Microsoft Word - 170-180 170 | Physics 2014) عام 3(العدد 27المجلد مجلة إبن الھيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014 Magnetic Lens Design Using Analytical Target Field Function Emad H. A. Al-Dawoody Dept. of Physics/ Collage of Sciences/ University of Al-Mustansyriyah Receivedin :16January2014, Accepted in :18March 2014 Abstract Analytical field target function has been considered to represent the axial magnetic field distribution of double polepiece symmetric magnetic lens. In this article, with aid of the proposed target function, the syntheses procedure is dependent. The effect of the main two coffectin optimization parameters on the lens field distribution, polepieces shape, and the objective focal prosperities for lenses operated under zero magnification mode has been studied. The results have shown that the objective properties evaluated in sense of the inverse design procedure are in an excellent correspondence with that of analysis approach. Where the optical properties enhance as the field distribution of the electron lens distributed along a narrow axial interval with high field peak and virsa. Keywords: magnetic lens , polepiece , aberrations , target function 171 | Physics 2014) عام 3(العدد 27المجلد مجلة إبن الھيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014 Introduction The basic goal of the optimization procedure for any system of a charged particle beam optics is to produce electrodes or pole pieces systems that provide giving focusing properties with minimum aberrations, i.e., the approach that aims to improve the quality of the electron device, in general, and any desirable procedure leading to optimum design of an electron optical system [1]. However, the effect of aberrations leads to a degradation for any system of charged particle beam optics. Indeed, generating aberrations is inevitable when the charged particle beams are extracted, accelerated, transmitted, and focused with electrostatic and magnetic fields. Strictly speaking, aberrations degrade the focused beam spot, limiting the spatial resolution of these instruments [2]. However, the most effective aberrations that deteriorate the efficiency of any electron optical instrument are the spherical and chromatic aberrations. Improvement of the resolution of electron microscope is limited by the aberrations of the objective lens. Spherical and chromatic aberrations are the most effective ones. Unfortunately, both spherical and chromatic aberrations cannot eliminate completely. But they can be minimized to improve the resolution of the electron microscope. Traditionally, effort in improving electron lenses has concentrated on minimizing these aberrations and searching for field distributions with minimum or zero spherical aberration [3]. As it is well known that, in the field of electron and ion optics, there are essentially two different optimization procedures to be followed in the design of charged particle lenses. The first is called analysis in which the process of trial and error is followed; while the second approach is the synthesis as in some times it is called the inverse design procedure. The main feature that characterizes the later approach is that there is a target function which has to be optimized with certain conditions. The axial electrostatic or magnetic field, potential, or the beam trajectory of the electron optical device can be presented by this target function. Mathematical formation Target Function Synthesis procedure of magnetic lenses followed in the present paper starts with suggesting the following target function to approximate the axial magnetic field distribution Bz (z).   ) 1 1 ( 2 2 1 2 a z z z o e a zB        (1) Where z is the axial coordinate along the optical axis, zo may be any axial coordinate taking different values, and an optimization parameter may represent the radius of the magnetic lens pole piece. The first derivative of the magnetic field given by equation (1) can be easily deduced to be as;     )1 2 ( 2 2 1 22 a z z z o e az z z           (2) Magnetic Scalar Potential As it is well–known that in the current-free region, the axial magnetic field distribution along the optical axis can be determined with aid of the magnetic scalar potential V(z) as    )( dz zdV zZ  (3) Where µo is the free –space permeability(4π ×10-7 H/m). 