Basic Relaxation Function Formalism

Relaxonomy --> here

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If the muon spin polarization [math]\displaystyle{ \vec{P} }[/math] is a function of time [math]\displaystyle{ t }[/math], then the autocorrelation function between the initial [math]\displaystyle{ i^{\rm th} }[/math] component and the [math]\displaystyle{ j^{\rm th} }[/math] component at time [math]\displaystyle{ t }[/math] is given by

[math]\displaystyle{ G_{ij}(t) \; \equiv \; \langle P_i(0) \, P_j(t) \rangle \; . }[/math]

Although there are interesting cases where [math]\displaystyle{ G_{ij} }[/math] is nonzero for [math]\displaystyle{ i \ne j }[/math],^{Turner 1985} most formalism in µSR is focussed on the transverse relaxation function [math]\displaystyle{ G_{xx} }[/math] or the longitudinal relaxation function [math]\displaystyle{ G_{zz} }[/math] , where the [math]\displaystyle{ x }[/math] and [math]\displaystyle{ z }[/math] directions are defined by the direction of the applied magnetic field [math]\displaystyle{ \vec{B} = B \hat{z} }[/math], if any. For the case of zero magnetic field (ZF), [math]\displaystyle{ \hat{z} }[/math] is defined by the initial direction of the muon spin polarization; for transverse magnetic field (TF), [math]\displaystyle{ \vec{P} }[/math] is taken to be initially in the [math]\displaystyle{ x-y }[/math] plane, or usually along [math]\displaystyle{ \hat{x} }[/math].

For TF the muon polarization is often expressed as a complex quantity, [math]\displaystyle{ \tilde{P}(t) = P_x(t) + i P_y(t) }[/math] so that simple precession without relaxation can be written [math]\displaystyle{ \tilde{P}(t) = P \exp[i (\omega t + \phi)] }[/math] where [math]\displaystyle{ P = |\tilde{P}(0)| }[/math], [math]\displaystyle{ \omega }[/math] is the precession frequency and [math]\displaystyle{ \phi }[/math] is the initial phase of the precession.