Nuclear Instruments and Methods in Physics Research A298 (1990) 389-395

389North-Holland

Pitfalls in the calculation of the field distribution of magnetic electronlenses by the finite-element method

Khadij a Tahir * and Tom MulveyDepartment of Electrical and Electronic Engineering and Applied Physics, Aston Umuersity, Birmingham B4 7ET, UK

In principle, a finite-element calculation of a magnetic or electrostatic structure has now achieved an accuracy that is difficult tocheck experimentally. However, incidental errors may occur m practice due to shortcomings in the computer itself, in the program orin the initial setting up of the data . Some common errors, culled from the literature, are reviewed ; ways of recognising their presenceare suggested. Some test structures and computing procedures are also put forward for estimating the accuracy of a particularcalculation . A software "B-H Tester" has been devised that is intended to help the operator to check default B- H curves that maybe stored m the computer . This can also be used to exercise a given program at its limits and hence reveal any weaknesses. Finally, aseries of tests has been carried out to assess the relative merits, in speed and accuracy, of the Gaussian elimination method of solvingthe matrix equations and that of the imcomplete Cholesky conjugate gradient (ICCG) method .

1 . Introduction

Just over twenty years ago, Munro [1] introduced thefinite-element method (FEM) into the calculation of themagnetic and electrostatic field distribution of electronlenses, guns, deflectors etc . This revolutionised the ap-proach to electron-optical design, the full significance ofwhich is only now being fully realised thanks to recentdevelopments in personal computers (PCs) and ad-vanced graphics facilities that allow a truly interactivedesign process.

The early computer programs in general use werelargely extensions of the original Munro programs, writ-ten by and for experienced operators of main-framecomputers who could be expected to sense when thecalculations were not entirely satisfactory, and would,in any case, check the results experimentally. By andlarge, this worked well with conventional designs, sincethere was a wealth of experimental electron-optical data ;theoretically based lens models were also available. Insome cases, however, significant discrepancies occurredbetween calculation and experiment, especially with newforms of lenses, such as single-pole-piece lenses or lenseswith highly saturated pole pieces. It was not alwaysclear, at the time, if the cause of the discrepancy lay inthe experimental or the computed result . In addition,the speed and accessibility of the early computers wasnot great. This meant that the number of meshes in thelayout had often to be unduly restricted, giving rise to

* Present address : Blackett Laboratory, Imperial College,London SW7 2BZ, UK .

Elsevier Science Publishers B.V . (North-Holland)

discontinuities in the axial flux density curves . Thesehad to be "smoothed" by an expert hand before at-tempting to calculate aberration coefficients . It was alsoknown empirically that the computed field could oftenbe "improved" by making ad hoc changes in the meshlayout . On the experimental side, it often proved dif-ficult to make accurate axial flux density measurementson high-quality objective lenses because of the smallbore and gap. Today, with fast personal computers,more people than ever before are using finite-elementprograms purely as a design tool, without reference toexperiment, and often without adequate diagnosticfacilities in the program for checking the accuracy ofthe results produced . Users of PCs, for example, operat-ing in single precision arithmetic to enable a largermesh layout to be employed, should be particularlycareful to devise independent checks on the accuracy ofthe calculations . In principle, the accuracy of the finite-element method is now much better than that of elec-tron-optical experiment. This has its dangers but itcould be a great advantage to manufacturers, if it canbe realised in practice, since it offers the attractivepossibility of building a complete electron-optical col-umn directly from a computer-aided design, therebyeliminating the construction and updating of costlyexperimental prototypes . The necessary programs forcalculating the performance of a complete column arenow available, (see for example ref. [2]) . Extra care isclearly needed in this case, to provide an independentway of checking the calculation . Fortunately, it is notdifficult to test the final result for consistency with thelaws of electron physics. It is also possible to devise testlenses and other structures whose characteristics are

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known accurately in advance and hence give usefulinformation about the operation of the computer pro-gram and the associated computer .

The present paper surveys various errors that canoccur between the creation of the sketched-out lens tobe calculated and the final printout from the computer,and suggests simple tests or auxiliary computations thatcan be carried out by the operator . Other tests are basedon a systematic investigation of some test structures,devised by the authors, that have been found useful inpractice .

In a previous paper [3], it was pointed out thatalthough great attention is now being paid in FEMprograms to the optimisation of the different parts ofthe magnetic circuit of high-performance lenses, this isnot necessarily being matched by the supply of ade-quate B-H data . The latest computer packages [4,51allow the incorporation, in critical parts of the iron

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circuit, for example in pole pieces, lens body and deflec-tors, of several different types iron and other ferromag-netic material .