172 | Physics 2014) عام 3(العدد 27المجلد مجلة إبن الھيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014 However, to determine the axial magnetic scalar potential distribution V(z) when the magnetic field is known along the optical axis, equation (3) can be integrated along the axial extension of the lens as follows[1]     dz)zB1( zf z z s   zV (4) Where Zs and Zf are the terminally of the field distribution. In concerning with the inverse design procedure, the problem of determining the magnetic scalar potential distribution V(z) is an important consequence for determining magnetic lens pole pieces. Therefore, with aid of equation (4), V(z) may be determined analytically when the field target function is easily to be integrated, or one can use the numerical integration techniques . Fortunately, the cubic spline integration technique that had been introduced by [4] for symmetric fields has been used in this paper. Accordingly, the magnetic field distribution Bz (z) for each spline interval k can be represented by the following cubic expression [4].          3 1 12 ZK 6 5.0B k kk kk kkkKK zz zz BB zzBzzBBz       (5) Where z is the axial coordinate along the spline interval such that 1 kk ZZZ . In the case of symmetrical magnetic field distribution, the potential values at the terminals Vzs and Vzf have the following property (Vzf=-Vzs=o.5NI ) (6) Where NI is the lens excitation. However, using equations (4) and (5), the magnetic scalar potential distribution V(z) along the solution domain [zs , zf ] can be determined as follows ( see, for more details in [4,5,6].   )(V 1 1 2 1 1 1 1kz        k j kk n k ako EdB (7) Where   ) 232 (E 3 1 2 k                              kkzzk k zkkzj dV V d VdV (8) and kk zz   1kd Polepiece Profile The final task of any synthesis procedure in the field of electron and ion optical systems is finding the electrodes or /and the pole piece profile which produces the suggested tragic function distribution (i.e, the distribution of the magnetic field in the present investigation). The solution of Laplace's equation in cylindrical coordinates system for rotationally symmetric magnetic fields can be expressed by the following power series expansion [1].       )(V 2 1 zr,V )2( 2 0 2 z r ni n n n n          (9) However, by using the first two terms of equation (9), the pole piece shape can be obtained from the following equation. 173 | Physics 2014) عام 3(العدد 27المجلد مجلة إبن الھيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014      zV VzV z p    2R p (10) Where  zR P is the radial height of the reconstructed polepiece shape, V"(z) is the second derivative of V(z) and Vp represents the potential at the lens terminal which is equal to (o.5 x total excitation of the lens). Electron Lens Aberrations In high-resolution transmission electron microscopy, lens aberrations play a key role in the imaging and the interpretation of object structures on an atomic scale .Aberrations are beneficial and detrimental to high-resolution imaging at tle same time. On the one hand, they introduce unwanted blurring in any imaginy plane, hence obscuring the finest object details. On the other hand, they are urgently needed to produce the desired phase contrast of the very thin objects required for high-resolution structure investigations [7]. Actually, any task in the field of the electron and ion optics aims to determine the aberrations of an optical instrument. To evaluate these aberrations, several axial functions must be determined. In this paper two relating figures of merit are considered to evaluate the objective focal properties, the spherical Cs and chromatic Cc aberration coefficients. The coefficients of these two important aberrations are represented by the following integral forms [8].                dzzrzrzBzrzBzrzB VV ZZZ z z rr S i c 2222224 88 3 128     (11)    dz zrB 8V η C 2α z z 2 z r C i o  z (12) where the limits of integrations, zo and zi are the object and image positions respectively, ƞ is the charge to mass quotient of the election and Vr is the relativistically corrected accelerating voltage. The axial functions r(z) and rʹ(z) are the election beam trajectory and its first derivative .The integrals of aberrations Cs and Cc can be totally determined when r(z) is known along the interval of the lens. Therefore, the beam trajectory r(z) can be evaluated by solving the following second –order differential equation [1]. 