Delayed Onset Relaxation

Relaxonomy --> here

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So far we have assumed that the relaxation mechanism acts on the muon spin polarization immediately upon its arrival in the sample and continues indefinitely. This idealization is never really correct, as many quite different processes occur as the muon slows down and the free ions produced in its track follow their own destinies; but usually all these events take place in times much shorter than the experimental time resolution, so we can safely ignore them.

Usually.

But what if there is some important change in the muon's environment that takes place significantly after its arrival? Well-documented examples are trapping at defects in crystals, chemical reactions, delayed muonium formation, etc. This can produce quite different "relaxation functions" as well as distributed changes of precession frequencies in TF.

For simplicity we will first assume ZF or LF with no precession, so that we can use a simple scalar <math> g_{zz}(t) </math> for each stage of the polarization's evolution. For now we will also assume that there are just two stages: the initial stage with <math> g_1(t) </math> followed by abrupt "formation" at time <math>\tau</math> of a later stage with <math> g_2(t-\tau) </math>.

If <math>\tau</math> is distributed as <math>D(\tau)</math> normalized to <math>\int_0^\infty D(\tau) d\tau = 1</math>, then we can write the outcome as

<math> G(t) \; = \; \left[ 1 - \int_0^t D(\tau) d\tau \right] g_1(t) + \int_0^t D(\tau) g_1(\tau) g_2(t - \tau) d\tau </math>

where the first term is from muons still in stage 1 and the second term is from those that are now in stage 2.

To simplify further (in order to produce an illustrative example) we may set <math>g_1(t) = 1</math> (no depolarization initially) and <math>g_2(t) = \exp(-\lambda t)</math> (exponential decay). Further simplifying the distribution of formation times, we can set <math>D(\tau) = k \exp(-k\tau)</math>, giving

<math> G_{ee}(t) = {1 \over \lambda - k} \left[ \lambda \exp(-k t) - k\exp(-\lambda t) \right]. </math>

This is plotted below for <math>\lambda = 1</math> and a variety of <math>k</math> values from 0.1 to 100.

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<math>G_{ee}(t)</math> for <math>\lambda = 1</math> and <math>k = 0.1, 0.2, 0.3, 0.4, 0.49, 0.51, 0.6, 0.7, 0.8, 0.9, 0.99, 1.01, 2, 3, 4, 5, 6, 10, 100, \infty</math>. Note that values of <math>k \sim \lambda</math> produce a convex curvature at early times, like the early part of a Gaussian relaxation, despite the fact that all "physical" time dependences are strictly exponential.

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<math>G_{ee}(t)</math> for <math>\lambda = 1</math> and <math>k = 1</math> versus a Gaussian relaxation, <math>G_g(t) = \exp(-t^2/2)</math>.

A similar derivation can be used for initially rapid relaxation followed by a nonrelaxing state; this was used to calculate the "residual polarization" after fast chemical reactions of muonium forming a diamagnetic compound. In that case both <math>g_1(t)</math> and <math>g_2(t)</math> are complex, with the real part representing the polarization component in the initial direction and the imaginary part representing its component perpendicular to both the original muon polarization and the applied magnetic field (a convenient way to treat precession).