xmpl2d14, 14th 'example' data file for CPO2D
A twotube lens with a linear voltage bridge across the gap.
The geometry of the lens is identical to that in the 'example' file xmpl2d01.dat, except that the gap between the cylinders is filled with an extra cylinder, the voltage of which varies linearly across the gap.
Usable only with the full, spacecharge version, CPO2DS (because the number of segments is greater than 50).
Detailed description:
The following data were obtained when the memory and speed of PC's was much more limited than at present, so the available number of segments was small and the requested inaccuracies were fairly high to give a quick demonstration.
The Finite Element and Finite Difference Methods (also known as 'grid' methods) are capable of treating this lens, allowing those methods to be compared with the present Boundary Charge Method.
Another difference is that a larger number (N) of segments is used than in xmpl2d01.dat, and the smallest available inaccuracy value for the potential evaluations. The cylinders have been subdivided to give higher accuracy for the axial potentials inside the first cylinder, and to ensure that the highest charge carried by any segment is not more than about 3 times larger than the average charge. Iterative subdivision (that is, 'adaptive segmentation) could also have been used, with almost the same results.
The following results are obtained for initial and final voltages of 1 and 10 respectively (and with a midplane voltage of 5.5):
f1,2, F1,2 = 
0.79819 
2.52416 
1.61306 
1.17757 

cs0,1,2,3,4= 
3.6705 
7.4862 
5.9445 
2.1875 
0.31744 
cc0,1,2,ptz= 
0.82675 
1.00126 
0.31561 
0.45394 

These results can be 'extrapolated to infinity', as described in section 3.4 of the User's Guide (and in the footnotes of file xmpl2d01.dat). The table below shows the results obtained from doing this, using values of N from 450 to 900. As in the lens treated in the file xmpl2d01.dat, plots of the lens parameters p versus 1/(N^2) are found to be approximately straight, and so the asymptotic values p(infinity) can be found. The values p(900) at N=900 are then used as the recommended values, and the errors are chosen so that they include p(825) and p(infinity). Plots of the parameters p versus various powers of acc, for acc from 0.00001 to 0.00004, are found not to be smooth in this example, but these values could still be used to give estimated additional contributions to the errors of the parameters. Instead we have used a version of the program in which inaccuracies of 0.000001 can be requested, and find that inaccuracies of 0.00001 and 0.000001 give results that differ by less than the errors already established from the extrapolation of N. The following results have been obtained with the more accurate version of the program (where the numbers in brackets represent the estimated errors in the last significant figures):
f1,2, F1,2 = 
0.79821(1) 
2.52415(2) 
1.61306(2) 
1.17758(1) 

cs0,1,2,3,4= 
3.6706(1) 
7.4866(3) 
5.9448(3) 
2.1876(2) 
0.31746(2) 
cc0,1,2,ptz= 
0.82676(3) 
1.00127(2) 
0.31561(2) 
0.45393(1) 

These values can serve as 'benchmark' values for future comparisons.
For interest, we reproduce the 'benchmark' parameters for the lens treated in the file xmpl2d01.dat:
f1,2, F1,2 = 
0.799682(4) 
2.528819(12) 
1.616288(7) 
1.179489(6) 

cs0,1,2,3,4= 
3.67422(5) 
7.49678(10) 
5.95583(8) 
2.19294(3) 
0.318416(5) 
cc0,1,2,ptz= 
0.828658(5) 
1.003617(6) 
0.316353(3) 
0.453212(4) 

It can be seen that the effect of using the extra electrode to fill the gap between the cylinders is to reduce all the lens parameters by approximately 0.2%, except the Petzval integral, which is increased by 0.2%