CPO2D 25th example -creation of secondaries with Maxwell or Ura energy distributions

See also xmpl2d48 for an example of secondaries that have a different mass and charge from the primaries.

Here the energies and directions of 10 rays are measured when they reach a test plane at z = 0.8

Here the line

rm 1111 1 1.5 1 2 seed, min en, kT, add, mult

triggers a lambertian angular distribution, 'r', and a maxwellian energy distribution, 'm'.

'1111' is the seed for random numbers

'1' is the primary minimum energy

'1.5' is kT for the distribution

'1' is a constant energy that will be added to the distribution

'2' is the multiplication factor for the current

The use of the lambertian angular distribution implies that the non-meridional mode will be switched on after the production of each secondary (this feature was added in March 2011).

A Ura energy distribution would be triggered by 'u', where '1.5' is now the most probable energy, see below.

The 'Ura' distribution is given by K Ura, H Fujioka and T Yokobashi, "Calculation of local field effect on voltage contrast in SEM", EM 1980, P Bredero (Ed) (Seventh European Congress on EM Foundation, Leiden 1980) 330-331. It has been quoted by J Zach, Optik 83 (1989) 30-40.

The formula is:

dn/dw = (4/3)*w*exp(-2*sqrt(w))*dw, where w = E/Emax and Emax is the most probable energy. The distribution has a long high-energy tail. The mean energy is 5*Emax.

A mathematics package can be used to analyse the energies given in the ray output file, to check that the energy distribution has the correct mean value etc (although the fluctuations tend to be large). The 100 equally probable values of E/Emax used by the CPO program are given below (and the 100th is adjusted to give the correct mean energy). Note that the last few of

these 100 values are large (due to the long high-energy tails in these distributions).

The 100 equally probable values of E/Emax used by the CPO program for the Maxwell energy distribution:

0.03586,0.07579,0.10790,0.13657,0.16317,0.18838,0.21257,0.23600,

0.25882,0.28116,0.30313,0.32478,0.34618,0.36738,0.38842,0.40933,

0.43014,0.45089,0.47158,0.49226,0.51292,0.53360,0.55430,0.57504,

0.59584,0.61671,0.63766,0.65871,0.67987,0.70114,0.72255,0.74409,

0.76579,0.78766,0.80970,0.83193,0.85435,0.87699,0.89985,0.92295,

0.94629,0.96989,0.99376,1.01792,1.04239,1.06716,1.09227,1.11772,

1.14354,1.16974,1.19634,1.22335,1.25081,1.27872,1.30712,1.33602,

1.36546,1.39545,1.42603,1.45724,1.48910,1.52164,1.55491,1.58896,

1.62381,1.65953,1.69616,1.73376,1.77239,1.81213,1.85305,1.89522,

1.93875,1.98373,2.03027,2.07851,2.12859,2.18065,2.23490,2.29153,

2.35079,2.41294,2.47832,2.54730,2.62033,2.69794,2.78080,2.86971,

2.96566,3.06994,3.18419,3.31065,3.45232,3.61357,3.80089,4.02475,

4.30348,4.67421,5.23252,6.78293

The 100 equally probable values of E/Emax used by the CPO program for the Ura energy distribution:

0.11297, 0.21629, 0.29695, 0.36848, 0.43484, 0.49789, 0.55866,

0.61779, 0.67572, 0.73278, 0.78920, 0.84517, 0.90085, 0.95634,

1.01177, 1.06721, 1.12275, 1.17845, 1.23437, 1.29058, 1.34712,

1.40405, 1.46141, 1.51924, 1.57758, 1.63649, 1.69599, 1.75613,

1.81694, 1.87848, 1.94077, 2.00385, 2.06777, 2.13257, 2.19829,

2.26497, 2.33265, 2.40139, 2.47123, 2.54221, 2.61439, 2.68783,

2.76256, 2.83866, 2.91619, 2.99520, 3.07576, 3.15795, 3.24183,

3.32749, 3.41500, 3.50446, 3.59595, 3.68958, 3.78545, 3.88368,

3.98437, 4.08766, 4.19370, 4.30262, 4.41458, 4.52976, 4.64834,

4.77053, 4.89653, 5.02660, 5.16099, 5.29998, 5.44388, 5.59304,

5.74784, 5.90868, 6.07605, 6.25046, 6.43249, 6.62279, 6.82211,

7.03127, 7.25125, 7.48313, 7.72820, 7.98793, 8.26406, 8.55865,

8.87416, 9.21355, 9.58046, 9.97939,10.41602,10.89765,11.43385,

12.03751,12.72660,13.52711,14.47868,15.64596,17.14498,19.21627,

22.50049,32.95967