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# DynamicKuboToyabe¶

## Description¶

Dynamic Kubo Toyabe fitting function for use by Muon scientists defined by

$G_z \left(t\right) = g_z\left(t\right) + \nu \int_0^t g_z\left(\tau\right) G_z\left(t-\tau\right) d\tau$

where $$g_z\left(t\right)$$ is the static KT function, and $$\nu$$ the muon hopping rate.

In zero field, $$B_0=0$$:
$g_z\left(t\right) = \mbox{A} \Bigg[ \frac{1}{3} + \frac{2}{3} \left( 1 - {\Delta}^2 {t}^2 \right) e^{-\frac{1}{2}\Delta^2 t^2} \Bigg]$
In the presence of a longitudinal field, $$B_0=\omega_0 /\left(2\pi \gamma_{\mu}\right)>0$$:
$g_z\left(t\right) = \mbox{A} \Bigg[ 1 - 2\frac{\Delta^2}{\omega_0^2}\Big(1-cos(\omega_0 t)e^{-\frac{1}{2}\Delta^2 t^2}\Big) + 2\frac{\Delta^4}{\omega_0^4}\omega_0\int_0^t \sin(\omega_0\tau)e^{-\frac{1}{2}\Delta^2\tau^2}d\tau \Bigg]$

DynamicKuboToyabe function has one attribute (non-fitting parameter), ‘BinWidth’, that sets the width of the step size between points for numerical integration. Note that small values will lead to long calculation times, while large values will produce less accurate results. The default value is set to 0.05, and it is allowed to vary in the range [0.001,0.1].

Name

Type

Default

Description

BinWidth

## Properties (fitting parameters)¶

Name

Default

Description

Asym

0.2

Amplitude at time 0

Delta

0.2

Local field

Field

0.0

External field

Nu

0.0

Hopping rate