From 136b80877667f3f33be3f4a1afc45a22f3cfd976 Mon Sep 17 00:00:00 2001
From: Dhevan Gangadharan <dhevanga@gmail.com>
Date: Thu, 3 Nov 2022 12:56:11 -0500
Subject: [PATCH] Improve readability of magnetic multipole formula.
---
doc/usermanuals/DD4hep/chapters/basics.tex | 10 +++++-----
1 file changed, 5 insertions(+), 5 deletions(-)
diff --git a/doc/usermanuals/DD4hep/chapters/basics.tex b/doc/usermanuals/DD4hep/chapters/basics.tex
index 65fbf6af6..6e1bfcd77 100644
--- a/doc/usermanuals/DD4hep/chapters/basics.tex
+++ b/doc/usermanuals/DD4hep/chapters/basics.tex
@@ -1663,12 +1663,12 @@ Magnetic Dipoles are defined as follows:
Magnetic Multipole Fields are developed according to their approximation using the multipole coefficients. The dipole is assumed to be horizontal as it is used for bending beams in large colliders
i.e. the dipole field lines are vertical.
-The different momenta are given by:
+The different field components are given by:
\begin{eqnarray*}
-B_n^{\mathrm{norm}}(x)&=& \frac{cp}{e} \sum_{m=1}^{\frac{n}{2}} \frac{(-1)^{m-1} C_n y^{2 m-1} x^{n-2 m}}{(2 m-1)! (n-2 m)!} \quad, \\
-B_n^{\mathrm{norm}}(y)&=& \frac{cp}{e} \sum_{m=0}^{\frac{n-1}{2}} \frac{(-1)^m y^{2 m} C_n x^{-2 m+n-1}}{(2 m)! (-2 m+n-1)!} \quad, \\
-B_n^{\mathrm{skew}}(x)&=& \frac{cp}{e} \sum_{m=0}^{\frac{n-1}{2}} \frac{(-1)^m y^{2 m} S_n x^{-2 m+n-1}}{(2 m)! (-2 m+n-1)!} \quad, \\
-B_n^{\mathrm{skew}}(y)&=& \frac{cp}{e} \sum_{m=1}^{\frac{n}{2}} \frac{(-1)^m y^{2 m-1} S_n x^{n-2 m}}{(2 m-1)! (n-2 m)!}\quad.
+B_{x}^{\mathrm{norm}} &=& \frac{cp}{e} \sum_{m=1}^{\frac{n}{2}} (-1)^{m-1} \; C_n \frac{ x^{n-2 m} \; y^{2 m-1} }{ (n-2 m)!(2 m-1)! } \quad, \\
+B_{y}^{\mathrm{norm}} &=& \frac{cp}{e} \sum_{m=0}^{\frac{n-1}{2}} (-1)^m \; C_n \frac{ x^{n-2 m-1} \; y^{2 m} }{ (n-2 m-1)! (2 m)! } \quad, \\
+B_{x}^{\mathrm{skew}} &=& \frac{cp}{e} \sum_{m=0}^{\frac{n-1}{2}} (-1)^m \; S_n \frac{ x^{n-2 m-1} \; y^{2 m} }{ (n-2 m-1)! (2 m)! } \quad, \\
+B_{y}^{\mathrm{skew}} &=& \frac{cp}{e} \sum_{m=1}^{\frac{n}{2}} (-1)^m \; S_n \frac{ x^{n-2 m} \; y^{2 m-1} }{ (n-2 m)! (2 m-1)! }\quad.
\end{eqnarray*}
With $C_n$ being ``normal multipole coefficients'' and $S_n$ the ``skew multipole coefficients''. The maximal moment used is the octopole moment. The lower four moments are used
to describe the magnetic field:
--
GitLab