CPO

The 'benchmark test' files.

The 'benchmark test' files deal with problems that have known analytical solutions, to demonstrate the speed, accuracy and capabilities of CPO-2D and CPO-3D. The 'comments' at the ends of the test and example files contain detailed information (including key formulas) on a wide variety of problems in charged particle optics. Look further below for short descriptions of each file.

The titles of the 'benchmark test' files are

CPO-2D:

test2d01 Idealised hemispherical deflection analyzer (HDA)

test2d02 Idealised cylindrical deflection analyzer

test2d03 Parabolic motion in uniform field (cylindrical symmetry)

test2d04 Parabolic motion in uniform field (planar symmetry)

test2d05 Relativistic motion in a uniform electric field

test2d06 Capacitance of a sphere

test2d07 Capacitance of coaxial cylinders

test2d08 Time-dependent potentials

test2d09 Coefficients of hemispherical deflection analyser (HDA)

test2d10 Space-charge limited current of a spherical diode (concave cathode)

test2d11 Space-charge limited current of a spherical diode (small convex cathode)

test2d12 Planar Pierce gun

test2d13 Spherical cold field emission diode

test2d14 Cylindrical diode, small inner cathode

test2d15 Field at a circular hole in an infinite sheet

test2d16 Ideal cylindrical mirror analyzer (CMA)

test2d17 Penetration of field and potential through a cylindrical mesh

test2d18 Field penetration through a flat mesh of flat strips

test2d19 Field at the tip of a cone

test2d20 Cylindrical Pierce gun

test2d21 Field at centre of a double cylinder lens

test2d22 Non-meridional rays, cylindrical symmetry electrodes.

test2d23 Non-xz-plane rays, planar symmetry electrodes.

test2d24 Motion in a User-supplied oscillating electric field.

test2d25 The '743' test.

CPO-3D:

test3d01 Idealised hemispherical deflection analyzer (HDA)

test3d02 Idealised cylindrical deflection analyzer

test3d03 Parabolic motion in uniform field

test3d04 Relativistic cyclotron motion

test3d05 Relativistic motion in a uniform electric field

test3d06 Capacitance of a cube

test3d07 Capacitance of a sphere

test3d08 Time-dependent oscillating potentials

test3d09 Coefficients of hemispherical deflection analyser (HDA)

test3d10 Space-charge limited current of a spherical diode (concave cathode)

test3d11 Space-charge limited current of a spherical diode (convex cathode)

test3d12 Spherical cold field emission diode

test3d13 Field at a circular hole in an infinite sheet

test3d14 Ideal cylindrical mirror analyzer (CMA)

test3d15 Capacitance of circular disc

test3d16 Motion in a User-supplied oscillating electric field

test3d17 Capacitance and singularities of a tetrahedron

test3d18 Capacitance of a unit square

test3d19 Capacitance of a unit triangle and summary of capacitance tests

test3d20 Ideal quadrupole ion trap

Further information and short descriptions of the test files:

CPO-2D:

test2d01 Idealised hemispherical deflection analyzer (HDA)

Here an idealised hemispherical deflection analyser is created out of two complete concentric spheres. Examples are shown of the potentials and fields in the space between the spheres. A ray is then started with the median energy along the median path. The final coordinate at the exit plane, which should be 1.0, can be seen to be 0.99996.

test2d02 Idealised cylindrical deflection analyzer

This deals with the 'perfect' cylindrical deflection analyzer, and is the analogue of test2d01.

test2d03 Parabolic motion in uniform field (cylindrical symmetry)

In this test a uniform electric field is set up between two circular discs. Examples of calculated potentials and fields are given. Two rays are traced, both of which should be parabolic. The coordinates of the first ray are recorded when it crosses a 'test plane' and again when it hits an electrode. At the test plane z should maximise at 0.5 and the energy should be 0.5 eV. In fact the calculated maximum z is 0.49995 and the energy is 0.4999. The errors are consistent with the requested ray inaccuracy of 0.05%. The second ray shows how the program deals with reflections, reducing the step lengths when the ray is near the reflection point.

test2d04 Parabolic motion in uniform field (planar symmetry)

This test is almost identical to that in test2d03, except that it is for a system of planar rather than cylindrical symmetry.

