Test2d13, 13th 'benchmark test' data file for CPO2D

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 current is given by the program with an error of 0.8%, in a total computing time of much less than a minute.


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.

In this example the cathode and anode have radii of 0.001mm and 10mm respectively, the cathode-anode voltage difference is 4000V, and so the field at the cathode is 4.0004E7 V/cm. The work function of the cathode is 4.5eV. The field at the cathode is 4.0004E7 V/cm.


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 Fowler-Nordheim equation (see below for details) is used, assuming a zero cathode temperature (thermal emission can be treated with CPO3DS).

The z = 0 plane is used as a plane of reflection symmetry and the cathode and anode are given 10 and 20 subdivisions respectively. The plane rx + z = 0.1 is used 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 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 0.05%, 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 current that should be given by the program is 1.4774 mA. In fact in this example it is 1.4669 mA, giving an error of 0.7%

Choice of step lengths:  The initial step length of a ray should obviously be  comparable to the radius of the cathode (if curved) but the step lengths need  to be much longer after a ray has left the vicinity of the cathode.  In this data file the initial step length has not been specified by the user, so the program makes it half the length of the cathode segment.  The program increases the length of later steps, up to the the specified maximum step length.

However it is better in general (and essential when there are space-charge effects) to use the 'advanced' option for increasing the step length in a 

controlled way as the ray tracing progresses.  Please refer to field emission crashes for further details. 


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 CPO2D 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 (in 2 planes) of the cathode.


Doubling the number of segments reduces the error in the current to 0.6% but increases the total computing time by a factor of approximately 3.

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, but in CPO2DS this is not taken into account. For a more careful treatment, including thermal effects, use CPO3DS.





The space-charge between the cathode and anode can be taken into account by using the ‘cathode iteration’ option, for example by

replacing the lines:

n calculate space-charges?

1 1 symmetries of rays in r (or x) and z planes

by the lines:

y calculate space-charges?

0.1 mesh spacing for space-charges

iterate previous cathode rays

0.5 2 damping factor, no of iterations


The current then quickly converges to 0.7269 mA, 1.0% lower than without space-charge (and it decreases by a further 0.3% if a much smaller mesh spacing is used). In fact 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.


In more detail:

It can be shown that at the radius r the ratio of the field due to space-charge to the field due to the surface charges is approximately

ratio = a*b,


a = (I*u)/(4*pi*epsilon0*v2*V2),

u = V2/(V2 + ei), where ei is a representative initial kinetic energy,

v2 = sqrt(2*e*V2/m) = velocity at r2,

b = c(x) - c(1),

c(x) = x*z/u - 0.5*ln((1 - z)/(1 + z)),

z = sqrt(1 - u/x),

x = r/r1.


b increases strongly with r and is a maximum at the outer radius r2.


Expanding b in terms of 1/x and (1 - u):

b/x = (V2 + 2*ei)/(V2 + ei) + ln(x)/(2*x) + O(1/x)

b is therefore approximately equal to x at r2.


max_ratio = (I*r2)/(4*pi*epsilon0*v2*V2*r1).

Using the parameters used above

max_ratio = 0.088

In practice the maximum ratio is unlikely to be larger than this.


Therefore the space-charge in the vicinity of the anode could be significant, but the space-charge near the cathode can be ignored.


The total charge inside a sphere of radius r is proportional to b.

An example of the value of b as a function of r, taking r1 = .00005m, u = .999:

r        0.00006        0.0001        0.001        0.01        0.1                1

       0.863        2.235        21.63        203.0        2006         20025

1000*b/r         14.38        22.35        21.63        20.30        20.06        20.03


So the total charge is approximately proportional to the radius.


We can therefore answer the question:

Can the size of the space-charge cells be larger (perhaps much larger) than the size of the cathode?

The answer is 'Yes', without introducing significant errors usually. (But the cells should of course be smaller than the size of the anode.)

The same applies to the diameter of the space-charge tubes (if the 'tube' method is used). In the vicinity of the cathode the tubes can be wider than the cathode and overlap, since the field due to the space charge is insignificant compared with the fields due to the surface charges on the cathode.