Authorea

Generalisation to normal and skew components

\label{sec:normalskew}

The previous section assumed an expansion at the mid-plane (\(y=0\)), symmetry around the mid-plane and considered only the vertical component of the field.

An extension using complex notation for the position (\(x+iy\)) and the field is given as

\begin{equation} B_{y}+iB_{x}=\sum_{n=0}^{\infty}(b_{n}\,+ia_{n})\frac{(x+iy)^{n}}{n^{n-1}}\\ \end{equation}

By introducing the normal and skew multipole coefficients \(KN\) and \(KS\) at order \(n\) as

\begin{equation} \label{eq:knn} \label{eq:knn}KN_{n}=q\,b_{n}/p_{s}=b_{n}/B\rho\\ \end{equation}

and

\begin{equation} \label{eq:kns} \label{eq:kns}KS_{n}=q\,a_{n}/p_{s}=a_{n}/B\rho\\ \end{equation}

the kicks received in each plane can be expressed as the summation over all components

\begin{equation} \Delta P_{x}-i\Delta P_{y}=\sum_{n=0}^{\infty}-(KN_{n}+iKS_{n})\frac{(x+iy)^{n}}{n!}\\ \end{equation}

Remark: need to add references to the \((a_{n},b_{n})\) field conventions in the magnet world.