Test2d16, 16th 'benchmark test' data file for CPO2D

Ideal cylindrical mirror analyzer (CMA)

The ideal cylindrical mirror analyzer is simulated by allowing the electrons to pass through the charge sheet at the position of the inner cylinder, thus avoiding apertures. The known conditions for axis to axis focussing are used. The second order focussing property of this analyzer is clearly demonstrated.

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 object and image have been chosen to be on-axis (other choices are also possible).

Theoretically (see for example J S Risley, Rev Sci Instrum, vol 43, p95-103, 1972) the launch angle should be

alpha0 = 42.307 degrees

the parameter

k = (E/U)ln(r2/r1)

should be 1.3098, where eE is the initial energy, U is the potential difference between the two cylinders and r1 and r2 are their radii, and then the distance between the object and image is

z0 = 6.1298*r1

With these conditions the analyzer has a second order focus, which means that the dependence of the axial crossing position z of a ray that starts with a launch angle alpha can be expressed as

z = z0 + A*(alpha - alpha0)^3 + ...

where A = -15.53. In other words the term in (alpha-alpha0)**2, which is present for most other types of energy analyzers, has disappeared, which allows the CMA to be used with a wider range of angles.

In the present example, r1 = 1, r2 = 2.5, the initial energy is eE = 1eV, the potential of the inner cylinder is 0 and so the potential of the outer cylinder is put at -0.6996.

The answer to the question in the data file:

stop ray if it crosses a segment?

is 'n' (for 'no'), and so the rays can pass through the sheet of charge at the position of the inner cylinder, which therefore simulates the behaviour of the 'perfect' CMA (that is, with no fringe-field effects from apertures in the inner cylinder).

5 rays are started, with values of (alpha-alpha0) of 0, +/- 2.5 and +/- 5.0 degrees. The values of z given by the program are compared in the table below with those given by the formula above:

alpha-alpha0 z theoretical z program

(degrees)

-5.0 3.075 3.072

-2.5 3.066 3.061

0.0 3.065 3.062

2.5 3.064 3.063

5.0 3.055 3.054

More accurate results can of course be obtained by using more segments, with the full version of CPO2DS.