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SpinDiffusion

Description

The Spin diffusion fitting function, models the diffusion of isotropic muonium as a function of applied field for 1D, 2D and 3D behaviour [1]. The data fitted using this fit function is assumed to be in units of Gauss.

\[\begin{split}\lambda(B) &= \frac{A^2}{4} J(\omega) \\ J(\omega) &= 2 \int_{0}^{+\infty} S(t)\cos(\omega t) \\ S(t) &= \prod_{i=1}^{3} \exp(-2 D_{i} t) I_{0}(2 D_{i} t) \\ \omega &= 2 \pi f = \gamma_{\mu} B\end{split}\]

where:

  • \(I_{0}(x)\) is the zeroth order modified Bessel function.

  • \(\omega\) is the angular momentum (\(MHz\)).

  • \(\gamma_{\mu}\) is the Muon gyromagnetic ratio (\(2 \pi \times 0.001356 MHz/G\)).

  • \(S(t)\) is the autocorrelation function, represented by an anisotropic random walk.

  • \(J(\omega)\) is the spectral density (\(MHz^{-1}\)). It is the Fourier Transform of \(S(t)\).

  • \(A\) is a parameter to be fitted.

  • \(D_{i}\) are the fast and slow rate dipolar terms. These are also fitting parameters.

Systems of different dimensionality \(d\) can simply be represented in terms of fast and slow rates \(D_{\parallel}\) and \(D_{\perp}\):

\[D_{1} = D_{\parallel}, D_{2}, D_{3} = D_{\perp} (d=1) D_{1}, D_{2} = D_{\parallel}, D_{3} = D_{\perp} (d=2) D_{1}, D_{2}, D_{3} = D_{\parallel} (d=3)\]

For the \(d=3\) case, the \(D_{\perp}\) parameter has no significance. It may be a good idea to fix this parameter to prevent the minimizer from performing unnecessary optimization steps in this case.

Attributes (non-fitting parameters)

Name

Type

Default

Description

NDimensions

Properties (fitting parameters)

Name

Default

Description

A

1.0

Amplitude, or Scaling factor

DParallel

1000.0

Dipolar parallel, the fast rate dipolar term (MHz)

DPerpendicular

0.01

Dipolar perpendicular, the slow rate dipolar term (MHz)

References

Categories: FitFunctions | Muon\MuonSpecific

Source

Python: SpinDiffusion.py