Muon polarization

Contributions of order a Za) m in muonic hydrogen generated by the two-loop muon form factors have almost exactly the same form as the respective contributions in the case of electronic hydrogen. The only new feature is connected with the contribution to the muon form factors generated bj insertion of one-loop electron polarization in the radiative photon in Fig. 7.9. Respective insertion of the muon polarization in the electron form factors in electronic hydrogen is suppressed as (mg/m), but insertion of a light loop in the muon case is logarithmically enhanced. 

The characteristic integration momenta in the matrix element of this perturbation potential between the Coulomb-Schrodinger wave functions are of the atomic scale mZa, and are small in comparison with the muon mass m. Hence, in the leading approximation the muon polarization may be approximated by the first term in its low-frequency expansion... 

In the case of the polarization insertions the calculations may be simplified by simultaneous consideration of the insertions of both the electron and muon polarization loops [18, 19]. In such an approach one explicitly takes into account internal symmetry of the problem at hand with respect to both particles. So, let us preserve the factor 1/(1 - - m/M) in (9.9), even in calculation of the nonrecoil polarization operator contribution. Then we will obtain an extra factor m /m on the right hand side in (9.12). To facilitate further recoil calculations we could simply declare that the polarization operator contribution with this extra factor m /m is the result of the nonrecoil calculation but there exists a better choice. Insertion in the external photon lines of the polarization loop of a heavy particle with mass M generates correction to HFS suppressed by an extra recoil factor m/M in comparison with the electron loop contribution. Corrections induced by such heavy particles polarization loop insertions clearly should be discussed together with other radiative-recoil... 

Note that in the parenthesis we have parted with our usual practice of considering the muon as a particle with charge Ze, and assumed Z = 1. Technically this is inspired by the cancellation of certain contributions between the electron and muon polarization loops mentioned above, and from the physical point of view it is not necessary to preserve a nontrivial factor Z here, since we need it only as a reference to an interaction with the constituent muon and not with the one emerging in the polarization loops. 

The contribution of the muon polarization operator was already considered above. One might expect that contributions of the diagrams in Fig. 10.8 with the heavy particle polarization loops are of the same order of magnitude as the contribution of the muon loop, so it is natural to consider this contribution here. Respective corrections could easily be calculated by substituting the expressions for the heavy particle polarizations in the unsubtracted skeleton integral in (10.3). The contribution of the heavy lepton t polarization operator was obtained in [37, 38] both numerically and analytically... 

In the external field approximation the skeleton integral with the muon polarization insertion coincides with the respective integral for muonium (compare (9.12) and the discussion after this equation) and one easily obtains [33]... 

This result gives a good idea of the magnitude of the muon polarization contribution since the muon is relatively light in comparison to the scale of the proton form factor which was ignored in this calculation. 

The total muon polarization contribution may be calculated without great efforts but due to its small magnitude such a calculation is of minor phenomenological significance and was never done. Only an estimate of the total muon polarization contribution exists in the literature [7]... 

Ionization losses, multiple scattering, and the deflection in the local magnetic field are considered. The decay of particles is simulated in exact kinematics, and the muon polarization is taken into account. 

Radiation chemical processes near the end of its thermalization track determine the environment in which the muon is found on an experimental time scale corresponding to its lifetime, = 2.2 ps. In experiments in transverse fields we distinguish between different components according to their signal amphtudes which are converted to fiactional muon polarizations Pi, with EPj = 1.0, if the fiill initial polarization in the beam is accounted for. 

In gases, sohd insulators, water, and in other chemically inert environments a Auction Pmu of muon polarization represents muonium. It is easily distinguished fi"om Pd by its characteristic precession fi"equency of 1.394 MHz/G in fields below ca. 20 G. The fi-equency is higher by two orders of magnitude than for diamagnetic environments because it is mainly the electron in the coupled p e" system that determines the Lannor fi-equency. 

Figure 3. Muon polarization in sodium nitrate solutions as a fimction of magnetic field (Reprinted from [7] with permission from Elsevier Science).

Fig. 7. Example of the pS Resonance of CCl,. Since the diamagnetic muon polarization of CCl is 1.0, it is frequently used as a reference material to calibrate the asymmetry. The upper spectrum is the evolution tiiiK spectrum of the diamagnetic muon at just-resonance (at the magnetic field corresponding to the resonance peak shown below). This corresponds to the term in the square bracket of 26.

This represents the time-dependence of the projection of the muon polarization on the axis of the positron detectors, effectively now for muons that are initially 100% polarized. This is the quantity provided by most theories, and is therefore what one wishes most to learn from the experiment. However, Aq can be divided out of the Fourier transform of the data as easily as from the signal, so it usually suffices to transform S(t). 

Recent zero-field studies of the free in longitudinal fields have revealed new and unique information on spin glass and other systems but here we concentrate on muonium and muonium-like states in zero magnetic field. In this case, precession is not observed in the classical sense of the word but rather a modulation of the muon polarization with time. This can be most easily understood in the case of muonium itself in terms of the isotropic Hamiltonian of Equation 30. As noted above, muonium is formed via the "capture" of an electron from the stopping medium. Since the fi is longitudinally polarizedl"3 (a ) but the captured e" is not (Qg or Pq), muonium forms initially in two spin states, defined by A> = lV°e> = I 1 1> and B> = I o /3e> = 1//2 10> +... 

The term Muon Spin Resonance defines a NMR-type technique. In the presence of a static external field one induces muon spin flips by the application of resonant radio frequency (Kitaoka et al. 1982, Kreitzman 1990, Hampele et al. 1990, Nishiyama 1992, Scheuermann et al. 1997, Cottrell et al. 1997) or microwave field (Kreitzman et al. 1994). The resonance condition is detected via a loss of muon polarization. As in NMR, firequency shifts and linewidth are the sensitive parameters. 

The first task is to formulate a theoretical signal fimction, which we denote as representing muon polarization as a fimction of time. For example, a transverse field measurement in the fast fluctuation limit should have the form... 

Fig. 40. Spontaneous ZF- xSR oscillation in single-crystal PrCuj (a) at 15K with muon polarization along the ft-axis (b) similar data, but with the muon polarization along the c-axis (c) temperature dependence of precession frequency (Schenck et al. 1998b).

Fig. 160. ZF spectra of CeCujAu above (top) and below (bottom) the Neel temperature (2.3 K). The initial muon polarization is parallel to the b axis (Amato et al. 1995b).

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