Kubo-Toyabe

KT Kubo-Toyabe (relaxation fimetion) ZF zero field... 

Fig. 20. Distribution of (a) field components and (b) field magnitude (c) shows die resulting static Kubo-Toyabe functions. Solid lines refer to Gaussian, dashed lines to Lorentzian field distributions. B is the most probable field and HWHM the half width at half maximum of the distributions.

Fig. 21. Zero-field dynamic Gaussian Kubo-Toyabe functions for different field fluctuation times. The curves are labeled by the value of t A (in rad). The calculation used the strong-collision model. In the Gaussian-Markovian approach G t) decays minimally slower.

Fig. 22. Zero field strong-coUision-dynamic Lorentzian Kubo-Toyabe function for different fluctuation times. The numbers are the value T a (in rad). Note the lack of decoupling as fluctuation rate —> oo. After Uemura (1981).

The term longitudinal field (LF) refers to an external field iqrplied along the initial direction of muon spin (the z axis). LF measurements are most useful in cases where a Kubo-Toyabe-like relaxation function is observed in ZF. The apphed LF competes with the internal field distribution, trying to hold the muon spin in its original direction (thus trying to prevent relaxation). If the longitudinal field Sl fulfills the condition... 

While eq. (37) is often presented as very generally applicable, it is in fact not correct when local field distribution is Lorentzian. As noted in the previous section, standard strong-collision dynamics do not decouple a Lorentzian Kubo-Toyabe (fig. 22, no motional narrowing). This odd behavior requires rather special circumstances to occur (triple- ordering in USb and DyAg, sect. 5.2, singlet ground state PrP, sect. 5.1.2). 

Fig. 25. Zero and longitudinal field = 104/y ) Kubo-Toyabe functions for local field fluctuating increasingly faster (fiom top to bottom). Left Gaussian field distribution right pure Lorentzian field distribution.

Fig. 41. ZF and LF-JiSR in nominally stoichiometric PrP at 4K. Solid lines are fits of strong-collision-dynamic Lorentzian Kubo-Toyabe relaxation (Noakes et al. 2000).

The ZF spectra of the Ik type I state in UAs take the form a weakly decaying Lorentzian Kubo-Toyabe function. An example is depicted in fig. 45. LF measurements show the... 

Fig. 45. ZF and LF spectra of different uranium NaCl-type compounds (UN, UP, UAs, USb) below their Neel temperatures. The fits to the data are Lorentzian Kubo-Toyabe functions (static for UN, UP and UAs, dynamic for USb (see discussion in text). Extended from Asch (1990).

Fig. 48. Comparison of the temperature dependence of width A (open symbols) and field fluctuation rate v (solid symbols) derived from fits to the Lorentzian Kubo-Toyabe spectra of UN (left) and USb (right). For UN the absolute values, and for USb the reduced values of parameters are plotted. At T -C width and rate for the two compounds are comparable. The lines are guides to the eye. Adapted ftom Mtinch et al. (1993) and...

Condition (ii) can be fulfilled in other crystal structures on occasion as well. An AFM state is usually a condition, since, as mentioned, the contact field will not vanish in a FM material (but is present only in conducting compounds). The important point is that in an AFM spontaneous spin precession can be absent although LRO of the spin system exists. The strict consequence of (ii) would be a non-depolarizing pSR signal in ZF. But in nearly all cases the field distribution (iii) exists and Lorentzian Kubo-Toyabe patterns are seen instead. It is important to realize that the width of the Lorentzian Kubo-Toyabe patterns is not simply connected to the size of the magnetic moments the concentration and nature of faults enters dominantly. A randomness of these faults (though probable) is not required since the muon positions woidd in any case be randomly distributed relative to them. We finally point out that the width of the field distribution is rather small (about 8 G for UAs) and in many cases not significantly different from that produced by nuclear dipoles. A distinction between the two can be cumbersome in some cases. 

The jxSR study by Grosse et al. (1999) used mosaics of oriented single-crystal platelets. LaS, the lower concentration limit (x = 0), is diamagnetic. The ZF spectra are static Gaussian Kubo-Toyabe patterns originating from the nuclear moments on La. Full decoupling needs only LF=10G. Between 300 K and 4K only a minute change in static width is seen, which can be accounted for by thermal contraction. These data show that effects of muon diffusion are not discemable. US, the upper concentration limit (x = 1) is a FM (7c = 177 K and fiu = 1.7 Ub). The reduction in moment was... 

Fig. 56. Left ZF- and LF-(iSR spectra of polycrystalline DyAg at 18K. The solid lines are fits to a nearly static Lorentzian Kubo-Tc abe fiinction. Right Temperature dependence of distribution width and fluctuation rate of the field at the muon site as obtained from fils to the Lorentzian Kubo-Toyabe patterns seen in ZF data of cr-DyAg up to 51K. From Kalvius et al. (1986, 1990)...

Fig. 97. Dynamic muon spin relaxation in a dilute spin glass as contrast to the behavior of the pure Lorentzian field distribution discussed in the section on longitudinal field measurements. Although a Lorentzian Kubo-Toyabe relaxation is observed in the fully spin-frozen state at lowest temperatures, motional narrowing occurs when the spins start to fluctuate. From Uemura (1981).

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