CPO
-- Advanced electrostatics for space-charge, cathodes, and high accuracy

CPO Publications and Applications

Titles of refereed publications from the Manchester group Finally, the CPO-2D, CPO-3D and MAN-2D programs have been used over many years by staff members and sabbatical visitors. This has resulted in one book and many papers. The 'standard' book on electrostatic lenses is:

Electrostatic lenses, by E Harting and F H Read, Elsevier Publishing Company, Amsterdam (1976).

The authors of the CPO programs continue to publish original work in charged particle optics. The papers, in reverse time order (the abstracts of some of the more recent papers are given separately below) are:

Abstracts of some of the more recent publications from the Manchester group, using the CPO-2D, CPO-3D and MAN-2D programs, in reverse time order:

62. Correction of the astigmatism of hairpin cathodes, by OEl-Mimi, FHRead and N J Bowring, in preparation.
61. Stability criteria for electrostatic low-energy electron storage systems, by P Harvey, P Hammond, F H Read and T Redish, in preparation.
60. Simulation of Thermionic Cathodes, by F. H. Read and N. J.Bowring, Nucl. Instrum. Meth. A531, 407-415 (2004).
A technique is described of simulating thermionic cathodes in the Boundary Element Method by replacing the space-charges in the cathode region by a set of thin sheets of charge. These sheets are subdivided into segments in the same way that the cathode itself is subdivided. This technique has been applied to cathodes that have zero and non-zero temperature T. It is shown that for T>0 the potential and position of the virtual cathode is well reproduced. Planar and non-planar cathodes are considered as well as unsaturated cathodes.
59. Field enhancement factors of one-dimensional and two-dimensional arrays of nanotubes, by FHRead and N J Bowring, Microelectronic Engineering 73-74, 679-685 (2004).
The field enhancement factors of flat arrays of randomly-spaced, aligned, closed-end nanotubes have been determined by computational modeling. Arrays that contain a large number of nanotubes have been considered as well as isolated clusters that contain a limited number of nanotubes. The standard deviation of the enhancement factors in large arrays has been found to be approximately proportional to the mean enhancement factor but not to be strongly correlated with the radius, length or spacing of the nanotubes. A simple procedure is given for estimating the density of high-emitting nanotubes, ie hot-spots. For isolated clusters the dependence of the average enhancement factor on the number of nanotubes in the cluster has been investigated.
58. The contributions of stochastic coulomb interactions and collective space-charge field aberrations to spatial spreading in charged particle projection systems, by FHRead and N J Bowring, Microelectronic Engineering 73-74, 97-105 (2004).
A new computational technique based on the nearest-neighbor method has been used to study the effects of stochastic electron-electron coulomb interactions on the blurring of the focus spots in a lithographic electron projection system. It has been found that the collective (ie global) space-charge introduces two other distinct forms of spatial spreading: these are collective space-charge field aberrations and pattern-dependent aberrations. We have been able to separately simulate the first two of these three forms of blurring and to study them in isolation from lens aberrations. It has been found that the blurring is dominated by stochastic coulomb interactions when the beam pencil angle is small but conversely is dominated by collective space-charge field aberrations when the pencil angle is large. The blurring due to pattern-dependent aberrations has been found to be relatively less important. Results are presented for a range of currents and pencil angles for a representative projection system.
57. Capacitances and singularities of the unit triangle, square, tetrahedron and cube, by F. H. Read, COMPEL 23, 572-578 (2004).
The boundary element method has been used to obtain the electrical capacitances and singularity exponents of the unit triangle, square, tetrode and cube.
56. The Parallel Cylindrical Mirror Analyzer: Axis-to-Axis configuration, by F H Read, D Cubric, S Kumashiro and A Walker, Nucl. Instrum. Meth. A519, 338-344 (2004).
The operation of the parallel cylindrical mirror analyzer (PCMA) in the axis-to-axis configuration is described. A wide range of energies can be covered in the parallel acquisition mode. A second-order focusing mode also exists for which a region of limited energy range can be viewed with higher energy resolution. The PCMA can easily be switched between the parallel acquisition and second-order modes. The acceptance full width in the transverse direction can be as large as 90. The effects of the meshes at the entrance and exit regions are considered. The PCMA is compared with the hyperbolic field analyzer (HFA).
55. Field enhancement factors of arrays of carbon nanotubes, by F H. Read and N J.Bowring, Nucl. Instrum. Meth. A519, 305-314 (2004).