Such computations also allow a manufacturer toassess the magnetic efficiency of the lens as awhole andits sensitivity to possible deviations of the relevant B-Hcurves from the supplier's specifications . This means,however, that care has to be taken to supply accurateand relevant B-H curves to the program. For thesereasons a software "B-H Tester", referred to below,was devised; this was found useful, for example, inchecking the default B-H curves, often of unknownpedigree, built into many programs and not readilyaccessible by the operator. As an unexpected byproduct,this tester subsequently proved useful in checking theoperation and inherent accuracy of a FEM program asa whole, including its algorithm, to the limit of perfor-mance of a given computer . Before describing this in

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"STAIRCASE POLEPIECE"

Fig . 1 . Upper left : quarter section of a double-pole lens of unknown mesh layout (cf . Cleaver [6]) . Lower left : mesh layout by Tahir[7] . Right: axial field distribution : (1) Unknown mesh, (2) recomputed by Tahir [71, (3) pole-piece surface replaced by a "stepped"

structure.

more detail, it may be useful to review some standardprocedures in calculating the field distribution in agiven lens and how they may be tested .

2. Choice of mesh layout

Once a proposed lens design has reached the stagewhere a simple sketch can be made showing its generalshape, it is possible to generate a mesh layout that willenable a preliminary calculation of its axial field distri-bution to be made . Subsequently it may be necessary torefine this mesh in order to improve the accuracy . Incritical work it is, of course, desirable to check at theoutset that the chosen mesh is near optimum, i .e . notsensitive to small changes . The effect on the axial fluxdensity of different choices of mesh is illustrated in fig .1 .

At the top left of the figure is a drawing of the upperquarter of the iron circuit and coil of a symmetricaldouble-pole-piece test lens [6] . The mesh layout wasunfortunately not revealed in the publication. Below isshown the mesh layout chosen by Tahir [7] for therecomputation of this structure . As a further check onthe mesh layout, the smooth surface of the pole piecewas replaced by a series of small rectangular steps(staircase method) . A rectangular mesh is known to befavourable mathematically [8] and tends to force theoperator to distribute the meshes smoothly withoutrapid changes in mesh size . On the other hand, itcertainly distorts the external shape of the pole pieces .On the right-hand side are shown the axial field distri-butions for the three mesh layouts as follows : (1) ascomputed by Cleaver [6], (2) as recomputed by Tahir[7], and (3) as recomputed by Tahir [7] by the "stair-case" method .

The good agreement of (2) and (3) suggests, withhindsight, that the mesh layout of (1) is not optimum.This could have been discovered by calculating the areaunder the curve, which is directly related to the ampèreturns generating the field, and should of course agreewith the excitation supplied to the computer. It shouldperhaps be mentioned that the staircase method hasbeen formalised by Podbrdsky and Krivanek [9], backedby a smoothly graded mesh . This is not difficult toimplement and forms a very useful check on variousalternative automeshing systems now available (e .g. ref.[5]).

3 . Experimental and internal consistency checks on asingle-pole-piece lens

In a single-pole-piece lens or, in fact, in any lenswith an "open" structure, the absence of a conventionaliron screen completely surrounding the pole pieces can

K Talur, T. Mulvey / Field distribution of magnetic electron lenses

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Fig. 2 . Experimental check of a single-pole-piece lens. Solidline : calculated by Christofides [10], crosses : measured byChristofides [10], open circles : recalculated by Talur [7] using

revised mesh layout.

create difficulties if the A = 0 boundary facing the polepiece is too near the pole piece. Fig . 2 shows thecalculated axial flux density distribution (solid line) ofan experimental single-pole-piece lens (HERMES II),designed by Christofides [10] as the probe-forming lensof an electron-probe fluorescence microanalyser . Theflux density values that he measured experimentally,indicated by crosses, are in good agreement with thosecalculated, except at the position of the peak . Unfor-tunately this is a critical region for the calculation ofelectron-optical properties. There is also a marked dis-crepancy in the area under the two curves ; this points toa possible computing error . The axial field distribution(open circles) was recalculated by Tahir [7] with thesame number of meshes but with a more evenly gradedmesh layout. This shows excellent agreement both in theshape of the curve and excitation . Such simple checksshould of course be carried out routinely at the time ofcalculation .

