5 The Time Dependent Spin Polarization in Zero Field

In this chapter we describe the time dependence of the spin polarization of the muon in a solid in zero applied field. Even in the absence of an applied field, there are typically local internal magnetic fields in a solid. The origin of these fields can be nuclear dipoles, electron orbital or spin moments, and generally the fields may be dynamic (either because of dynamics of the magnetic system or because of muon motion).

First, consider the situation of completely static fields, i.e. the correlation time for the field experienced by a muon $\tau$ is much greater than the muon lifetime $\tau_\mu$.To make the following discussion general, we allow for multiple muon sites in the material and index the site by i. There are generally of the order N sites of type i, and they are all crystallographically equivalent (N is the number of crystalline unit cells in the sample). The magnetic field distribution function (for site i) $\Phi^i({\bf B})$.is defined as the probability that the magnetic field at the site i is ${\bf B}$. For a perfectly ordered magnetic structure in which the all the sites of type i are magnetically equivalent, i.e. the field at each site is the same ${\bf B_0}$, $\Phi^i({\bf B})= \delta({\bf B}-{\bf B_0})$.In practice, this situation is approximately realized in crystallographically dense magnetic systems at temperatures much less than the magnetic ordering temperature. A single muon experiencing a local field ${\bf B}$,simply precesses in that field, so the ensemble average time evolution of the muon spin polarization is

where the initial muon spin polarization is in the $\hat{z}$ direction, $\hat{B}$ is the direction of the field ${\bf B}$, and the probability of any given muon ending in a site of type i is f_i. For the delta function field distribution, then P_z(t) will then be simply the sum of a constant term and an oscillating term. At the opposite extreme (from the ordered field distribution) is the situation of complete disorder. The field distribution in this situation can be modelled in several ways. If the local field is disordered in orientation, its components will be uncorrelated, and the field distribution will factor:

where the probability that ${\bf B}$ in site type i will have a component B_j is $\Phi^i_j(B_j)$.Following this assumption, one can now postulate a form for the random field distribution, which will depend on the particular situation under consideration. A common situation is that of randomly oriented nuclear dipoles. In the situation where these moments are relatively dense, it is found that a gaussian field distribution (centred at zero) is appropriate, i.e.

where $\sigma$, the width of the field distribution, is correlated with the size of the nuclear dipole moment and the average distance between the muon site and the nuclei. If the nuclear moments are more dilute, there will be a wide variety of distances between the average muon and the nearest moment. Because of the strong spatial dependence of the magnetic field due to a point dipole, muons that stop far from any moment will experience a much lower field than muons that stop near a moment. In this situation, the field distribution is broader than a gaussian, and is better modelled by a lorentzian (centred at zero), i.e.

where w is a parameter which characterizes the width of the $\Phi^i$ and is correlated with the size of the dilute moments and their volume density.

The integral (5.1) can be done analytically for both of the above field distributions. The results, known as the gaussian and lorentzian Kubo-Toyabe functions (respectively), are:

These functions with their associated field distributions are plotted in Fig. 5.33. Note that if one assumes that an observed P_z(t) is due to a static random field distribution, then one can find the distribution by using an approximate inversion of Eq. (5.1), e.g.

where $\Theta$ is the unit step function which is zero for negative argument.

For any field distribution, there will, on physical grounds, generally be a maximum value of the field at site i, B_maxⁱ. This fact together with the form of Eq. (5.1) implies that, at t=0, $\frac{d}{dt}P_z(t)=0$. This is clearly not the case for the lorentzian Kubo-Toyabe, Eq. (5.6). This is because the lorentzian field distribution is unphysically broad, and is merely an approximate model. All real field distributions will have a high field cutoff that is sharper than the lorentzian, and P_z(t) will be flat at early times. However, the turnover to zero slope at early times may occur on a timescale which is experimentally inaccessible. The time t_min before which P_z(t) is not observed depends on the details of the experiment, but t_min is typically in the range 1ns-100ns. The lorenztian Kubo-Toyabe may be a reasonable model for P_z(t), but only for t>t_min.

The general features of P_z(t) which are characteristic of random field distributions centred at zero are: i) the early time fall off, whose rate is proportional to the width parameter of the distribution, ii) the dip, whose depth depends on how broad the distribution is and whose position is inversely proportional to the width parameter, and iii) the late time recovery to 1/3 of the full polarization. The source of the 1/3 ``tail'' is simply that on average 1/3 of the muons polarization will lie parallel to the local field and consequently not precess (time independent term in Eq.(5.1)). Note that the 1/3 tail is robust to the average over multiple sites, but that the dip is not. If the variation of the width parameter of the field distributions between different sites varies the position of the dip by an amount on the order of the breadth of the dip, the dip will be obscured by the site averaging.

Between the extremes of the delta function and random field distributions, there is a broad range of behaviour. For a magnetic system, if the ordering wavevector is away from the extremes of the Brillouin zone (centre or corners) or if there are multiple ordering wavevectors, all sites of type i will not be magnetically equivalent, and some broadening of the field distribution from the ideal delta function will occur. Similarly, broadening will occur because of the coexistence of any disorder, for example, random nuclear diploes + some ordered magnetic state, or simply disorder in the magnetic structure. Generally, oscillations due to sharp peaks in field distribution will occur in magnetically ordered systems, but they will be damped by the effects mentioned above. The work of Kalvius [] provides an interesting general review of $\mu {\cal SR}$ in metallic magnets. There is some more specific discussion of the effects of disorder in a spin density wave state in section 6.3.

