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# Meier¶

## Description¶

Time dependence of the polarization function for a static muon interacting with nuclear spin [1].

$A(t)=\frac13(2P_x+P_z)$

,where

$P_z(t) = \frac{1}{2J+1}\left\{1+\sum^J_{m=-J+1}[\cos^2(2\alpha_m)+\sin^2(2\alpha_m)\cos(\lambda^+_m-\lambda^-_m)t]\right\},$
$P_x(t) = \frac{1}{2J+1}\sum^J_{m=-J} \{ \cos^2\alpha_{m+1}\sin^2\alpha_m\cos(\lambda_{m+1}^+-\lambda_m^+)t +\cos^2\alpha_{m+1}\cos^2\alpha_m\cos(\lambda_{m+1}^+-\lambda_m^-)t +\sin^2\alpha_{m+1}\sin^2\alpha_m\cos(\lambda_{m+1}^--\lambda_m^+)t +\sin^2\alpha_{m+1}\cos^2\alpha_m\cos(\lambda_{m+1}^--\lambda_m^-)t\},$
$\lambda_m^\pm = \frac{1}{2}[\omega_Q(2m^2-2m+1)+\omega_D\pm W_m],$
$W_m = \{(\omega_D+\omega_Q)^2(2m-1)^2+\omega_D^2[J(J+1)-m(m-1)]\}^\frac{1}{2},$
$\tan(2\alpha_m)=\frac{\omega_D[J(J+1)-m(m-1)]^\frac{1}{2}}{(1-2m)(\omega_D+\omega_Q)},$

$$\omega_D$$ is the angular frequency due to dipolar coupling,

$$\omega_Q$$ is the angular frequency due to quadrupole interaction of the nuclear spin $$J$$ due to a field gradient exerted by the presence of the muon,

$$J$$ is the total angular momentum quantum number,

and $$m$$ is the z-component of the total orbital quantum number.

## Properties (fitting parameters)¶

Name

Default

Description

A0

0.5

Amplitude

FreqD

0.01

Frequency due to dipolar coupling (MHz)

FreqQ

0.05

Frequency due to quadrupole interaction of the nuclear spin (MHz)

Spin

3.5

J, Total angular momentum quanutm number

Sigma

0.2

Gaussian decay rate

Lambda

0.1

Exponential decay rate

Categories: FitFunctions | Muon\MuonSpecific

Python: Meier.py