# Muon g-2 measurement

Helpful diagrams to explain the measurement of the anomalous moment g-2 of the muon, via its precession in the magnetic field of a storage ring at FNAL.
Also see general magnetic moment diagrams, or the “decay” tag for more similar figures.

Uniform circular motion of a positively charged particle in a uniform magnetic field with cyclotron frequency ωC:

Uniform circular motion of a negatively charged particle:

Muon spin and momentum are aligned at all times in the storage ring if g = 2 exactly.

If g > 2, the muon spin will rotate faster due to anomalous precession.

Anomalous frequency ωa with some initial phase shift:

Edit and compile if you like:

% Author: Izaak Neutelings (March 2020)
\documentclass[border=3pt,tikz]{standalone}
\usepackage{amsmath} % for \dfrac
\usepackage{mathabx} % for \Earth
\usepackage{bm} % \bm
\usepackage{physics}
\usepackage{tikz}
\usetikzlibrary{angles,quotes} % for pic (angle labels)
\usetikzlibrary{arrows.meta}
\usetikzlibrary{calc}
\usetikzlibrary{decorations.markings}
\tikzset{>=latex} % for LaTeX arrow head
\usepackage{xcolor}
\colorlet{Bcol}{violet!90}
\colorlet{BFcol}{red!50!black}
\colorlet{vcol}{red!50!black}
\colorlet{veccol}{green!50!black}
\colorlet{Icol}{blue!70!black}
\colorlet{wcol}{orange!90!black}
\colorlet{metal}{blue!30!black!12}
\colorlet{Scol}{green!60!black}
\tikzstyle{BField}=[->,very thick,Bcol]
\tikzstyle{current}=[->,Icol] %thick,
\tikzstyle{force}=[->,very thick,BFcol]
\tikzstyle{velocity}=[->,thick,vcol]
\tikzstyle{spin}=[->,very thick,Scol]
\tikzstyle{charge+}=[very thin,draw=black,top color=red!50,bottom color=red!90!black,shading angle=20,circle,inner sep=0.2]
\tikzstyle{charge-}=[very thin,draw=black,top color=blue!50,bottom color=blue!80,shading angle=20,circle,inner sep=0.2]
\tikzstyle{vector}=[->,thick,veccol]
\tikzset{
BFieldLine/.style={thick,Bcol,decoration={markings,mark=at position #1 with {\arrow{latex}}},
postaction={decorate}},
BFieldLine/.default=0.5,
pics/Bin/.style={
code={
\def\RB{0.12}
\draw[pic actions,#1,line width=0.6] % ,thick
(0,0) circle (\RB) (-135:.7*\RB) -- (45:.7*\RB) (-45:.7*\RB) -- (135:.7*\RB);
}},
pics/Bout/.style={
code={
\def\RB{0.12}
\draw[pic actions,#1,fill=white,line width=0.6] (0,0) circle (\RB);
\fill[pic actions,#1] (0,0) circle (0.3*\RB);
}},
pics/Bout/.default=Bcol,
pics/Bin/.default=Bcol,
pics/spin/.style={
code={
\def\L{0.38}
\draw[-{Latex[length=3,width=2.5]},pic actions,rotate=#1,line width=0.8,Scol] (0,0) -- (\L,0);
\draw[pic actions,rotate=#1,thin,white]
(0.35*\L,0)++(170:{0.16*\L} and {0.22*\L}) arc (170:190:{0.16*\L} and {0.22*\L});
\draw[-{Latex[length=1.2,width=1]},pic actions,rotate=#1,very thin]
(0.35*\L,0)++(25:{0.16*\L} and {0.22*\L}) arc (25:305:{0.16*\L} and {0.22*\L}) --++ (50:0.09*\L);
}},
pics/spin/.default=90,
}

\begin{document}

% CYCLOTRON FREQUENCY
\begin{tikzpicture}
\def\R{1.2}
\def\v{0.6*\R}
\def\F{0.55*\R}
\coordinate (O) at (0,0);
\coordinate (Q) at (\R,0);

% MAGNETIC FIELD
%\foreach \i [evaluate={\y=(\i-1)*\ymax/(\NBy-1);}] in {1,...,\NBy}{
%  \foreach \i [evaluate={\x=(\i-1)*\xmax/(\NBx-1);}] in {1,...,\NBx}{
%    \pic[rotate=-90] at (\x,\y) {Bin};
%  }
%}
\pic at (15:1.25*\R) {Bout};
\node[Bcol,right=3] at (15:1.25*\R) {$\vb{B}$};

% ROTATION VECTOR
\pic at (O) {Bin={wcol}};
\node[wcol,left=2] at (O) {$\vb*{\omega}_\mathrm{C}$};

% CHARGE
\draw[dashed] (O) circle (\R);
\node[charge+,scale=0.9] (Q) at (Q) {$+$};
\draw[vector] (Q) --++ (-90:\v) node[below=-1] {$\vb{v}$};
\draw[force] (Q) --++ (180:\F) node[above=-1] {$\vb{F}$};

\end{tikzpicture}

% CYCLOTRON FREQUENCY, negative
\begin{tikzpicture}
\def\R{1.2}
\def\v{0.6*\R}
\def\F{0.55*\R}
\coordinate (O) at (0,0);
\coordinate (Q) at (\R,0);

% MAGNETIC FIELD
%\foreach \i [evaluate={\y=(\i-1)*\ymax/(\NBy-1);}] in {1,...,\NBy}{
%  \foreach \i [evaluate={\x=(\i-1)*\xmax/(\NBx-1);}] in {1,...,\NBx}{
%    \pic[rotate=-90] at (\x,\y) {Bin};
%  }
%}
\pic at (-15:1.25*\R) {Bout};
\node[Bcol,right=3] at (-15:1.25*\R) {$\vb{B}$};

