Transverse Field Relaxation

Relaxonomy --> here

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By convention we define the muon spin polarization [math]\displaystyle{ \vec{P} }[/math] to be initially in the [math]\displaystyle{ \hat{x} }[/math] direction and the applied magnetic field [math]\displaystyle{ \vec{B} }[/math] to be in the [math]\displaystyle{ \hat{z} }[/math] direction. This generally implies precession of [math]\displaystyle{ \vec{P} }[/math] in the [math]\displaystyle{ x-y }[/math] plane at a frequency [math]\displaystyle{ \omega_\mu = \gamma_\mu B }[/math], where [math]\displaystyle{ \gamma_\mu/2\pi \approx 0.01355 }[/math] MHz/G. The relaxation of [math]\displaystyle{ \vec{P} }[/math] is then measured in the rotating reference frame (RRF) precessing with the muon at [math]\displaystyle{ \omega_\mu }[/math] where [math]\displaystyle{ \vec{P} }[/math] is always in the [math]\displaystyle{ \hat{x} }[/math] direction:

[math]\displaystyle{ G_{xx}(t) \; \equiv \; \langle P_x(0) \, P_x(t) \rangle \; . }[/math]

Note that the assumption of a single precession frequency nominally precludes the main cause of "relaxation" in TF, namely dephasing of muons precessing at different frequiencies in different local fields. This formalism can still be applied as long as the distribution of local fields [math]\displaystyle{ B_\mu }[/math] is symmetric about some average field [math]\displaystyle{ \langle B_\mu \rangle }[/math]. However, this is not true for a number of important cases, such as the field distribution in the vortex lattice of a type II superconductor. In those cases this description is strictly invalid, but people use it as an approximation anyway.

That is, when the precession signal relaxes, it is usually assumed that [math]\displaystyle{ G_{xx}(t) = G_{yy}(t) = G_{\rm sym}(t) }[/math] so that

[math]\displaystyle{ \tilde{P}(t) = G_{xx}(t) + i G_{yy}(t) = G_{\rm sym}(t) \exp[i (\omega t + \phi)] }[/math];

but this is equivalent to assuming that the distribution of precession frequencies is symmetric about some average [math]\displaystyle{ \omega }[/math]. For cases like superconductors in the mixed state this is invalid; the lineshape is asymmetric and if one wishes to describe the complex polarization in this form one must use an explictly complex [math]\displaystyle{ \tilde{G}(t) = G_{xx}(t) + i G_{yy}(t) }[/math] that may mix [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] components, effectively introducing other frequencies besides the average [math]\displaystyle{ \omega }[/math]. Even in those cases, it is almost always assumed that [math]\displaystyle{ G_{ij}(t) = 0 }[/math] for [math]\displaystyle{ i \ne j }[/math]. In particular, it is almost always assumed that the muon polarization never begins to approach thermal equilibrium.