Graef M. Introduction to conventional transmission electron microscopy

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168 The transmission electron microscope

axis. The point P

has coordinates (x

, y

) in the object plane, and is represented

by the complex number u

= x

+ iy

. The distance between P

and the optical

axis is then given by r

= u

∗

. The point P

is similarly described by the complex

number u

= x

+ iy

. We will assume that the image rotation along the optical

axis has been absorbed into the deﬁnition of the respective coordinate frames. It can

then be shown that the image point P

is located at the position u

+ u

, where u

is the Gaussian position, and

u

= Cr

← spherical aberration

+2(K + ik)r

+ (K − ik)u

∗

← coma

+(A + ia)u

∗

← astigmatism

+ Fr

← ﬁeld curvature

+(D + id)r

← distortion. (3.42)

The eight constants C, K , k, A, a, F, D, and d are the real

†

aberration coefﬁcients;

they describe the ﬁve primary or Seidel aberrations. The magniﬁcation factor M

has been omitted for clarity. The primary aberrations can be grouped into three

categories: the aperture aberrations (terms independent of the object position u

in equation 3.42), the chromatic aberrations (terms independent of u

, discussed

in Section 3.7.4) and the parasitic aberrations (in general these are linear in u

linear in u

). Geometric aberrations are caused by the properties of the magnetic

ﬁeld. Chromatic aberrations are caused by the instabilities in the lens current and/or

instabilities in the acceleration potential; they give rise to slightly different electron

trajectories in the lens and hence to a blurring of the image. Finally, parasitic

aberrations are caused by inhomogeneities or imperfections in the lens pole pieces,

deviations from perfect circular symmetry, etc.

The ﬁrst term in equation (3.42), spherical aberration, is of third order in the

coordinates u

, and hence also of third order in the angle α between the trajectory

and the optical axis. The next distortion, coma, is of second order in α, and is

described by two numbers K and k, which can be converted into an amplitude and

angle, i.e. a single complex number. The next two aberrations, astigmatism and

ﬁeld curvature, are linear in α; astigmatism is also represented by two numbers

A and a. Finally, the ﬁfth aberration, known as distortion, only depends on the

position in the object plane, and is also described by two numbers D and d. Each of

the aberrations describes a particular way in which the wave front can depart from

the spherical shape. The true wave front shape is a linear superposition of the ﬁve

aberrated wave fronts. Next, we will describe brieﬂy the nature of the ﬁve Seidel

aberrations.

†

One can also deﬁne asymptotic aberration coefﬁcients for projector lenses.

3.5 Basic electron optics: lens aberrations 169

Circle of least

confusion

Fig. 3.13. Spherical aberration of rays far from the optical axis.

3.5.2.1 Spherical aberration

The nature of spherical aberration can be understood by considering the ﬁrst term

in equation (3.42) and rewriting the position vector u

in polar coordinates for a

circle of radius r

in the lens plane: u

= r

(cos θ + i sin θ), where θ ranges from

0to2π. The correction term then becomes

x

+ iy

= MCr

(cos θ + i sin θ),

which is a circle with radius proportional to the cube of the distance r

. The circle

does not depend on the location u

of the object point. Figure 3.13 shows schemati-

cally how electrons leaving an axial object point are affected by spherical aberration:

rays passing through the outer zone of the lens (i.e. through a circle with a large

radius r

) are focused more strongly than the paraxial rays, and will hence come

to a focus before the Gaussian image plane.

†

The spherical aberration coefﬁcient

(which is proportional to the coefﬁcient C deﬁned above) can be computed in

terms of the axial magnetic ﬁeld distribution B(z) by:

‡





(z)



(z) − B

(z)





(z)

h(z)





(z)dz, (3.43)

where h(z) is a paraxial trajectory, which leaves an axial object point with unit

slope, i.e. h



) = 1 [HK89b]. The presence of powers of the magnetic ﬁeld in this

expression makes the spherical aberration particularly sensitive to small variations

in the ﬁeld. In the Gaussian image plane, a point object is imaged as a disk, with

radius

= MC

. (3.44)

†

The Gaussian image plane is the plane conjugate to the object plane in the absence of aberrations; it can be

computed from Newton’s lens equation, if the object position is known.

‡

There are many different ways to write down the spherical aberration coefﬁcient; the expression above is special

in that it clearly shows that the spherical aberration of a round lens is always a positive quantity. This is known

as Scherzer’s theorem.

170 The transmission electron microscope

The narrowest diameter of the electron beam corresponds to the so-called cir-

cle of least confusion. The spherical aberration coefﬁcient C

is typically ex-

pressed in millimeters; modern microscopes have an objective lens with C

the range 0.5–1.5 mm, while for older models the objective C

can be several

millimeters.

