in semiconductors,” Rev. Mod. Phys. 60, 69–159 (1988).
  * Yao  _et al._ (2021) Z.-F. Yao, Q.-Y. Li, H.-T. Wu, Y.-F. Ding, Z.-Y. Wang, Y. Lu, J.-Y. Wang, and J. Pei, “Building crystal structures of conjugated polymers through x-ray diffraction and molecular modeling,” SmartMat 2, 378–387 (2021).
  * Kadono  _et al._ (1989) R. Kadono, J. Imazato, T. Matsuzaki, K. Nishiyama, K. Nagamine, T. Yamazaki, D. Richter, and J.-M. Welter, “Quantum diffusion of positive muons in copper,” Phys. Rev. B 39, 23–41 (1989).
  * Mashita  _et al._ (2016) R. Mashita, H. Kishimoto, R. Inoue, A. Koda, R. Kadono, and T. Kanaya, “Dynamics of polybutadiene reinforced with unsaturated carboxylate studied by muon spin relaxation ($\mu$SR),” Polymer 105, 510–515 (2016), this paper contains errors in eq. (5) that affect the results of the analysis.
  * (44) F. Pratt, private communication.
  * Beck, Meier, and Schenck (1975) R. Beck, P. F. Meier, and A. Schenck, “On the change of polarization of positive muons in alkali halides,” Z. Phys. B Condens. Matt. 22, 109–115 (1975).
  * Storchak  _et al._ (1996) V. Storchak, S. F. J. Cox, S. P. Cottrell, J. H. Brewer, G. D. Morris, and N. V. Prokof’ev, “Muonium localization in solid krypton,” Phys. Rev. B 53, 662–667 (1996).
  * Yaouanc and Dalmas de Réotier (2010) A. Yaouanc and P. Dalmas de Réotier,  _Muon Spin Rotation, Relaxation and Resonance: Applications to Condensed Matter_ , International Series of Monographs on Physics (Oxford University Press, Oxford, 2010).
  * Kadono  _et al._ (1990) R. Kadono, R. F. Kiefl, E. J. Ansaldo, J. H. Brewer, M. Celio, S. R. Kreitzman, and G. M. Luke, “Delocalization of muonium in NaCl,” Phys. Rev. Lett. 64, 665–668 (1990).
  * See  _et al._ (2020) See, for example, H. Takahashi, and Y. Tanimura, “Open quantum dynamics theory of spin relaxation: Application to $\mu$SR and low-field NMR spectroscopies,” J. Phys. Soc. Jpn. 89, 064710 (2020).
  * Debye (1929) P. Debye,  _Polar Molecules_ (Dover Publications, New York, 1929).
  * Bloembergen, Purcell, and Pound (1948) N. Bloembergen, E. M. Purcell, and R. V. Pound, “Relaxation effects in nuclear magnetic resonance absorption,” Phys. Rev. 73, 679–712 (1948).
  * Edwards and Anderson (1975) S. F. Edwards and P. W. Anderson, “Theory of spin glasses,” J. Phys. F: Metal Phys. 5, 965 (1975).
  * Edwards and Anderson (1976) S. F. Edwards and P. W. Anderson, “Theory of spin glasses. II,” J. Phys. F: Metal Phys. 6, 1927 (1976).
  * Fleming  _et al._ (1996) D. G. Fleming, J. J. Pan, M. Senba, D. J. Arseneau, R. F. Kiefl, M. Y. Shelley, S. F. J. Cox, P. W. Percival, and J.-C. Brodovitch, “Spin relaxation of muonium-substituted ethyl radicals (MuCH2ĊH2) in the gas phase,” J. Chem. Phys. 105, 7517–7535 (1996).
  * McKenzie (2011) I. McKenzie, “Spin relaxation of a short-lived radical in zero magnetic field,” Phys. Chem. Chem. Phys. 13, 1168–1173 (2011).
  * Risch and Kehr (1992) R. Risch and K. W. Kehr, “Direct stochastic theory of muon spin relaxation in a model for trans-polyacetylene,” Phys. Rev. B 46, 5246–5257 (1992).
  * Mabboux  _et al._ (1995) P.-Y. Mabboux, J.-P. Travers, Y. Nicolau, E. Samuelsen, P. Carlsen, and B. François, “Spin dynamics in poly-(3-alkylthiophenes) and in a block polystyrene - polythiophene copolymer,” Synth. Met. 69, 361–362 (1995).
  * Nosov and Yakovleva (1963) V. G. Nosov and I. V. Yakovleva, “Depolarization of $\mu^{+}$ mesons in solids,” Sov. Phys. JETP 16, 1236–1246 (1963).