SPHERICAL ABERRATION
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS DR. IR. C. J. D. M. VERHAGEN, HOOGLERAAR IN DE AFDELING DER TECHNISCHE NATUURKUNDE, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN
OP WOENSDAG 5 NOVEMBER 1969 TE 14.00 UUR
DOOR
NICOLAAS HENRICUS DEKKERS
natuurkundig ingenieur geboren te Dordrecdt
Noyt heeft 'et iemant hier soo klaer| Of 't hapert loch al hier of daer.
(Cats)
Aan allen die hebben bijgedragen tot het tot stand komen van dit proefschrift.
CONTENTS.
TABLE OF CONTENTS 5
INTRODUCTION 7
1. MAGNETIC INDUCTION AITO VECTOR POTENTIAL 9
2. EQUATIONS OF MOTION AND RAY EQUATION OF AN
ELECTRON IN A MAGNETIC FIELD IT
3. THE SPHERICAL-ABERRATION CONSTANT C 22 s
3«1» Derivation of the general equation 22 3.2. Spherical aberration in thin lenses 25 It. THE POSSIBILITY OF CORRECTION OF SPHERICAL
ABERRATION 2? lt.1. General proof 27
it,2. Correction of weak lenses 29
5. A SUPERCONDUCTOR AS CORRECTOR OF SPHERICAL
ABERRATION . SU 5.1. General considerations 3^*
5.2. Fields and currents 37 5.2.1. Field of a current-carrying disk 37
5.2.2. Currents induced in a superconducting disk 1*0
5.2.2.1, Total trapping of the flux UO 5.2.2.2. Incomplete trapping. Computer program 1*7
5.3. Estimate of the current density necessary
6 . A SUPERCOITOUCTING ZONE PLATE AS A CORRECTOR
OF SPHERICAL ABERRATION 53 6.1. Scattering by a correcting film 53
6.2. A superconducting zone plate 53
T. ALIGiniENT ERRORS 60 7.1. General considerations 60
7.2. Errors due to a shift of the corrector 60 7.3. Errors due to tilt of the corrector 62
8. ADMISSIBLE HEAT FLOW TO THE CORRECTOR 6h
9. CORRECTION BY MEAITS OF AXIAL CURRENTS 6 T
10. THE EXPERIMENTS 72 1 0 . 1 . The p r e p a r a t i o n of t h e c o r r e c t o r 72
1 0 . 2 . The c r y o s t a t l6 1 0 . 3 . Arrangement f o r c o r r e c t i o n 82
1 1 . CONCLUSIONS AND POSSIBLE APPLICATIONS 8 9
SUI.ffilARY 91
SAI'IENVATTING 9h
CONCISE LIST OF SYMBOLS 97
INTRODUCTION.
Similar to optical lenses, electron lenses show aberrations. In electron microscopes it is the spherical aberration that ultimately limits the resolving power.
Unfortunately it is impossible to correct the spherical ab-erration in rotationally symmetrical lenses by simply chang-ing the shape of the pole pieces producchang-ing the focuschang-ing field whether this is an electrostatic or a magnetic field.
This was first proved by Scherzer in 1936'.
However, in these rotationally symmetrical lenses the spher-ical aberration can be corrected by introducing currents or charges within the imaging beam.
Several efforts have been made to bring charges inside the electron beam.
Later Scherzer^ and independently Le Poole found that the ef-fects of the holes in gauzes coxild be greatly reduced by min-imizing the total charge. Experiments were carried out in this laboratory by Rus^ and Earth**.
Haufe^ tried to produce a correctional space charge by in-jecting electrons from aside in electrostatic lenses.
The use of currents is included in the suggestion of Marton^ to put a superconducting ellipsoid on the axis of a lens to obtain zonal correction.
Independently Le Poole and the author^ got the idea to intro-duce a superconducting thin film inside a lens, perpendicular to the axis, in which persistent currents could be induced. In the first chapters of this thesis the theoretical back-ground of this idea and the theoretical possibilities of its application in electron microscopes will be considered.
It is shown that a correcting superconducting ring can give positive phase contrast for object details of dimensions between 2.3 and 2.9 A and between 3.6 and 8.U A .
In chapter 7 the effect of tilt of this corrector is inves-tigated and it is proved that a very accurate positioning
is necessary.
Chapter 10 contains a description of the equipment used to prove the correction experimentally.
A second idea of currents inside the electron rays consist-ed of an electron beam along the axis of a conventional lens directed opposite to the imaging electrons. A simple calcu-lation showed that a correcting current of 80 mA would be more than sufficient. Le Poole pointed out that the electro-static effect of this current would be larger by a factor of the order of c/v , so that a much smaller current would be sufficient. Already in his first experiments he reduced the spherical aberration by a factor of 5 by injecting the e-lectrons in the proper way into the lens.
In chapter 9 some attention will be given to this correction by space charges.
1, MAGNETIC INDUCTION AND VECTOR POTENTIAL.
In this first chapter a series expansion of the magnetic vector potential and magnetic induction are derived. They differ from the well-known series expansions as fields including electric currents will be considered. This is done, because in this thesis a current-carrying disk placed in the field is proposed as a corrector of spherical aberration.
We will deal with magnetic fields only and as normally time-independent fields are used, the Maxwell equations can be simplified to
rot H = J (1.1) div B = £ (1.2)
in which II is the magnetic field strength, B the mag-netic induction and J the current density, which causes the magnetic field. The connection between H and B in vacuo is given by
B = po H (1.3)
\iQ being the permeability of the vacuum. Equation 1.2 can be solved by posing
B = rot A (1,1*)
by this equation, except for the gradient of a scalar func-tion X because rot grad x = 0 .
Combination of equations 1.1, 1.3 and 1.1+ gives
rot rot A = y 0 £ (1.5)
from which A can be solved.
In order to describe the problems involved more concretely a cylindrically symmetric coordinate system, with coordinates
r, the radial coordinate (fi, the tangential coordinate z, the axial coordinate
is chosen.
In this system equation 1.5 splits into the following three equations:
r-direction:
, S^rA, , ^^A S^A S^A _1_ r _1_ <^ J_ r r ^ z i_ r r " T T " O J O * . 3(t.9r 3(||2 8z2 3 r 8 z (1.6a) - t z - d i r o c t i o n : d^A a ^ r A , . , 3rA^ 3A ^ 3(t)3z 3z2 3 r 3(t) ^ ( 1 . 6 b )
z-direction:
1 . 3A 3A - , 3A 3A^
1 ( d^ I r Z 1 d r 1 Z (p 1 •) .f
7 ' - T ? l ' " • 9 ^ " ' ' " 3 ^ ' " " 3 ? l 7 " 3 r " ' 9 r " ' ^ " ^° z
( 1 . 6 c )
Two assumptions will be made now:
a. All quantities are independent of (|), because of the ro-tational symmetry.
b. Only the component J^ of the current density exists, which means that only fields of concentric circular cur-rents in planes perpendicular to their axes will be con-sidered.
Thus
J^ = J, = 0 (1.7) r z
With t h e s e two assumptions 1.7 t h e e q u a t i o n s 1.6 a r e s i m p l i -f i e d t o 32A a^A r - d i r e c t i o n : - — — + r—:r- = 0 ( l . S a ) 3 z2 3^3== 32A g ^ 3rA (fi-direction: - •'• - -r— I -* ^' '' } = VnJ, ( l . 8 b ) 2 3 r ^ r 3 r ' ^ <i> a Z 1 3 ^ S ^^z z-direction: - - . { r ^ - r ^ - - } = 0 (1.8c) From the equations 1.8 it is seen that A and A are
in-^ r z dependent of J and vill not change either by changing J
which implies that A and A have to be zero, because
r z ' they do not exist when J is zero.
So
A = A = 0 (1.9) r z
For simplicities sake A and J will be written as A and J. From equations 1.9 and 1.1* it follows that B is zero too.
So
B, = 0 (1.10) 9
To solve A the (Ji-component of equation 1,8 is now written in the following form:
1 32rA ^ 1 32rA J_ ^rA _ j ,^ ^^^
^ 3 z2 ^ 3 r2 " r2 ^^ ' " ""^
This equation will be solved by inserting for A the series expansion A = j; r" f (z) n = 0,1,2, (1.12) n and for PQJ PQJ = I r" g^ (z) n = 0,1,2 (1.13) n
Then the recurrence relation f "(z) + s (z) ""^^ 1 - (n+2)2
is found, in which " denotes a second derivative. As both A and J are zero for r equal zero, the series expansion for A becomes
A = r fi(z) - §^i"(z) + - ^ fi""(z) + ....
r3 r^ , ei"(^)
- -^ 6i(z) + -^ { — g g3(z) I + ••• (1.15)
the coefficients of the terms with f being those of the Bessel function Ji(r).