0rB rV dz rd 2 z2 2 8 η  (13) Results and Discussion Effect of the parameter a In the present work five various values of the parameter a have been taken under consideration (i.e.,a=1,2,3,4,and 5mm) when the parameter zo is kept constant at 0.5mm .The effect of this parameter on the axial magnetic field distribution and their related axial functions such as V(z) ,Rp(z) as well as the objective properties of a symmetric double polepiece magnetic electron lens has been investigated. Figure1 shows different axial magnetic field distributions corresponding to the five various values of the parameter a. One can see that, as the parameter a is increased the physical and the geometrical parameters Bmax the halfwidth W, and NI are varied as shown in figure2. However, as a is increased the axial 174 | Physics 2014) عام 3(العدد 27المجلد مجلة إبن الھيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014 field would be distributed along a large axial interval,i.e, the halfwidth W is increased with increasing a, while the peak axial flux density value Bmax and the excitation of the lens NI are decreased. The axial magnetic scalar potential distribution V(z) corresponding to the field distributions is plotted in figure 3. It is noted that the outer region of V(z) curves about the symmetry plane (i.e., z=0) represents a field-free region, since the potential is approximately constant in these regions. This behavior can be explained according to the equation (Bz(z)=-µo dV(z)/dz).This means that the electron beams travel in straight lines in these regions. Therefore, the main effect of the parameter a is confined in the region between the polepieces, i.e.,the effective (air gap) region in which the charged partical beams suffer from refraction effect. The reconstructed polepieces for each value of the parameter a are shown in figure 4. Actually, one may note that the increasing of a will significally change the polepiece profile. The coincidence between this synthesis result and the well known conventional counterparts is that both of them announce the increasing of the bore diameter will redacting the ability of the polepiece in localizing and confining the magnetic flux lines. The objective properties the focal length fo,cs,and cc are affected by varying the parameter a as shown in figure 5. It is noted that as the parameter a decreases, the lens will be of good performance, hence, the aberrations cs and cc as well as the focal length fo have small values for fields (lenses) of small values of the parameter a. It should be mentioned that figure 5 illustrates the values of cs, cc, and fo at NI/Vr1/2=20 for a lens operated under zero magnification mode. The values Bmax ,W,and NI as well as the objective properties fo,cs,and cc at NI/Vr1/2=20 for different values of the parameter a are listed in table I. o Effect of the parameter z Five values of the parameter zo (0.5, 0.7, 0.9, 1.1, 1.3 mm) have been chosen to investigate the effect of this parameter on the axial functions distributions, polepiece profiles, and the objective focal properties. The parameter a has been kept constant at 1mm, while the length of the lens which has no effect on the design of the magnetic lens is kept constant at 20mm. Figure 6 shows different field distributions at different zo having different half width, Bmax, and NI. However, variation of Bmax,W and NI with the parameter zo is shown in figure7. The magnetic scalar potential distributions and the polepiece of the magnetic lens for different values of zo are shown in figures 8 and 9 respectively. The values of the objective focal length f0 ,the spherical cS and chromatic cC aberration coefficients for various values of zo are shown in figure 10. It is clear that these optical properties enhanced as the parameter zo decreases, i.e, the small zo values makes the field distribution more localized in the air gap region. Table II shows the computed parameters Bmax,W,and NI and values of the lens objective properties fo,cS,and cC at NI/Vr1/2=20. Conclusion Results in present article have shown that the objective focal properties cs, cc,and fo can be minimized to a suitable values under the effect of the bore polepiec rabius and a specific axial extension value. Also, the results shown that the behavior of the aberration in sense of synthesis procedure is in excellent with that of analysis one. References 1- Szilagyi,M.(1988) Electron and ion optics .Plenum Press Newyork and London 2- Miyamoto,K.and Hatayama,A(2009). Theoretical study of the electrostatic lens aberration of a negative ion accelerator for a neutral beam injector. Plasma and Fusion Research,4,1-17. 175 | Physics 2014) عام 3(العدد 27المجلد مجلة إبن الھيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014 3- Al-Batat,A.H.H.(2013). Modeling and Design For Objective Charged Partical Lens, (IJAIEM),2.(10).25-32. 4- Al-Obaidi,H.N.