From the expression above, we can see that the spherical aberration coefﬁcient

of a round magnetic lens cannot become a negative quantity, i.e. rays affected

by spherical aberration intersect the optical axis before reaching the image plane.

This also means that it is impossible to create a round magnetic lens with zero

spherical aberration, since one can show that a lens with C

= 0 does not have

a real image [Cre77]. Spherical aberration correction has been a hot research

topic ever since Scherzer’s discovery of the theorem that now bears his name.

Only recently has technology reached the point where C

-correctors are becoming

available commercially, and perhaps the microscopes of the future will have such

built-in correctors as a standard component. The interested reader is encouraged

to consult the following sources: [HK89b, Chapter 41], [CK80], [Ros71], [HU97],

[KDL99]. C

reduction and/or correction remains an area of very active research and

development.

3.5.2.2 Coma

The second term in equation (3.42) can be rewritten without loss of generality

by considering the following special case: the object point is ﬁxed at the location

= x

, the aberration coefﬁcient has only one non-zero component (k = 0), and

the electrons pass the lens plane in concentric circles of radius r

centered on the

optical axis:

x

+ iy

= MKx



+ u



;

= MKr



2 + e

i2θ



;

= MKr

[

(

2 + cos 2θ

)

+ i sin 2θ

]

Figure 3.14(a) shows the aberration ﬁgure for the conditions: x

= 1, M = 1, K =

0.1, and u

= 1 + i0. The images of the ﬁve concentric circles with increasing

radii r

are displaced along the positive x-direction in such a way that the entire

aberration ﬁgure is located between two straight lines. This aberration only affects

object points that are not located on the optical axis. The coma coefﬁcients can also

be expressed in terms of special electron trajectories and the axial ﬁeld B(z); for

explicit expressions we refer the reader to [HK89a].

A more general example of coma is shown in Fig. 3.14(b); eight points are

selected in the object plane at the locations (±1, 0), (0, ±1), and (±1, ±1). The

3.5 Basic electron optics: lens aberrations 171

(a)

(b) (c)

Fig. 3.14. Aberration ﬁgures due to coma for the parameters described in the text.

coma coefﬁcient is taken to be 0.05 − i0.1, and the concentric circles are centered

on the optical axis. The resulting aberration ﬁgures are computed for the same eight

points in the image plane, ignoring the magniﬁcation factor. If the concentric circles

are centered on an off-axis point in the lens plane, say the point u

= 0.4 + i0.2,

then the aberration ﬁgures become more complex and are different for different

object points, as shown in Fig. 3.14(c). Note that the aberration ﬁgures remain

located between the same two straight lines, regardless of the position of the point

. The orientation of the lines is determined by the value of K + ik.

It is clear from the examples in Fig. 3.14 that coma is an undesirable lens aberra-

tion, since it generates extended, asymmetric images of point objects. It is possible

to perform a coma-free alignment of an electron microscope, and we will return to

this in Chapter 10 on phase contrast microscopy.

3.5.2.3 Astigmatism and ﬁeld curvature

Both astigmatism and ﬁeld curvature are linear in u

, so it is convenient to treat

them together. In the image plane the combined effect of astigmatism (represented

in polar notation as A + ia = Ae

iθ

) and ﬁeld curvature is given by

u

= Ar

i(θ

−θ)



− y

+ 2ix



+ Fr

iθ

, (3.45)

where the angle θ once again describes a circle of radius r

in the lens plane. The

effect of the ﬁeld curvature term is illustrated in Figs 3.15(a) and (b) for F = 0.05

and 0.15, respectively. The aberration coefﬁcients are shown in the form [A,θ

, F].

Note that the angle θ is only varied over three-quarters of a circle; this will be useful

to illustrate the effect of the astigmatism coefﬁcients. The further the object point

is located from the optical axis, the larger its image will be. This is to be contrasted

with the effect of spherical aberration, which results in disks of the same size,

regardless of the position in the object plane.

172 The transmission electron microscope

(a) [0, 0, 0.05] (b) [0, 0, 0.15] (c) [–0.15, 0, 0.15]

(d) [–0.05, 0, 0.15] (f) [0.15, 0, 0.15](e) [0.05, 0, 0.15]

(g) [0.05, 45, 0.15] (h) [0.05, 90, 0.15] (i) [0.05, 180, 0.15]

Fig. 3.15. Illustration of the effect of astigmatism and ﬁeld curvature (represented in the

form [A,θ

, F]) on the image of eight object points aligned on a square. The circle segments

correspond to ﬁve circles of increasing diameter in the lens plane.