The function fi(z) and its derivatives can be expressed in the induction on the axis B (o,z) , which will be de-noted simply by B. By calculating the induction from equa-tion 1.1*, which is in this coordinate system
B
=1^
, B = - M (1.16)
z r 3r ' r 3z
we f i n d t h a t B = 2 f i ( z ) , which t u r n s e q u a t i o n 1.15 i n t o
r ^ r^ .„„ . r^ A ( r , z ) = i B - i ^ B" + ^ B"" + r 3 r 5 S l " ( z ) - -g- g i ( z ) + gij' {-^^ S 3 ( z ) } + . . . . (1.1T) and t h u s f o r B and B we f i n d r e s p e c t i v e l y z r _ 2 „*• n = n - i l - -R" + i - •R"" + and r 2 r** S l " ( z ) - - ^ S l ( z ) + •^- { —g g 3 ( z ) } + . . . . (1.18) B_ = - -^ B' + I ^ B ' " - - £ r B ' " " + . , , ,
T^^ -I5ir
3 5 g ' " ( z )+ f - 6 i ' ( z ) -fir {-h5 g3'(^)}+
( 1 . 1 9 )These formulae differ from the well-known series expansions for B and B in as much as they contain the functions
z r "^ g(z) and their derivatives,
Although i t may seem t h a t t h e c u r r e n t s only a f f e c t t h e h i g h e r - o r d e r t e r m s of B and B , i t must be remembered t h a t t h e c u r r e n t s change t h e i n d u c t i o n on t h e a x i s t o o and t h u s t h e terais c o n t a i n i n g B,
In the next chapters the influence of these currents on the derivation of the paraxial ray equation and of the third-or-der aberrations will be investigated.
Concluding this chapter an expression for the magnetic flux * will be derived.
* is defined by
$ = ƒ B . dS (1.20) S "
in which S denotes a surface. Choosing for S a circular disk perpendicular to the axis, having a radius r , and using equation 1.1* and Stokes' theorem, equation 1.20 can be changed into 2TT $ = ƒ A r d<() 0 which means $ = 2 Ti r A (1.21)
This equation is useful because, when $ is known, it can provide boundary conditions for rA to solve the current-free case of equation 1.11.
If for instance rA is known on the disk to be used for correction of spherical aberration, the field around this disk can be calciilated.
For a much quicker and more generally useful solution a com-puter program was made. In chapter 5 this program will be
2. EQUATIONS OF MOTION AND RAY EQUATION OF AN ELECTRON IN A MAGNETIC FIELD.
To find the equations of motion of an electron moving in a magnetic field, in which a current-carrying disk is placed, we start with the equations of motion of Lagrange:
dt 3qj^ 3qj,
in which L denotes the function of Lagrange and q a generalized coordinate. In our description a cylindrically symmetric coordinate system is used, so that q is
succes-je sively r , ((> and z.
In the case of a magnetic field only, the Lagrangian is given byS
L = ^ m (r2 + rH^ + i.^) - e r A ^ (2.2)
in which A and A are supposed to be zero, and in which A denotes again A,, -e is the electron charge being -I6x10~2° C , m the electron mass being 9.11x10"^' kg. Substitution of equation 2.2 in equation 2.1 gives for
\ = ^ for qj^ m r - m r | 2 + e $ | ^ = 0 (2.3) OT •Ir (m r2 $ - e r A) = 0 (2,1*) at
for q, = z ^k
m z + e ^ 1 ^ = 0 (2.5)
dZ
Equation 2.1* can be integrated separately and yields
m r 2 $ - e r A = m ro2 ^Q - e rQ AQ ( 2 . 6 )
The r i g h t h a n d p a r t of t h i s e q u a t i o n i s zero i f t h e . e l e c -t r o n s e i -t h e r i n -t e r s e c -t -t h e a x i s somewhere ( r o = O) o r run i n a m e r i d i o n a l p l a n e b e f o r e e n t e r i n g t h e m a g n e t i c -f i e l d r e g i o n (JIQ = 0 , AQ = O). Both c o n d i t i o n s c l e a r l y have t h e same p h y s i c a l meaning. ^ can t h e n be s o l v e d
from e q u a t i o n 2.1* as , _ £ A
m r (2.7)
The conditions can be proved^ to be no real restriction, as each sagittal ray can be composed by a linear combina-tion of two meridional solucombina-tions.
Substitution of equation 2.7 makes it possible to simplify equation 2,3 to / e A xo , e A ^ 3rA m r = m r ( )'^ - e { ) -— m r m r 3r e2 „ 3A - m ^ ~ or
(f)2Af (2.8)
and i n t h e same way e q u a t i o n 2.5 t o e A N 3rA o r m z = - e [ — — I ^ ra r 3z
^ = - ( I )'
A H(2.9)
m dzStriking is the resemblance between equation 2.8 and 2,9» The advantage of these equations is that they are compact, although they do not contain any approximation.
As it is more useful to know r as a fimction of z in-stead of t , we pass from r = r(t) and z = z(t) to the system r = r { z(t) } and z = z(t).
For this transition the following equations hold:
^=1
(2.10a)
dz z
.. dr ..
^.LIJLI (2.10b)
dz2 (i)2
Furthermore the kinetic energy of the electrons equals the product of their charge and the potential V related to the cathode:
i m ( P + r2|2 + ^2) = eV (2.11)
From the last six equations the five derivatives to t can be eliminated resulting in the next equation in more general form derived by several authors " :
dr 2
dz 2 2 — V - A 2 ^^ "^^ ^'^ e
We like to call attention to this equation because it is rather simple and still contains no approximations at all, for which reason it can be used in computer programs to find r as a function of z ,
Moreover, it is valid in fields both without and with space currents,
To find the paraxial ray equation the first term of the series expansion 1,17 of A must be inserted into the equa-tions 2,6 and 2,12, yielding
M f (2,13)
and
d2r eB2r r^ •,) \
7 7 = --s;r (2,11*)
These are the paraxial ray equations for electrons in magnet-ic f i e l d s . To find r as a function of z equation 2.ll* should be integrated from the object t o the image plane. This integration can be applied as B is continuous in s p i t e of the c u r r e n t s , as w i l l be shown in chapter 5| and thus the only effect of the currents on the f i r s t - o r d e r properties of the lens i s , t h a t they change the magnitude of B somewhat
and thus cause a small change of the focal distance of the l e n s . The effect of the currents on the higher-order terms i s quite d i f f e r e n t , however,
After s u b s t i t u t i n g equation 1,17 in 2.12 and neglecting terms in r with order higher than the t h i r d one, we find
d2r eB2r , e r 2 i-n^t j . dz'^
+ i r^B { B" + 2gi(z) } - - ^ r3B'* - rr'2B2} (2.15) This equation differs from t h a t of the current-free case by t h e term containing g i ( z ) . As g i ( z ) i s the only
coeffi-cient of the current appearing in the t h i r d - o r d e r equation, t h e t h i r d - o r d e r aberrations are mainly affected by current densities proportional t o r . In deriving the t h i r d - o r d e r aberrations an integration from object t o image plane i s applied t o expressions containing the t h i r d - o r d e r terms of equation 2.15. We have t o be careful in performing these i n -t e g r a -t i o n s , as d i s c o n -t i n u i -t i e s due -t o -the curren-t-carrying
disk are present in the t h i r d - o r d e r terms. As w i l l be derived in chapter 5, B and B' caused by the currents are contin-uous but B" i s discontincontin-uous at both surfaces of the cur-rent-carrying disk. Furthermore gi(z) i s discontinuous at
these surfaces t o o . The influence of these d i s c o n t i n u i t i e s
on the value of the spherical-aberration constant w i l l be investigated in chapter 1*. In chapter 3 the general i n t e g r a l expression for the spherical-aberration constant w i l l be
3. THE SPHERICAL-ABERRATION CONSTANT C .
3.1. DERIVATION OF THE GENERAL EQUATION.
Whilst in the next chapter the influence of a current-carry-ing disk on the spherical aberration will be investigated, we will first derive the integral expression of the spherical-aberration constant C ^^ in a form in which partial inte-grations have not yet been applied. As mentioned earlier in chapter 2 discontinuities in the integrand caused by the current-carrying disk require special attention. These closer investigations will be described in chapter h.
VJe will consider the third-order tenns as a perturba-tion on the paraxial equa-tion 2.ll*.
