Graef M. Introduction to conventional transmission electron microscopy

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138 The transmission electron microscope

Table 3.1. Partial chronology of the history of the electron and the electron

microscope, and other events which have had a signiﬁcant impact on the

microscopy ﬁeld, with references to fundamental papers.

Year Event

1871 Cromwell Fleetwood Varley suggests that the carriers of electricity are

corpuscular, with a negative charge [Var71]

1876 Eugene Goldstein studies discharges in gases, and coins the name cathode rays,

starting a long debate about their nature [Gol76]

1891 George Johnstone Stoney coins the word electron for the unit of charge [Sto91]

1897 Emil Wiechert is the ﬁrst to obtain reasonable bounds on the magnitude of e/m

(January) [Wie97]

1897 Walter Kaufmann and J.J. Thomson

independently measure

e/m (April)

[Kau97, Tho97]

1899 J.J. Thomson determines the value of e, which makes him the discoverer of the

electron [Tho99]

1905 Albert Einstein publishes the special theory of relativity and establishes the

equivalence of mass and energy [Ein05]

1913 Niels Bohr introduces a model for the structure of the hydrogen

atom [Boh13a,

Boh13b,

Boh13c]

1923 Maurice de Broglie establishes the principle of wave–particle duality [de 23]

1925 Wolfgang Pauli discovers the exclusion principle; Werner Heisenberg

develops matrix quantum mechanics [Pau25, Hei25]

1926 Erwin Schr¨odinger develops quantum mechanics based on differential

equations; Hans Busch develops the theory of magnetic lenses [Sch26,

Bus26]

1927 Clinton Davisson and Lester Germer discov

er electron diffraction [DG27]

1928 Paul Dirac formulates the relativistic theory of the electron; Hans Bethe

develops the ﬁrst dynamical theory of electron diffraction [Dir28a, Bet28]

1931–4 Ernst Ruska and Max Knoll build the ﬁrst electron microscope [KR32a, KR32b,

Rus34a, Rus34b]

we refer the interested reader to the book The Growth of Electron Microscopy,

edited by T. Mulvey [Mul96].

3.3 Overview of the instrument

Since the ﬁrst electron microscope was constructed by Ruska and Knoll, TEMs have

evolved into complex computer-controlled machines, and many of the underlying

physical principles are hidden from the user in modern machines. Students who

learn to operate the latest (digital) model as their ﬁrst machine may ﬁnd it hard

to operate some of the earlier models, especially the ones where lens and gun

alignment are mostly mechanical. An inexperienced user should be aware of the

fact that different microscope models may have a different layout of the most

3.3 Overview of the instrument 139

120°

Specimen rod

Fig. 3.1. Schematic top view of the section removed from the JEOL 120CX microscope

column; the specimen holder location is also indicated.

commonly used controls and that some “mental adjustments” may be needed when

moving from one instrument to another. The contents of this and the following

chapter should provide some common ground for the operation of all microscope

models.

An optical microscope, used in transmission mode, typically consists of ﬁve

parts: the light source; the condensor lens, which focuses the light beam onto the

sample; a transparent sample, usually a thin section of rock or tissue; the objective

lens, sometimes in contact with the sample through a contact medium (oil); and

the magnifying lens(es), often combined with the ocular or eyepiece. The number

of lenses may vary but most optical microscopes have all ﬁve components. A TEM

can be divided into similar sections. The ﬁve sections of a TEM are (from top

to bottom): the electron gun, the illumination stage, the objective lens, with an

electron transparent sample immersed into its magnetic ﬁeld, the magniﬁcation

and projection system, often with three or more lenses, and the detector (a viewing

screen, photographic camera, charge-coupled device (CCD) camera, etc.

A cross-section of a JEOL 120CX microscope column is shown on the right in

Fig. 3.2.

†

A corresponding schematic drawing, taken from the microscope manual,

is shown on the left, along with an outline of the ﬁve main components.