The field enhancement factors of flat arrays of aligned, closed-end nanotubes have been determined by accurate computational modelling. Square and random arrays have been considered. The dependence of the enhancement factor on the radius, length and density of the nanotubes has been found for square arrays and the results have been parametrized by a simple empirical function. The density fluctuations of random arrays have been simulated to determine the distribution of enhancement factors. A simple procedure is given for estimating the number of high-emitting nanotubes that have enhancement factors much higher than the mean value for the array.
54. Monte-Carlo Calculation Of Boersch Energy Spreading, by F.H. Read and N.J.Bowring, Nucl. Instrum. Meth. A519, 196-204 (2004).
The effects of electron-electron coulomb interactions have been simulated by a general computational technique that can be applied to almost any system, independently of the shape of the beam or the regimes that might be present, provided that the closest-encounter approximation is valid for the system. The technique has been applied to the benchmark test of the energy spreading in a converging round beam. The results are compared with previous analytical results.
53. Computational simulation of an electrostatic aberration corrector for a low-voltage scanning electron microscope, by L.A.Baranova, F.H.Read and D Cubric, Nucl. Instrum. Meth. A519, 42-48 (2004).
Two aspects of the design of electrostatic aberration correctors for low voltage scanning electron microscopes are considered. The first is that of optimizing the geometry and scale size so that the fields at the surfaces of the electrodes do not exceed the breakdown value. The second aspect is that of providing an accurate computational simulation of the paraxial fields of the lens system.
52. Accurate Monte-Carlo calculation of Boersch energy and angle spreading, by F.H. Read and N.J.Bowring, Rev. Sci. Instrum. 74 2280-2287 (2003).
The effects of stochastic electron-electron interactions have been studied by a computational technique that attempts to avoid all systematic errors, leaving only the random errors that are inherent in a Monte-Carlo calculation. The technique has been applied to the energy spreading in a converging round beam. The results are compared with previous analytical results.
51. The parallel cylindrical mirror electron energy analyzer, by FHRead, Rev. Sci. Instrum. 73 1129-1139 (2002).
50. Aberrations caused by mechanical misalignments in electrostatic quadrupole lens systems, by LABaranova and FHRead, Optik 112, 1331-138 (2001).
49. An iron-cored coil system for the measurement of angular distributions of charged particles, by D Cubric, R Ward, G C King and FHRead, Rev. Sci. Instrum. 71 3323-3325 (2000).
48. Extrapolating the number of segments to infinity in the Boundary Element Method, by FHRead, International Series on Advances in Boundary Elements, vol 8 (Boundary Elements XXII, Eds. C A Brebbia and H Power, WIT Press, Southampton, UK), pp 139-144 (2000).
47. A fast parallel acquisition electron energy analyzer, by C G H Walker, A Walker, R Badheka, S Kumashiro, M Jacka, M El Gomati, M Prutton and F H Read, SPIE Vol 3777, 252-257 (1999).
A new type of electrostatic electron energy analyzer is described that can acquire an electron energy spectrum in one shot. It uses a hyperbolic field to focus electrons emitted from a solid in the energy range 50 eV to 2500 eV into a dispersive plane of about 50 mm length. An expression for the energy resolution is given and the effect of side and base plates on the behaviour of the device is discussed. The main intended area of application for this type of analyzer is parallel data acquisition in Auger electron spectroscopy (AES) and X-ray photoelectron spectroscopy (XPS). The potential to acquire a spectrum in just a few seconds is possible with this device.
46. The charge-tube method for space-charge in beams, by F H Read, A Chalupka and N J Bowring, SPIE Vol 3777, 184-191 (1999).
The charge-tube method is an accurate and efficient way of assigning the space-charge of a beam in computational simulations of charged particle systems. The method makes use of the trajectory steps that result from the process of trajectory integration. The space-charge associated with each step of each trajectory is assigned to a narrow cylindrical tube that surrounds the step. The total space-charge of a beam is then the sum of the charges in all the resulting the tubes. In systems of 2-dimensional axial symmetry the charge tubes become conical sheets of charge, and for some purposes these need to be given a finite thickness. The charge-tube method is particularly useful for simulating the space-charge of beams that are very narrow compared with their length. The implementation of the method is described and results obtained with it are compared with those obtained by the traditional charge-cell method.
45. The charge-tube method for space-charge simulations, by F H Read, A Chalupka and N J Bowring, COMPEL (Int. J. for Computation and Mathematics in Electrical and Electronic Engineering), 18, 548-555 (1999).
The charge-tube method is an accurate way of assigning the space-charge of a beam in computational simulations of charged particle systems. The method is described and is compared with the traditional charge-cell method.
44. Short and long range penetration of fields and potentials through meshes, grids or gauzes, by F H Read, N J Bowring, P D Bullivant and R R A Ward Nucl. Instr. Meth. A427, 363-367 (1999).