This equation being linear, every solution of it can be written as a linear combi-nation of two independent solutions and
Fig 3.1. Two independent solutions of the paraxial ray equation.
a Y which are usually fixed by
(see fig 3.1): r = 0 ao r' = 1 ao and (3.1) r = 1 YO r = 0
r = a r + Y r (3.2) a Y
in which a and Y are constants independent of z. To find the solution of the third-order equation 2.15, which is written as
ê - - f v f * = ( ^ l (3.3)
i n which
£(z) = j~Y { r 2 r ' B B' + i r^B { B" + 2 g i ( z ) } +
- ^ - | ^ r 3 B ' ^ - r r'2B2 } (3.1*)
the constants a and Y ai"e now considered to be dependent of z . In a conventional way ^2 the total solution is found: r = a r + Y r + ; r x (3.5) a Y r 'r - r r ' o Y 01 Y i i X { r ƒ r e(z) dz - r ƒ r E ( Z ) dz } o o
in which t h e i n t e g r a t i o n s must be performed from t h e ob-j e c t plane o t o the image plane i .
D i f f e r e n t i a t i n g r ' r - r r ' and using t h e fact t h a t r
and r are solutions of the paraxial equation we can easily prove that r 'r - r r ' is a constant and thus
a Y 01 Y
equal to its value in the object plane where it is unity. The disk of confusion in the image plane of an axial point in the object plane caused by spherical aberration is found for Y = 0.
Then a is the angle at the object side of an actual ray leaving the object. As r is zero in the image plane the radius r of the disk of confusion is given by
r^ = - r / r^e(z) dz (3.6) In literature r is expressed mostly in the constant
C ( M ) , which is a function of M , and in a as s
r^ = M C^(M) a3 (3.7)
s s
corresponding to a blurr in the object plane of C (M) a^,
Comparison of 3.6 and 3.7 shows, that s
1 r C (M) = 1 r e(z) dz (3.8) ^ Ma3 J " o As in our case and r = a r^ (3,9) r^= M (3.10)
i C (M) = - ^ j { r ^r •BB'+ i r '•B{B"+ 2 g i ( z ) } + s 8mV ' '• a a a •' o --rrrr r ' * B ' * - r 2 r ' 2 B 2 | dz omV a a a ' ( 3 . 1 1 ) where i n l i t e r a t u r e only C i s s e e n , always C («>) i s meant, t h u s p e r d e f i n i t i o n
C = C ( = 0 )
s s
This C is the important constant in electron microscopes, because there the magnification of the objective is
generally large.
3.2. SPHERICAL ABERRATION IN THIN LENSES.
For thin lenses the angle of deflection Y can be expressed easily in C .
If the object distance is denoted by D and the image distance by Z their connection is given by
l4 = 7 (3.12)
and Z = M D (3.13)or
D=ii-J^f
Mf being the focal distance.
For small angles the angle of deflection Y being the sum of a and a/M is
Y = ^ ~ - ^ a (3.11*) and the extra deflection angle A Y « caused by spherical
aberration is
:i C (M) a^
AY = — h (3.15)
Thus the total angle of deflection Y is given by
Y = ^ ^ - ^ a + 1 — a3 (3.l6) u il ^ or using a = § (3.17) we find Y = 7 + ( M 4 T ) ' C (II) ^ (3.18) U I - 1 + 1 S 4
For thin lenses it is a good approximation to consider the extra deflection A Y as independent of M , so for thin lenses only
( i f f r ) 0^00 = C3(»)
As C («=) was defined as C , the angle of deflection s s Y can be written as
u
Y = 7 + C^ - (3.19)
1*. THE POSSIBILITY OF CORRECTION OF SPHERICAL ABERRATION.
1*.1. GENERAL PROOF.
In this paragraph we will perform a partial integration on equation 3.11 for C . As mentioned already in chapter 3 we must give attention to discontinuities in the integrand
caused by the current-carrying disk.
As will be derived in chapter 5» currents can cause a dis-continuity in B " . Figure 5.1+ gives the part of the induction and its first and second derivative caused by these currents running in a disk of thickness d placed at z = 0. The first derivative has a break at both sides of the disk, causing a discontinuity in B".
However, as B' is continuous a partial integration of i r "^B B " is peimissible and thus the integral over this teim from object to image appearing in 3.11:
C (M) =-ltv] { r 'r 'BB'^ IryiB"^ 2gi(z)} + o e ^ 4j34_ ^ 2^ .2B2 I az ömV a a a ' is given by 1 1 I ƒ B B"r ** dz = - i ƒ ( B'2r "*+ 1* BB'r ^r • ) dz ° ° (1*.1)
with which C ( M ) is changed into i
C (M) = Trr~T ƒ {vB'r 2+ Br r •)2+ r ^r '^B^+ -rrTr r '*B'*} dz + s 16mV ^ ^ a a a a a 4niV a '
o
The f i r s t i n t e g r a l i s the well-known i n t e g r a l f i r s t derived by Scherzer ^, which shows t h a t normally the spherical
aberration i s positive by lack of the second i n t e g r a l . The induction B and i t s derivative B' in the f i r s t i n t e g r a l consist of two p a r t s , one from the main f i e l d B and B ' and one of the correcting currents b and b ' .
m
I f the current-carrying disk i s small the fields b and b ' w i l l differ from zero only in a small region compared t o the
region of B and B ' and cause l i t t l e change of the f i r s t ° m m i n t e g r a l ; thus t h i s i n t e g r a l can be denoted by C (M) , the spherical-aberration constant i f no correction i s applied, and 1*.2 i s written as
C^(M) = C ^ ( M ) ^ - ^ } r^'*Bgi(z) dz (1*.3) o
The i n t e g r a l of 1*.3 i s supposed to cause a decrease of C ( M ) . s To evaluate t h i s i n t e g r a l the current-carrying disk w i l l be
considered t o be so t h i n t h a t r and B can be taken con-a
stant in the disk. Thus the i n t e g r a l is
•j»S- r •* B ƒ g (z) dz (l*.l*)
8mV ac c ^ **!
the index c denoting the corrector plane. With 3,9 and 1*,1* equation 1*,3 i s written as
r '^
C ( M ) = C (M) - Tj—, - £ _ B ƒ g ( z ) dz
s s u 8mV L c ^ ° i
a ^ '•
(1*.5) This equation indicates that by inducing the proper
c u r r e n t d e n s i t y c o r r e c t i o n can be a c h i e v e d .
The way t o do t h i s w i l l be e x p l a i n e d i n c h a p t e r 5. We w i l l f i r s t g i v e a simple proof of t h e p o s s i b i l i t y of c o r r e c -t i o n of s p h e r i c a l a b e r r a -t i o n i n -t h e nex-t p a r a g r a p h .
1*.2. CORRECTION OF VffiAK LENSES.
For a weak l e n s a simple proof of t h e p o s s i b i l i t y of c o r -r e c t i o n of s p h e -r i c a l a b e -r -r a t i o n can be g i v e n .
I n a weak l e n s r i s supposed t o be c o n s t a n t i n t h e r e g i o n of t h e f i e l d . Only r ' i s supposed t o change some-what i n t h e l e n s . See f i g u r e 1*.1. The s l o p e of t h e ray at t h e o b j e c t s i d e i s r ' and at t h e image s i d e r . ' . As t h e a n g l e of d e f l e c t i o n Y i s s m a l l , i t can be c a l c u -l a t e d from Y = a r c t g r ' - a r c t g r . ' = - ƒ r" dz o (l*.6)
i n which r" i s given by t h e g e n e r a l ray e q u a t i o n i n A 2 . 1 2 .
object image
To get the deflection of the main field and the correcting currents, we will split A and B into two parts: one caused by the main field only (A and B ) and one caused
"^ '^ m m by the corrector (a and b ) , thus
and A = A + a (1*.7) m B = B + b (1+.8) m S u b s t i t u t i o n o f 1*.7 i n t h e ray e q u a t i o n 2 . 1 2 g i v e s d r 2 ^2 ^^ * ^ d l ^ ^ 3(A + a ) 3(A + a ) 1^= (;A + a) 2 r'(A + a) 2 ' dz2 2 ^ - (A + a ) 2 ' ^ 3r "" 8z e n (1*.9) i n w h i c h a i n t h e d e n o m i n a t o r can b e n e g l e c t e d , a s e v e n 9 . m A ^ IS u s u a l l y s m a l l c o m p a r e d t o 2 —V .