†

Several of the photographs in this chapter were obtained from a cross-section of a real TEM, a JEOL model

120CX, built in 1978 (serial number EM 156049-17). Throughout the second half of 1999 the author, with

the help of Mr N.T. Nuhfer and Mr G. Biddle, removed a 120

◦

lengthwise wedge from the column, using a

high-capacity band saw. One edge of the wedge cut is along the leftmost (when facing the microscope) side of

the goniometer stage, the other cut is at 60

◦

from the opposite side, as illustrated in the schematic top view in

Fig. 3.1. The cut extends from the top of the electron gun down to the top of the viewing chamber, over a length

of approximately 1060 mm; the viewing chamber itself was left intact, to provide sufﬁcient mechanical support

to the column. The surfaces of the cuts were sanded to a smooth ﬁnish and covered with a protective layer of

polyurethane, to minimize corrosion. After completion of all the cuts, the microscope column was reassembled

at the bottom of the stairwell on the ground ﬂoor of Roberts Engineering Hall, Carnegie Mellon University

campus. Additional photographs of the cross-section are available from the

website.

140 The transmission electron microscope

electron gun

illumination

objective

lens

magnification

observation

optical axis

Fig. 3.2. Schematic drawing

and photograph of a real cross-section of a JEOL 120CX

transmission electron microscope, with the main stages outlined. (Drawing reproduced

with permission.)

3.3 Overview of the instrument 141

In addition to the ﬁve main stages listed above, a TEM contains a number of

subsystems that are vital for proper operation, but are of secondary importance to

the operator during routine use of the microscope.

†

The high voltage system. This system is usually located in a separate unit and is connected

to the electron gun by a thick cable. For acceleration voltages of 400 kV and higher, this

type of cable connection is no longer possible and the accelerator is placed just behind

the microscope column or on top of the column.

The vacuum system. Electrons need a good vacuum (in the range of 10

−7

–10

−10

Pa (or

−5

–10

−8

torr, where 1 torr = 133.322 Pa) to travel the typically 1–2 m length of the

column, hence every TEM is equipped with a high vacuum system. The system generally

consist of one or more roughing pumps, possibly with a vacuum buffer tank, one or more

oil diffusion pumps and/or ion getter pumps and/or turbomolecular pumps, and a num-

ber of independent vacuum circuits with manual or automatic (pneumatic or electronic)

valves. Some microscopes are modiﬁed for ultra-high vacuum conditions, which means

that the partial pressures of all gases in the column are better than 10

−8

torr. In older

microscopes, such as the Siemens Elmiskope 102, the operator had to be familiar with

the pumping diagrams and cycles and often manually open or close valves. Nowadays,

this task is fully automated and interlocks on the high voltage system and the ﬁlament

emission system will prevent the operator from using the machine if the vacuum level is

inadequate.

The operator should always take note of the proper vacuum level at which the micro-

scope must be operated. This should be part of the prework checklist. The vacuum is

usually best in the electron gun and slightly worse at the specimen stage. Some micro-

scopes allow observations at near ambient pressures in a variety of gas mixtures through

the use of an environmental specimen cell; in those cases the sample region is separated

from the rest of the microscope column by differential pumping apertures.

In older microscopes, such as the JEOL 120CX, the vacuum region of the microscope is

separated from the rest by about two-dozen O-rings. In modern instruments the electrons

travel through so-called liner tubes, long cylindrical sections with a small inner diameter

(a few millimeters), which extend across several lenses and reduce the total number of

vacuum seals needed.

The cooling system. Large electric currents can generate a lot of heat (Ohm’s law), hence

the lens coils, which can carry several amperes of current, must be water-cooled. This

system is fully automatic and the operator should never have to intervene at any point.

The cooling system is interlocked with other systems, so that the microscope cannot be

used if insufﬁcient cooling water is available.

Radiation shields. All electrons emitted by the ﬁlament must eventually ﬁnd their way

back into the electrical circuit, and must get rid of their kinetic energy. All surfaces

†

A TEM operator should be aware of the meaning of the warning signs and indicator lights should any of the

subsystems described in this section malfunction or cause problems. We refer to the microscope manual for a

detailed description of all subsystems.

142 The transmission electron microscope

inside the column that may be exposed to electrons are therefore made of a conductive

material. A more serious problem is posed by the conversion of kinetic energy into x-ray

photons wherever the electrons enter solid matter. All areas of the microscope where

x-rays may be generated are therefore surrounded by a double shield: the ﬁrst layer of the

shield is a low atomic number material, typically aluminum, which converts most of the

kinetic energy into low-energy x-ray photons. The second layer is a high atomic number

material, typically lead, which efﬁciently absorbs the low-energy x-ray photons. The

viewing window is made of a special lead containing glass and can be rather expensive.

