Muon spin precession

Now the field is turned on. Using the coordinate system presented in fig. 8, we have Sy, II (-z) and By, x (since By is assumed to be the external field only). It follows that Sy processes in the ( v,z)-plane. The angular distribution W(0) of the emitted positrons is coupled to the motion of At time zero, Sy points away from Dp and Nf (t) oc (1-ao). Half a precession period later, Sy points towards Dp and Nf(t) oc (1 -I- ao). The count rate Nf(t) will thus be modulated with the amplitude ao and the muon spin precession frequency fy. One finds... 

We now discuss the information obtainable from the muon spin precession frequency / i. The strong-field limit means that the external field is large compared to internal field contributions and is mainly determined by 5app. Furthermore, for a truly random distribution of internal fields we expect their mean to be zero. The applied field, however, causes (a perhaps very weak, but still finite) magnetization of the sample and thus destroys the full randomness of moment orientation. As a result, a small internal field adds to the applied field. The resulting spin precession frequency is slightly shifted from... 

Fig. 19. Angular dependence of the muon spin precession frequency Vjj of a single crystal of paramagnetic CeBj in transverse field. Such data can be used to fix the muon stopping site (see text). v i is the spin precession frequency for a bare i. From Amato et al. (1997a).

Fig. 27. Temperature dependence of the spontaneous muon spin precession frequency in single-crystalline TbNij with a Curie temperature of 23 K. The dashed curve is the appropriate fiee-ion Brillouin function. Inclusion of CEF interaction in the ground-state multiplet gives the solid curve which fits the data well. The behavior around the critical temperature is typical for a second-order magnetic phase transition. After Dalmas de Reotier et al. (1992).

The spontaneous muon spin precession frequency below 7n was observed by Hofinaim et al. (1978) in polycrystalline material and more recently by Ekstrom et al. (1996, 1997) in a sin e-crystal specimen. The amplitude of the precession signal is larger in the c geometry, meaning that the local field lies predominantly in the basal plane as well. Typical data for the temperature dependence of the precession frequency are shown in fig. 33. The stopping site of the muon is not known, but calculations of the dipolar fields... 

Fig. 82. Left Temperature dependence of spontaneous muon spin precession frequency and of relaxation rate of the associated non-oscillating signal in magnetically ordered PrCojSij. Right Field dependence of the paramagnetic relaxation rate near the Neel temperature (top) and fit to a critical law (bottom) for ZF and LF measurements as indicated. The measuring geometry is always c. After Gubbens et al. (1997a, 1998).

Fig. 86. Temperature dependences of the spontaneous muon spin precession iiequency (in the ferromagnetic state) and the muon spin relaxation rate (both in the ferromagnetic and paramagnetic states) in EuO. The data were analyzed with an exponential decay of muon spin polarization throughout. After Hartmann et al. (1996).

Fig, 96, [xSR study on RuSr2GdCu20g, Left ZF spectra in the FM state Tq = 133K) above and below the superconducting transition T = 16K), Right Temperature dependence of the spontaneous muon spin precession frequency. The sharp rise at the low-temperature end is due to the AFM ordering of the Gd sublattice. 

ZF-pSR spectra for T < 7n on polycrystalline material (Kalvius et al. 1994, 1995b) revealed a rather well defined (i.e., comparatively weakly damped) spontaneous muon spin precession pattern containing two distinct frequencies (fig. Ill, left). The variation of the precession frequencies with temperature is in agreement with a second-order phase transition at Tn (fig. Ill, right). But why two well-defined frequencies are observed, when the spin structure is allegedly incommensurate, is a serious problem. We return to this issue below within the discussion of CePtSn. 

Fig. 116. Left Temperature dependence of the rate A of muon-spin depolarization caused by electronic dipole moments on the Ce ions of CeRhSb (squares) and Lao,Ce 9RhSb (circles). Right Transverse field (lOmT) muon spin precession firequency of CeRhSb as a function of temperature. From Lidstrom et al. (1997).

Amato et al. (1995b) studied CeCusAu (x = 1) and CeCu5.9Auo.i (x = 0.1, the critical concentration) by [iSR using single-crystalline specimens. In the former material, the ZF spectra clearly establish the AFM transition at Tn 2.3K by the appearance of spontaneous muon spin precession (see fig. 160). The depolarization of the spontaneous precession signal can well be described by Jq Bessel function. As discussed in sect. 3.7, (see eq. 53) this spectral shape is typical for incommensurate magnetic order. The ordered... 

Fig. 166. piSR data on single-crystalline Ce,Nij. Left Measurements above Top Temperature dependence of the ZF-relaxation rate for two orientations. Bottom Angular dependence of the muon spin precession frequencies (two signals are seen) at 20 K in a transverse field of 6 kG. The marks refer to crystalline orientations parallel to the applied field. Right Measurements below T, . Top ZF spectra at 1.8K for c 5 and c (inset). The fits are explained in text. Bottom Temperature dependence of the spontaneous muon spin precession frequency. (Bottom left Schenck et al. 2001 all others Kralzer et al. 2001.)... 

Figure 15.26. The temperature dependence of the muon spin precession frequency in j8-phase crystals of/ -NPNN in zero applied field. The frequency is nearly proportional to the spontaneous magnetization. (Reprinted with permission from ref. 31)...

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