5.2.3 Discussion

We compare in Table 3 the ordering temperatures of the Sr(Ca)-Cu-O series of compounds, which is a one-dimensional chain system for Sr₂CuO₃ [55], and a two-dimensional layer system for Ca_0.86Sr_0.14CuO₂ (n$\rightarrow$$\infty$ structure of the spin-ladder series) [56]. The ordering temperature of the 3-leg ladder system is between those of the chain system (Sr₂CuO₃) and the layer system (Ca_0.86Sr_0.14CuO₂). Suppression of $T_{\rm N}$ in the chain system has been explained as an effect of the low-dimensionality [55]; the intermediate ordering temperature for the 3-leg system is consistent with this idea, because the 3-leg ladder structure is the smallest extension of the one-dimensional chain toward two-dimensionality.

 

compound structure magnetic order

$$T_{\rm N} \sim ^{\rm a}$$ Sr₂CuO₃ CuO chain (1D)

$$\sim \pm ^{\rm b}$$ Sr₄Cu₆O₁₀ 3-leg ladder

$$T_{\rm N} \sim ^{\rm c}$$ Ca_0.86Sr_0.14CuO₂ CuO₂ plain (2D)

$^{\rm a}$Obtained by $\mu$SR [55].
$^{\rm b}$Obtained by $\mu$SR: this work [1].
$^{\rm c}$Obtained by neutron scattering [57] and $\mu$SR [55].

Another feature of the magnetic order in the 3-leg ladder system is the absence of critical slowing down of the Cu moments in the $\mu$SR time window ($\sim 10\mu$s). In Fig.31, we show $\mu$SR spectra at long times. No enhancement of the relaxation rate was observed around the transition temperatures. This `abrupt' onset of magnetic order has been observed in other one-dimensional systems, such as antiferromagnets Sr<sub>2</sub>CuO<sub>3</sub> [55], Ca<sub>2</sub>CuO<sub>3</sub> [58] and a spin-density-wave system (TMTSF)<sub>2</sub>-PF<sub>6</sub> [59]. In these systems, the absence of critical slowing down have been explained as follows; in one-dimensional systems, the ordering temperature is determined by the small inter-chain interactions, and therefore, there may be preformed short-range ordered regions before the three-dimensional order takes place [60]. Then the ordering process is a phase-lockings of these short-range ordered regions, which do not involve a slowing down of the moments. The absence of the critical slowing down in the 3-leg ladder system may also reflect its low-dimensionality, if the `phase-locking' picture of the magnetic order is relevant.

  

Figure 31: LF=100 G $\mu$SR spectra around the transition temperatures.