Merge branch 'main' of https://github.com/ajsteinmetz/plasma-partition

Author: ajsteinmetz

plasma-partition.tex

@@ -212,7 +212,7 @@ \section{Theory of thermal matter-antimatter plasmas}
  \label{kgp}
  E^{n}_{\sigma,s}(p_{z},{\cal B})=\sqrt{m_{e}^{2}+p_{z}^{2}+e{\cal B}\left(2n+1+\frac{g}{2}\sigma s\right)}\,,
 \end{align}
-where $n\in0,1,2,\ldots$ is the Landau orbital quantum number. \req{kgp} differentiates between electrons and positrons which is to ensure the correct non-relativistic limit is reached; see \rf{tab:1}. The parameter $g$ is the gyro-magnetic ($g$-factor) of the particle. Following the conventions found in \cite{Tiesinga:2021myr}, we set $g\equiv g_{e^{+}}=-g_{e^{-}}>0$ such that electrons and positrons have opposite $g$-factors and opposite magnetic moments.
+where $n\in0,1,2,\ldots$ is the Landau orbital quantum number. \req{kgp} differentiates between electrons and positrons which is to ensure the correct non-relativistic limit is reached; see \rf{fig:schematic}. The parameter $g$ is the gyro-magnetic ($g$-factor) of the particle. Following the conventions found in \cite{Tiesinga:2021myr}, we set $g\equiv g_{e^{+}}=-g_{e^{-}}>0$ such that electrons and positrons have opposite $g$-factors and opposite magnetic moments which is schematically shown in \rf{fig:schematic}.
 
 As statistical properties depend on the characteristic Boltzmann factor $E/T$, another interpretation of \req{tbscale} in the context of energy eigenvalues (such as those given in \req{kgp}) is the preservation of magnetic moment energy relative to momentum under adiabatic cosmic expansion.
 
@@ -250,7 +250,7 @@ \section{Theory of thermal matter-antimatter plasmas}
     \cline{2-3}
     \end{tabular}
     \caption{Organizational schematic of matter-antimatter $(\sigma)$ and polarization $(s)$ states with respect to the sign of the non-relativistic magnetic dipole energy $U_{\rm Mag}$ (obtainable from \req{kgp}) and the chemical $\mu$ and polarization $\eta$ potentials as seen in \req{partitionpower:2}.}
-    \label{tab:1}
+    \label{fig:schematic}
 \end{figure}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
@@ -283,22 +283,24 @@ \subsection{Spin and spin-orbit partition functions}
     +\exp\left(k\frac{-\mu-\eta}{T}\right)\exp\left(-k\frac{{\tilde m}_{-,+}\varepsilon_{-,+}^{n}}{T}\right)\\
     +\left.\exp\left(k\frac{-\mu+\eta}{T}\right)\exp\left(-k\frac{{\tilde m}_{-,-}\varepsilon_{-,-}^{n}}{T}\right)\right)
 \end{multline}
-We note from \rf{tab:1} that the first and forth terms and the second and third terms share the same energies via
-\begin{align}
+We note from \rf{fig:schematic} that the first and forth terms and the second and third terms share the same energies via
+\begin{gather}
     \label{partitionpower:3}
     \varepsilon_{+,+}^{n}=\varepsilon_{-,-}^{n}\,,\qquad
     \varepsilon_{+,-}^{n}=\varepsilon_{-,+}^{n}\,.\qquad
-    \varepsilon_{+,-}^{n}<\varepsilon_{+,+}^{n}\,,
-\end{align}
+    \varepsilon_{+,-}^{n}<\varepsilon_{+,+}^{n}\,.
+\end{gather}
 
-\req{partitionpower:3} allows us to reorganize the partition function with a new magnetization quantum number $s'$ which characterizes paramagnetic flux increasing states $(s'=+1)$ and diamagnetic flux decreasing states $(s'=-1)$. This recasts \req{partitionpower:2} as {\color{red}(maybe we should show $\epsilon_{s^\prime}=\epsilon_{++},\epsilon_{+-}$ clear )}
+\req{partitionpower:3} allows us to reorganize the partition function with a new magnetization quantum number $s'$ which characterizes paramagnetic flux increasing states $(s'=+1)$ and diamagnetic flux decreasing states $(s'=-1)$. This recasts \req{partitionpower:2} as
 \begin{multline}
     \label{partitionpower:4}
     \ln{{\cal Z}_{e^{+}e^{-}}}=\frac{e{\cal B}V}{(2\pi)^{2}}\sum_{s'}^{\pm1}\sum_{n=0}^{\infty}\sum_{k=1}^{\infty}\int_{-\infty}^{+\infty}{\rm d}p_{z}\frac{(-1)^{k+1}}{k}\\
     \left[2\xi_{s'}\cosh\frac{k\mu}{T}\right]\exp\left(-k\frac{{\tilde m}_{s'}\varepsilon_{s'}^{n}}{T}\right)
 \end{multline}
 with dimensionless energy, polarization mass, and polarization redefined in terms of $s'$
 \begin{gather}
+    \epsilon_{s'=+1}^{n}=\epsilon_{+,-}^{n}\,,\qquad
+    \epsilon_{s'=-1}^{n}=\epsilon_{+,+}^{n}\,,\\
     {\tilde m}_{s'}^{2}=m_{e}^{2}+e{\cal B}\left(1-\frac{g}{2}s'\right)\,,\\
     \eta\equiv\eta_{+}=-\eta_{-}\qquad\xi\equiv\xi_{+}=\xi_{-}^{-1}\,.
 \end{gather}