Dynamic Relaxation
Relaxonomy --> here
So far we have treated only the behavior of the muon polarization at fixed locations where the local magnetic fields are static. This is often not the case, and situations in which the muons are hopping between different sites and/or the local fields at those sites are fluctuating comprise many of the most interesting µSR studies.
Therefore it behooves us to model the resultant relaxation functions.
The simplest cases are when either the muon hops between sites with different static magnetic fields and/or the muon is at a fixed site with a randomly fluctuating local magnetic field. In either case we usually assume that the field at each hop or after each fluctuation is selected randomly from an easily modeled distribution <math>{\cal D}(\vec{B})</math> -- e.g. a Gaussian distribution of field magnitudes about an average <math>B_0</math> with a width <math>\Delta</math> :
<math>{\cal D}(|B|) = {1 \over \sqrt{2\pi} \; \Delta} \exp\left[- {1\over2} \left(B-B_0 \over \Delta \right)^2\right] </math>,
with a direction randomly selected from an isotropic distribution. (In ZF, <math>B_0 = 0</math> is usually assumed.)
It is easy to imagine many plausible situations in which these assumptions are violated; but one has to start somewhere. The trick is not to get stuck there.
Next we assume that the muon hops and/or the field fluctuates at a rate <math>\nu</math> in a history-independent way.
One approach, at least in ZF, is to assume that the static relaxation function <math>G_{zz}^{\rm stat}(t) = g(\Delta, t)</math> simply restarts at every hop/fluctuation, using the muon polarization at the time of the hop as a new starting point. This is called the strong collision model and can be modeled using Kehr's recursion relation:
<math>G^{\rm dyn}(\Delta,\nu,t) = g(\Delta,t) \; e^{-\nu t} + \nu \int_0^t G^{\rm dyn}(\Delta,\nu, t-\tau) \; g(\Delta, \tau) \; e^{-\nu \tau} d\tau </math> ,
which is sometimes solvable using Laplace transforms. (Numerical methods work too.)
In the limit of <math>\nu \to 0</math>, the Gaussian distribution above, centered on <math>B_0 = 0</math>, gives a static function
<math>g_G(\Delta, t) = {1\over3} + {2\over3} \exp\left[-{1\over2}\Delta^2 t^2\right] \left(1 - \Delta^2 t^2\right)</math>
and the corresponding dynamic relaxation functions look like this:
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Dynamic Gaussian Kubo-Toyabe function for various hop or fluctuation rates. |
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Of course, the assumptions that go into calculating <math>g(\Delta, t)</math> are idealizations of the real field distributions at lattice sites and/or the actual processes involved in muon "hopping" and field "fluctuations". They also omit all effects of quantum entanglement between the muon spin and the nearby nuclear or electronic spins, which need not remain static during their dipolar or hyperfine interactions with the muon. As has been demonstrated in copper crystals and (more dramatically) in fluoride crystals, such coupled spin systems often evolve together to produce "static" <math>g(\Delta, t)</math> functions bearing little resemblance to that described above.
Fortunately, Kehr's recursion relation usually works (numerically) on those "static" functions as well, although of course it is still an approximation.
Note that, for hop or fluctuation rates <math>\nu</math> much larger than the static width <math>\Delta</math>, the dynamic relaxation looks more and more exponential, <math>G^{\rm dyn}(t) [\nu \gg \Delta] \to e^{-\lambda t}</math> where <math>\lambda</math> decreases as <math>\nu</math> increases. This corresponds to the phenomenon of "motional narrowing" in NMR.