Then e q u a t i o n U.9 c a n e a s i l y b e s p l i t i n t o two p a r t s , o n e c o n t a i n i n g o n l y t e r r a s o f t h e m a i n f i e l d a n d t h e o t h e r one h a v i n g t e r m s w i t h a t o o : d ^ r dz 1 + (4^)2 3A 3A {A - ^ - r' A - ^ } + dz2 2 ^ V - A 2 ' "1 3 r '^ 3z e m
' '
^S^'
U 3a ^
{A T— + a + a -r-- +'\^ 3a ,
„ m . 2 m 3 r . 3 r 2 —V - A "^ 3 r (1+.10) e m ^ 3A . I/A ° a . m , oa \1 - r ' ( A - r - + a + a - r - ) | m dz . dz ' dZThe i n t e g r a l over t h e f i r s t p a r t i s t h e normal d e f l e c t i o n w i t h h i g h e r - o r d e r terms i n t h e u n c o r r e c t e d c a s e , t h e second p a r t g i v e s t h e e x t r a d e f l e c t i o n angle AY caused by t h e c o r r e c t o r . To f i n d t h e l o w e s t - o r d e r term of t h i s AY c o m p a r a t i v e l y small terms a r e n e g l e c t e d ; so t h e c o e f f i c i e n t
Q
can be reduced t o - - r - r and t h e terms w i t h r ' and 3a ^ a •—— between t h e a c c o l a d e s can be n e g l e c t e d as t h e y a r e s m a l l w i t h r e s p e c t t o t h e o t h e r t e r m s , ^^^^ i , 3A o 3r By i n s e r t i n g f o r A i t s f i r s t - o r d e r expansion on t h e a x i s given by 1.17 as s^'B , t h e i n t e g r a l of 1*.11 i s reduced t o
] IB ^&z
J m 9r o which i s u s i n g 1.16 i ƒ i B r b dz (1*.12) ' m oAlthough B is the magnetic induction of the main field on the axis, according to this derivation b is the induc-tion caused by the corrector along the real electron path. This will turn out to be essential for the evaluation of the integral 1*.12. In working out this integral r is considered to be equal to r and B equal to B in
c m ^ c t h e r e g i o n where b d i f f e r s from z e r o . Thus B and r
can be put i n f r o n t of t h e i n t e g r a l and only i
ƒ b dz (1*.13) o
has to be solved.
Fig It,2. Two integration paths; p^ along a real trajectory, p„ in the field free region.
Therefore picture 1+.2 i s drawn in which two possible i n t e g r a t i o n paths from object t o image are given. Paths p and p are the integration paths respectively along
1 2
a r e a l t r a j e c t o r y and in the field-free region.
I denotes the current in the corrector within a c i r c l e r with radius r » "t the t o t a l current in the corrector. Making a contour i n t e g r a l from object t o image plane
along path p and back over path p
1 2 ƒ b dz = PO ( I^ - I^)
i t i s seen t h a t (U.ll*)
And s o , f i n a l l y , equation 1*.9 becomes
^y - V^ I" B pn ( I. - I ) 4mV c c " t r (1^.15)
Although I i s constant, I i s a function of r and i f I is proportional t o r2 , according t o a current density proportional t o r , a t h i r d - o r d e r term i s in-troduced whose magnitude and sign can be adjusted t o
create the desired change of the spherical aberration, I t i s also seen from equation 1*,15 t h a t paraxial rays
get the largest deflection and thus in case of correction all rays are converged to the focus of the marginal rays, Combination of 1*.15 with 3.19 shows that in case of cor-rection the relation between C and I is given by
s J, o J
C = ^ f"* B ^^^-^ (1*.16)
s '*mV c o
which is in complete agreement with the general equation 1^.5.
5. A SUPERCONDUCTOR AS CORRECTOR OF SPHERICAL ABERRATION.
5.1. GENERAL CONSIDERATIONS.
In the previous chapters i t has been demonstrated t h a t a current-carrying disk perpendicular t o the z-axis can correct the spherical aberration.
This current-carrying disk should have a current density proportional t o r and thus a current proportional t o r 2 . The problem i s , how such a current-carrying disk can be made. If a normal r e s i s t i v e metal is used t o malie those c i r c u l a r currents around the a x i s , these currents must be fed in from outside. This v/ould render i t d i f f i c u l t t o make I continuous and proportional t o r2 . Moreover the win-dings would stop the imaging electrons.
Furthermore i t i s doubtful whether sufficient r o t a t i o n a l symmetry can be obtained.
To overcome the problem of feeding in the currents from outside the use of a superconducting film i s proposed. A superconductor could t r a p or shield an amount of flvix by means of p e r s i s t e n t c u r r e n t s . To t r a p the right amount of flux the following procedure can be followed.
A. Suppose (see fig 5.1) one has a homogeneous f i e l d of strength H in which a film of superconductive material i s placed. The f i e l d strength i s lower than the upper c r i t i c a l f i e l d H of the material of the
film. Above H no superconductivity would be pos-s i b l e at a l l .
diskofsupercpnducting I nr^oteriQl T > T c H < He superconducting disk T<^Tc H < He T < Tc H < He
Fig 5.1, 2 and 3; 5.1. Film of superconductive udterial perpendicular to a magnetie field; 5.2. After lowering the temperature below T . Flux
c freezing in. 5.3. After changing the field. Flux trapping.
The temperature T is higher than the critical perature T of the film considered. Below this tem-perature the film will be superconducting. Above this temperature, however, the field penetrates the material completely apart from a small diamagnetic or paramag-netic behaviour of the normal material of the film, which will be neglected.
B. If now (see fig 5.2) the temperature is lowered below T , the film will split into very small regions of superconducting and normal material.
By the large demagnetization coefficient of a thin film'^ ^, a film of type-I superconducting material will not be able to expel the flux as it would normally do ^"^j and it will come into the intermediate state,
If a film of type-I superconductor is thinner than a certain value, depending on the material, it will behave like a type-II superconductor. It will then not
expel the flux, but concentrate it to so called super-current vortices each containing a unit of magnetic flux,
Thus the field will never be expelled when the film is thin and perpendicular to the field. For a more extensive and detailed theory of superconductivity see
literature ^^.
C. A change of the field strength (see fig 5.3) will now induce persistent circular currents in the film keeping the flux through the film constant. As there is inter-action between the flux and the currents the flux must be pinned in the superconductor. A pure superconductor cannot pin the flux, but dislocations, crista! boundaries, holes and impurities can provide this pinning.
A superconductor which pins the flux is called hard, Any soft superconductor can be made more or less hard ^^ and thus be used as corrector of spherical aberration. Most useful as correctors are very hard superconductors with a high upper critical field e.g. Nb3Sn or NbTi. They have a critical temperature of '^18 K and upper critical fields of more than 10^ A/m at helium temperature.
In the next paragraphs we will derive the field due to currents proportional to r^ in a disk of finite thick-ness d . Furthermore the currents induced in a circular disk will be calculated.
5.2. FIELDS AND CURRENTS.
5.2.1. FIELD OF A CURRENT-CARRYING DISK.
As was proved in chapter 1 and 2 only the f i r s t - o r d e r term of c i r c u l a r currents running in planes perpendicular t o the axis come into the t h i r d - o r d e r terms of the ray equation. Furthermore i t was assumed t h a t these currents cause a
continuous f i e l d d i s t r i b u t i o n and continuous f i r s t derivative on the a x i s , whereas the second derivative i s discontinuous. To prove t h i s we suppose t o have a c i r c u l a r disk of radius a and thickness d placed perpendicular t o the z-axis and having i t s center at z = 0.
The current density be given by
U[jJ = gi(z) r (5,1) gl(z) is obviously zero outside the disk,
To find the induction on the axis we have to integrate a l l contributions dB caused by the current elements dl in the disk given by
PO <il = SlC?) r <ir d? (5.2) PO <il ^^
dB = (5.3) 2 { r2 + (z-i)2}3/2
t, denoting the axial position of the current plane imder
consideration.
So we find for the t o t a l induction
id a2+ 2 ( z - ü ^ g i ( Ü B = ƒ ( , _ _ 2 | z - ? | ) dc
- i d {a2+ (z-?)2}l/2 2 (5.1^)
in which the integration over r was achieved already. To find the induction in the region of the disk a i s con-sidered t o be large compared t o z and ^ , so t h a t the term between the brackets of equation 5.1* approaches a. Thus the i n t e g r a l approaches ppl/a, I being the t o t a l current in the disk. This means t h a t B i s approximately constant in the region of t h e disk.
