Comparison between Two Types of Electrodes for Electrostatic Quadrupole Lens

Transcription

1 Comparison beteen To Tpes of Electrodes for Electrostatic Quadrupole Lens Sura A. Obaid Department of Phsiolog and Medical Phsics, College of Medecine, Al-Nahrain Universit, Baghdad, Iraq. Abstract This ork aims to find the optimum design of electrostatic quadrupole lens and make the comparison beteen to different electrodes shapes hich are computed ith the aid of transfer matrices. To tpes of electrostatic quadrupole lens, clindrical conve electrodes, and clindrical concave electrodes ere used to find optimum field model, hich as close to the field distribution for each design of the proposed lens. The path of charge-particles beam is traverse the field model has been determined b solving the trajector equation of motion in Cartesian coordinates. Then, take the comparison of the optimum values of spherical aberration coefficients and resolution limits in both convergence and divergence plans for each proposed tpe. Keords: Electrostatic, quadrupole Lens, spherical aberration, resolution limits, trajector equation. الخالصة تهدد ه هددلد ال ايددة دد التصددم ا ا م ددد ل يددة اه كيددت ت ا ة ق ا ددة ا رادد ق مدد ا دد اي مم هددة قدد هددكا مدد هددلد ال يدد ب ق يددت هة قم دد ه ا اهتمدد د المصدد ك ب قطدد اهم لاب قرادد ق ايدداكاه ة قيدداك مط قددة كا خدد لاب ارادد ق ايدداكاه ة قيدداك مم دد د أ دد قهمددكلل الم دد د الاه كيددت ت ا ا م ددد لاددد تصددم ا مدد ال يدد ب الم ت عددة قاددالد ددتا طيدد ق ميدد طامددة ال يدد م ب الممددطكهة المدد د خددالد قهمددكلل الم دد د قكايدداة طددد م لددة الط اددة ل مدد ة ق أطدد ا ب الا ت دا ه كر تا مم هة الم ا الم لى لم مالب الا غ الا كي كط ك التطل د اال الهمكل الم ت ع د

2 Introduction Electron lenses are one of the principle components of the electron optics devices. Both electrostatic lens electric field and magnetic lens magnetic field are used to focuses charged particles; each tpe has its advantages and disadvantages. The focusing poer of electrostatic lenses independent of the mass of the charged particles and depends onl on their energ; therefore to be preferred over magnetic lenses hen heav particles of moderate energ must be focused. There are numerous tpes of electrostatic lenses, each one of them classif according to electron-optical Lens Design This ork is concerned ith the design of electrostatic quadrupole lens consist of four identical smmetric parts are cut out from clinder. These four parts represent the concave and conve clindrical quadrupole lens. The effect of changing the geometrical shape of electrodes for each design like; the angle of gape beteen electrodes Г, here the ratio of clindrical electrode radius R to aperture radius a is constant, electrode voltage ratios V/V, electrode thicknesses and spacing beteen them, as ell as the radial and longitudinal dimensions of the properties and domain of application. The first published ork on electrostatic fields ith rotational smmetr dates back to 96 []. The quadrupole lenses are second in importance onl to the aismmetric electron lenses. Quadrupole lenses began to be used in electron-optical devices in the 95 s. These lenses have a high focal poer and are used to focus the beams in high-energ. The produce fields ith to mutuall perpendicular planes of smmetr and to planes of asmmetr b means of four identical electrodes arranged smmetricall about the ais. Therefore, if the -z plane of the lens collects the charged particles, the - z plane ill cause them to diverge, i.e., the quadruple lens is astigmatic [,]. electrode. The particular electrode configuration ma be found in the value of K electrode shape and then the ecitation parameter of lens as active parameter is changing [,6]. Quadrupole lens is described as converging lens if the particles moving in z plane are deflected toard the z-ais and hile described as diverging lens if the particles moving in -z plane are deflected aa from the z-ais [5]. In the current ork, the electrodes shapes hich taking into consideration are the clindrical conve and clindrical concave electrodes that generall used instead of hperbolic electrodes due to the difficulties of fabricate.

