Discussions

Comparing the temperature dependence of the muon T₁ relaxation rate (Fig.33) with that of ⁶³Cu-NMR (Fig.28), one question may arise: why the temperature dependence is opposite between $\mu$SR and NMR. In the NMR measurements, the T₁ relaxation increased at higher temperatures, as has been ascribed to the excitation over the spin gap, while in $\mu$SR, it decreased as shown in Fig.33. As discussed below, the qualitatively different temperature dependence of the T₁ relaxation rates may be attributed to the different scales of the experimental time windows.

The time window of the $\mu$SR method is typically $1 \ {\rm ns}- 10\ \mu{\rm s}$, and for NMR, it is $100\ \mu{\rm s}- 10^3\ {\rm s}$. Therefore, in the paramagnetic fluctuation regime, NMR is more capable of detecting faster fluctuations than $\mu$SR. As has been shown by gap excitation type temperature dependence [52], the ⁶³Cu nuclear spin relaxation is most likely caused by magnetic excited states which produce fast field fluctuations. From a simple scaling argument below, these excited states do not cause fast enough muon spin relaxation detectable in the $\mu$SR time window.

The only difference between the muon spin relaxation and the ⁶³Cu nuclear spin relaxation is the gyromagnetic ratio of the probe spins ($\gamma_{\mu}$ and $\gamma_{_{\rm Cu}}$) and the electron-nuclear spin coupling strength, which reflects the probe spin site. From previous Knight-shift and susceptibility measurements, the hyperfine coupling parameter between a ⁶³Cu nuclear spin and electron moments has been obtained as [52]:

where the suffix (c and ab) indicates the crystalline orientation of the parameter. The coupling between a muon spin and the electron moments is probably a dipolar coupling, and its magnitude can be estimated from the static field-width of the ordered 3-leg ladder system (Fig.29):

The gyromagnetic ratio of the two probe spins are [28,61]:

Using these parameters and the T₁ relaxation formula [34], the scaling factor of the muon/⁶³Cu T₁ relaxation rates is estimated as:

Since the ⁶³Cu T₁ relaxation rate in the 2-leg ladder system is $\stackrel{<}{\sim}4\times 10^2$ [sec^-1] (Fig.28), the corresponding muon spin relaxation rate should be $\sim 2\times 10^{-6}$ [$\mu s^{-1}$] at room temperature, and smaller at lower temperatures (see the right axis of Fig.28). These relaxation rates are too small for the $\mu$SR time window, and therefore, the magnetic excited states do not contribute to the muon spin relaxation.

  

Figure 34: Possible muon spin relaxation mechanism. The unpaired spins associated with the defects may cause muon spin relaxation.

Then, what would be a relevant relaxation mechanism for the muon spins? One possible scenario is that the muons detect the unpaired spins which are related to vacancies and defects in the system (Fig.34). From susceptibility measurements, the amount of native unpaired moments has been estimated as $\sim$0.26 at.% of the copper ions [51]. The idea of the relaxation from the unpaired moments qualitatively explains the general temperature and longitudinal field dependence of the muon relaxation rate; since the couplings between the unpaired spins are presumably small, these spins should remain paramagnetic down to the milli-Kelvin regime, giving a temperature-independent relaxation rate ($T\stackrel{<}{\sim}40$ K in Fig.33). At higher temperatures, the unpaired spins may have additional fluctuations related to the magnetic excited states; in this situation, the muon spin relaxation rate should decrease, and respond less to the longitudinal fields (see data at $T\stackrel{\gt}{\sim}40$ K in Fig.33), reflecting more dynamic local fields.

As shown in the inset of Fig.33, we analyzed the LF dependence of muon spin relaxation rate $\lambda (T\!\rightarrow\! 0)$ with the T₁ relaxation formula in dilute spin systems (eq.33). The resulting Lorentzian field width (a) and the field fluctuation rate ($\nu$) are shown in Table 4. In the same table, estimates of these parameters are given, which are based on the unpaired spin concentration from the susceptibility measurements (see the caption of Table 4 for detail.) Although the quantitative agreement between the estimates and the experimental results is poor, it is worth while to point out that the unpaired spins assumed here will induce a muon spin relaxation rate ($\lambda=4a^2/\nu\approx 5.7\times 10^{-3}\ \mu{\rm s}^{-1}$) which is in the time range of $\mu$SR measurements. Therefore, the scenario of muon spin relaxation from the unpaired spins is better than that from the excited states. See the concluding remark at Chapter 7 for more discussions.

 

Parameter Experiment Estimate (native spins)

$$a\ (\mu s^{-1}) 0.085^{\rm a}$$ 3.1(1)

$$\nu 5.08^{\rm b}$$ 51(6)

^a An estimate using eq.26. For the calculation, the hypothetical Gaussian width ($\Delta_{100\%}$) with all the spin-cites filled with static moments was taken as the Gaussian width observed in the ordered 3-leg ladder system ($\Delta(T\!\rightarrow\! 0)=26\ \mu{\rm s}^{-1}$; see Fig.29b). The unpaired spin concentration ($c=0.26\%$) has been estimated from the susceptibility measurements [51]. ^b The dipolar fluctuation rate of the unpaired spins, which is $\sim 60\ (\approx \sqrt{10/3}/2\pi\times\gamma_e/\gamma_\mu)$ times larger than the Lorentzian width (a). See the discussion in Chapter 5 for more details of the estimation procedure.