CPO2D -30th example file, enhancement factor for a 'hemisphere on post'.
The 'post' has a height h = 1 micron (ie 0.01mm) and a radius a = 1nm. To create a field an anode is placed at an arbitrary 9 micron away, with a potential of 0.01V, giving a 'far field' of 1V/mm.
The field enhancement factor depends only on h/a
The ray output file contains the field at 10 points at distances of 0.05 to 0.5 nm from the surface of the cap.
Results, fields in V/mm:
|
r |
z |
er |
ez |
etotal |
|
0.000E+00 |
5.000E-08 |
0.000000E+00 |
-5.464995E+02 |
5.464995E+02 |
|
0.000E+00 |
1.000E-07 |
0.000000E+00 |
-4.984520E+02 |
4.984520E+02 |
|
0.000E+00 |
1.500E-07 |
0.000000E+00 |
-4.567765E+02 |
4.567765E+02 |
|
0.000E+00 |
2.000E-07 |
0.000000E+00 |
-4.203965E+02 |
4.203965E+02 |
|
0.000E+00 |
2.500E-07 |
0.000000E+00 |
-3.884485E+02 |
3.884485E+02 |
|
0.000E+00 |
3.000E-07 |
0.000000E+00 |
-3.602371E+02 |
3.602371E+02 |
|
0.000E+00 |
3.500E-07 |
0.000000E+00 |
-3.351977E+02 |
3.351977E+02 |
|
0.000E+00 |
4.000E-07 |
0.000000E+00 |
-3.128681E+02 |
3.128681E+02 |
|
0.000E+00 |
4.500E-07 |
0.000000E+00 |
-2.928671E+02 |
2.928671E+02 |
|
0.000E+00 |
5.000E-07 |
0.000000E+00 |
-2.748781E+02 |
2.748781E+02 |
Analysis of the results:
Extrapolate field to z=0 by quartic extrapolation of E(z)*(z+1e-6)**2: gives E(a) = 602.29. Therefore enhancement factor, gamma = 602.29
Vary N = number of segments, dz = z_anode - z_baseplate, r = radius of bridge to ensure that inaccuracy is less than about 0.01 percent. All dimensions in mm, radius a of cap = 1E-6mm
Vary length h of post:
|
h |
l/a |
N |
dz |
r |
gamma |
Edgcombe |
Fit |
difference% |
|
.003 |
3001 |
1650 |
.06 |
.03 |
1644.99 |
1643.6 |
1643.29 |
0.10 |
|
.001 |
1001 |
750 |
.01 |
.01 |
602.82 |
597.5 |
603.33 |
0.09 |
|
.0003 |
301 |
950 |
.01 |
.01 |
202.21 |
200.5 |
202.88 |
0.33 |
|
.0001 |
101 |
750 |
.01 |
.01 |
76.994 |
76.37 |
76.76 |
0.30 |
|
.00003 |
31 |
950 |
.01 |
.01 |
28.405 |
28.18 |
28.43 |
0.07 |
Here l = h + a, the total length of the nanotube.
The empirical fit is to 1.0782*(l/a + 4.7)**0.9152
The column labelled 'Edgcombe' are those given by C J Edgcombe and U Valdre, Solid State Electronics 45 (2001) 857-863. These authors use the fit
1.125*(l/a + 2)**0.91
Dependence of field strength on angle (ie angle to axis):
|
theta |
Er |
fit |
difference% |
|
0 |
602.84 |
601.65 |
0.2 |
|
10 |
601.54 |
600.25 |
0.2 |
|
20 |
597.51 |
596.14 |
0.2 |
|
30 |
590.61 |
589.24 |
0.2 |
|
40 |
580.54 |
579.33 |
0.2 |
|
50 |
566.78 |
566.12 |
0.1 |
|
60 |
548.45 |
549.20 |
0.1 |
|
70 |
523.98 |
527.28 |
0.6 |
|
80 |
489.68 |
497.15 |
1.5 |
|
90 |
428.29 |
422.59 |
1.3 |
Fit is 422.6 + 179.1*sqrt(cos(theta))