Subject:
Grid transparency calculations
From:
Jim Crittenden <critten@lepp.cornell.edu>
Date:
Wed, 12 Mar 2008 16:15:33 -0400
To:
Mark Palmer <map36@cornell.edu>
CC:
David Rubin <DLR@CESR10.LNS.CORNELL.EDU>, Joe Calvey <jrc97@cornell.edu>, Jesse Livezey <jal245@cornell.edu>, Jerry Codner <gcodner@lepp.cornell.edu>, Shlomo Greenwald <sg@lepp.cornell.edu>

Mark

Using OPERA tracking, I have performed some
preliminary calculations of grid transparency for
electrons of 1, 10 and 100 eV in grids with varying
degrees of etching-related hole angles.

To give an idea of the model, I made some pictures.
Here is a view of the holes in the exaggerated "45-degree" version,
where the diameters of the holes on the surfaces run into each other:
http://www.lepp.cornell.edu/~critten/cesrta/notes/12mar08/rfa_10_grid.png

Here is a view of the exit side of the grid, including a few 1-eV electron
tracks and arrows indicating field direction and strength.
One can see the focusing effect on the electron trajectories and
the strong field resulting from the 50 V on the collector with the
grid at ground.
http://www.lepp.cornell.edu/~critten/cesrta/notes/12mar08/rfa_10_tr1.png

Now zoom in a little so as to see the field arrows turning to point
perpendicularly into the steel wall of the grid.
http://www.lepp.cornell.edu/~critten/cesrta/notes/12mar08/rfa_10_tr2.png

Zoom back out to get a clearer view of the electron trajectories.
http://www.lepp.cornell.edu/~critten/cesrta/notes/12mar08/rfa_10_tr3.png

Now turn to the entrance side of the grid, where the field is much weaker.
One can see it turning sideways to be perpendicular to the hole wall, which
is what gives the electrons the sideways focusing kick.
http://www.lepp.cornell.edu/~critten/cesrta/notes/12mar08/rfa_10_tr4.png

With this evidence that the model is doing something reasonable, I decided
to see if I could make an estimate of the grid transparency and its dependence
on the grid hole shape. To do this, I used a 21x21 grid of transverse grid entrance
positions extending from the center of the central hole out to half the distance
to the next hole. Here is an elevation view of the electron tracks in the X=0 plane.
http://www.lepp.cornell.edu/~critten/cesrta/notes/12mar08/rfa_8_1ev_2.png

Let's begin with the cylindrical holes. I found that 278 of the 441 1-eV e- tracks did not
make it to the grid. For 10 eV and 100 eV, 277 tracks did not make it. So we have:
Outer grid hole diam = Inner grid hole diam
--------------------------------------------------------
 1-eV:  37.0%
10-ev: 37.2%
100-ev: 37.2%

This seems reasonable given the 36-38% optical grid transparency numbers mentioned
during yesterday's meeting. Now lets look at the effect of conical holes. Starting
with the exaggerated "45-degree" holes.
Outer grid hole diam = Inner grid hole diam + Grid thickness
------------------------------------------------------------------------------
 1-eV:  44.9%
10-ev: 41.5%
100-ev: 41.5%
Even at 100 eV, there is still enough focusing to save a few tracks
relative to the cylindrical holes. More focusing raises the transparency, of course.

Here are the results for the two other outer hole diameter values I
calculated.
Outer grid hole diam = Inner grid hole diam + 1/2 Grid thickness
------------------------------------------------------------------------------
 1-eV:  41.5%
10-ev: 40.4%
100-ev: 39.7%

Outer grid hole diam = Inner grid hole diam + 1/4 Grid thickness
------------------------------------------------------------------------------
 1-eV:  40.6%
10-ev: 39.2%
100-ev: 39.2%

So a model and some analysis tools exist.
We should check these results for consistency with Joe's calculations.

-- Jim


========================================================
James Crittenden                   Tel. (607) 255-9424
Wilson Synchrotron Laboratory      Fax  (607) 255-8062
Cornell University
Ithaca, New York 14853-8001
========================================================