FIG. 7. (Color online) (a) ZF-\( \mu \) SR time spectra in the temper-ature range between 4.2 K and 10 K. (b) \( \mu ^ { \circ } \) SR time spectra measured at 30 mK in the presence of various longitudinal fields (LFs) up to 100 mT. The inset shows the earlier part of the spectrum at ZF. The solid lines are the fitted results by using Eq. (6)..

the base temperature. The initial fast depolarization ex-hibits a gradual suppression with increasing LF and is nearly quenched at \( ^ { \dag } \) F = 100 mT. Surprisingly, we also observed a finite slope due to the residual magnetic fluc-tuation in these spectra at a large time, which is almost independent of the LF, even at 30 mK. These features can be observed from the fast damped 2/3 amplitude followed by a slowly relaxing 1 \( / \) 3 component originating from the magnetic ordered state with slow persistent spin dynamics.

In a previous \( \mu \) SR study, Sugiyama et al. analyzed their ZF-\( \cdot \mu \) SR spectra by using a combination of an expo-nentially relaxing cosine oscillation for the damped 2 \( / \) 3 component and an exponential function for the 1 \( / \) 3 tail signal.11 They showed that the ratio of the relaxation rate of the 1 \( / \) 3 tail component \( \lambda _ { \mathrm { t a i l } } \) to the precession frequency \( \omega _ { \mathrm { A F } } \) is close to unity, \( \lambda _ { \mathrm { t a i l } } / \omega _ { \mathrm { A F } } \, \sim \, 1 \), where \( \lambda _ { \mathrm { t a i l } } / \omega _ { \mathrm { A F } } \) corresponds to the field distribution normalized to the mean value of \( B _ { \mathrm { l o c } } \). However, their fitting function is based on an unique precession signal at a spontaneous local field for the conventional magnetic ordered state, \( \lambda _ { \mathrm { t a i l } } / \omega _ { \mathrm { A F } } \ll 1 \) . The observed broad \( B _ { \mathrm { l o c } } \) distribution and residual magnetic fluctuation strongly suggest that a highly inhomogeneous magnetic state appears in the ground state of \( \mathrm { S r } _ { 2 } \)\( \ldots \) 4.

The inset of Fig. 7(b) shows the earlier part of the ZF-\( \mu \) SR spectrum at 30 mK. As mentioned above, the GKT and LKT functions are widely attributed to the \( \mu \) SR spectra for random magnetism in dense and diluted spin systems. However, the ZF spectrum at 30 mK lacks a decrease that usually precedes the asymptotic tail for the GKT function, although it exhibits a Gaussian-like initial depolarization. We attributed this to a broad dis-tribution of \( B _ { \mathrm { l o c } } \), which is not characterized by a sin-gle root-mean-square width, and adopted the Gaussian-broadened Gaussian (GbG) function as a recursive func-tion for the curve fitting.20–22

The GbG function is described as one of the general-ized forms of the GKT function, where \( \Delta \) in Eq. (2) has a distribution with the mean value \( \Delta _ { 0 } \) and root-mean-square width W. \( G _ { \mathrm { G b G } } ( t ) \) for ZF is written as

\[ \begin{array} { c c } { G _ { \mathrm { G b G } } ( t ) = \displaystyle \frac { 1 } { 3 } + \frac { 2 } { 3 } \bigg ( \displaystyle \frac { 1 } { 1 + R ^ { 2 } \Delta _ { 0 } ^ { 2 } t ^ { 2 } } \bigg ) ^ { 3 / 2 } \bigg ( 1 - \displaystyle \frac { \Delta _ { 0 } ^ { 2 } t ^ { 2 } } { 1 + R ^ { 2 } \Delta _ { 0 } ^ { 2 } t ^ { 2 } } \bigg ) } \\ { \times \exp \! \left( - \displaystyle \frac { \Delta _ { 0 } ^ { 2 } t ^ { 2 } } { 2 ( 1 + R ^ { 2 } \Delta _ { 0 } ^ { 2 } t ^ { 2 } ) } \right) \! , \! \! \! \! } \end{array} \]

where R is the ratio of the distribution width, and \( R = W / \Delta _ { 0 } \). For \( R \) = 0, the GbG function is identical to the GKT function with a deep minimum after the ini-tial depolarization. The depth of the minimum decreases with increasing R and becomes almost independent of R for \( R > \) 1. Thus, we imposed the condition 0 \( \le R \le \) 1 in the following analysis. We note that the GbG function does not approach the LKT function, even for \( R \gg \) 1.

For the curve fitting, we presume that G(t) is primar-ily described by the product of the GbG function and a slow exponential decay due to the persistent spin dynam-ics in addition to a supplementary term for exponential damping:

\[ G ( t ) = f G _ { \mathrm { G b G } } ( t ) e ^ { - \lambda t } + ( 1 - f ) e ^ { - \lambda _ { \mathrm { e x t } } t } , \qquad ( 6 ) \]

where f is the volume fraction exhibiting inhomoge-neous magnetism, and \( \lambda \) is the relaxation rate reflecting the spin dynamics. The second exponential term with the relaxation rate \( \lambda _ { \mathrm { e x t } } \) is for extrinsic contributions originat-ing from the magnetically ordered state region with rel-atively fast persistent dynamics, which makes the above muon spin depolarization model unreasonable. The T dependence of f is plotted in the inset of Fig. 8. The in-homogeneous magnetism develops over the almost entire volume of the sample below 6 K. We obtained \( R = 0 . 7 ( 1 ) \) below 4.2 K, whereas R is unity at higher temperatures. This discrepancy might be caused by the lack of data for \( t \sim 0 . 1 \)\( \cdot \ \mu \) s that is masked by the limited time resolu-tion of the pulsed muon beam (∼100 ns). We note that the dynamical LKT function and dynamical spin glass models19 could not reproduce the spectra over the early time range.

The absence of decrease in the \( \mu \) SR time spectra tells us important aspects of the magnetic ground state. Noakes et al. reproduced the shallow decrease in the GbG