test2d05 Relativistic motion in a uniform electric field

The set-up is the same as in test2d03, except that the energy is 1 MeV and the voltage is 1 MV. At the turning point z should be 0.38312, and is found to be 0.38309 (an error of 0.008%)

test2d06 Capacitance of a sphere

The total charge on a sphere is reproduced in a few seconds in this benchmark test, with an error of 0.04%. By extrapolating to an infinite number of segments it is obtained essentially exactly.

test2d07 Capacitance of coaxial cylinders

A cylinder of radius 1mm is coaxial with another of radius 2mm. The potential difference is 1V. The total charge is found to be 8.02522E-14, compared with the correct result 8.02607E-14 (=2*pie*epsilon0*0.001/ln(2)). The error is therefore 0.01%, but the total number of segments is only 49.

test2d08 Time-dependent potentials

This test reproduces the sinusoidal motion of an electron in a sinusoidal electric field. A uniform electric field is set up between 2 flat parallel plates, as in file test2d04. The value of z at the crossing point of a particular plane should be 0.12452 mm. The value given by the program is 0.12441, an error of 0.09% (which is consistent with the requested ray inaccuracy, 0.05%, and the total number of segments, 50).

test2d09 Coefficients of hemispherical deflection analyser (HDA)

This file is almost identical to test2d01, except that the 'coefficients of lens' option has been used. The User asks for three coefficients (the energy dispersion, the linear magnification and the second order angle coefficient), and specifies the exit plane and the starting conditions of the median (ie reference) ray. The program then automatically sets up and traces the minimum number of rays, each with a different starting condition, to deduce these coefficients.

test2d10 Space-charge limited current of a spherical diode (concave cathode)

The concentric spherical cathode and anode have radii of 2mm and 1mm respectively, and the cathode-anode voltage difference is 10V. The space-charge limited current is given by the program with an error of 1.8% in the last (ie the fourth) iteration, with a total computing time of much less than a minute. The error remains at this level in later iterations.

test2d11 Space-charge limited current of a spherical diode (small convex cathode)

The concentric spherical cathode and anode have radii of 0.01cm and 1cm respectively, and the voltage difference is 1V. The theoretical space-charge limited total current is 8.031 microamp. After 4 iterations the value given by the program has an error of 1.9%, in a total computing time of about a minute. The result converges after a total of 10 iterations to give an error of 1.3%

test2d12 Planar Pierce gun

The cathode has a width of 4mm in the x direction and is infinite in the y direction, the cathode-anode spacing is 10mm in the z direction and the cathode-anode voltage difference is 10V. The focus electrode that surrounds the cathode is inclined at the well-known 'Pierce angle' of 67.5 degrees. Childs Lawformulas8 is used in the immediate vicinity of the cathode surface, assuming a zero cathode temperature (thermal emission can be treated with CPO-3DS). The current density should be 0.73808 microamp/mm**2, and is given by the program with an error of 0.5% after 7 iterations, in a total computing time of much less than a minute.

test2d13 Spherical cold field emission diode

test2d14 Cylindrical diode, small inner cathode

The coaxial cathode and anode have radii of 0.01cm and 1cm respectively, and the voltage difference is 1V. The theoretical space-charge limited total current is 0.013601 mA/mm. After 5 iterations the value given by the program has an error of 5%, in a total computing time of less than a minute. The result converges after a total of 15 iterations to give an error of 1.4%

test2d15 Field at a circular hole in an infinite sheet

The problem is simulated by putting a hole of diameter 1 in an electrode of diameter 5, and enclosing the regions on both sides of the electrode with cylinders of length 10. With 50 segments the errors are less than 0.0006 for the potential and field on the axis, but go up to 0.003 for the more difficult region near the aperture edge (where the fields become infinitely large at the edge itself). The values extrapolated to an infinite number of segments are much more accurate.

test2d16 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.

test2d17 Penetration of field and potential through a cylindrical mesh

Planar symmetry is used to simulate an infinitely long cylindrical mesh composed of thin parallel wires. The mesh has a transparency of 97.5 percent and is surrounded by a complete cylinder at a different potential. It is found that the external field penetrates only a short distance through the mesh, but that there is a finite change in the potential throughout the mesh.

test2d18 Field penetration through a flat mesh of flat strips

test2d19 Field at the tip of a cone

test2d20 Cylindrical Pierce gun

test2d21 Field at centre of a double cylinder lens

test2d22 Non-meridional rays, cylindrical symmetry electrodes.

test2d23 Non-xz-plane rays, planar symmetry electrodes.

test2d24 Motion in a User-supplied oscillating electric field.

test2d25 The '743' test.