Differentiation of equation 5.1* t o z yields
id (z-ü{3a2+ 2 ( z - ü ^ } | z - 5 | g i ( d
B'= I [ ; 2 ) dc
- i d {a2+ ( z - ü 2 } 3 / 2 z-c 2 ( 5 . 5 ) The first term between the brackets of equation 5,5 is now neglectible compared to the second term in the region a >> z, To finish the integration we will suppose an exponential decrease of the current density from the edges of the disk to its centre, or
g (z) = g exp(-id/x) cosh(z/X), -id < z < id
(5.6)
in which d denotes the thickness of the disk, x the
penetration depth and g a constant depending on the current density at the face of the disk.
By achieving the rather complicated integration of equation 5.5 B' can be shown to be approximately:
- 2 g X s i n h ( z / X ) exp(-5d/X) i n t h e d i s k , g X { 1- e x p ( - d / X ) } f o r - a << z < - i d , and - g X { 1- exp(-d/X)} f o r id < z << a. Thus B' i s continuous a t b o t h s i d e s of t h e d i s k . F u r t h e r d i f f e r e n t i a t i o n shows t h a t B" has f i n i t e d i s -c o n t i n t i i t i e s a t t h e f a -c e s . B" i s - 2 g c o s h ( z / x ) e x p ( - i d / x ) i n t h e d i s k , and 0 f o r i d < | z | << a. So t h e minimim v a l u e of B" i s - g . ,B . B' / B"
y^,---4
-^N \ ^ \ . . ' - " :Fig 5.^. Induction on the axis of a current carrying disk with first and second derivative,
The curves of B, B' and B" a r e drawn i n f i g u r e 5•l^. They c l e a r l y show t h a t t h e f i r s t d e r i v a t i v e B' i s con-t i n u o u s con-t h r o u g h con-t h e d i s k . T h e r e f o r e con-t h e p a r con-t i a l i n con-t e g r a con-t i o n a p p l i e d i n c h a p t e r 1* on t h e t e r m of e q u a t i o n 3.11
5.2.2. CURRENTS INDUCED IN A SUPERCONDUCTING DISK.
5.2.2.1. TOTAL TRAPPING OF THE FLUX.
In this paragraph we will investigate the currents induced in a hard-superconducting disk if the external field is changed and the field penetrating the superconductor in its normal state is completely trapped. This will happen if the pinning forces which keep the flux lines at fixed positions are larger than the force tending to move the electrons which maintain these vortices. This force seems to be ^^ proportional to the gradient of the induction. As a thin disk of thickness d and radius a has a demagnetization coefficient n given by
the field gradient at the edge of the disk will be large even with a relatively weak external field and thus some flux will penetrate into or leave the disk if the change of the applied field exceeds a value 1-n times that for a long rod parallel to the field ^^.
To find the external field of a disk which traps the flux completely the disk will be considered to be the border-line case of an infinitely thin oblate spheroid.
If a homogeneous field with magnetic induction Bi is frozen into a superconducting spheroid and the field is decreased to B2 , currents are induced generating a field with an induction b of magnitude
b = Bi - B2 ( 5 . 8 ) t h u s keeping t h e i n d u c t i o n at t h e s u r f a c e of t h e d i s k , b e i n g B2+b on i t s former v a l u e B^. To f i n d t h e f i e l d d i s t r i b u t i o n around t h e f l a t s p h e r o i d t h e c u r r e n t - f r e e case of e q u a t i o n 1.11 32rA 32rA I + 1 l^^=o_ (5.9) ^ 3z2 ^ 3r2 ^ ' 3 ^
must be s o l v e d . Since t h e magnetic f l u x i s given by e q u a t i o n 1.21 as <t> = 2 Ti r A t h e boundary c o n d i t i o n s f o r rA a r e z = 0 : rA = i r2Bi f o r 0 < r < a and z = ±00 : rA = i r2B5 2 r - i 5 2 ( 5 . 1 0 ) a i s t h e r a d i u s of t h e circiiLar d i s k , which i s p l a c e d at z = 0. The s o l u t i o n of e q u a t i o n 5,9 must be adapted t o t h e boundary c o n d i t i o n s 5 . 1 0 . On t h e b a s e of t h i s adapted s o l u t i o n t h e component of t h e magnetic i n d u c t i o n i n t h e r - d i r e c t i o n B can be c a l c u l a t e d along t h e d i s k from B = r o t A. This component B i s n e c e s s a r y t o c a l c u l a t e
t h e c u r r e n t i n t h e d i s k .
I t i s not e a s y , however, t o adapt t h e g e n e r a l s o l u t i o n of
equation 5«9 t o t h e boundary c o n d i t i o n s 5 . 1 0 , b e c a u s e rA i s known only i n a p a r t of a c o o r d i n a t e p l a n e (z = O). To
oblate spheroidal coordinates n , 6 and tj) .
The transformation formulae for t h i s transformation are ^^:
X = a cosh n sin 0 cos (j) y = a cosh n sin 9 sin tj) z = a sinh n cos 6
(5.11)
The coordinate surfaces are
a2 cosh2n a2 cosh2n a2 sinh2n = 1 and a2 cos29 = 1 oblate spheroids n = const. (5.12) hyperboloids of one sheet 9 = const. ( 5 . 1 3 ) 'ip_ const.
Figure 5,5 gives an impres-sion of the s p a t i a l position of the coordinate surfaces, As the system i s r o t a t i o n a l l y
Fig 5.5. Planes of constant coordinates in s y m m e t r i c a p i c t u r e Of t h e oblate spheroidal coordinate system.
intersections with a plane of (fi = constant (see fig 5,6) will be sufficient. The coordinate planes with 6 = constant are confocal hyperboloids of one sheet, the coordinate planes with n = constant are confocal ellipsoides. The disk at z = 0 is now the coordinate plane n = 0,
n = O rA = i a2 s i n 2 9 c o s h 2 n Bj ri = ±00 rA = i a2 s i n 2 0 c o s h 2 n B2
(5.1M
rA being totally known on two coordinate surfaces. In this new coordinate system the equation 5.9 or originally
(rot rot A ) ^ = 0 becomes 3-'rA 3n2 tghn 3rA . S'^rA 3n - cotg9 39' 3rA '39 (5.15) 0 (5.16)
which has a quite simple form. This equation can «.const be solved by separation
of variables. As both boundary conditions con-i tain sin29 the follow-ing solution is tried:
rA = H ( n ) s i n 2 6 ( 5 . I T )
[ »«con»t[
a n d f o r t u n a t e l y s u b s t i t u Fig 5.6. Intersection with a plane of constant • . t i o n Of e q u a t i o n 5 , 1 0 i n t o 5 . 9 c a u s e s t h e d i s a p -p e a r a n c e o f 9 from t h e e q u a t i o n a n d r e s u l t s i i l d2H . , dH o TT - r, —— - t g h n " r - - 2 H = 0 dn2 ^"^ (5.18) A g a i n on t h e b a s e o f t h e b o u n d a r y c o n d i t i o n s a s s o l u t i o n o f
equation 5.11
Hl = cosh2n (5.19)
is tried and proves to be a solution. The second solution H2 of equation 5.11 can be found from 2
H2 = Hi {A + B ƒ ^ ^ } (5.20)
Hi2
in which A and B are arbitrary constants.
From equation 5.17, 5.19 and 5.20 the solution of 5-9 is composed
rA = sin20 cosh2n{Ci+C2 ~^^^ '^ +C2 arctg(sinh n)}
cosh2n (5^21) The constants Ci and C2 are found, when rA of
equa-tion 5.21 is adapted to the boundary condiequa-tions, and appear to be
Ci = i a2 Bi
and (5.22) • C2 = 7 - (B2 - Bi)
changing finally equation 5.21 into Bi B2-B1 sinh n
rA = a2 sin^ö cosh2n{-T— + " [ + arctg(sinh n)}}
cosh^ (5,23) The magnetic induction can be found from B^ = rot A, In the
( r o t A)„= 3rA a2(cosh2n-3inh29)^/2coshri s i n 9 3n (5.21*) B = ( r o t A) = "^ "^ a2(cosh2n-sinh29)^/2j,Qgi.^f^ ^I^Q 39 ( 5 , 2 5 ) The r a d i a l component B of t h e i n d u c t i o n along t h e d i s k
can be found by c a l c u l a t i n g t h e d e r i v a t i v e of e q u a t i o n 5.2l* a f t e r s u b s t i t u t i n g t h e s o l u t i o n 5 . 2 3 , because Bg(n = 0) = B^(z = 0) = B^^ This y i e l d s B = - t g 9 (Bi - Bo) r e ÏÏ '^ ^ "^ Brc Bi-Bj " i 1 T 1 1 • 1 // '
y
-i u u ID 0.6 02 02 0 0£a ( 5 . 2 6 ) ( 5 . 2 7 ) On t h e s i d e where z i s p o s i t i v e i . e . 0 < 0 < gir , e q u a t i o n 5.27 becomes w i t h t h e t r a n s f o r m a -t i o n formulae 5.11 B = - • r e IT (B1-B2) 7177Fig 5.7, Radial induction along a superconducting disk with conplete flux trapping,Full line: analytical cal-culation, Dotted line: computer calcal-culation,
a'-r (5.28) However, on the side where z is negative i.e. iir < 9 < TT, B has the oppo-site sign.