Consider the point u

= 1 + i0, and take θ

= 0; the expression above then

simpliﬁes to

u

= r



−iθ

+ Fe

iθ



;

= r

[

(

F + A

)

cos θ + i

(

F − A

)

sin θ

]

It is clear from this expression that for F =±A the aberration ﬁgure collapses into

a straight line; this is illustrated graphically in Fig. 3.15(c)–(f).

Another way to represent the effect of astigmatism is through a trajectory dia-

gram. Figure 3.16 shows that the focal length of an astigmatic lens becomes depen-

dent on the azimuthal angle, and varies continuously between two extreme values,

indicated by arrows in the ﬁgure. This lens aberration can be completely canceled

by means of quadrupole stigmators. Most microscopes have two sets of stigma-

tors: one for the condensor lens, to remove astigmatism from the incident beam,

and one for the objective lens. Manual astigmatism correction requires considerable

3.5 Basic electron optics: lens aberrations 173

Fig. 3.16. Effect of astigmatism on paraxial rays for a 400 kV electron in Glazer’s bell-

shaped model with B

= 0.5 T and a = 3.0 mm; the magnitude of the astigmatism corre-

sponds to a 25% azimuthal change of the magnetic ﬁeld strength.

(a) (b)

D<0

D>0

D<0

D>0

d= - 0.05

d=0

Fig. 3.17. Schematic illustration of the distortion aberration for D = [−0.3 ...0.3] with

(a) d = 0 and (b) d =−0.05. The original square is outlined in (a).

practice and is generally regarded as one of the more difﬁcult aspects of transmission

electron microscopy. With modern computer-controlled microscopes, astigmatism

correction can be performed entirely and very accurately by a computer algorithm.

3.5.2.4 Distortion

The last Seidel aberration does not depend on the position u

in the lens plane and is

illustrated in Fig. 3.17. The distortions for d = 0 are known as pin-cushion distor-

tion (for D > 0) and barrel distortion (for D < 0). For d = 0 an additional torsional

distortion is superimposed on the barrel or pin-cushion shapes, as illustrated in

Fig. 3.17(b). The distortion aberration is only important in the low-magniﬁcation

174 The transmission electron microscope

spherical aberration

0.4

field curvature

-0.1

astigmatism

0.1, 45∞

distortion

0.1, 10∞

coma

0.1, 0∞

= 0+0i

-2 -1

01 2

-2

-1

Fig. 3.18. Illustration of the output of the ION routine seidel.pro for the lens aberrations

stated.

range of a microscope and can be ignored safely for most higher-magniﬁcation

work.

3.5.2.5 Combination of the ﬁve Seidel aberrations

It is now straightforward to combine all ﬁve Seidel aberrations into a single illustra-

tion (Fig. 3.18), so that the total effect of the aberrations can be studied. The reader

may access the ION routine

seidel.pro from the website to enter various combi-

nations of primary aberration coefﬁcients. The routine will return the aberration

ﬁgures for nine points on a square in the object plane, with ﬁve circles of linearly

increasing radii around an arbitrary point in the lens plane.

3.6 Basic electron optics: magnetic multipole lenses

The example trajectory computations of Section 3.4 illustrate a rather simple way

to deal with electron optical elements. In the absence of currents inside the ﬁeld

region, one starts from expressions for the magnetic scalar potential ω obtained

as solutions of Laplace’s equation. By making judicious use of the symmetry of

the problem, signiﬁcant simpliﬁcations can often be introduced. For instance, the

general expression for the magnetic scalar potential in cylindrical coordinates is

3.6 Basic electron optics: magnetic multipole lenses 175

given by

ω(r,ϕ,z) =

∞



m=0

[

(r, z) cos mϕ + b

(r, z) sin mϕ

]

For a round magnetic lens only the term in m = 0 survives, and this leads to

the radial series expansion in equation (3.2). There are many other magnetic or

electrostatic elements which do not have rotational symmetry, but instead display

a number N of symmetry elements; such elements are known as multipole lenses.

In the presence of N symmetry planes, the magnetic scalar potential must satisfy

the relation [Szi88]

ω(r,ϕ,z) = ω



r,ϕ±

2πl

, z



from which we can derive that

ω(r,ϕ,z) =

∞



n=0

(r, z)r

cos(nNϕ).

For a detailed discussion of this expression and relations derived from it we refer

to [Szi88, Chapter 3] and references therein. For the purposes of this textbook it

will sufﬁce to discuss brieﬂy the standard magnetic quadrupole, and electrostatic

deﬂectors, because they are common components in a microscope column.