More detailed checks on the accuracy of a calculatedaxial field distribution are also possible . The axial fielddistribution created by the coil, which can be readilycalculated analytically from the Biot-Savart law, con-tains valuable independent diagnostic information, sinceit is not influenced by the finite boundary (A = 0)

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imposed in a FEM calculation . The area under thiscurve is an independent measure of the excitation ap-plied to the lens, and should therefore be equal to thearea under the axial flux density curve calculated by theFEM. In practice, the area under the FEM calculationshould be slightly less than that under the Biot-Savartcurve, since the finite size of the FEM mesh effectivelycuts off the tails of the Biot-Savart curve, which ex-tends to infinity . In practice it can sometimes be larger ;this indicates a computing error . An error less than 1%should be aimed at but 0.01% is possible with greatattention to detail. A further check can also be made bymaking use of the fact that the magnetised iron cannotcontribute to the total lens excitation ; it can only alterthe distribution of the axial flux density . Hence thepositive component of the flux density due to the ironmust exactly balance the negative one. The total areaunder the curve should therefore be zero in the absenceof computing errors . These points are illustrated in fig .3, which shows a single-pole-piece lens with a squarepole piece. Curve 1 (crosses) shows the FEM calculationof the axial field distribution of the lens and curve 2(squares) shows that of the coil alone calculated fromthe Biot-Savart law. The agreement is within 1% . Curve3 (open circles) shows the flux density produced by theiron as obtained by subtracting the contribution fromthe coil from the total flux density, i .e. curve 1 minus

K. Tiber, T. Mulvey / Field distribution of magnetic electron lenses

Fig . 3 . A detailed check on the accuracy of a FEM calculation :upper half of a single-pole-piece test lens . Crosses : Total axialfield distribution ; squares: axial field distribution from thecoil ; open circles : axial field distribution from the iron. Area

under this curve should be zero.

curve 2. The area 3A is equal and opposite to that of 3Bto an accuracy of better than 1%, an additional checkon the satisfactory placing of the boundary.

4, The finite-element method as an aid in the speculativestudy of nonconventional lens arrangements

Since the FEM can solve the field distribution ofprospective lenses that might not be easy to manufac-ture with present technology, it is possible to considerlens designs that at first sight would not appearworthwhile to develop experimentally, but might pro-vide useful ideas for improving conventional lenses . Asan example, consider the tentative design of a high-flux-density (9 T) TEM objective lens with saturatedsoft iron pole pieces of square cross section, as shown infig . 4. Since the optimisation of the lens bore can beconsidered as a separate problem, for computationalsimplicity a vanishingly small bore is assumed. In con-ventional lenses, it is usually assumed that the design ofthe coil does not significantly affect the axial fielddistribution. However, this is not universally true, sotwo notably different coil designs are incorporated. Coil1 is a narrow pancake coil, while coil 2 is a long thinsolenoid, representative of a conventional winding.

In order to ensure no loss of ampère turns in theA =0 boundary in the initial design stage, the lens issurrounded by an iron box of 5 mm thickness and highpermeability. This allows the loss, if any, of ampèreturns in the lens circuit due to excessive external strayfields to be detected and quantified at an early stage. Ifsuch fields become apparent they can readily be re-moved by thickening the lens shell. The main point ofinterest here is the shape of the axial field distributionand its peak value, rather than the absolute size andexcitation of the model lens . In a particular situation,for example, by Kelvin's rule, if the magnetic fielddistribution is known for one combination of dimen-sions and excitation (2 x 105 A turns in fig . 4), the lensand its excitation may be scaled by a factor k. The fluxdensity in corresponding points of original and scaledmodel are the same . The peak field in the scaled modelis therefore unaltered but the half-width of the fielddistribution decreases by a factor k. This is true even inthe presence of saturation. Fig. 4 shows that the thinexcitation coil produces a narrow field distribution witha peak of 9 T. The conventional solenoidal windingproduces a broad distribution with a peak value of 4.2T. It should perhaps be pointed out that the currentdensity needed to produce the required excitation in alens of normal size would require a superconductingwinding. The high leakage field outside the lens showsthat the shroud is clearly inadequate, especially whenthe long solenoid is employed . These stray fields can bereduced by using a greatly increased size of yoke .

5. A software B-H tester

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K. Tahir, T" Mulvey / Field distribution of magnetic electron lenses

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Fig . 4. Speculative design for a high-flux-density (9 T) lens of simple construction with two types of coil.