When a conducting partially-transparent mesh is used to separate two regions of different electrostatic field strength there are short range changes to the field and potential distributions in the vicinity of the mesh and also long range changes far from the mesh. The forms and magnitudes of these changes are investigated computationally for meshes that consist of parallel or crossed round wires or crossed flat strips.
43. Minimisation of the aberrations of electrostatic lens systems composed of quadrupole and octopole lenses, by L A Baranova and F H Read, Int. J. Mass Spect. 189, 19-26 (1999).
Aberrations are investigated and compared for two types of multiplets based on electrostatic quadrupole and octupole lenses: mid-acceleration systems in which an accelerating potential is applied to the middle lenses of a set of quadrupole lenses and systems in which some of the quadrupole lenses are replaced by combined quadrupole-octupole lenses. It is shown that for systems consisting of three lenses the mid-acceleration type has the smaller aberrations. For systems consisting of four or five lenses the third order aperture aberration can be eliminated in the quadrupole-octupole type, while the mid-acceleration type has the advantages of lower levels of chromatic and fifth order aperture aberrations and also relative simplicity of construction and voltage adjustment.
42. Edgeways electrostatic deflectors with reduced aberrations, by F H Read, Nucl. Instr. Meth. A427, 177-181 (1999).
Electrostatic deflection systems in which the deflector plates are edgeways to the beam -ie the plates are flat sheets that lie in a plane perpendicular to the beam- are investigated. A deflection system that can deflect in both transverse directions is considered and the geometry is optimised to minimize the aberrations, which are found to be from 4 to 30 times smaller than the aberrations of conventional deflection systems.
41. Comparison of FDM, FEM and BEM for electrostatic charged particle optics, by D Cubric, B Lencova, F H Read and J Zlamal, Nucl. Instr. Meth. A427, 357-362 (1999).
The accuracy and speed of three programs based on different methods for solving problems in electrostatic charged particle optics are compared, using a set of benchmark tests. The methods are the Finite Difference, Finite Element and Boundary Element Methods (FDM, FEM and BEM respectively).
40. Reduction of the chromatic and aperture aberrations of the stigmatic quadrupole lens triplet, by LABaranova and FHRead, Optik 109, 15-21 (1998).
39. Penetration of electrostatic fields and potentials through meshes, grids or gauzes, by F H Read, NJBowring, PDBullivant and RRAWard, Rev. Sci. Instrum. 69, 2000-2006 (1998).
A theoretical, computational and experimental study has been carried out of the penetration of electrostatic fields and potentials through meshes. For a mesh that is used to shield an inner region from an external field it is found that there is a finite change in the potential throughout the internal region. Similarly it is found that for a flat mesh used to shield the region on one side of a mesh from a field on the other side, a finite change in potential exists in the field-free region. Meshes consisting of parallel wires and of crossed wires that form square holes are considered. Excellent agreement is obtained between the theoretical and computational results, enabling the theoretical formulae to be used to calculate the potential penetration for meshes of any transparency. It is found that for the calculation of electrostatic fields or potentials the mesh can be replaced by a coincident surface which has an 'equivalent potential' that depends on the field difference at the mesh.
38. A study of photoelectron angular distributions in xenon using a new magnetic angle changing technique, by D Cubric, D B Thompson, D R Cooper, G C King and F H Read, J Phys B 30, L857-864 (1997).
37. Aberrations due to localized potential defects on apertures, by FHRead, LABaranova, NJBowring, JLambourne and TCWard, Electron Microscopy and Analysis 1997, Institute of Physics Conference Series 153, 101-104 (1997)
36. Comparison of Finite Difference, Finite Element and Boundary Element Methods for electrostatic charges particle optics, by DCubric, BLencova and FHRead, Electron Microscopy and Analysis 1997, Institute of Physics Conference Series 153, 91-94 (1997)
35. Aberrations due to localized potential defects on apertures, by FHRead, LABaranova, NJBowring, JLambourne and TCWard, Rev. Sci. Instr. 69, 84-90 (1998).
The aberrations caused by localised potential defects near the edges of apertures that are used to define charged particle beams are investigated. Such defects arise in practice from the charging-up of insulating layers deposited by the beam. In the weak-field limit of small aberrations it is established that the change in direction of a ray passing through an aperture depends on the position of the ray at the plane of the aperture but not on the slope of the ray. It is found that the resulting changes to the spot size of a converging beam cannot be characterised in terms of aberrations coefficients if the beam fills the aperture. A new characterisation of the aberrations is therefore defined, and a numerical evaluations are carried out for a representative types of potential defect. Examples are given of the increases in spot size caused by such defects.