In an analogous way t h e i n d u c t i o n along t h e z - a x i s can be c a l c u l a t e d from 5 . 2 5 . This w i l l not be done h e r e as we a r e only i n t e r e s t e d i n t h e c u r r e n t s i n t h e d i s k which can be
r=a c a l c u l a t e d from B . T h e r e -r e f o r e we make a contour i n t e -C g r a l along t h e contour -C of f i g u r e 5.8 and by t a k i n g ƒ H dc we e a s i l y f i n d f o r a z.o C
" small contour close to the
Fig 5,8. Integration path for current
calculation, s u r f a c e s of t h e d i s k
UO J d = - 2 B^^ ( 5 , 2 9 )
i n which B denotes B along t h e s i d e of t h e d i s k where r e r ^
z i s p o s i t i v e , and t h u s w i t h 5.23 l*(Bi - B2) r
Po J = " 7 = ; = ^ (•?-30) TI d / a 7 - r 2
From equation 5.29 it is clear that the full line in figure 5.7 represents the current density too. It is proportional to r for values up to about ia .
By expansion of the square root of equation 5.30 the first-order term of PQJ having the coefficient gi (cf. equa-tion 1.13) turns out to be
l*(Bi - B2)
gl = (5.31) IT d a
Thus even when no homogeneous f i e l d i s frozen i n , o r i f t h e f l u x cannot be t r a p p e d completely t h e c o n s t a n t t e r m of t h e a p p l i e d f i e l d B2 and t h e c o n s t a n t term of t h e t r a p p e d
f i e l d Bi , determine the f i r s t - o r d e r term in the current density. And as t h i s term was proved t o cause the t h i r d -order correction (chapter 2 ) , trapping of an inhomogeneous f i e l d w i l l give correction t o o .
5.2.2.2. INCOMPLETE TRAPPING. COMPUTER PROGRAM.
Before the exact analytical solution of the field around the superconducting disk vras found, a computer program was written in Algol, which provides a solution of equation 5.9
1 3'^^ , 1 ''^^ 1 3rA , ^ 3z2 r 3r2 ^'3^
by field relaxation. As equation 5.9 is not the Laplacian equation, the available programs could not be used. However programs solving the Laplacian equation would have to be changed slightly as the the Laplacian equation for rA is
'Jit,fit,1^.0
The component B of the f i e l d at the disk was calculated for the case of a t o t a l l y trapped f i e l d , which was calculated a n a l y t i c a l l y in the former paragraph. The r e s u l t s of t h i s example are represented in figure 5.T by the dotted l i n e . For r / a < 0.75 the differences between computed and t h e -o r e t i c a l value are l e s s than ^%,
Compared t o the t h e o r e t i c a l s o l u t i o n , the computed program has the advantage t h a t i t can also be used for the relaxa-t i o n of relaxa-the f i e l d in orelaxa-ther geomerelaxa-tries.
along a s u p e r c o n d u c t i n g d i s k , which does not completely t r a p t h e f l u x . As an example t h e measurements of R C A - s c i e n t i s t s on s u p e r c o n d u c t i n g NbsSn d i s k s were u s e d . In a s p e c i a l i s -sue of t h e RCA-Review 2 0 f^i^Q normal component B of a
z
f i e l d t r a p p e d i n c o m p l e t e l y by such disks i s given as a func-t i o n of func-t h e r a d i u s . This enables us func-t o give func-t h e a c func-t u a l bound-ary c o n d i t i o n s f o r rA , which i s d i r e c t l y connected t o t h e f l u x $ by e q u a t i o n 1 . 2 1 . .... I f an e x t e r n a l f i e l d " 9 * Be in i s a p p l i e d t o such a d i s k , and i s changed as shown i n f i g u r e 5.9 t h e f i e l d d i s t r i b u t i o n i n t h e d i s k w i l l b e -come an almost l i n e a r f u n c t i o n of t h e r a d i u s r . The s i t u a t i o n of an e x t e r n a l f i e l d of 0,1 V/b/m2 was fed i n t o t h e computer and t h e r e s u l t f o r B i s shown i n f i g u r e 5.10. For r up t o about O.l+a t h e curve of B does not d e v i a t e much from a s t r a i g h t l i n e .
In t h e l i t e r a t u r e quoted t h e c u r r e n t d e n s i t y i n t h e d i s k s was c a l c u l a t e d from
Fig 5.9. Induction B of hard superconducting disk in decreasing external field B ^ as measured in RCA laboratories. 3B 3B Po J = Po r ° t H = 3z 3r 3B — r << _ - z 3z 3r ( 5 . 3 2 )
, which was claimed however, w i t h t h e assumption
t o follow from t h e e x p e r i m e n t a l s e t - u p of a p p l y i n g a homo-geneous e x t e r n a l f i e l d . We b e l i e v e t h a t t h i s assumption i s wrong. From t h e f i g u r e s 5.9 and 5.10 i t can be seen t h a t
•r-^ is at l e a s t two orders of magnitude larger than —^ , as the thickness of t h e i r superconducting layers was 36 pm. I t means t h a t the current can be calculated from formula 5.29 being 2 B J = 3B, re Po <i in which -r—^ is neglected.
From figure 5.10 a current density at r = 0.6a of approxi-mately 0.75 ^ 10^" A/m2 is calculated, whilst a calculation based on -r-^ >> -r—^ would result in 10^ A/m2 , which is
3r 3z ' really far below the usual performance of these
superconduc-tors. By contrast, our current-density calculations are in striking agreement with results reported by
oth-21 ers'^ ,
The conclusion of this paragraph is that it will be practically impossible to get a current distri-Fig 5,10, Radial induction B as calculated from b u t i o n p r o p o r t i o n a l t O
asa
the data of figiire 5,9.
, but t h a t only a part of the disk should be used in which deviations from r2 are not l a r g e . The higher-order terms in that pari; of the disk w i l l be small and unable t o cause large fifth-and higher-order effects in the ray equation of electron o p t i c s .
If, f i n a l l y , the current density proportional t o r can be represented by the dotted l i n e in figure 5.10» the average value gl of gi(z) which was defined in equation 1.13 i s
If we calculate gj from equation 5.31, which was derived for the case of complete flux trapping we find 2x106 vrb/m**, So we see that the first-order term of the current density does not change much if the field is not completely trapped, From equation 5.31
l+(Bi - B2) Si =
ir d a
we see, furthermore, that gi is inversely proportional to the radius of the disk, thus gi can be enlarged by decreas-ing the radius a of the disk,
In the next paragraph we shall investigate the thickness and radius of these disks to get complete correction of spherical aberration in strong magnetic electron lenses,
5,3 ESTIMATE OF THE CURRENT DE^ISITY NECESSARY FOR CORRECTION.
In this paragraph the current necessary for correction of spherical aberration will be calculated. As was found earlier (cf. equation 1*.5)
&'&l--^Y-T-^ C^(M)^ (5.33)
c c
in case of complete correction, gi is the average value of gl(z). ¥e shall calculate the required product d'gi for strong lenses with large magnification.
To do this the following assumptions will be made.
1. The corrector is supposed to be situated close to the cen-tre of the field. Thus B is the maximum value of the
in-c
2. As the magnification is large, C (M) = C , ' s u s
3. 8mV/e B 2 = f2^ a rotigh approximation, which holds exact-ly for a homogeneous field. The focal length will be as-sumed to be 1,6 mm which is the value for the Philips E M 300 electron-microscope objective lens,
1+, C s f , which holds approximately for strong lenses, s
5, a = r/f, in which r denotes the radius of the aperture, 6, As the corrector is placed in the centre of the field
r n I ^/2 r, which again applies strictly only to a homo-geneous field,
With these assumptions we find
d.gi = l*.Gxlo3 Vs/m3
and as gj f o r t h e d i s k w i t h r a d i u s 1,25xl0~2m was found t o be about 2,2xlo^Wb/m'*, t h e t h i c k n e s s r e q u i r e d amounts t o
d s 2x10"3 m '
This thickness is much too large for use in an electron mi-croscope, It is likely, however, that the disk can be scaled down. The amount of flux, which can be trapped will become smaller but the current density will increase from zero in the centre to its maximum with a much larger gradient gi/po However, the minimum radius of a is r a 0.7r and thus if
• c a is reduced from 1.25xlO~2m to 0.7r the thickness of
2x10~^m can be reduced to 0.12r.
