Test3d12, 12th 'benchmark test' data file for CPO3D

Spherical cold field emission diode


The cathode and anode have radii of 1m and 1cm respectively, and the cathode-anode voltage difference is 4000V. The work function of the cathode is 4.5eV. The Fowler-Nordheim equation is used. The initial velocity components are randomised, assuming a room-temperature cathode and the low-temperature limit. The current that should be given by the program is 1.4774 mA. In fact it is 1.4632 mA, giving an error of 1%, in a short total computing time.

This benchmark test is of course artificial  -real field emission cathodes are sharp points mounted usually on cones.  Here we are only illustrating the use of the Fowler-Nordheim equation.  When simulating real cathodes it is important to follow the advice on setting up the ray paramerters  Please also see general information on field emission.


This is a 2-dimensional simulation and so of course can be solved with higher accuracy by CPO2DS, see test2d12.dat.


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.

As explained in field emission crashes  it is usually important to use the option that changes the step length gradually.  

The energy is used here to define the changes (rather than the time or z).  This retains accuracy near the cathode but gives longer step lengths later in order to  reduce the computing time.  If the option is not used then crashes can occur.  This happens because at the beginning of a step (before the accurate trajectory  integration) the program uses the initial velocity, acceleration and step length to estimate the final velocity, which might then be unphysical.

To determine the parameters used in this option requires some experimentation, using a high printing level (temporarily) to inspect the  step lengths and energies of the first few steps of the first ray.  Note that if a large change in the step length is requested then the program automatically makes the change more gradual.

 Here the step length is set 0.000005 for an energy up to 1000eV, then 0.0005 up to 3keV, and 0.5 thereafter.


The Fowler-Nordheim equation (see below for details) is used, assuming a zero cathode temperature.

All 4 planes of reflection symmetry are used, and so the minimum sector is 1/16 of the complete sphere.

The cathode and anode are given 20 subdivisions each, but the program changes the number to 22 (as can be seen from the information put on the screen and in the processed data file), and so the number of cathode segments is entered as 22.

The potential should be a/r + b, and the field should be -a/r^2, where a = -4/.9999, b = 4000/.9999 for the present radii and voltages. The potentials and fields are explicitly requested for r = 0.002 and 0.1 mm, and it can be seen that the errors in these are approximately 0.2%, which is higher than the requested inaccuracy because the number of subdivisions is small. The user might want to ask for the field at the surface of the cathode, but (1) account must be taken of the change in the radius due to the application of the 'inscribing' correction, (2) the field must be sampled along a line that passes through the centre of curvature and the centre of the cathode segment (which is the starting point for a ray), and (3) the field must be sampled near to, but not on, the surface of the segment, because the field changes sign at the surface, and is indefinite on the surface itself. (The program takes careful account of all these things.)

The outer corners of the minimum sector are at

(10,0,0,), (7.07,7.07,0), (0,0,10),

which are in the plane x+0.414*y+z = 10, which has been used above as the test plane (so that rays which fail to reach this plane do not contribute to the space-charge). No iterations are used in this example to improve the calculation, because the space-charges of the rays have only a small effect on the field emission current (but see the comments below).

The total current should be 1.4774 mA (see below), after reflections in 4 symmetry planes. In fact in this example it is 1.473 mA, giving an error of 0.3%


The current is extremely sensitive to the magnitude of the field at the surface of the cathode -a change of 1% in the field gives a change of 17% in the current, for the present example. Also it should be noted that the most difficult field evaluation for CPO3D is the field near to the surface of an electrode, and that the calculated value of such a field is often significantly less accurate than for a field or potential at a point further away from a surface. This problem could be overcome, easily but artificially, for a cathode that is a complete sphere, because the field is given by the surface charge density, which is accurately calculated, but this cannot be done for cathodes of arbitrary shape (and so is not done by the program). To alleviate this problem the program automatically uses the highest available accuracy for calculating the surface field, independently of the ray accuracy requested by the user. The field is sampled a short distance away from the cathode surface and a small correction is applied to take account of this distance and of the size of the segment and the local curvature of the cathode.


The computing time for the boundary charges is very short, and for the rays is a few seconds. Increasing the number of segments reduces the error in the total current.

It can be noticed in the output data file that there is a randomised component in the energies. This is present because as explained in the relevant note the velocity components are randomised even if kT is zero. The energy spread is characterised by the acceleration energy d gained over a distance lambda/(4*pi), where lambda is the de Broglie wavelength corresponding to the energy eW and W is the work function -see R H Good and E W Muller, Handbuch der Physik, Vol 21, 176 (1956) and P W Hawkes and E Kasper, Principles of Electron Optics (Academic Press, 1989). For the present conditions, d = 0.37 eV.


An 'artificial' option is also available (in CPO3DS) in which the electrons start with fixed energy (chosen by the user), normal to the surface.



The space-charge between the cathode and anode can be taken into account, by replacing the lines:

n n calculate space-charges?

1 1 1 1 symmetries of rays in yz, zx, xy and x=y planes

by the lines:

y n calculate space-charges?

0.1 distance parameter for space-charges

iterate previous cathode rays***********************

0.5 2 damping factor, no of iterations

The current then quickly converges to 0.09068 mA, 0.8% lower than without space-charge. More accurate results can be obtained by using CPO2DS -see file test2d12.dat. The space charge in the immediate vicinity of the cathode surface does not directly change the field at the surface (unlike the situation for a thermal cathode governed by Childs Law) but the space charge in the region between the cathode and anode does produce changes of the order of a few volts in the potentials in that region, and so does have a small indirect effect on the field at the cathode surface. Because the effect is indirect it is (fortunately) not essential to use a small mesh spacing for the space charge cells.

The Fowler-Nordheim equation has been taken from R. H. Good and E. W. Muller, Handbuch der Physik, vol 21 (1956) and P. W. Hawkes and E. Kasper, Principles of Electron Optics (Academic Press, 1989). Distances are in mm, the field F is in V/mm, and the current density j is in mA/mm**2. The work function of the material of the cathode is P eV.


y = 0.0012000*sqrt(F)/P,

t(y) = 1 + 0.1107*y**1.33,

v(y) = 1 - y**1.69 + 0.0028*sin(2*pi*y),

(the 3rd term being a new empirical correction term)

c = 6.8309E6*P**1.5*v/F,

d = 9.7596E-8*F/(sqrt(P)*t),

p = kT/d,


 j = 0.0015414*(F/t)**2*(1/P)*exp(-c)*(pi*p)/sin(pi*p).


This is valid for y < 1 and p < 0.7. To quote Hawkes and Kasper ''For larger values of p no satisfactory simple theory is known''.

Note that for cpo2ds kT and p are zero and so the factor (pi*p)/sin(pi*p) = 1.


The most probable energy, ei, of the electrons is given by

 ei = -(P + d).

Note that this energy is negative and that it corresponds to an energy slightly below the Fermi energy of the cathode (in the present example d is 0.176 eV). This energy is used as the initial energy for the ray tracings.


The field emission electrons have an energy spread of width 2.44*d.