As mentioned above, the question of supplying accu-rate and relevant B-H data to the various parts of themagnetic circuit is now of crucial importance in elec-tron-optical design, especially if it is desired to comparealternative designs critically before manufacture . In

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many programs, default B-H curves are provided, whichmay or may not match the data of the magnetic materialto be used in the lens in question, and in any case aredifficult to access from the keyboard . It was thereforedecided to devise a software analogue of a typical B-Htester used to measure the B-H curve of an iron sam-ple . Fig . 5 shows the mesh layout of an analogue of a

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Fig . 5 . Software B-H tester mesh layout . Left-hand layout as in Munro programs ; right-hand layout as in Lencovd programs .Hatched area: iron bar under test surrounded by thin (black) solenoid ; dotted area : high-permeability shroud .

IV . ELECTRON/ION OPTICS

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Fig . 6 . Flux density relative distribution from the B-H tester along a uniformly magnetised iron bar (100 mm long) (Munro's"Material 1" (1971), permeability 2500) . Lower curve : ICCG method in single precision (10% low) . Middle curve : Gaussianelimination method m single precision (S% low). Top curve : ICCG and Gaussian elimination m double precision superimposed

(error < 0.01%) .

magnetic tester for obtaining B-H curves of ferromag-netic material. The iron bar (hatched region) of length100 mm under test is surrounded by a long solenoid of30 mm inner diameter and coil thickness 0.1 mm.

The external iron shroud has a relative permeabilityp, r of 106 to make iron losses in the outer shell negligi-ble . The aim was to achieve an accuracy in calculationof better than 0.1% .

On the left-hand side of the centre line, the meshlayout is set up as for a Munro program whilst on theright the mesh is set up as for a Lencovâ program, sincethese programs operate with the mesh layout in oppo-site quadrants, although the mesh indexing follows thesame pattern . The B-H tester was found to be veryuseful in showing up small errors in listings of B-Hcurves and made it easy to extrapolate incompletecurves . It was also noted that very different B-H curvesshown in the literature could be referred to under thesame trade name by different authors making compari-sons of designs . During the course of these investiga-tions, it was found that the combination of the bigvariations of relative permeability in the bar under testwith the applied field H and the high total flux andhigh relative permeability Ft . = 10 6 in the outer shroudproved to be a more searching test of the FEM programthan any of our previous tests . It also had the greatadvantage that the extremely high values of B and H inthe bar, and its uniformity are known in advance toprecisely the accuracy of the data supplied (0.01%) .Preliminary studies of the standard Munro programsand the BMAGPC of Lencovâ with the aid of the B-Htester, revealed some differences in the computed re-sults. These programs use different algorithms for solv-

ing the mesh equations, but it was realised that no firmconclusions could be drawn since it was not possible tokeep all other factors constant . At this point, the availa-bility of the Munro M12 (linear) and M13 (nonlinear)programs at Imperial College, which operate in Gaus-sian elimination and ICCG mode respectively, offeredthe chance of comparing, under controlled conditions,the behaviour of the Gaussian elimination method andthat of ICCG by means of the B-H tester . The basicadvantage of Gaussian elimination is that the resultantmatrix is guaranteed positive definite, and therefore hasa firmer mathematical basis than that of the ICCGmethod, which cannot give such assurance, although inpractice it appears to be entirely satisfactory with elec-tron lens problems . The advantage of the ICCG methodis that it is faster, to an increasing extent as the numberof meshes in the layout is increased .

6 . Testing a complete program with theme-H tester

Whichever method is used, the strategy of the calcu-lation is the same. In the first calculation, the ironcircuit is considered to have a uniformly high, constantrelative permeability l1 r (linear calculation) . If this is settoo low, large magnetic leakage effects will arise, if toohigh, arithmetical problems may arise in terms that areto be divided by Frr. In the B-H tester, it was necessaryto set the relative permeability of the outer shell to avalue of 106 in order to prevent ampère turns being lostin this part of the iron circuit, thereby introducing asystematic error into the axial flux density distribution .