3 Figure. Electrodes of quadrupole lens: a clindrical conve electrodes, b clindrical concave electrodes [,]. The Potential Distribution Variation of the parameter K according to electrode shape gives different modes of potential distribution. In clindrical coordinates, the potential a long the optical ais is given b the formula;[]. V r,, z D z V r / a cos here V is the electrode potential take volt in the current stud, r is the of the radial displacement of the beam from the optical ais measured in mm, θ is the angle beteen electrodes and z-ais measured in degree, and Dz is the electrodes field the quadrupole field components hich is equal to; [6]. D z K f z Thus, f z refers to function mode hich is normalized to unit at the center z. The parameter K K,K named the electrode shape, determined b the electrode geometr from K and K hich is represented the coefficients of the electrode shape for clindrical conve K and clindrical concave electrodes K respectivel are described as equations; [,6] K sin / ln R/ a K sin / Where R is the radius of clindrical conve electrode equal to [] R. a 5 The effective lengths of the electrodes L for conve electrodes, L for concave electrodes are given b relations []. L. 56 a 6 L. 5 a 7

4 Where l is geometrical length of electrodes. The Trajector of Electron Beam In the present ork the polarities of each electrode of the and aes as shon in figure. Since the positive potential +V is applied on the ais electrodes of the lens, then the positivel charged particles ill be repelled along the ais and ill be directed toards the z ais. The positivel charged particles ill be attracted b the negative potential V along the ais of the lens. The trajector equations in cartesian coordinates for the charged particles beam traversing the field of a quadrupole lens are given b []: '' f z 8 these boundaries it terminates in the form of a half bell - shaped field represented b the folloing function []. f z d / [ z z / ] L hen z > L f z d / [ z z / ] L hen z < L z z Where d is the aial etension of the field beteen the to points. Figure. Field distribution of a quadrupole lens Modified bell shaped model [5]. '' f z 9 Where β is the ecitation parameter given b the folloing relation []. V K / a V Where V acceleration voltage, '' '' and are the second derivatives ith respect to z [7,8]. Where f z is the modified bell shaped field model as in figure represents the intermediate case beteen the rectangular section of constant maimum value f z ma in the region z L z z L and beond The Spherical aberration The Spherical aberration is the name given to the effect here the focal length of a lens ill var depending on ho far ou are from the centre of the lens. What this means in realit is that a parallel ras of light entering the lens near its centre, ill be focused less or more than a parallel ra entering near the edges of the lens. In a quadrupole lens, the spherical aberration in the gaussian image plane can be epressed as [,9] and [].

5 p s C C M zi s p D D M zi Where α and are the image side semi aperture angles in the z and z plane, respectivel. The coefficients C and D characterize the aberration in the convergence and divergence plane, retrospectivel. The coefficient C s determines the aberration of the real idth image in the plane =, and D s is that for the imaginar image in the plane of = []. The spherical aberration coefficients C and D are given b the forms [7]. ]] sin sin [ sin sin 7 [ sin 5 5 s n n d C 5 And [7]. ]]]] sin cos cos sin [ sin sin sin [ ]] sin cos cos sin [ ] sin sin sin [ [ ] sin cos sin cos [ [[ sin p n n k d C 6

6 D s d [ [ n sin ncos 5 sin sin sin cos sin sin cos [ ] cos [5 ] sin cos sin cos sin ]] cos cos sin cos sin cos n n sin sin ] sin sin [ sin cos sin 6 sin sin cos 7 Also [7]. D p d C p [ cos sin cos 8 sin 8 sin ] Where is the value corresponds to the position of the object and to that of the image, and n for electrostatic quadrupole lens.the properties of the quadrupole lens are characterized b the parameter [7]. W d for the convergence plane 9 W d for the divergence plane. The Limiting Resolution The resolving poer of an optical device is the abilit to form to separate images of to point objects ver close together. The theoretical limit of resolution of microscope is determined b the combined effect of chromatic and spherical aberrations. The practical resolution of an instrument also depends on the supplementar aberrations that arise from constructional defects, or the observer himself, or from mechanism of interaction beteen the beam

