6.4

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TFMuonium

Description

General case TF muonium rotation

\[A(t)=\frac{A_0}{4}\{(1+\delta)a_{12}\cos(\omega_{12}+\phi)+ (1-\delta)a_{14}\cos(\omega_{14}+\phi)+(1+\delta)a_{34}\cos(\omega_{34}+\phi)+(1-\delta)a_{23}\cos(\omega_{23}+\phi)\}\]

and,

\[\delta= \frac{\chi}{\sqrt{1+\chi^2}},\]
\[\chi = (g_\mu+g_e)\frac{B}{A},\]
\[d = \frac{(g_e-g_\mu)}{g_e+g_\mu},\]
\[E_1=\frac{A}{4}(1+2d\chi) \qquad E_2=\frac{A}{4}(-1+2\sqrt{1+\chi^2})\]
\[E_3=\frac{A}{4}(1-2d\chi) \qquad E_4=\frac{A}{4}(-1-2\sqrt{1+\chi^2}),\]
\[\omega_{ij}= 2 \pi (E_i - E_j),\]
\[a_{ij}=\frac{1}{(1+(\omega_{ij}/(2\pi f_\text{cut}))^2)},\]

where,

\(A_0\) is the amplitude,

A (MHz) is the isotropic hyperfine coupling constant,

\(\phi\) (rad) is the phase at time \(t=0\),

\(g_\mu = 0.01355342\) , the gyromagnetic ratio of muon,

\(g_e = 2.8024\) , the gyromagnetic ratio of electron,

and \(f_\text{cut} = 10^{32}\).

(Source code, png, hires.png, pdf)

../../_images/TFMuonium-1.png

Properties (fitting parameters)

Name Default Description
A0 0.5 Amplitude
Field 5.0 B-field (G)
A 600.0 Isotropic hyperfine coupling constant (MHz)
Phi 0.0 Phase

Source

Python: TFMuonium.py