% ROTATION VECTOR
\pic at (O) {Bout={wcol}};
\node[wcol,left=2] at (O) {$\vb*{\omega}_\mathrm{C}$};

% CHARGE
\draw[dashed] (O) circle (\R);
\node[charge-,scale=0.9] (Q) at (Q) {$-$};
\draw[vector] (Q) --++ (90:\v) node[above=-1] {$\vb{v}$};
\draw[force] (Q) --++ (180:\F) node[above=-1] {$\vb{F}$};

\end{tikzpicture}

% CYCLOTRON g = 2
\begin{tikzpicture}
\def\R{1.4}
\def\v{0.4*\R}
\def\F{0.55*\R}
\def\N{8}
\coordinate (O) at (0,0);
\coordinate (Q) at (\R,0);

% CYCLOTRON
\draw[fill=metal,even odd rule] (O) circle (0.84*\R) circle (1.16*\R);

% MAGNETIC FIELD
\pic at (15:1.4*\R) {Bout};
\node[Bcol,right=3] at (15:1.4*\R) {$\vb{B}$};

% SPINS
\foreach \i [evaluate={\ang=90-(\i-1)*360/\N;}] in {1,...,\N}{
\draw[velocity,-{Latex[length=4,width=3]}] (\ang:1.02*\R) --++ (\ang-90:\v);
\pic at (\ang:0.94*\R) {spin={\ang-90}};
}
\node at (0,0) {$g = 2$};

\end{tikzpicture}

% ANOMALOUS PRECESSION
\begin{tikzpicture}
\def\R{1.4}
\def\v{0.4*\R}
\def\F{0.55*\R}
\def\N{8}
\coordinate (O) at (0,0);
\coordinate (Q) at (\R,0);

% CYCLOTRON
\draw[fill=metal,even odd rule] (O) circle (0.84*\R) circle (1.16*\R);

% MAGNETIC FIELD
\pic at (15:1.4*\R) {Bout};
\node[Bcol,right=3] at (15:1.4*\R) {$\vb{B}$};

% SPINS
\draw[-{Latex[length=3,width=2.5]},thick,Scol!40] (90:0.94*\R) --++ (-3*\N:0.30*\R);
\foreach \i [evaluate={\ang=90-(\i-1)*360/\N; \dang=3*(\i-1);}] in {1,...,\N}{
\draw[velocity,-{Latex[length=4,width=3]}] (\ang:1.02*\R) --++ (\ang-90:\v);
\pic at (\ang:0.94*\R) {spin={\ang-90-\dang}};
}
\node at (0,0) {$g > 2$};

\end{tikzpicture}

% MOMENTUM & SPING
\begin{tikzpicture} %[scale=0.5]
\def\v{0.7}
\draw[velocity,-{Latex[length=4,width=3]}] (0,0) --++ (\v,0);
\pic[scale=1] at (0,0.2) {spin={0}};
\node[left,scale=0.5] at (0,0) {momentum};
\node[left,scale=0.5] at (0,0.2) {spin};
\end{tikzpicture}

% ANOMALOUS PRECESSION
\begin{tikzpicture}
\def\v{0.7}
\def\N{8}
\def\ang{-50}
\coordinate (O) at (0,0);

% SPINS
\draw[very thin,dash pattern=on 1pt off 1pt] (O) -- (\ang:0.95*\v);
%\draw[-{Latex[length=3,width=2.5]},thick,Scol!40] (90:0.94*\R) --++ (-3*\N:0.30*\R);
\draw[velocity,-{Latex[length=4,width=3]}] (O) --++ (\v,0);
\pic[scale=1] at (O) {spin={\ang}};
%\draw[-{Latex[length=2,width=2]},scale=0.5]
%  (\ang-30:0.8*\v) arc (\ang-30:\ang-100:0.8*\v)
%  node[midway,below,scale=0.8] {$\omega_\mathrm{a}$};
\draw[-{Latex[length=2,width=2]}]
(0:0.65*\v) arc (0:\ang:0.65*\v)
node[midway,right,scale=0.7] {$\omega_\mathrm{a}t + \phi$};

\end{tikzpicture}

% ANOMALOUS PRECESSION - 10
\begin{tikzpicture}
\def\v{0.7}
\def\ang{-10}
\coordinate (O) at (0,0);
\draw[velocity,-{Latex[length=4,width=3]}] (O) --++ (\v,0);
\pic[scale=1] at (O) {spin={\ang}};
\end{tikzpicture}

% ANOMALOUS PRECESSION - 50
\begin{tikzpicture}
\def\v{0.7}
\def\ang{-50}
\coordinate (O) at (0,0);
\draw[velocity,-{Latex[length=4,width=3]}] (O) --++ (\v,0);
\pic[scale=1] at (O) {spin={\ang}};
\end{tikzpicture}

% ANOMALOUS PRECESSION - 120
\begin{tikzpicture}
\def\v{0.7}
\def\ang{-120}
\coordinate (O) at (0,0);
\draw[velocity,-{Latex[length=4,width=3]}] (O) --++ (\v,0);
\pic[scale=1] at (O) {spin={\ang}};
\end{tikzpicture}

% ANOMALOUS PRECESSION - 150
\begin{tikzpicture}
\def\v{0.7}
\def\ang{-150}
\coordinate (O) at (0,0);
\draw[velocity,-{Latex[length=4,width=3]}] (O) --++ (\v,0);
\pic[scale=1] at (O) {spin={\ang}};
\end{tikzpicture}

% CYCLOTRON FREQUENCY
\begin{tikzpicture}
\pic at (0,0) {Bout};
\node[Bcol,right=3] at (0,0) {$\vb{B}$};
\end{tikzpicture}

\end{document}