CPO-3D:

test3d01 Idealised hemispherical deflection analyzer (HDA)

test3d02 Idealised cylindrical deflection analyzer

This deals with the 'perfect' cylindrical deflection analyzer, and is the analogue of test3d01.

test3d03 Parabolic motion in uniform field

In this test a uniform electric field is set up inside a cube by applying potentials +1 and -1 to two opposite 'end' faces, and the appropriate linearly varying potentials to the other 'side' faces. All four planes of reflection symmetry are used, which increases the number of sudivisions by a factor of 16. Examples of calculated potentials and fields are given on the screen. Two rays are traced, both of which should be parabolic. The coordinates of the first ray are recorded when it crosses a 'test plane' and again when it hits an electrode. The second ray shows how the program deals with reflections, reducing the step lengths when the ray is near the reflection point.

test3d04 Relativistic cyclotron motion

This provides a test of the motion of a relativistic electron of energy 10 MeV in a uniform magnetic field of strength 1 Tesla. The radius of the cyclotron orbit is accurately reproduced.

test3d05 Relativistic motion in a uniform electric field

The set-up is the same as in test3d03, except that the energy is 1 MeV and the voltage is 1 MV. At the turning point z should be 0.38312, and is found to be 0.38308, an error of 0.01%

test3d06 Capacitance of a cube

This benchmark test calculates the capacitance of a unit cube. All faces of the cube are set to voltage +1, and the capacitance is deduced from the total charge on the faces. The capacity is obtained in a few seconds with an error of 0.4%, and by extrapolating to an infinite number of segments it is obtained with an estimated inaccuracy of 0.008%.

test3d07 Capacitance of a sphere

The total charge on a sphere is reproduced in a few seconds in this benchmark test, with an error of 0.04%. By extrapolating to an infinite number of segments it is obtained essentially exactly.

test3d08 Time-dependent oscillating potentials

This test reproduces the sinusoidal motion of an electron in a sinusoidal electric field. The field is applied to a cube, as in file test3d03.

test3d09 Coefficients of hemispherical deflection analyser (HDA)

This file is almost identical to test3d01, except that the 'coefficients of lens' option has been used. The User asks for three coefficients (the energy dispersion, the linear magnification and the second order angle coefficient), and specifies the exit plane and the starting conditions of the median (ie reference) ray. The program then automatically sets up and traces the minimum number of rays, each with a different starting condition, to deduce these coefficients.

test3d10 Space-charge limited current of a spherical diode (concave cathode)

The concentric spherical cathode and anode have radii of 1.0 and 0.1 respectively, and a voltage difference of 1V. The total space-charge limited current (which is independent of the scale length) should be 1.005 m A, and is given by the program as 1.000 m A, an error of 0.5%, in a total computing time of the order of a minute.

test3d11 Space-charge limited current of a spherical diode (convex cathode)

The cathode is a complete sphere of radius 0.1 mm, inside a concentric anode of radius 10 mm. The space-charge limited current should be 8.0304 m A, and is reproduced by the program with an eror of 0.1% -a notable achievement.

test3d12 Spherical cold field emission diode

The cathode and anode have radii of 1m m 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 total computing time of less than a minute.

test3d13 Field at a circular hole in an infinite sheet

The problem is simulated by putting a hole of diameter 1 in an electrode of diameter 5, and enclosing the regions on both sides of the electrode with cylinders of length 10. With 192 segments the errors are less than 0.004 for the potential and field on the axis, but go up to 0.014 for the more difficult region near the aperture edge (where the fields become infinitely large at the edge itself). With N = 768 these maximum errors become 0.0014 and 0.009 respectively.

test3d14 Ideal cylindrical mirror analyzer (CMA)

test3d15 Capacitance of circular disc

The capacitance of a thin circular disc is used to illustrate the process of extrapolating the number of segments to infinity, and to illustrate the options of automatic subdivision of segments and concentration of subdivisions, and also to illustrate the process of extrapolating different regions separately.

test3d16 Motion in a User-supplied oscillating electric field

test3d17 Capacitance and singularities of a tetrahedron

test3d18 Capacitance of a unit square

test3d19 Capacitance of a unit triangle and summary of capacitance tests

test3d20 Ideal quadrupole ion trap