In order to assess the behaviour of the critical linear

computation, two programs were used, the standardMunro M12 using Gaussian elimination (GE) and asimilar program, but using ICCG, which we will refer toas M12(ICCG). They were both used on a PC with anNDP compiler, that allowed a 100 x 60 mesh layout ineither single or double precision. It was immediatelyclear, see fig. 6, that both programs produced significanterrors in single precision . M12(ICCG) gave results thatwere 12% low [lowest curve] and M12 [middle curve]was only slightly better, 8% low. On changing to doubleprecision [top curve] both programs gave an accuracy of0.01% as set by the program-end criterion of 0.01% inboth programs . The ICCG matrix iteration limit crite-rion is set to 10 -s in the MUNRO(ICCG) program .For an accuracy of 0.01%, the linear program took 6min in GE and 2 min in ICCG . The M13 andM13(ICCG) programs (nonlinear mode) were then runin double precision and produced the same high compu-tational accuracy (0.01%). The calculation took 15 minand three iterations in GE and 8 min and four iterationsin ICCG. The ICCG is thus faster by a factor of about2 for this mesh size . Similar results were obtained withthe Lencovâ BMAGPC program, which uses ICCG,once the default (ISELIN) limit and the end-programaccuracy criterion values were reset to the above values .The above illustrates the value of having two indepen-dent algorithms available and the options of single anddouble precision .

7. Conclusions

If a finite-element program has to be capable oftackling a wide range of electron-optical problems, in-volving a variety of ferromagnetic materials of widelydifferent permeability, it is essential that diagnosticinformation should be fed back to the operator if trulyinteractive, innovative design work is to be undertakenon a PC . Several of the diagnostic tests for the con-sistency of the data supplied discussed here are alreadyavailable in standard packages and can play an im-portant part in increasing confidence in the computedresult.

The software-based B-H tester, originally intendedas an aid to checking default B-H curves in computerprograms, has proved useful in testing the numericalaccuracy achievable in finite-element programs underconditions that are, if anything, more exacting thanmost of those that arise in routine lens calculations .These tests show that the FEM is capable of an accu-racy of some 0.01%, well beyond the accuracy of experi-mental measurements on actual lenses and ferromag-netic material . This test is not necessarily definitive forall computing situations, but it seems to be a simple anduseful test method . Modern software helps enormouslyin reducing operator-induced errors such as unsatisfac-tory meshing, but other errors have still to be guarded

K. Tahir, T. Mulvey / Field distribution of magnetic electron lenses

Acknowledgement

References

39 5

against . Computer error can arise through the inap-propriate use of single precision, where the lack ofadequate diagnostic facilities in the program may leavethe operator unaware that anything is amiss. In criticalwork, for example, in the computer-aided design of acomplete electron microscope or an electron beamwriter, it may be desirable to use two independentalgorithms, as a check on accuracy . Clearly the GE andICCG methods are complementary but different . This isan ideal situation for checking a computed result . Whatis perhaps striking about this investigation is the obvi-ous shortcoming of both methods in the handling of thelinear calculation and in deciding on the best value forthe relative permeability, so as to avoid too large andtoo small numbers being produced in the calculation . Ina real material, the B-H properties of the iron reduceconsiderably the permeability in parts of the iron circuitcarrying high flux densities. In the linear part of thefinite-element calculation, this does not happen and soarithmetical problems can arise unnecessarily. Perhapsfurther thought should be given to replacing the linearcalculation by a "quasilinear" algorithm that wouldsuppress these unwanted very large and very smallnumbers.

The authors are grateful to Dr . E . Munro for provid-ing us with his M12 and M13 programs set up withindependent GE and ICCG algorithms . They also wishto thank Dr. B . Lencovâ for stimulating discussions andsuggestions for making critical calculations .

[1] E . Munro, Computer design methods in electron optics,Ph.D. Thesis, Cambridge University (1971) .

[2] B . Lencovâ and M . Lenc, these Proceedings (3rd Int .Conf . on Charged Particle Optics, Toulouse, France, 1990)Nucl. Instr . and Meth. A298 (1990) 45 .

[3] T . Mulvey and K. Tahir, Ultramicrosc. 33 (1990) 1 .[4] B . Lencovâ, Inst . Phys . Conf . Ser. no . 93, 1 (1988) 75 .[5] B . Lencovâ and G . Wisselink, these Proceedings (3rd Int .

Conf . on Charged Particle Optics, Toulouse, France, 1990)Nucl. Instr . and Meth. A298 (1990) 56 .

[6] J.R .A . Cleaver, Optik (Stuttgart) 49 (1978) 413 .[7] K. Tahir, A critical assessment of the finite-element

method for calculating magnetic fields m electron opticsPh.D. Thesis, Aston University (1985) .

[8] B . Lencovâ and M. Lenc, Scanning Electron Microsc . 3(1986) 897 .

[9] J. Podbrdsk~ and O.L. Krivanek, Optik (Stuttgart) 79(1988) 177 .

[10] S . Christofides, Electron optical and related design con-siderations for micro-X-ray sources, Ph.D. Thesis, AstonUniversity (1982) .

IV . ELECTRON/ION OPTICS