34. Improved Extrapolation Technique in the Boundary Element Method to find the Capacitances of the Unit Square and Cube, by FHRead, J. Computational Physics, 133, 1-5 (1997).
Extrapolation of the number N of subdivisions to infinity in the Boundary Element Method involves fitting computed results to a power series in 1/N. A technique is described of choosing the sizes of the subdivisions in such a way that the terms of lowest order in 1/N are eliminated, giving a more accurate extrapolation. The technique is used to establish the capacitances of the unit square and cube with relative errors of 3.10-7 and 10-6 respectively.
33. The production and optical properties of an unscreened but localised magnetic field, by FHRead and JMChanning, Rev. Sci. Instrum. 67, 2372-2377 (1996).
A technique is described of using current carrying coils to produce a magnetic field that is well localised, falling off quickly with distance. The basis of the technique is that the first few magnetic multipole moments of the system are arranged to be zero. The electron optical properties of this type of field are then investigated.
32. Ultimate numerical accuracy of the Surface Charge Method for electrostatics, by FHRead and NJBowring, Computation in Electromagnetics (Institution of Electrical Engineers, Conference Publication 420, 1996), p57-61.
The highest accuracy that can be obtained in practice with the Surface Charge Method is considered for various types of problem, including (1) capacitance calculations, (2) the calculation of potentials and fields, (3) trajectory integrations, and (4) cathode and space-charge problems.
31. Crossed aperture lenses for the correction of chromatic and aperture aberrations, by LABaranova, SYaYavor and FHRead, Rev. Sci. Instrum. 67, 756-760 (1996).
The chromatic and aperture aberrations of crossed five-aperture lenses are analysed by direct ray tracing. The apertures are rectangular and the voltages are applied in such a way that the first-order properties of the crossed lens are similar to those of a quadrupole doublet. It is shown that in astigmatic modes the chromatic and aperture aberrations of one of the linear images can be simultaneously eliminated or made negative. It is also shown that stigmatic modes exist in which the magnification is different in two perpendicular planes and in which the image blurring caused by the chromatic and aperture aberrations in the direction of smaller magnification is ten times smaller than that given by a round lens of the same focal length and the blurring in the other direction is at least two times smaller. The stigmatic crossed lens also gives a larger working distance than the equivalent round lens. The crossed lens will therefore be preferable for many probe forming systems.
30. Reduction of energy non-linearity of hemispherical deflection analyzers when used with multidetectors, by SCPage and FHRead, Nucl. Instrum. Methods A363, 249-253 (1995).
A '3-dimensional' computational study has been carried out on the energy non-linearity of a hemispherical deflection analyzer (HDA) used in conjunction with a multidetector. It is found that the energy non-linearity can be minimised by keeping the detector and its mounting at a single potential and not attempting to correct the fringing field to reproduce the field for the 'perfect' HDA. The values of the energy aberrations and the position of the focal plane are computed for a range of values of the ratio of the radii of the two hemispheres.
29. Defocusing of charged particle beams transmitted through meshes, by DLWilliams, FHRead and NJBowring, Nucl. Instrum. Methods A363, 120-123 (1995).
A mesh that separates two regions of different electrostatic field strength exerts a 'micro-lensing' effect on charged particles that are transmitted through the mesh, causing the directions of the particles to change. A computational study has been carried out to establish the magnitudes of the directional changes for four types of mesh that are commonly used, square holes in thin and thick plates, welded and woven.
28. Aberrations of an electrostatic three-cylinder system used to simultaneously focus and deflect charged particles, by ICDowker, FHRead, NJBowring and PHammond, Nucl. Instrum. Methods A363, 54-58 (1995).
The properties have been studied of an electrostatic three-cylinder lens-deflector in which the middle cylinder is split longitudinally into four parts, allowing the system to be used to simultaneously focus and deflect a beam of charged particles. The aberrations are found to be of the first order in the deflection.
27. Electrostatic electron-optical crossed lens with controlled astigmatism, by LABaranova and FHRead, Rev. Sci. Instrum. 65, 1993-1997 (1994).
A crossed lens consisting of square and rectangular apertures in four equally spaced electrodes has been studied. The two inner apertures are rectangular with their longer directions mutually at right angles and the two outer apertures are square.. The strength and astigmatism of the lens can be separately controlled through the voltages applied to the rectangular apertures, enabling line images of variable position and length to be produced from point objects.