7 and the object to be observed. The resolution limit is given b equation []. R.6 C Where is the electron avelength hich is given b the form[]..5 V Results and Discussion The aial potential distribution ratio V / V represented in equation is plotted as a function of theta, as shon in figure,it is found that ver close to modified bell-shaped model b use quadrupole lenses of clindrical conve and clindrical concave electrodes. This result agrees ith the results mentioned in various references such as [6,,], and []. Figure. The aial potential distribution ratio V / V as a function of θ for electrostatic quadrupole lens ith clindrical aconve and b concave electrodes. The values of K and computed from equations,, and are represented as a function of as shon belo in figure hich is illustrate in the increment in leads to firstl increasing in K and arriving to maimum values of K =.97 and. mm - at 5.55 for concave electrodes and has the same behavior in conve electrodes ith K =.7 and.8 mm - at Then drop to lo values decreases. The ecellent agreement shon ith other literatures Baranova and Yavor in [].

8 Figure. a The coefficient of electrode shape K, b the ecitation parameter β as a function of gap angle Г. The trajector equation of charge particles has been solved for the case of modified bellshaped model b using simplified transformation for equations 8 and 9. When =.mm -, =.8 mm -, and z = 5- mm in case modified bell-shaped model. These equations describe the paths of charge particles in convergence and divergence plane respectivel. In electrostatic quadrupole lens charge particles incident in the -z plane different paths from that in the -z plane, as shon in figure 5. From figure 5 shos the charge particles in the convergence plane -z plane is deflected toard the optical ais z, hile in the divergence plane -z plane is aa from the optical ais. i.e. the quadrupole lens is astigmatic, and the results is identif ith that published b Baranova and Yavor in []. Figure 5. Trajector of charge particles in electrostatic quadrupole lens ith clindrical a conve andb concave electrodes for both convergence -z plane and divergence -z plane.

9 The estimation of optimization for spherical aberration coefficients in both convergence and divergence plane improved the effects of changing the electrode angle and the electrode shape of these lenses on the objective properties spherical aberration coefficients, as shon in figures 6 and 7, for convergence and divergence plane, respectivel. Figure 6 shos the proportionall relation beteen spherical aberration coefficient C p, C p,c s,c s, and Г for the convergence plane in to electrode shapes, here the spherical aberration coefficient C p for the electrostatic quadrupole lens of clindrical conve electrodes and C p the electrostatic quadrupole lens of clindrical concave electrodes are decreasing in negative value as Г is increasing and the have a minimum values C p = and C p =-.8 at = The loest absolute value of spherical aberration coefficients C p,and C p are found after and before these minimum values. The spherical aberration coefficients C s, and C s have the same behavior in all rang of gap angle Г and the loest absolute accurate values of these coefficients should be at range The minimum absolute values of C s =.885 and C s =.885 at Figure 6. The spherical aberration coefficients a C p and C p, b C s and C s as a function of gap angle Г.

10 The spherical aberration coefficients D p and D p in divergence plane represented as a function of Г as shon in figure 7-a. According to figure 7-a e notice the values of D p are alas positive and increasing ith Г up to maimum value. mm at Г = 5 then happen slo drop then up to the same maimum value at Г = 6.. Therefore drop sharpl ith increasing Г. While, the values of D p are alas negative and decreases in its absolute values ith increment of Г until 7.6 mm at Г= 5.5.then dropping ith increasing Г. The last parameter of spherical represented as a function of Г shon in figure 7-b, hich has the same behavior for all values of Г. The values of D s and D s are alas negative, and these values increases ith Г increases. These parameters be have as constant values at Г.5. The D s give us the better absolute value than D s hich is. mm at Г =5.55. One can be concluded from figures 6 and 7 that the variation of the four spherical aberration coefficients as a function of Г, give us the best values for the clindrical conve electrodes for hole range of Г. aberration coefficients are D s and D s Figure 7. The spherical aberration coefficients of electrostatic quadrupole lens a D p, D p, and b D s, Ds as a function of gap angle Г.

