CPO3D 64th example: enhancement factor for a square array of nanotubes, all having the form of 'hemisphere on post'.

The 'posts' have a height h = 0.5 micron (ie 0.0005mm) and a radius a = 1nm.

They are placed here at a distance s = 160nm apart.

To create a field an anode is placed at an arbitrary 4 micron away, with a potential that gives a 'far field' of 1V/mm.

3 symmetry planes are used.

There are 25 tubes (an array 5 times 5). The central tube has more segments than the other tubes.

The total number of segments (after reflections) is 13120, of which 8944 are for the nanotubes (1920 for the central tube).

A change made in July 2019 was to retain the spherical end cap on the central tube, but to remove the caps on the other tubes, to avoid

overlapping of segments caused by the high level of symmetry. This is not expected to affect the results significantly.

The total number of segments (after reflections) is 13120, of which 8944 are

for the nanotubes (1920 for the central tube).

A 'boundary plate' surrounds the top ends of the tubes, in the z = 0 plane. The potential of this plate is V3 = 0.000123, which is aproximately the mean potential in the plane z = 0 in the region of the tubes. This potential is added to the anode potential to give the required far field. The value of V3 is determined iteratively to ensure that the potential above the central tube at (x,y,z) = (0,0,.0003) is approximately equal to that above the boundary plate at (0,0.001,0.0003).

The system is enclosed in a square shield of side 0.0016.

Results (before the removal of the end caps mentioned above):

x y z ex ey ez etotal (V/mm)

0.00E+00 0.00E+00 5.00E-08 0.00000E+00 0.00000E+00 -9.04431E+01 9.04431E+01

0.00E+00 0.00E+00 1.00E-07 0.00000E+00 0.00000E+00 -8.22335E+01 8.22335E+01

0.00E+00 0.00E+00 1.50E-07 0.00000E+00 0.00000E+00 -7.52763E+01 7.52763E+01

0.00E+00 0.00E+00 2.00E-07 0.00000E+00 0.00000E+00 -6.92330E+01 6.92330E+01

0.00E+00 0.00E+00 2.50E-07 0.00000E+00 0.00000E+00 -6.39435E+01 6.39435E+01

0.00E+00 0.00E+00 3.00E-07 0.00000E+00 0.00000E+00 -5.92846E+01 5.92846E+01

0.00E+00 0.00E+00 3.50E-07 0.00000E+00 0.00000E+00 -5.51584E+01 5.51584E+01

0.00E+00 0.00E+00 4.00E-07 0.00000E+00 0.00000E+00 -5.14844E+01 5.14844E+01

0.00E+00 0.00E+00 4.50E-07 0.00000E+00 0.00000E+00 -4.81978E+01 4.81978E+01

0.00E+00 0.00E+00 5.00E-07 0.00000E+00 0.00000E+00 -4.52450E+01 4.52450E+01

Analysis of these results:

Extrapolate field to z=0 by extrapolation of E(z)*(z+a)**2. But this extrapolation is not smooth at small z. So take the lowest value of E(z)*(z+a)**2 (which is at z=1E-7) and divide it by a**2, which gives E(a) = 9950.3. Therefore enhancement factor, gamma = 99.50.

Please see the footnotes to the data file for a considerable amount of further information.

Please also see papers 55 and 59 in the publications list.