The aperture radius r depends on the required resolving power. If a radius of 25 pm would be used, corresponding to
a resolved object of 2 5 the thickness of the superconductor has to be about 3 pm.
The result can probably be somewhat more favourable, as for very thin disks (2.5 ym) a relatively large shielded field was reported22j i.e. more than twice the value expected.
Evidently geometrical effects play a large role and only further measurements can give a definite answer to what will really be the minimum thickness for some special geometry.
6. A SUPERCONDUCTING ZONE PLATE AS A CORRECTOR OF SPHERICAL ABERRATION.
6.1. SCATTERING BY A CORRECTING FIU-I.
In the previous chapter it was pointed out that for correc-tion of spherical aberracorrec-tion in a strong field a supercon-ducting film of Nb3Sn, which together with NbTi is probably the best material up till now for a superconducting correc-tor, should have a thickness of at least 3x10 ^m.
As can easily be verified in the literature 23^ g, film of this thickness is practically opaque to electrons. It means that a superconducting film cannot be used as corrector of spherical aberration and the striving for a continuous cur-rent distribution in the corrector must be given up.
If a gauze or zone plate is used as a corrector there may be enough transmission of electrons, however, the current dis-tribution will become discontinuous, which makes it impossi-ble to perform complete correction of spherical aberration. In the next paragraph we shall investigate whether a super-conducting zone plate can be designed with better perform-ance than a normal Hoppe plate. Special attention will be paid to the question, whether such a zone plate can carry the induced currents required for correction, because its thickness cannot be much larger than the width of the rings.
6.2. A SUPERCONDUCTING ZONE PLATE.
To find the wave front at the image side of an electron-mi-croscope objective lens we use a simple derivation.
As the rays are normal to the wave front, the direction of the wave front can be found from the direction of the imag-ing rays and thus from the deflection angle.
If the object is placed in the focal point all rays will leave a perfect lens parallel to the axis. A perfect lens has a deflection Y ~ I'/f , which means that rays from an object placed in the focal point will leave the lens paral-lel to the axis. If the lens is made somewhat weaker, -so that the focus is removed over a distance Az from the ob-ject, the angle at the image side is given by - rAz,/f2. In reality, however, the lens is not perfect and the speri-cal aberration causes an extra deflection given by C r^/f'*.
s If, moreover, a corrector is inserted into the lens, the de-flection angle is increased again by an amount
given by '+.15, in which I denotes the current outside the circle with radius r.
The total angle Y S't the image side of the lens is there-fore :
Y = - ^ . C ^ f ' H - T ^ r B p o I (6.1)
drAs -Y i s the slope of the rays, 1/Y = -T" for the wave front and thus the z-coordinate of the wave front as a func-t i o n of r i s found by infunc-tegrafunc-ting Y func-t o r r e s u l func-t i n g in
z = _ ^ - 1 r2 + J - 2 . r»* + f2 f"^ or simplified
z = K r"* - (1-k) r2 in which and K = C _ 1 s f"^ e Az„ 1 = + i ( 6 . 3 ) (6.1+) ( 6 . 5 ) (6.6)
I t raust be noted, t h a t k can be varied as a function of r by choosing the right d i s t r i b u t i o n of I , whereas a change of 1 gives a defocusing independent of r . The wave front i s drawn in figure 6.1 for several values of
k-1 .
-5X
-3X
l - k > 0 In case k is zero
(with-out correcting currents) 1-k i s independent of r and only one of the curves of figure 6.1 represents the actual case.
In case there are correct-Fig 6.1. Wave fronts of a lens with spherical i n g c u r r e n t s , h O W e v e r , k
aberration with different amounts of . , , ,
IS no longer zero and 1-k
defocusing,
i s a function of r , which means t h a t we can s h i f t from one curve of figure 6.1 t o an-other , depending on the s i t e and strength of the currents, As very t h i n objects give mainly phase c o n t r a s t , i t would b(
nice, i f the wave front could be modified in such a way thai z = (n + l)\ over a very large region, so that a positive phase contrast i s obtained for object d e t a i l s having a largi range of s i z e s .
Therefore we t r y t o use t h e t o p s of t h e a p p r o p i a t e curves of f i g u r e 6 . 1 . By simple a r i t h m e t i c we find from e q u a t i o n 6 . 3 f o r t h e v a l u e s r of r a t which a t o p i s s i t u a t e d r ^ 2 = ^ ( l - k ) ( 6 . 7 ) and f o r t h e t o p v a l u e s z^ = - K r^'» ( 6 . 8 ) At a s e p a r a t i o n A from r , t h e change i n z i s Az = + 1* K r 2 A2 ( 6 . 9 ) Xf
We want the maximum values of z to be
z^ = - (n - |)X n = 1, 2, (6.10)
and p e r m i t only
Az = aX, (6.11)
a c c o r d i n g t o R a y l e i g h ' s c o n d i t i o n .
I t i s c l e a r t h a t t h e v a l u e s of r are f i x e d by e q u a t i o n 6.10 and 6 . 8 , because K i s independent of t h e c o r r e c t i n g c u r r e n t s , and when r i s known t h e a d m i s s i b l e width of a
Xf
s l i t can be c a l c u l a t e d from e q u a t i o n 6.9 and 6 . 1 1 .
The r e s u l t s a r e l i s t e d i n t a b l e 1; t h e v a l u e s of C and
3
f were t a k e n t o be t h o s e of t h e P h i l i p s E M 300 microscope i . e . C = 1.57x10"2m and f = 1.59xl0~3ra.
n 1 2 3 1* 5 6 10 z^ (in X) - 3/8 -11/8 -19/8 -27/8 -35/8 -1*2/8 -75/8 r^ (in pm) 12.1* 17.2 19.7 21.5 23.0 2l*.l 27.8 2A (in pm) 10.1 7.3 6.1* 5.8 5.5 5.3 1+.5 If we use the tops for re-spectively n=l and n=5 a zone plate of only one ring can be used giving rise to a prop-er phase shift in the regions of r from Table 6.1 7,U to I7.I+ pm and from 20.2 to 25.8 pm (see fig 6.2). In this case the ring width becomes 2.8 ym.
It should be noted that these values of r denote virtual radii because of the curvature of the electron paths in the lens region, For the EM 300 the actual val-ues of r in the aperture plane are small-er by a factor of 0,87. Thus the actual in-ner and outer radius of the
operture diafrogma Fig 6 . 2 . Selection of s u i t a b l e wave fronts for phase contrast,
The corrector only transmits the parts of the wave fronts indicated by the heavy l i n e s ,
ring become respectively 15.1 and 17,6 pm and the ring width is 2,5 ym,
If the tops for n = 1 and n = 1* were used the ring width would become only 1,0 pm which is probably too small to be manufactured,
The inner region is left open to transmit the undeflected beam and to produce amplitude contrast for the larger details,
Once the tops are selected, both the defocusing 1 and the . correction currents required to get the tops at the proper positions, can be calculated from equation 6,7, 6,5 and 6,6,
Outside the correction current k is zero, because of the definition of I , and 1 is calculated from the simplified equation 6,7
2 1 (6,12) t 2 K
thus in the case of figure 6,2
1 = 2 K r52 (6.13)
Using equation 6,6 we find for the defocusing
Az^ = 328 nm (6,11*)
Inside the ring 1 has the same value 2 K r52 and we find for k from equation 6,17 and 6,13
k = 2 K (r52 - ri2) (6,15)
Substitution of k in equation 6,5 and inserting again the values of the E M 300 finally yields for I :
I = 0,105 A (6,16)
Supposing a critical current density in the superconducting ring of 10^0 A/m2 , which is normal for Nb3Sn and taking into account that the width of the ring is 2,5 ym, we find for the required thickness d of the ring a minimum value of
d = 1*,2 pm (6,17)
It shovild be possible to make a zone plate of the dimensions calculated above, as much more complicated zone plates were already manufactured2'+.
The zone plate designed here would provide positive phase contrast for details between 2.3 and 2.9 A and between
3.6 and 8.1* K,
Larger details will be imaged only with poor phase contrast. However, they can be expected to give ampli-tude contrast. Figure 6.3 gives a plot of the pha^se contrast transfer function for the E M 300 objec-tive lens equipped with the corrector just described.