11 The resolution limit computes according to equation and. Representing the resolution limits R cs and R cs as a function of Г shon in figure 8-a, hich has the same curve behavior ith minimum values R cs =. nm at Г = 5.5, and R cs =.9 nm at Г = While, the resolution limits R cp and R cp represented as a function of Г shon in figure 8-b. The values of R cp increasing ith increment Г up to maimum values R cp =. nm, and the values of R cp increasing slol ith increasing Г up to maimum values R cp =. nm, these maimum values happen at Г = 5.55 and then drop ith increasing Г. Representing the resolution limits R Ds and R Ds as a function of Г shon in figure 9-a, hich has approimatel the same behavior ith minimum values R Ds =.7 nm, and R Ds =.9 nm at Г = While the resolution limits R Dp and R Dp represented as a function of Г shon in figure 9-b. The values of R Dp increasing ith increment Г up to maimum values R Dp =. nm at Г = 5.5, then happen slo drop and up to the same maimum value at Г =6.5, therefore, drop sharpl ith increasing Г. Thus, the values of R Dp decreasing ith increasing of Г until minimum values R Dp =.6 nm at Г = 5.55, then increasing these values ith increasing Г. Figure 8. The resolution limit of electrostatic quadrupole lens ar cs, R cs, and b R cp, R cp as a function of gap angle Г.

12 Figure 9. The resolution limit of electrostatic quadrupole lens ar Ds, R Ds, and b R Dp, R Dp as a function of gap angle Г. Conclusions The quadrupole lens sstem has man variable geometrical and operational parameters; thus conclusive result is rather difficult. Hoever, from the present investigation one ma conclude the folloing: a. It appears that the modified bellshaped field model ver close to the aial field distribution of the electrostatic quadrupole lens of concave and conve electrodes and it gives good properties. b. In general, the electrostatic quadrupole lens of concave electrodes gives the best result of all spherical aberration parameters and resolution limit or resolving poer than the electrostatic quadrupole lens of clindrical conve electrodes. References. Baranova, L. A. and Yavor, S.Ya. 98, Electrostatic lenses, Sov.Phs.Tech.Phs., 9 8, S.Ya. Yavor, 968,p.6 "Focusing of charged particles b Quadruple lenses ".. P.W. Hakes, 97, "Quadrupoles in electron lens design", Adv.Electronics and Electron Phs.Supplement 7, ed. L. Marton, Academic Press, Ne York and London.. M. Szilagi, 988,"Electron and ion optics", Plenum Press, Ne York. 5. Grime, G.W. and Watt, F. 988,"Focusing protons and light ions to micron and submicron dimensions "Nuclear Instruments and Methods in Phsics Research, B, 7-.

13 6. A. Kiss and E. Kolta, 97,"Investigations on the effective length of asmmetrized quadrupole lenses", Nucl.Instrum. Meth.78, A. D. Dmnikov, T. Ya. Fishkova and S.Ya. Yavor, 965, "Spherical aberration of compound quadrupole lenses and sstems ", Nucl. Instrum. Meth., 7, P. Grivet, 97,"Electron Optics", Pergamon Press, Oford and Ne York. 9. Okaama, S. 989, "Electron beam lithograph using a ne quadrupole triplet", SPIE, Electron-beam, X-ra, and Ion-beam Tectnolog: Submicrometer Lithographies VIII, 89, PP S. Okaama and H. Kaakatsu, 978, "Potential distribution and focal properties of electrostatic quadrupole lenses", J. Phs.E: Sci. Instrum.,, 6.. Oda A. Hussein, 9, "Determination of the most favorable shapes for the electrostatic Quadrupole ", J. Al-Nahrain Universit, Science,, Hakes, P.W. 965/966, "The electron optics of a quadrupole lens ith triangular potential", Optik,, L.R. Burns et al. 7, "Evaluation of a novel design for an electrostatic quadrupole triplet ion beam lens", Nucl. Instr. and Meth. B..5. Fishkova, T. Ya., Baranova, L. A., and Yavor, S. Ya. 968, "Spherical aberration of stigmatic doublet of quadrupole lenses" rectangular model.sov.phs.tech.phs.,, P.W. Hakes, 97, " Electron optics and electron microscop", Talor & Francis LTD, - Macklin Street London WCB5NF.. T. Haashi and N. Sakudo, 968, " Quadrupole field in circular concave electrodes ", Rev. Sci. Instrum., 9 7,

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