3.4 Summary of the Kubo-Toyabe theories

 As a concluding remark concerning the zero/low-field spin relaxation theories, I will mention the importance of the longitudinal field measurements when using the $\mu$SR technique as a local magnetic probe. In Fig.24, the simulated muon spin relaxation in two different spin systems are shown. One is a dense spin system with fast fluctuations, and the other is a dilute spin system with slow fluctuations. In zero-field, the two systems presents almost identical exponential relaxation, and the two situations are indistinguishable. In longitudinal field measurements, however, these two systems become distinguishable. In the slowly fluctuating system, the relaxation is decoupled with relatively small longitudinal fields, which are comparable to the field distribution width (Fig.24b). In the fast fluctuating system (Fig.24a), decoupling requires much larger fields. These qualitatively different responses to the applied longitudinal fields allow one to experimentally distinguish between slow and fast fluctuations of the local fields.

  

Figure 24: (a) Muon spin relaxation in a fast fluctuating dense spin system: $G^{\rm DGKT}(t;\Delta,H_{\rm LF},\nu)$, and (b) in a slowly fluctuating dilute spin system: $G^{\rm DL}(t;a,H_{\rm LF},\nu)$. These two systems are distinguishable in longitudinal field measurements, but not in zero-field.

In the following chapter, the $\mu$SR technique is applied to a spin-ladder material, Sr_n-1Cu_n+1O_2n. The set of measurements presented there will exhibit how Kubo-Toyabe theories help us investigate the magnetism of this material.