16 20 2i 26
3 e i 2.9& 2.3A object dttoil size
Fig 6,3, Theoretical phase contrast function PCF for the
Philips E M 300 objective lens equipped with the annular corrector,
7 . ALIGNMENT ERRORS.
7 . 1 . GENERAL CONSIDERATIONS.
I n t h e c a s e of t h e s u p e r c o n d u c t i n g r i n g as c o r r e c t o r t h e c a l -c u l a t i o n of t h e misalignment e r r o r s i s r a t h e r s i m p l e .
Misalignment means i n t h i s c a s e t h a t wrong p a r t s of t h e wave f r o n t s a r e u s e d , so t h a t t h e phase d i f f e r e n c e of t h e wave w i t h r e s p e c t t o t h e u n d e f l e c t e d beam does not obey t h e r e l a -t i o n s 6.10 and 6 . 1 1 , r e q u i r e d f o r op-timum phase c o n -t r a s -t . There a r e two p o s s i b i l i t i e s of misalignment:
a. The c o r r e c t o r i s p e r p e n d i c u l a r t o t h e a x i s but o f f - c e n t r e by a d i s t a n c e 6 .
b . The c o r r e c t o r i s t i l t e d over an angle e .
I n t h e next p a r a g r a p h s b o t h e r r o r s w i l l be determined.
7 . 2 . ERRORS DUE TO A SHIFT OF THE CORRECTOR.
Suppose t h e c o r r e c t o r i s s h i f t e d through a d i s t a n c e 6 along an Xaxis p e r p e n d i c u l a r t o t h e zajcis out of i t s p r o p e r p o s i -t i o n ( s e e f i g 7 . 1 ) .
I n t h e shaded r e g i o n i n s i d e t h e r i n g a t o o l a r g e p a r t of t h e lowest curve of f i g u r e 6 . 2 i s u s e d , whereas i n t h e shaded r e g i o n o u t s i d e t h e r i n g a t o o l a r g e p a r t of t h e f i f t h curve i s u s e d . As t h e f i f t h curve i s much s t e e p e r t h e l a t t e r e f f e c t p l a y s t h e l a r g e s t r o l e .
As t h e phase d i f f e r e n c e between t h e t o p of t h e curve and t h e o u t e r margin of t h e r i n g , i f p r o p e r l y a l i g n e d , i s given by
I X = 1* K r. 2 A2 (7.1)
the difference is given by
1* K r^2 (^ + 5)2 (7.2)
in case of misalignment.
The extra phase shift is thus given by the difference of 7.2 and 7.1 being
^ U ^ ) .
(7.3)If we require this difference to be smaller than for instance 20^ of Jx , then 6 has to be smaller than A/10 , which
means, if A equals 2.8 ym , that the shift of the corrector must be smaller than 0.28 pm . It is obviously difficult to perform proper align-ment. Perhaps the
sim-plest way to achieve this is to put the microscope in the diffraction mode, and to adjust the sharply imaged corrector until it is as symmetrical as possible around the central beam.
Fig 7,1, Shifted corrector. In the shaded region wrong parts of the wave front are trans-mitted,
7.3. ERRORS DUE TO TILT OF THE CORRECTOR.
Suppose the corrector i s t i l t e d around the y-axis thro\igh an angle e (see figure 7 . 2 ) .
Then an effect a r i s e s , which was not included in previous calculations as a l l currents were supposed t o run in planes perpendicular t o the z - a x i s .
In case of t i l t of the corrector t h i s condition i s not f u l f i l l e d and an extra deflection w i l l be caused by the t r a n s -verse component b of the magnetic f i e l d (see figure 7 . 2 ) .
At a t i l t through a small an-gle e t h i s component i s
b
(7.5)
The Lorentz force F on the imaging electrons will cause a deflection of
AY = ^ ƒ F^^ dt (7.6) mz ' y
Fig 7,2, Tilted corrector. The current in the o
ring causes a transverse component
b of the induction,
» . . .
In t h i s i n t e g r a l dt i s r e -placed by dz/z and z w i l l be eliminated by eV = mi^/2 which i s a good approximation as r and r$ are compara-t i v e l y small; compara-thus
1 1
^^ = ^ 2irv ^' ƒ b^ dz . e (7.7)
ocorrec-tor current (cf. paragraph 1*.2). So, finally. AY is given by
AY = poi irbr )'• " ^'^'^^
By i n s e r t i n g for I 0.105 A as derived in chapter 6 and for V 105 V we find
AY = 1.2x10"'*. e (7.9)
This deflection causes a f i c t i t i o u s s h i f t of the object through a distance fAY and i f we suppose t h a t t h i s s h i f t must be smaller than 1 A , e has t o be smaller than
5x10"'* radians. Thus a t i l t of the corrector should be r e -moved very accurately.
I t w i l l probably be very d i f f i c u l t t o find an indication of proper alignment. Perhaps the best indication can be obtain-ed from asymmetries in the image, using image amplification.
I t should be noted here t h a t an effect analogous t o t h a t of the former paragraph, i . e . that wrong parts of the wave front are transmitted by the c o r r e c t o r , gives an error proportional t o e2 which i s an order of magnitude smaller than the ef-fect described above,
8 , ADMISSIBLE HEAT FLOW TO THE CORRECTOR. I f a c o r r e c t o r as d e s i g n e d i n c h a p t e r 6 i s a p p l i e d i n an e -l e c t r o n - m i c r o s c o p e o b j e c t i v e -l e n s t h e r e w i -l -l b e a h e a t f -l o w t o t h e c o r r e c t o r c a u s e d b y t h e i m a g i n g e l e c t r o n s s t o p p e d b y t h e c o r r e c t o r . The i m a g i n g e l e c t r o n s a r e s c a t t e r e d i n t h e o b j e c t a n d t o e v a l u a t e t h e p a r t o f t h e e l e c t r o n s t h a t j u s t h i t s t h e c o r r e c -t o r we u s e -t h e c u r v e s o f Von B o r r i e s 2 3 g i v i n g -t h e r e l a -t i v e number o f e l e c t r o n s s c a t t e r e d e l a s t i c a l l y o r i n e l a s t i c a l l y i n t o a d i r e c t i o n e . T h e y a r e drawn i n f i g u r e 8 . 1 f o r s c a t t e r i n g a t a c a r b o n o b -j e c t l a y e r o f m a s s t h i c k n e s s 10~^ k g / m 2 . The a c c e l e r a t i n g v o l t -a g e i s 100 kV . F o r t h e r i n g o f c h a p t e r 6 0 = 1.2x10~2 r a d i a n s . N io-\. ^ 10-1 -^i in rod. Fig 8,1, Angular distribution of inelastic and
elastic scattering in a carbon layer of mass thickness 10~^ kg/n^,
At t h i s a n g l e d — /d£l II0
öü s u b t e n d e d by t h e r i n g o f N
No 9x10"^* and Uxio"^ re-is about 7 for inelastic scattering and about 30 for elastic scattering. By integra-tion over the solid angle
v/idth 2.8 ym , we find for spectively.
Thus the ratio of electrons scattered just to the corrector
— o
and the nonscattered electrons is some 5^10 .
The unaided eye observing an extended object of high con-trast on the fluorescent screen of the electron microscopes needs after several minutes of dark adaption a current den-sity of 2x10~^ A/m2 to produce acceptable images2 5.
We estimate that objects of low contrast need some 100 times more. Thus with a screen of area 2x10 2 ]3i2 a current of l*x10~^ A is required. At an accelerating voltage of 100 kV this current corresponds to an energy of 1* mW.
So there is a minimum heat flow to the ring of some 20 yVJ. In practice a much larger area of the object is illiminated, so that the actual heat flow may be 10 - lOQx this value.
To make a rough es-timate of the maxi-mum temperature of the ring we suppose that the temperature of the ring is high-est at the points A of figure 8.2, and that the total heat flow to a part of the ring between two neighbouring points
Fig 8,2, Corrector suspended in the aperture by 5 heat ^ ^^^ t o b e C O n -conducting bars, d u c t e d from t h e s e
points to B as in-dicated by the arrows. We suppose a copper substrate of 1 ym thickness to be under the superconducting ring which has a width of 2.5 ym. The radial bars may have the same dimensions, The thermal conductivity of the superconductor can be neg-lected, The thermal-conductivity coefficient X of the cop-per is 5^1o2 W/m'K in this temcop-perature region. If the ring is connected to the aperture by 5 bars the temperature dif-ference between the points A and B is some 3x10~2 K.