It is interesting to consider the effect of application of a longitudinal field (in the $\hat{z}$ direction) which is on the order of the internal fields. In this case the same model (Eq.(5.1)) for the time dependent average polarization can be used, but the field distribution is shifted because the net local field is the vector sum of the applied field (${\bf B_{app}}$)and the internal field. As the applied field is increased, it is clear that the net field at all sites will approach the direction of the applied field $\hat{z}$, and hence the time-independent part of the integral (5.1) will dominate, and the time-dependence will be ``quenched''. For the delta function field distribution, the amplitude of the oscillating part of P<sub>z</sub>(t) will simply decrease continuously to zero as B<sub>app</sub> increases. For the random distributions, the ``relaxation'' quenches when B_app is of the order of the width of the random field distribution, for example see Fig. 5.34. There are thus two ways to measure the breadth of a static random field distribution: i) the magnitude of a zero field relaxation rate (converted with the appropriate factor $\gamma_\mu$), and ii) the LF at which the zero field relaxation is quenched. Note that, in some exotic systems, these two measures do not agree.

So far, we have discussed the magnetic field distribution as if it were simply a classical field, which the muon samples randomly in space. However, the sources of these fields are electronic currents and electronic and nuclear spins, and should be treated quantum-mechanically. Instead of simply treating the muon spin quantum mechanically in the local field, one should correctly consider the full hamiltonian of the solid + muon + interactions. Practically, the only entities that need to be treated quantum mechanically are those that interact strongly with the muon, i.e. the nuclei and electrons in the immediate vicinity. For most situations, even the local environment of the muon can be treated in an effective classical picture. One exception is, however, muonium. In zero field the time dependent muon spin polarization in muonium is determined by the hyperfine spectrum, i.e. it contains oscillating components at well defined frequencies: the zero field splittings of the hyperfine levels. The spectrum of frequencies can be complicated further by the indirect nuclear hyperfine coupling of Mu to neigbouring nuclear dipoles or by anisotropy of the hyperfine interaction. An example of this occurs in pure fullerite at low temperature, where because of the anisotropy of the hyperfine interaction, the spectrum of the exohedral C₆₀Mu radical contains several frequencies in zero field which are low enough to observe, see Fig. 5.35. Note that, in the presence of magnetism, the large electronic moment in Mu will cause the time-dependence of any Mu signal to be of unobservably high frequency. Quantum mechanical treatment of the full dipolar interaction can be found in . Note that for muons, the dipolar relaxation is always due to ``unlike'' spins, since the muons are perfectly dilute. The situation in a magnetic system is even more extreme: the moments are electronic (see Table 2.3), and through their magnetic levels are strongly split by the crystal field and exchange couplings to their neighbours.

The time-dependent polarization P_z(t) observed in zero field is related to the polarization function in high ($B_{app} \gg$the internal fields) transverse field P_x(t). In this case the envelope of the oscillations at the Larmor frequency of the muon (we ignore the case of Mu here) measures nearly the same thing as P_z(t), since the muon is simply precessing in the net field. However, in TF, the two directions perpendicular to the muon spin are inequivalent (since one is along the applied field), whereas in ZF they are equivalent. Thus the relaxation rate is reduced geometrically by a factor of $\sqrt{2}$. In addition for the case of relaxation from the full dipolar interaction (including non-secular terms), the relaxation rate in TF may, in some cases, be reduced by another factor of $\sqrt{(5/2)}$ [201]. This extra enhancement in zero field, from terms in the interaction which in ZF cause the muon and nuclear moments to flip-flop, does not occur in the case of magnetically ordered moments, because the energy required to flip the ordered moment is non-zero (it is given by an exchange energy, which is generally quite large). Because of this enhancement (and also because of the large frequency shifts in TF experiments in magnetic materials), zero field $\mu {\cal SR}$ is the preferred $\mu {\cal SR}$technique for the study of magnetism.

Dynamics in the magnetic field introduces a further complication in P_z(t). The spectral density $J(\omega)$ of the local field is simply defined as the Fourier transform $J^i(\omega) = {\cal F}[{\bf B^i(t)}]$. Time variation of the local field can induce muon spin flip transitions. These flips occur randomly at a constant rate, leading to a characteristic exponential relaxation in P_z(t). The rate of this relaxation is determined, for example in time dependent perturbation theory, by the value of J at the Larmor frequency of the muon (in the time averaged local field). We note that if the fluctuations in the field are either too slow or too fast, they will be ineffective in causing relaxation. Dynamics can be included in a number of ways, which will not be discussed here. The following qualitative features arise from dynamic features. Quenching of the dynamic relaxation by application of a longitudinal field (e.g. Fig. 5.34) is much less effective. In the dynamic case, quenching occurs only for fields such that $\gamma_\mu B$exceeds the highest frequency $\omega_{max}$ with appreciable spectral density J of the fluctuations. The quenching field thus depends sensitively on the particular dynamics involved, but is generally much larger than the field esimated from the zero field relaxation function (assuming it is of static origin). In the case of slow dynamics, relaxation of dynamic origin may only be apparent as a slow exponential decay of the 1/3 tail of P_z(t), and may be modelled as an exponential relaxation ($\exp{(-\lambda t)}$) times a static Kubo-Toyabe function.