Student projects and publications

The CPO-2D and CPO-3D programs, and their forerunner the MAN-2D program, were used for project work in the Physics Department at the University of Manchester in the period 1970-2002. Many publications have resulted (see below). This project work was very much enjoyed by the students (and the supervisors!), and it occasionally lead to published papers. The projects were supervised by the two authors of the programs, who are now employed elsewhere.

Projects that last three half-days:

The demo and standard versions of CPO-2D and CPO-3D were used by about 24 students each year, for project work that forms part of a lecture course on 'Charged Particle Dynamics'. Most of these students were in the final-year of a four-year undergraduate degree course, while a few of them were first-year masters students or doctoral students. The choice of topics included:

  • Behaviour of charged particles in crossed electric and magnetic fields.
  • Confinement of electrons in a magnetic bottle.
  • Relativistic motion in a uniform electric or magnetic field.
  • Investigation of the focussing properties of a two-cylinder electrostatic lens.
  • Investigation of the energy resolution of a hemispherical deflection analyzer.

Projects that last 20 days:

The full editions of CPO-2D and CPO-3D were used by 8 final-year undergraduate students each year, for extended project work. The topics that have been studied in the past include:

  • Penetration of electrostatic fields through wire meshes.
  • Effects of parasitic 'patch fields'.
  • Changes to the properties of the cylindrical mirror analyzer due to the apertures in the inner cylinder.
  • Optimisation of the geometry of simple plate deflectors.
  • Effect of stray magnetic fields on the performance of energy analyzers.
  • Effects of mechanical misalignment of a three-cylinder lens.
  • Effects of mechanical misalignment of a hemispherical deflection analyzer.
  • Space-charge effects in a field emission source.
  • Why are the space-charge fields radial for converging beams of finite length?
  • Accurate evaluation of capacitance of the tetrahedron and other solids of regular shapes.
  • Elliptical lenses.
  • Trapping of electrons.

Projects that last 6 months:

The full editions were also used by a small number of students for research projects that formed part of the masters or doctoral courses. Some of the resulting publications are listed below. The topics included:

  • The astigmatism of hair-pin cathodes.
  • The effect of thermal velocity spread on the properties of the Pierce gun.
  • The 'micro-lensing' effects of wire meshes.
  • Aberrations of split-lens deflectors.
  • Charge neutralisation of conducting surfaces in the presence of magnetic fields.
  • Aberrations and optimisation of electrostatic deflectors.
  • Restoration of second order focussing of the cylindrical mirror analyzer with apertures in the inner cylinder.
  • Optimisation of space-charge for a system of electron gun and high resolution energy selector.
  • Use of a localised magnetic field for initial energy selection of an electron gun.
  • Effect of space charge on the energy resolution of the hemispherical deflection analyser.
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