<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\subsection{NCRF: Grid of Tubes Alternative to Foil Windows.}
\label{option-grid}
\begin{table}[!htb]
\begin{center}
\caption{Performance dependency on rf cavity apertures.}
\begin{tabular}{|ccc|}
\hline
Maximum aperture  &amp; $\mu$/p&amp; $\mu$/p   \\
  (cm)    &amp; Be foil&amp; Al tubes \\
\hline
21 &amp; .174 &amp; 0.189 \\ 
25 &amp; 0.19 &amp; 0.204 \\
30 &amp; 0.195 &amp; 0.21 \\
\hline
\end{tabular}
\end{center}
\label{deprtb}
\end{table}

If the radii of the Be foil rf windows could be increased without
increasing their thickness, then the performance can be
improved. Table~\ref{deprtb} shows results for the baseline window
thicknesses, and for $80\mu$m Al windows that correspond in material 
thickness  to a grid
of tubes (see below). In both cases there appear to be significant
gains. However, for edge-cooled Be foils, their thickness must be increased 
as the fourth power of radius to avoid excessive temperature rise. If
this is done, the performance falls instead of rising. For a gas-cooled grid 
of thin-walled tubes, however, the pipe thickness is
independent of aperture radius, and no degradation would be expected as the aperture increases.
 
Tracking with 5~cm diameter pipes has shown that the field
 non-uniformities lead to increases in emittance, but these problems
 can be avoided if the pipe diameter is reduced and the number of pipes 
  is increased. A second advantage of many small tubes is that, for a
 given pressure, the wall thicknesses can be reduced.

For 1-cm diameter pipes, spaced on 2-cm centers, with wall
thicknesses of 25 $\mu$m) the tension in the walls with 1 atmosphere of gas
in the pipes would be only 3000 psi, which
should be acceptable. For this diameter, the non-uniform field
effects appear small.  When a pair of such grids (at right
angles to one another) is simulated by a plane foil with the same
average material thickness (80~$\mu$m Al), then the performance gain with respect to the baseline, as seen in Table~\ref{deprtb}, is 17\% for a 25~cm aperture and
approximately 20\% for a 30~cm aperture (see Section~\ref{APP-NCRF:grid} for a discussion of the grid tube approach).</pre></body></html>