(1995) Determination of the design of Magnetic electron lenses operated under ressigned magnification condition, Ph.D. Thesis, University of Baghdad, Baghdad-Iraq. 5- Al-Saadi, S.R.A.(2007) Improvement In Application of The Theory of Charged Beam Optics,Ph.D.Theses,College of Education, , the University of Al-Mustansiriyah, Baghdad,Iraq. 6- Al-Jubori,W.J.(2001) Inverse Design of Asymmetrical Magnetic Lenses in the Absence of Magnetic Saturation ,Ph.D. Thesis, College of Science, the University of Al-Mustansiriyah, Baghdad,Iraq. 7- Lentzen,M.(2006). progress in aberration-corrected hight-resolution transmission electron microscopy using hardware aberration correction .Microsc.Microanal 12,191-205. 8- El-Kareh,A.B.and El-Kareh,J.C.J,(1970). Electron beams,lenses,and optics.(Academic Press). Table No.(1): Some lens parameters and the objective properties for various values of the parameter a. a(mm) Bmax(T) NI(A-t) W(mm) CC(mm) CS(mm) fo(mm) 1 0.4472 471 1.246 0.3181 0.2447 0.4455 2 0.2425 347 1.69 0.4324 0.3328 0.6050 3 0.1644 285 2.054 0.525 0.4044 0.7348 4 0.1240 248 2.3644 0.5890 0.4416 0.8348 5 0.0995 222 2.6398 0.6580 0.4939 0.9323 Table No. (2): Some lens parameters and the objective properties for various values of the parameter Zo. (mm)oF (mm)SC (mm)CC W(mm) NI(A-t) (T)maxB (mm)oZ 0.4455 0.2447 0.3181 1.246 471 0.4472 0.5 0.5508 0.3029 0.3936 1.540 747 0.5735 0.7 0.6557 0.3607 0.4687 1.832 1038 0.6690 0.9 0.7620 0.4194 0.5449 2.130 1334 0.7399 1.1 0.8586 0.4544 0.6058 2.431 1632 0.7926 1.3 176 | Physics 2014) عام 3(العدد 27المجلد مجلة إبن الھيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014 -10 -5 0 5 10 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 z(mm) B z( T ) a=1mm a=2mm a=3mm a=4mm a=5mm B m a x (T ), N Ix 1 0 ^ 3 (A ‐t ), W (m m ) a(mm) Bmax NI(A‐t) W(mm) -10 -8 -6 -4 -2 0 2 4 6 8 10 -250 -200 -150 -100 -50 0 50 100 150 200 250 Z(mm) V (z )( A -t ) a=5mm a=4mm a=3mm a=2mm a=1mm Figure No.(1): The axial magnetic field distribution Bz(z) for various values of the parameter a at zo =0.5mm Figure No.(2): Variation of Bmax, W, and NI with a Figure No.(3): The axial magnetic scalar potential distribution V(z) for various values of a at z0=0.5mm 177 | Physics 2014) عام 3(العدد 27المجلد مجلة إبن الھيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 0 1 2 3 4 5 6 7 8 9 10 z(mm) R p( m m ) a=1mm a=2mm a=3mm a=4mm a=5mm -10 -8 -6 -4 -2 0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Z(mm) B z( T ) zo=0.5mm zo=0.7mm zo=0.9mm zo=1.1mm zo=1.3mm Figure No.(4): Different polepiece shapes for different values of the parameter a Figure No.(5): Variation of fo, cs, and cc with the parameter a at NI/Vr1/2=20 values at a=1mm oFigure No.(6): Different field distributions for different z cC (m m ), cs (m m ), fo (m m ) a(mm) Cc(mm) Cs(mm) fo(mm) 178 | Physics 2014) عام 3(العدد 27المجلد مجلة إبن الھيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014 B m a x N Ix 1 0 * 3 (A ‐t ), W (m m ) Bmax(T) NI(A‐t) W(mm) Zo(mm) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 0 1 2 3 4 5 6 7 8 9 10 Z(mm) R p( m m ) zo=0.5mm zo=0.7mm zo=0.9mm zo=1.1mm zo=1.3mm -10 -8 -6 -4 -2 0 2 4 6 8 10 -1000 -800 -600 -400 -200 0 200 400 600 800 1000 Z(mm) V z( A -t ) zo=1.3mm zo=1.1mm zo=0.9mm zo=0.7mm zo=0.5mm . for a=1mm oW,NI with the parameter z maxFigure No.(7):B for oFigureNo.(8): The magnetic scalar potential distribution for different values of z a=1mm oFigure No.(9): The reconstructed polepieces at different values of z 179 | Physics 2014) عام 3(العدد 27المجلد مجلة إبن الھيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014 =201/2rat NI/V owith the parameter z o,and fS,cCFigureNo.(10): Variation of c 180 | Physics 2014) عام 3(العدد 27المجلد مجلة إبن الھيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014 تصميم عدسة مغناطيسية باستخدام دالة ھدف مجالي تحليلي عماد حميد احمد الداودي الجامعة المستنصرية/كلية العلوم /قسم الفيزياء 2014اذار 18، قبل البحث :2014كانون الثاني 16استلم البحث : الخالصة ة وباالعتماد على تم في ھذا البحث تمثيل المجال المغناطيسي المحوري لعدسة ثنائية القطب المتناظرة بدالة ھدف تحليلي زيع مجال اسلوب التوليف تم استخدام تلك الدالة. حيث تم دراسة تاثير اھم عاملي امثلية في دالة الھدف على كل من تو وصلت اليھا باستخدام واص البصرية الشيئية باستعمال نمط التشغيل الصفري . والنتائج التي تالعدسة و شكل اقطابھا و الخ لخواص البصرية اسلوب التصميم العكسي بينت مدى التطابق الكبير مع التي درست باستخدام اسلوب التحليل. حيث ان ا لك الخواص ضيقة بقمة مجال عالية وتسوء ت تتحسن عندما يكون مجال العدسة االلكترونية موزع على طول منطقة محورية اذا كان المجال موزع على منطقة محورية واسعة. العدسة المغناطيسية, االقطاب, الزيوغ, دالة الھدفالكلمات المفتاحية: