design of an electrostatic lens using the charge

DESIGN OF AN ELECTROSTATIC LENS USING THE CHARGE DENSITY METHOD A Thesis Submitted to the College of Science of Al-Nahrain University in Partial Fulfillment of the Requirements for the Degree of

Master of Science in

Physics by

Basma Hussian Hamad Al-Shammary (B.Sc.2001) in

Thul-Qu’da 1424 A.H.

January 2004 A.D.

Examination Committee Certification We certify that we have read the thesis entitled “Design of an Electrostatic Lens Using the Charge Density Method” and as an examination committee, examined the student Miss Basma Hussien Hamad Al-Shamary on its contents, and that in our opinion it is adequate for the partial fulfillment of the requirements of the degree of Master of Science in Physics.

Signature: Name: Title:

Date: / /2004

Signature: Name: Title: Date: / /2004

Signature: Name: Title: Date: / /2004

Signature: Name: Dr. Ahmad k. Ahmad Title:Assistant Professor (Supervisor) Date: / /2004

Signature: Name: Dr.Sabah M. Juma Title:Professor (Supervisor)

Date: / /2004 Signature:

Name:Dr.Laith Abdul Aziz Al-Ani (Dean of the College of Science) Date: / /2004

DEDICATION TO MY FAMILY

I would like to express my sincere thanks and deep gratitude to Prof. Dr. Sabah M. Juma and Dr. Ahmad K. Ahmad for supervising the present work and for their support and encouragement throughout the research. I would like to thank Dr. Fatin Abdul Jalil Al-Mudarris for her valuable assistance during the work. I am grateful to the Dean of College of Science and the staff of the Department of Physics at Al-Nahrain University for their valuable support and cooperation. My special thanks are also due to Dr. Akram, Mr. Moshtaq, and Ms. Zina (Department of Mathematics). The assistance given by the staff of the central library at Baghdad University specially (Ms. Hind, Ms. Heiam and Ms. Shaima) is highly appreciable. Last but not least, I would like to record my deep affection and thanks to my parents specially my sister Samera for their moral support and patience throughout this work.

Bassma

Contents Synopsis List of Symbols

1. 1.1 1.2 1.3 1.3.1 1.3.2 1.3.3 1.4 1.5 1.6

Electrostatic Lenses Types of Electrostatic Lenses Numerical Determination of Electrostatic Field Finite-difference method Finite-element method Boundary element method Charge Density Method Advantages of Charge Density Method (CDM) Aim of Project 2.

2.1 2.2 2.3 2.4 2.5 2.6

3.2

4.2 4.3

COMPUTER PROGRAMS

Computer Programs for Determining the Axial Potential Distribution of a Two-Cylinder Electrostatic Lens Computer Program for Computing the Beam Trajectory And the Optical Properties 4.

4.1

THEORETICAL CONSIDERATION Potential at a Point Due to a Charge Ring The Charge Density Technique The Trajectory Equation Lens Parameters Lens Aberrations Two-Electrode Immersion Lenses

3. 3.1

INTRODUCTION

RESULTS AND DISSCUSIONS

The Charge Density on the Electrode of an Immersion Lens and the Corresponding Axial Potential Distributions at Different Gaps Electron Beam Trajectory Under Zero Magnification Condition Relative Aberration Coefficients Under Zero Magnification Condition

4.3.1

Relative spherical aberration coefficient 4.3.1 Relative chromatic aberration coefficient

4.3.2

Comparison between the relative spherical and chromatic aberration coefficients 4.3.4 Relative focal length

4.4

4.4 Electron Beam Trajectory Under Finite Magnification Condition Relative Aberration Coefficients of Finite Magnification Condition 4.5.1 Relative spherical aberration coefficients 4.5.2 Relative chromatic aberration coefficients 4.5.3 Comparison between the minima of the relative spherical and chromatic aberration coefficients

4.5

The Charge Density on the Electrodes of an Immersion Lens and the Corresponding Axial Potential Distributions at Different Radii 4.7

Electron Beam Trajectory Under Zero Magnification Condition

4.8

Relative Aberration Coefficients Under Zero Magnification Condition

4.8.1 Relative spherical aberration coefficient 4.8.2 Relative chromatic aberration coefficient 4.8.3 Comparison between the relative spherical and chromatic aberration coefficients 4.8.4 Relative focal length 4.9 Electron Beam Trajectory Under Finite Magnification Condition 4.10

Relative Aberration Coefficients Under Finite Magnification Condition 4.10.1 Relative spherical aberration coefficients

4.10.2 Relative chromatic aberration coefficients 4.10.3Comparison between the relative spherical and chromatic aberration coefficients 5. CONCLUSIONSANDRECOMMENDATIONS FOR FUTURE WORK 5.1

Conclusions 5.2 Recommendations For Future Work Appendices References

List of Symbols A-1

Inverse of matrix A

a

Radius of ring

Aji

Square matrix element

Cc

Chromatic aberration coefficient in the image-and object-side

Cs

Spherical aberration coefficient in the image-and object-side

D

Diameter of the electrode

dV

Potential at the point P due to charge element S on a ring

e

Electron charge E(z)

Electric filed along the axis z

fi,fo

Focal lengths in the image and object side respectively

G

Air gap separating the two electrodes

H

Dimensionless factor

K(k2)

Elliptic integral of the first kind

k2

Elliptic integral modulus

ki

Elliptic integral modulus of ith ring due to any point of z

kji

Elliptic integral modulus of jth ring due to ith ring

L

Length of each electrode

M

Linear magnification

m

Electron mass

N

Number of rings

PF1,PF2

Focal points on object and image side respectively

Q

Total charge on a ring

Qi

Charge on ith ring

R

Radial displacement of the beam from the optical axis

r

Distance between the charged element and the observation point P on a ring

rc

Radius of electrode (hollow cylinder)

Ro

Radial displacement of the beam in object plane

R’

Derivative of the radial displacement with respect to z i.e.dR/dz

S

Charged element on a ring

Uo

Axial potential in object plane

U(r,z)

Radial-axial potential distribution

U=U(z)

Axial potential distribution

U’(z),U”(z)

First and second derivative with respect to z i.e.dU/dz and d2U/dz2

V Vj

Potential of a point P due to a ring of charge Q Voltage applied on the two cylindrical electrodes of the immersion lens

z

Optical axis

Zi, Zo

Image- and object- side position

z i , zj

Mid-point of the ith and jth rings

β

An angle equal to φ/2 where φ is the angle between the radius of the ring and the x-axis

Δzi

Width of the ith ring

εo

Electrical permitivity in vacuum (ε0=8.85E-19 F.m-1)

λ

Linear charge density, (cm-1)

υ

Electron velocity

υ1

Electron velocity when it enters the region of the lens

υ2

Electron velocity when it leaves the region of the lens

σi

Surface charge density of the ith ring (cm-2)

Φ

Electrostatic potential (scalar potential)

Synopsis A computational investigation has been carried out in the field of non-relativistic charged-particle optics using the charge density method as a boundary value problem with the aid of personal computer under the absence of space-charge effects. This work has been concentrated on designing a two-electrode electrostatic immersion lens whose electrodes are cylindrical in shape separated by an air gap. The variable parameters of the two electrodes are the voltage ratio applied on them, the air gap separating them, and their radii. The immersion lens has been investigated under finite and zero magnification conditions. The axial potential distribution of an electrostatic immersion lens has been computed by taking into consideration the distribution of the charge density due to the voltages applied on the two cylindrical electrodes. Potentials have been determined anywhere in space by using Coulomb’s law. The paraxial ray equation has been solved with the aid of the computed axial potential distribution in order to determine the trajectory of the charged-particles beam along the lens field using Runge- Kutta method. The axial potential and its first and second derivatives have been used for computing the optical properties of the lens under consideration. The lens focal length and its spherical and chromatic aberrations have taken into account due to their importance from the optical point of view. The computed aberrations have been normalized in terms of the focal length under zero magnification operational conditions and in terms of the magnification under finite magnification conditions. The computational results showed that it is possible to design and hence construct various types of electrostatic lenses with the aid of the charge density method.

CHAPTER ONE

1. INTRODUCTION 1.1

Electrostatic Lenses Electrostatic lenses are used in many scientific instruments to maneuver ion or electron beams by acceleration, focusing, deflection, or by a combination of these three actions. Furthermore, electrostatic lenses have been used to control beams of electrons or ions, and needed in several different fields, principally those of electron microscopy, cathode ray tubes, thermionic valves, ion accelerators, and electron impact studies (Mulvey and Wallington 1973).

The virtues of electrostatic lenses

include their small size and weight and their low power requirements, which makes lighter, more stable power supplies reusable and greatly simplifies fabrication. Since the focal power of electrostatic lenses is independent of the charge-to-mass quotient of the charged particles and depends only on their energy, they are therefore to be preferred over magnetic lenses when heavy charged particles of moderate energy need to be focused (Read 1978).

1.2

Types of Electrostatic Lenses : There are different types and geometries of electrostatic lenses depending on how their field region is generated. The distribution of the electric field in front of and beyond the lens (i.e. in the object and image region respectively) depends on the electrodes shape. The axial behavior of the potential  (z) is generally taken as the criterion for classification. Thus one may classify electrostatic lenses into the following main groups.

a-The immersion lens: It may consist of as few as two electrodes as illustrated in figure 1.1. It has two different constant voltages at its sides i.e. the electrostatic field E(z)=dU(z)/dz is zero at the two terminals.

Immersion lenses accelerate or retard the particles while the beam is focused (Baranova and Yavor 1984).

Figure 1.1. An electrostatic immersion lens and the corresponding axial potential distribution U(z) . b-The cathode lenses : It may be called an immersion objective with a field abruptly terminated on the object side by the source of charged particles as shown in figure 1.2.

Figure 1.2. A cathode lens and the corresponding axial potential distribution.U(z) .

c-The unipotential (einzel) lens: It is usually consists of three electrodes. This kind of lens has the same constant potential at both the object and image sides i.e. the charged particle energy remains unchanged. Figure 1.3 illustrates two geometries of an einzel lens and their corresponding axial potential distribution.

The voltage V2 applied on the middle

electrode is either higher or lower than the voltage V1 on the terminal

electrodes.

Figure 1.3. An einzel lens and the corresponding axial potential distribution.U(z) . d-The diaphragm (single-aperture) lens: In this lens there is a homogeneous field on at least one side i.e. the potential on one side or both is not constant but increases or decreases linearly (see figure 1.4). This means that there is an electrostatic field of constant intensity in the immediate vicinity of the lens.

Figure 1.4. A diaphram lens and the corresponding axial potential distribution U(z) . The field of an electrostatic lens changes the velocity of electrons both in magnitude and direction. If υ1 is the velocity of the electron when it enters the region of the lens and υ2 is the velocity of the electron leaving the region, the lenses are classified into the following categories (Grivet 1972).

a-

Accelerating when υ2> υ1

b-

Neutral (unipotential) when υ2= υ1

c-

Decelerating when υ2< υ1

d-

Electron mirror when υ2=0 and then reverses its direction.

1.3

Numerical Determination of Electrostatic Field Determination of electrostatic fields usually requires the solution of a complicated boundary value problem. Most electrostatic field calculations involve charge distribution on the surface of the conductors only i.e. the space charge effect is neglected, therefore, the field equation to be solved will be Laplace’s equation (Bonjour 1980). The solution of Laplace’s equation ( =0) with specified boundary conditions makes it 2

possible to determine the potential as a function of coordinates from which the components of the field intensity can be calculated (Szilagy 1988).

Many computer programs have been developed for solving problems in electrostatic charged particle optics. Nearly all of the programs are based on one of the following methods (a) Finite Difference Method (FDM) (b) Finite Element Method (FEM) and (c) Boundary Element Method (BEM). The BEM is also known as charge density method (CDM) or boundary charge method (BCM); see for example Cubric, et al (1999).

1.3.1

Finite-Difference Method According to Burden and Faires (2000), the finite-difference method (FDM) was first introduced by Liebmann in 1918 and thus it is often called “Liebmann’s Method”. The basic idea of the procedure is to subdivide the space within the field under consideration into finite squares or rectangular grids. There are many ways for excuting this procedure. Among the most often used are the five-point relaxation and the nine-point relaxation methods; the former relates the potential at any point to the potential of four nearest neighbors while the latter uses also

the four points that are the next nearest along the radial and axial direction. There are two different ways of deriving the finite-difference formula that replaces Laplace’s equation namely, the Taylor series method and the integral method (Bonjour 1980). The continuous differential equation is replaced by a system of algebraic equations that can easily be solved. One can assume that the axial potential distribution is known from the solution of the paraxial-ray equation; other boundary conditions are needed for finding the solution to the system of linear algebraic equations (Szilagyi 1988). Two different numerical techniques, the direct and the iterative methods, can solve the system of linear algebraic equations. The best known direct method is the gaussian elimination and backward substitution (Gerald and Wheatley 1985). There are many iterative methods to solve linear and non-linear system of equations. For more specific details on this method one may see Forsythe and Wasow (1966).

1.3.2

Finite-Element Method The finite-element method (FEM) was first used in electron optics by Munro (1973) who applied it to the computation of the magnetic field in round lenses. The idea of this method is to subdivide the field into triangles, each node being the common vertex of the adjacent triangles. It is assumed that the potential varies linearly across each triangular element then the linear functions are pieced together so that the resulting potential distribution would be continuous over the entire region (see for example Barth et al 1990, Hawkes and Kasper 1989). With this approximation the potential throughout each element is uniquely determined by the potential at its vertices. Hence the contribution from each element to the value of the functional can be expressed in terms of the vertex potential.

The minimization of the functional then yields a system of linear algebraic equations in terms of the node potentials (Szilagyi 1988). This set of linear equations can be solved to give the potential at each meshpoint by means of the direct method or the iterative method, which have been outlined in section (1.3.1). For more specific details on the finite element method one may see Munro (1973, 1975).

1.3.3

Boundary Element Method The boundary element method (BEM) is based on a simple fact that in the static case any region occupied by a conductor is free of field. If potentials are applied on the conductors (e.g. electrodes) the charges distribute themselves on the surfaces which become equipotentials. This is equivalent to forcing definite charge distribution on the electrodes (Renau et al 1982). These charge distributions are considered to be the sources of the electrostatic potential distribution in the space surrounding the electrodes including the electrode potentials themselves. If the electrode potential can be replaced by these surface charge distributions on the electrodes, the value of the potential may be easily calculated anywhere in the space by simply using the superposition principle without employing any sophisticated computational grids as in the finitedifference or finite-element methods (Hartring and Read 1976, Mautz and Harrington 1970, and Van Hoof 1980).

This method has been found to give accurate results, efficient in the use of computer time and storage, and applicable to a wide range of lens configurations. The charge density method is a particular example of BEM.

1.4

Charge Density Method The charge density method for solving Laplace’s equation was first applied in electron-optical systems by Cruise (1963) who computed the potential distribution in an axially symmetric electrostatic lens, which contained no insulators. Mautz and Harrington (1970) gave the equations relating to rotationally symmetric Laplacian potentials in the absence of insulators, and described a computer program for the numerical solution of these equations. Almost simultaneously Singer and Braun (1970) presented a similar treatment applicable to electrostatic systems of any arbitrary configuration containing open boundaries and dielectric media.

This charge density method has been used to compute aperture lenses (Read 1970), two-cylinder lenses (Read et al 1971), and three-cylinder lenses (Adams and Read 1972a,b). In these papers, they gave computed cardinal element and spherical aberration coefficients. Natali et al (1972) used this method to calculate the electric field and trajectory for thick walled two-tube electrostatic lens. These authors call the procedure the integral equation method for computing potentials, but to avoid confusion with other methods in which an integral equation is solved numerically preferred to call it the charge density method (CDM) for computing potentials Mulvey and Wallington (1973). Also CDM was used to determine the cylinder lens potentials and focal properties (Cook and Heddle 1976).

A data book by Harting and Read (1976) employed the CDM method to calculate focal and aberration properties of a wide range of electrostatic lenses such as unequal diameter two-cylinder lenses, doubleand triple-rectangular tube lenses, double- and triple-slit lenses and thick

walled two cylinder lenses. Viljoen (1976) and Hoch et al (1978) have applied this method for the purpose of designing electrostatic lenses of various types. Van Hoof (1980) applied the method to axially symmetric electrostatic lenses, consisting of coaxial cylinder lenses. Renau et al (1982) have been discussed the singularities and approximations that occur in the charge density method of solving electrostatic problems in which space-charge is absent or present.

Fung (1998) used a

reformulated and extended CDM to compute the axial potential and then used this potential to calculate the trajectory of positron beam in symmetric einzel, asymmetric einzel, and two- tube electrostatic lenses. The accuracy and speed of this method has been studied by Cubric et al (1999) and compared it with two other methods FEM and FDM for solving problems in electrostatic charged particle optics using a set of bench mark tests.

With the aid of CDM a computer program was written by Read and Bowring (2000) for finding electrostatic potentials and fields for systems of conducting electrodes with and without space charge; they called it surface charge method (SCM) because of its aptness and clarity. It should be mentioned, however, that in most of the work given above the lenses that are used for this purpose has been divided into N-rings; these rings are of variable width and are made narrower near the gap, where the charge density changes most rapidly. But in the present work the system of cylinders under applied potential have been replaced by a system of charged rings which have the same width as illustrated in figure 1.5.

Figure 1.5. Replacing a series of cylinders under applied potentials with a series of charged rings (Fung 1998)

1.5 Advantages of the Charge Density Method (CDM) The method has obvious advantages over the FDM and FEM. (a)

The potentials and fields are obtained indirectly in CDM, in two

stages, firstly obtaining the surface charges and then using these charges to find the potentials and fields at any point. This gives a fast and accurate way of solving electrostatic problems, and is ideally suitable for space-charge problems (Read and Bowring 2000).

(b)

It works for systems with open boundaries without any

approximations about the potential distributions along those boundaries. In addition it requires much less memory because the potential is only calculated in the regions where it is really needed (Mautz and Harrington 1970).

1.5

Aim of the Project Various computational methods have been applied in electron optics for evaluating the focal properties of electrostatic lenses. The focal properties include parameters such as focal length and aberration coefficients, which are usually functions of various parameters such as electrodes voltage ratio. The present computational work applies the charge density technique for evaluating lens properties. The lens optical properties will be investigated in detail for the purpose of designing lens systems, which have electron-optically acceptable parameters. Therefore in the present work, the charge density method, being an accurate method of solving Laplace’s equation for equidiameter coaxial cylinders separated by a finite distance would be used to calculate the axial potential distribution anywhere in the space. The results would be used to obtain the focal lengths and aberration coefficients over a wide range of voltage ratios and cylinder separations.

Two of the various magnification conditions that are well known in electron optics have been taken into account in the present investigation, namely, the finite and the zero magnification conditions due to their resemblance to the lens trajectory. It is important to note that in this project only the axial potential distribution is taken into consideration. Furthermore, because of the complex nature of the present problem under investigation, the following assumptions have been made: (a) The thickness of the material from which the lenses are constructed is very much less than the radii of the cylinders. (b) The space charge effects are neglected in order to satisfy exactly the following Laplace’s equation.

2=0

(1-1)

where  is called the Laplacian operator and  is the electrostatic potential (scalar potential ) measured in volts. 2

Equation (1-1) determines the function  in charge-free region. (c) Non-relativistic velocities (low energy beams) of electrons will be considered i.e. the velocity υ of the electrons at any point inside the field is determined from the following equation:

υ=[(-2e/m )  ]1/2 where e and m are the charge and mass of the electron.

(d) Aberrations associated with the source have been neglected.

(1-2)

CHAPTER TWO

2. THEORETICAL CONSIDERATIONS 2.1 Potential at a Point Due to a Charged Ring With the charge density method one may replace a system of cylinders under applied potentials with a system of rings of charges. In this case one would need to find the potential at a point due to a charged ring. To implement this, consider a ring uniformly charged as shown in figure 2.1. The ring has a radius a and carries a total charge Q; the potential at the point P due to a charge element at S is given by (Fung 1998),

dV 

1 ad 4 r

(2-1)

d is the angle subtended by the element at the center of the ring carrying a linear charge density λ, and r is the distance between the where

charge element S and the observation point P.

For a uniformly

distributed charge, the linear charge density of the ring is given by, λ= Q / 2 π a

(2-2) z

P(ρ,z)

r z y

φ P(ρ,o)

a S

Figure (2.1) .Circular ring of charge of negligible thickness. In polar coordinates x  a cos , y  a sin and z  z ; therefore

r is

given by,



r    a cos   a sin   z 2 2

2



1

2

(2-

3) Summing over the whole ring gives,

x

Q 2 ad V 4 0 2a 0 (   a cos ) 2  a sin 2  Z 2 1





1

2

(2-

4)

2 Since sin ( / 2) 

cos  1 and   2 , hence it can be show that, 2

cos  2sin2   1

(2-

5) Differentiating equation (2-5) with respect to β gives,

 sin d  4sin  cos d

(2-

6) Now



sin  1  cos2 

show that [see Appendix A-I],



1

2

and by using equation (2-6) one can

sin  2sin  cos

(2-

7) From equations (2-6) and (2-7) it can be deduced that d  2d . When

  0,2 ,..., cos  1 and    / 2 . Since d and d are of opposite sign thus when the angle

 increases from 0 to 2  , the angle

 decreases from  / 2 to -  / 2 . Taking this into consideration, the integral of equation (2-4) becomes [see Appendix A-II],

1

Q V 40 2a

 2

a(2d )

   a 



2

2

 z2

 1  k 1

2

sin 2 

2



1

2

(2-

8) 2 where k 

4 a    a 2  z 2

Equation (2-8) can be simplified into form,

V 

Q 2

1



40     a 2  z 2





1

2

It can be recognized that

 1  k 0

2

 1  k

2

0

d 2

d

2

sin2 



1

sin2  2



1

2

is an elliptic integral of

the first kind with modulus k 2 . One can write the following equation, which represent the potential of a point due to a ring of charge Q (Weber 1950).

V

Q 2



1

4o    a 2  z 2



1

K (k 2 ) 2

(2-

9) 2

where K(k ) represents the elliptic integral of first kind.

2.2 The Charge Density Technique The first step in the present method for calculating the axial potential distribution of a two-cylinder electrostatic lens is to find the charge density on each surface of the conducting sheets from which the lens is constructed. In the absence of dielectrics the electrostatic potential at any point in space is determined by the free surface charges on the conductors in the space (Read et al 1971). The second step is, therefore, to use the determined charge density for computing the potential distribution in the space of the lens.

In applying this method for equidiameter coaxial cylinders separated by a finite distance G it has been assumed that the cylinder walls have negligible thickness so that the potential in regions which are not very close to the cylinders is determined simply by the algebraic sum of the inner and outer charge sheets (Bonjour 1980). To solve the problem, the cylinders have been divided into N-ring, each ring carries a charge Qi (i =1,2,3,…,N) which contributes to the potentials of all the rings. The potential of the ith ring can be expressed as a combination of the contributions from all charge rings (Fung 1998).

Consider the lens cylinders shown in figure 2.2 of radius rc and length 10 rc (Mulvey and Wallington 1973). If the combined charge densities

 i  Qi / 4rc zi ,

on the surfaces of the cylinders are

where

zi represents the width of ith ring. If there are no other charges present then the potential at any point z in space is given by,

U (rc , z ) 

N

1

o

2  k K ( k  i i i )zi

(2-

i 1 j i

10) where

ki 

4r

c

2rc 2

 z i  z 

2



1

2

and K (ti ) is the complete elliptic integral of the first kind which can be evaluated by the use of the following polynomial approximation (see Szilagyi 1988),

K (ki )  a0  a1H  a2 H 2  a3H 3  a4 H 4  (b0  b1H  b2 H 2  b3H 3  b4 H 4 ) ln(1/ H ) (2-11) where H =1-ki2 which is a dimension less factor.

G

∆zi

rc Z zj zi 10rc

Figure (2.2). Simple coaxial two-cylinder lens consisting of a large number of circular strips in order to obtain the potential distribution by CDM. The potential

V j at a point C in figure 2.2 on the ith element is due

to a constant charge density σ on each element, which is uniformly distributed around a circle of radius rc . The potential

V j is given by the

following expression (Harting and Read 1976),

Vj 

N

A  i 1

ji

(2-

i

12) where

Aji

is a square matrix element. The above set of equations may

be reduced to the following simple matrix equation,

V  A

(2-

13) The charge density σ is mathematically considered a column vector. In applying this procedure to the cylinder problem one may take different values of the voltage applied on the first and second electrodes, the column vector σ is then obtained by inverting the matirx

A (Renau et al

1982, Shirakawa et al 1990). Hence, from equation (2-13),

  A1  V

(2-

14) In the present work an iterative procedure is used to get the inverse of matrix

A with the aid of a computer program based on LU-Factorization

method (see for example, Kolman 1985).

To evaluate the elements of

A one needs to know the potential at the

strip j caused by a uniform charge density σi in the strip i. The matrix element

Aji

is given by (Mulvey and Wallington 1973),

Aji 

k ji zi

0

2

K (k ji )

(2-

15)

k ji 

where

4r

c

2

 z ji

2



1

2

z ji  zi  z j

and

zi

2rc

and

z j being the mid point of the ith and jth ring respectively; they

are given by,

z i  ( z i 1  z i 1 ) / 2 , z j  ( z j 1  z j 1 ) / 2 . It should be

noted that when j is equal to i the elliptic integral (equation 2-11) will be infinite and a singularity in the potential not in

V (equation 2.8) is caused but

Aii itself.

2.3 The Trajectory Equation The equations of motion of a charged particle travelling at a nonrelativistic velocity in an electric field near the axis of a cylindrically symmetric system can be reduced to the following paraxial ray equation (Grivet 1972, Paskawski 1968),

d 2 R U  dR U    R0 2 dz 2U dz 4U 16)

(2-

where

U

and

U  are the first and second derivatives of the axial

potential U respectively. R represents the radial displacement of the beam from the optical axis z, and the primes denote a derivative with respect to z. Equation (2-16) is a linear homogeneous second order differential equation. If the potential distribution along the z-axis is known, that is if the function U (z) is known the factors in front of

dR / dz and R is also known. It should be noted that three important deductions could be made from equation (2-16) (El-Kareh and ElKareh 1972).

(a)The quotient of charge to mass ss(Q/m) does not appear, indicating that the path is the same for any charged particle no matter what may be its Q/m provided it enters the field with the same constant kinetic energy. (b)The equation is homogeneous in

U (z) ; i.e. the trajectory remains

unaltered when the voltage is increased proportionally for all the electrodes. (c)The equation is homogeneous in R and z which means that n-fold change in the dimensions of the field region and the electrodes produce corresponding change in the dimensions of the trajectory since the equipotentials don’t alter the shape of the path.

If the electrode is

doubled in size, the image will be doubled in size, the ratio between the two remaining constants.

2.4 Lens Parameters With the aid of computer programs, the optical properties of an electrostatic lens are calculated. The following is a brief description of the optical parameters.

Object Side: It is the side of the lens at which the charged particles enter the electric field. Image Side: It is the side of the lens at which the charged particles emerge from the field. Object Plane: It is the plane at which the physical object is placed, or a real image is formed from a preceding lens, on the object side of the lens as shown in figure 2.3. Image Plane: It is the plane at which a real image of the object is formed on the image side of the lens as shown in figure 2.3. Focal Points: A focal point is the image of a bundle of rays incident on a lens parallel to the optical axis. If these rays arrive at the lens from the object side, then these rays are collected at the image focal point PF 2. If these parallel rays are incident from the image side, the lens at the object focal point PF1 will collect them. The plane perpendicular to the optical axis and passing through either PF1 or PF2 is known as the object or the image focal plane respectively (Wallington 1971).

Figure (2.3) The cardinal elements of an axially symmetric lens. Principal Points: The incident and refracted rays are extended until they intersect the planes passing through the intersections and perpendiculars to the axis are called principal planes. The points of intersection of the principal planes with the optical axis are called principal points P 1 and P2 as shown in figure 2.3. The distances from the principal points to the corresponding foci are called the focal lengths f 1 and f2.

Also the

distances from the reference plane to the focal points are called the midfocal lengths F1 and F2 (Lencova and Lenc 1994). Magnification (M): In any optical system the ratio between the transverse dimension of the final image and the corresponding dimension of the original object is called the linear magnification M, i.e. M = image height / object height

(2-

17) There are three magnification conditions under which a lens can operate, namely:

a-

Zero Magnification Condition: In this case the operating condition is Zo   , and the ray enters the lens field parallel to the optical axis.

b-

Infinite Magnification Condition: The operating condition is Zi   , and the ray leaves the lens field parallel to the optical axis.

c-

Finite Magnification Condition: The operating condition is that in which Zo and Zi are at finite distances from the lens.

2.5 Lens Aberrations The electron paths, which leave points of the object close to the axis at small inclinations with respect to the axis, intersect the image plane in points forming a geometrically similar pattern. This ideal image is called gaussian image, and the plane in which the image is formed is called the gaussian image plane. If an electron leaving an object point a finite distance from the axis with a particular direction and velocity intersects the gaussian plane at a point displaced from the gaussian image point, this displacement is defined as the aberration (Hawkes 1972, Klemperer and Barnett 1971).

The quality of any electron optical system depends not only upon the wavelength of the electrons, but also upon the aberrations from which it

may suffer. If the accelerating potential and the lens excitations fluctuate about their mean values, chromatic aberration will mar the image. If the properties of the system are investigated, using a more exact approximation to the refractive index than is employed in the gaussian approximation, one would find that geometrical aberrations affect both the quality and the fidelity of the gaussian image (Grivet 1950, Renau and Heddle 1986). When the properties of the system are analyzed using the non- relativistic approximation, the disparities between the relativistic and non-relativistic results can be conveniently regarded as a relativistic aberration.

Space-charge is another source of image defects. In the presence of high concentration of charged particles, the electrostatic repulsion forces between particles of the same charge makes it almost impossible to focus these particles exactly into a point (Heddle 1991,Yamazaki 1977, Zhigarev 1975).

The most important aberrations in an electron-optical system are spherical and chromatic aberration. Thus the present work has been focused on determining these two aberrations for an immersion electrostatic lens operated as an objective lens.

Spherical aberration is one of the most effective geometrical aberrations. It is defined in both light and electron optics, as the change in focal properties of a lens with radial height of the ray. Off-axis electrons spend less time in the field than do paraxial electrons because they do not penetrate as far into the field, and as a result the focal distance

is longer for electrons whose trajectories make larger angles with the optical axis than for paraxial electrons (Meisburger and Jacobsen 1982, Szilagyi 1986).

Chromatic aberration in light optics refers to the change in focal properties with the wavelength of light. In electron optics it refers to the change in focal properties with kinetic energy of the electrons; lower energy electrons, and therefore spend less time in the field and are less strongly converged by the field.

Consequently the focal distance is

longer for lower energy electrons than for high energy electrons (Read 1969 a, b, 1971, Wollink 1987).

The spherical and chromatic aberration coefficients are denoted by

Cs and Cc respectively. In the present investigation the values of Cs and Cc are normalized in terms of the image side focal length, i.e. the relative values of Cs / f i and Cc / fi are investigated as figures of merit which are dimensionless.

The spherical aberration coefficient Cs and the chromatic aberration coefficient

Cc referred

to the image side are calculated from the

following equations (Szilagyi 1978).

U 1 / 2 Cs  16R 4

 5  U   2 5  U   4 14  U   3 R 3  U   2 R 2  4 Zo 4  U   24  U   3  U  R  2  U  R  U R dz   Zi

(218)

U 1 / 2 Zi  U  U   1 / 2 Cc  2   RR  R U dz 4U  R Zo 2U

(2-

19) where U  U (z) is the axial potential, the primes denote derivative with respect to

z , and U i  U ( zi ) is the potential at the image where z  zi .

It should be noted that Cs and Cc in the object plane can be expressed in a similar form of equations (2-18) and (2-19) where U 1 / 2 and R are 4

replaced by U o

1 / 2

and Ro

4

respectively. The integrations given in the

above equations are executed by means of Simpson’s rule (Hawkes 1980, Szilagyi et al 1987). In the present work, equations (2-18) and (2-19) have been used for computing Cs and

Cc in the image side under

various magnification conditions.

2.6 Two-Electrode Immersion Lenses

A two-electrode immersion lens is the simplest among the various types of electrostatic lenses where the potentials are different at both

U(z) V2

V1 a

zm

b

sides of the lens. The axial potential distribution of a two-electrode immersion lens has a very simple form shown in figure 2.4. It is a monotonous function of the axial coordinate z and asymptotically approaches the electrode potentials V1 and V2 at the two sides of the lens. The potential distributions do not contain constant potential regions and has one inflection point at z

m

where the axial component of the field

reaches its maximum absolute value. This may or may not happen at the geometrical center of the lens, therefore, one can talk about symmetric or asymmetric lenses ( Szilagyi 1988).

Figure (2.4). Potential distribution of a two-electrode

immersion lenses The focusing action of these lenses can be realized by considering the simple case of two equidiameter cylinders placed coaxial as shown in figure 2.5. If the ray enters from left to right it experiences an inward force because the gradient of the potential has a radial component directed toward the axis. In general, when a ray proceeds in the direction

of an increasing potential i.e. V1 < V2 a concave equipotential has a convergent effect.

The opposite is true, if the ray proceeds in the

direction of a decreasing potential i.e. V1 > V2. Basically, the same happens for more complicated cases too, with the only difference being that there will be more than two focusing or defocusing regions. If the field is bounded at both sides, the net effect is always focusing (Edwards 1983, Zhigarev 1975).

Figure (2.5) The action in electrostatic immersion lens

CHAPTER THREE

3.COMPUTER

PROGRAMS

3.3Computer Programs

for Determining the Axial Potential Distribution of a Two- Cylinder Electrostatic Lens Computer programs have been written in Fortran 77 language, which have been sequentially executed. The final result would be the axial potential distribution, which would be generated anywhere in the space for an immersion electrostatic lens consisting of two cylinders.

A

Pentium 3 personal computer has been used for executing these programs.

Figure 3.1 shows a flowchart of the computer program P1, which divide the two cylinders of the electrostatic lens into N rings and prints the mid-point of each ring. Figure 3.2 shows a flowchart of the computer program P2, which uses the value of the mid-point of the ith ring and the jth ring that has been computed, by a program P1. Program P2 computes the elliptic integral of first kind K (t ji) given by equation (2-11) at the strip j caused by the strip i. The modulus of the computed elliptic integral is tji ; it is determined by two loops, one for the ith ring and the other for the jth ring. When the mid-point at the ith ring equals to that at the jth ring the elliptic integral goes to infinity. In order to avoid this problem one may take the value at the mid-point of the ith ring to be larger than that at the jth ring by 0.1mm. This value of K(t ji) is used in equation (215) to compute the elements of matrix Aji with the aid of program P2. The importance of determining the matrix Aji is in finding the values of the charge density σi on each ring.

Start

Input No. of rings N,ZI(1)=Mid point of first Ith ring,ZJ(1)=Mid point of first Jth ring

I = 1 to N

Is

Yes

I=N/2

N J= 1 o to N

There is a gap G ZI (I)=G+ZI (I-1)

ZI (I+1)=ZI (I)+1

Is

J=N/2

N ZJ (J+1)=ZJo(J)+1

Yes There is a gap G ZJ (J)=G+ZJ (J-1)

Print, ZJ (J)

Print, ZI (I)

END

Figure (3.1) Flowchart for the Fortran program P1which divides two cylinders into N rings.

Start

Input N, ∆zi,R,ZI(I),ZJ(I )J)

I = 1 to N

J = 1 to N

Yes

Is ZI(I)=ZJ(J)

ZI(I)=ZI(I)+0.1 No

ZJ(J)=ZJ(J)+0.1

t= 2rc/ [4rc2+(ZI(I)-ZJ(J))2]1/2 H= 1-t2 K= ao+a1H+a2H2+a3H3+a4H4+(bo+ b1H+b2H2+b3H3+b4H4) ln(1/H)

A(I,J)=(t*∆zi*K)/ πεo

Print A(I,J)

END

Figure(3.2) Flowchart for the Fortran program P2 to evaluate the elliptic integral and the element of matrix Aji for all rings.

Figure 3.3 shows a flowchart of the computer program P3 that is based on LU-Factorization method. The inverse of matrix Aji is determined with the aid of this program. The inverted matrix A ji-1 is used to compute the charge density on the ring i by applying different voltages on the two electrodes. Figure 3.4 shows a flowchart of the computer program P4 which determines the charge density by simply multiplying the matrix Aji-1 by the vector V(j). The axial potential distribution is the only unknown. Thus to find this distribution anywhere in the space one should evaluate the elements of matrix C (i,j) using the formula given in equation (2-15) by putting any point in the space instead of the value of mid-point of the jth ring (see figure 3.5).

Figure 3.6 illustrates a flowchart for the Fortran program P6 for the evaluation of the axial potential distribution U(I). It is the product of multiplying the new matrix C(i,j) by the vector of charge density σ(j) for each point within the gap separating the two electrodes.

Start

Input A(I,J) BAS= A (1,1)

J = 2 to N

A(1,J)=A(1,J)/BAS

K=2 to N

J= K to N Sum = A (J, K)

I = 1 to K1 Sum =Sum- A (J, I)*A(I,K)

A (J,K)=Sum BAS=A (K,K)

0

Figure (3.3) Flowchart for the Fortran program P3 to find the inversion of matrix Aji using LU-Factorization method.

0

J= K+1 to N Sum =A(K,J) I= 1 to K-1 Sum = Sum-A(K,I)*A(I,J)

A(K,J)=Sum/BAS

K=N to 2 Step -1

J= K-1 to 1 Step -1

Sum = -A(J,K)

I=J+1 to K-1 Sum =Sum-A(J,I)*A(I,K)

A(J,K)=Sum

K=1 to N-1 A(K,K)=1/A(K,K)

J=K+1 to N Sum=0.0 1

1

I= K to J-1 Sum =Sum-A(J,I)*A(I,K)

A(J,K)=Sum/A(J,J) A(N,N)=1/A(N,N) K= 1 to N J= 1 to N-1

Yes

Is

J>K

No LBAS=K

Sum=0.0

I=LBAS to N Sum = Sum + A(J,I)*A(I,K) A(J,K)=Sum Print A(J,K) END

LBAS=J+1

Sum =A(J,K)

Start

Input A(J,K),N

J= 1 to N Sum=0.0

K= 1 to N

Is

J