Basic Relaxation Function Formalism

Relaxonomy --> here

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

<math> G_{ij}(t) \; \equiv \; \langle P_i(0) \, P_j(t) \rangle \; . </math>

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

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