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\title{Notes on 2+1D Space Charge}
\author{Stephen Brooks}
\maketitle

% \section{Introduction}

A useful formula is that if the full charge density is separable
\[ \rho_\mathrm{3D}=\rho(x,y)\lambda(z), \]
then the space charge equation $\nabla^2\phi = \rho_\mathrm{3D}$ can be solved exactly by the infinite sum
\begin{eqnarray*}
\phi &=& \sum_{n=0}^\infty (-1)^n\ \nabla^{-2(n+1)}\rho\ \lambda^{(2n)} \\
&=& \sum_{n=0}^\infty (-1)^n\ \phi_{2D,n}\ \lambda^{(2n)},
\end{eqnarray*}
where we may calculate $\nabla^2\phi_{2D,0}=\rho$ and $\nabla^2\phi_{2D,n+1}=\phi_{2D,n}$ via Poisson solvers.
The associated force law is
\[ \B F = \B\nabla\phi = \sum_{n=0}^\infty (-1)^n\vectwo{
\B\nabla\phi_{2D,n}\ \lambda^{(2n)}
}{
\phi_{2D,n}\ \lambda^{(2n+1)}
}. \]
This series converges provided the repeated integrations transversely more than cancel the repeated derivatives longitudinally, which happens if $\rho$ varies spatially faster than $\lambda$.

It is also true that a non-separable density may always be expanded into a sum of separable terms
\[ \rho_\mathrm{3D}=\sum_{n=0}^\infty \rho_n(x,y)\lambda_n(z), \]
each of which may be treated by the previous series and linearly combined.  The convergence of this sum depends on the complexity of the charge distribution (the degree of longitudinal-transverse coupling), whereas the convergence of the other depends on the aspect ratio of detail in a long beam.

The traditional 2+1D formula corresponds in part to the $n=0$ term of both series.  Its force is proportional to 
\[ \B F = \vectwo{\B\nabla\phi_{2D,0}\ \lambda}{\phi_{2D,0}\ \lambda'}. \]
$\phi_{2D,0}$ is the solution to the usual transverse-only space charge problem.  The transverse force term is exactly as expected, but the longitudinal term has $\lambda'$ multiplied by the potential instead of a constant.  Which other terms in the expansion make this resemble a constant times $\lambda'$ is unclear, in fact it is not clear that a constant times $\lambda'$ is a better approximation than this first term.

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