A round magnetic lens does not exert a direct radial force on an electron traveling

nearly parallel to the optical axis. The small radial force which brings the electron

closer to the optical axis is entirely due to the azimuthal velocity component ac-

quired as a consequence of the cross product nature of the Lorentz force. Multipole

elements can be used to exert mostly radial forces on the electron beam, and the sim-

plest multipole ﬁeld is that present in a beam deﬂector. The main purpose of a beam

deﬂector is to provide a lateral shift of the entire beam, with as little disturbance of

the beam structure as possible [HK89a]. In other words, the ideal deﬂector simply

moves the beam around, without changing its shape. The currents required to run

magnetic deﬂectors and quadrupoles are substantially less than those required for

round magnetic lenses. While round lenses require anywhere between 1 and 15 A,

deﬂection coils typically use 10–500 mA and are reversible, i.e. the direction of

the current can be reversed. Correction coils use even less current, typically in the

range of 1–200 mA [Szi88].

There are several beam deﬂection systems in a standard TEM. On the micro-

scope console they are commonly labeled as beam shift, beam tilt, image shift, and

so on. In addition, many microscopes have a scanning attachment as part of the

illumination system; the incident beam can then be scanned periodically along a

line or a rectangular area on the sample, and various detectors measure a scattered

176 The transmission electron microscope

signal (e.g. secondary or back-scattered electrons), fully synchronized with the

incident beam. The scan coils used for this purpose are essentially high-speed

beam deﬂectors, somewhat similar in nature to the deﬂection system in a television

tube.

While a good deﬂector does not appreciably change the shape of the electron

beam, a quadrupole lens (or rather a pair of quadrupole lenses) is used to adjust

the beam shape, in particular to correct astigmatism. In the next two sections we

will discuss brieﬂy the main characteristics of deﬂectors and quadrupole lenses.

There is also an extensive literature on the aberrations of multipole elements. This

is far beyond the scope of this book, and we refer the interested reader to [HK89a,

Chapters 29 and 32] and references therein.

3.6.1 Beam deﬂection

Beam deﬂectors can consist of electrostatic or magnetic components. The main

purpose of a deﬂector is to change the direction of the electron beam with re-

spect to the optical axis. Electrostatic deﬂectors usually consist of two pairs of

electrodes oriented perpendicular to each other, one providing deﬂection in the x-

direction, and the other in the y-direction. More complex designs with multiple elec-

trodes are also commonly used. Magnetic deﬂectors are usually used for scanning

applications.

While the electron optical characteristics of deﬂectors are interesting in their

own right, it is not necessary to discuss them in detail. For the “average micro-

scopist” it sufﬁces to point out that there are two different operation modes for

deﬂectors: beam tilt and beam shift. They are shown schematically in Fig. 3.19

for a converged beam. In the beam shift mode, the ﬁrst set of deﬂectors changes

the trajectory without changing the beam shape; the second deﬂector changes the

(a) (b)

Beam shift Beam tilt

(c)

deflector

optical

axis

Fig. 3.19. Schematic illustration of (a) beam shift mode and (b) beam tilt mode. Part (c) is

a photograph of the beam deﬂector housing of the JEOL 120CX microscope.

3.6 Basic electron optics: magnetic multipole lenses 177

trajectory by precisely the opposite angle, so that the electron beam is translated

or shifted across the sample. The ﬁeld of the individual deﬂector elements is rather

short along the beam path, so that the deﬂection can be represented as an abrupt

change of direction. In beam tilt mode the second deﬂector provides a deﬂec-

tion twice as large as that of the ﬁrst deﬂector, so that the illumination spot does

not move. Only the angle between the beam and the optical axis at the illumina-

tion spot changes. The same pair of deﬂectors can be used for both beam tilt and

beam shift by changing the relative excitations of the various deﬂector surfaces or

coils. It goes without saying that the deﬂection system must be properly aligned so

that beam tilt and beam shift can operate independently. Figure 3.19(c) shows the

beam deﬂector of the JEOL 120CX microscope; the round housing contains the

deﬂector plates and is located below the condensor stigmator which is shown in

Fig. 3.20(c).

(a) (b)

1 cm

(c)

(d)

Fig. 3.20. (a) Equipotential contours for an ideal quadrupole ﬁeld with a potential pro-

portional to xy. The symmetry planes are inclined at 45

◦

with respect to the coordinate

axes; (b) magnetic ﬁeld conﬁguration corresponding to the potential in (a). (c) and (d) are

photographs of the condensor and objective stigmators of the JEOL 120CX microscope,

respectively.