Absence of precession in the Zn-doped samples

In the Si-doped system, antiferromagnetic order was clearly exhibited as the spontaneous muon precession (Fig.59). In the Zn-doped systems, Néel order was not clear, although an existence of a static order below $T_{\rm N}$ was presented from the `1/3-component' (Fig.56) and the decoupling behaviors in longitudinal-fields (Fig.58). The absence of muon spin precession suggests that our Zn-doped sample has more randomness than the Si-doped crystal.

One apparent randomness of the Zn-doped samples appeared as the distribution of the Néel temperatures (Fig.57). The spread of $T_{\rm N}$, which probably originates from inhomogeneity of the sample, may have smeared out the $\mu$SR spectral structures, as discussed in the following.

We may suppose that the spread of $T_{\rm N}$ was caused by an inhomogeneity of the Zn concentration (x). Using the $x\!-\!T$ phase diagram shown in Fig.53, the distribution of the Néel temperatures ($\delta T_{\rm N}$) may be mapped to a fluctuation of the Zn concentrations. The result yields $\delta x\sim 0.5, 1$and 0.5 % for the x=2, 4 and 8 % systems, respectively. These variations of the Zn concentrations may be mapped to the spread of the field-width $\delta\!\Delta$ using the inset of Fig.57; the result becomes $\delta\!\Delta/\Delta= 0.16, 0.2$ and 0.04 for the x=2, 4 and 8 % samples, respectively. The above-mentioned spreads are all Gaussian standard deviation.

The $\mu$SR spectrum with the spread of internal fields can be obtained from a convolution:

where, we assumed the ideal muon relaxation $P_\mu^{\rm ideal}(t;f,\Delta)$ to be the relaxation observed in the Si 2% single crystal (eq.59). In the small inhomogeneity limit ($\delta\!f/f, \delta\!\Delta/\Delta\ll 1$), the integral is approximately performed as:

where the precession suffers extra damping as $\Delta^{\rm osc}\rightarrow\Delta^{\rm osc}+\pi\,\delta\!f$, due to the distribution of the frequencies.

In Fig.61, we show a simulated $\mu$SR spectrum for the Zn 4% doped system ($\delta\!f/f\approx\delta\!\Delta/\Delta= 0.2$), obtained from a numerical integration of eq.60. The precession became less obvious than the Si-doped crystal; this result implies that the macroscopic sample inhomogeneity may be one of the reasons for the absence of muon spin precession in the Zn-doped systems.

The effects of more microscopic randomness, such as substitutions to the spin site with the non-magnetic Zn ions, are not clear at the present stage.

  

Figure 61: A simulated $\mu$SR spectra for the Zn 4% system. Muon spin precession in our Zn-doped samples should be suppressed because of the trivial sample inhomogeneity. See the text for the simulation procedure.