The radiation shields contribute signiﬁcantly to the weight of the column, and are thicker

for higher-energy microscopes.

The photographic camera system. A photographic camera, consisting of two removable

boxes, is mounted below or behind the viewing screen: one box contains unexposed

plates and the other one receives the exposed plates (depending on the microscope type

the number of available plates can vary anywhere from 20 to 50). The exposure time and

ﬁlm sensitivity can be set

from the microscope console, although one should not change

the latter unless the type of ﬁlm is changed. All microscopes indicate the magniﬁcation

(or camera length for diffraction) and the plate serial number near the edge of the negative;

some models also allow for the

inclusion of 30 or so characters of text, to identify the

operator and sample.

This concludes the enumeration of the more common microscope subsystems.

Possible additional subsystems include energy dispersive analyzers for x-rays or

electron energy loss spectroscopy (EDS and EELS), cameras (TV rate, CCD, or

imaging plate), scanning attachments, various detectors before and after the sample,

imaging energy ﬁlters, etc. With the exception of electron detectors, we will not

address these additional subsystems in this book.

3.4 Basic electron optics: round magnetic lenses

3.4.1 Cross-section of a round magnetic lens

In a light optical microscope, glass lenses are used to ﬁrst focus a light beam onto the

sample, and then magnify the image to the desired magniﬁcation. Each lens is a care-

fully shaped piece of glass. The radius of curvature of each surface along with the

refractive index of the glass determine the optical properties and quality of the lens.

Glass lenses have been used since the early compound (multi-lens) microscopes

built by Zacharias Jansen in 1595 and the high-quality single-lens microscopes

built by Anton Leeuwenhoek in the second half of the seventeenth century [Jon97].

Light optics has become a highly developed ﬁeld, with numerous applications that

affect us in our daily lives. Cameras, CD players, lens and mirror-based telescopes

are only a few examples of the widespread availability of high-quality optics; light

3.4 Basic electron optics: round magnetic lenses 143

optical axis

upper polepiece

lower

polepiece

lens coil

yoke

(a)

(b)

Fig. 3.3. (a) Schematic representation of the magnetic ﬁeld lines around a circular current-

carrying coil; (b) a magnetic yoke concentrates the ﬁeld lines near the center of the coil and

provides a particular geometry to the ﬁeld.

optical systems can be designed and built to any speciﬁcations, with virtually no

aberrations at all.

Deﬂection of electrons requires magnetic or electrostatic lenses. All modern

transmission electron microscopes contain at least six and often more than six

round magnetic lenses. The main purpose of a round magnetic lens is to focus the

electron beam towards the optical axis. The basic principle behind such a lens is

shown in Fig. 3.3(a): a circular current running through a coil produces a magnetic

ﬁeld that is nearly axial at the center of the coil. We know from electromagnetism

that the strength of the ﬁeld obtained at the center of the coil is proportional to the

magnetic permeability of vacuum (or air) [Jac75]. We can signiﬁcantly increase the

attainable ﬁeld strength by using a solid yoke with a large magnetic permeability

to guide the ﬁeld lines and increase the ﬁeld strength near the center of the coil

(Fig. 3.3b). The details of the shape of the pole pieces near the center will determine

the precise geometry of the magnetic ﬁeld.

A cross-section of a real magnetic lens (the ﬁrst condensor lens of a JEOL120CX)

is shown in Fig. 3.4; the large rectangular area to the right is the tightly packed

copper winding, as shown in the inset. The upper and lower portions of the yoke

are separated by a brass spacer. The lens pole pieces, typically made of a high-

permeability Fe–Co alloy, are visible to the left of the winding. The magnetic ﬁeld

strength is highest near the optical axis in between the upper and lower pole pieces.

The distance between the upper and lower pole pieces is known as the lens gap.

While the geometrical details may vary from one lens to the next, all round magnetic

lenses are basically identical in design to the lens shown in Fig. 3.4. Next, we will

144 The transmission electron microscope

1st condensor

coil

brass spacer ring

upper

lower

pole

piece

optical

axis

brass spacer ring

yoke

Fig. 3.4. Cross-section of the ﬁrst condensor lens of a JEOL 120CX microscope. The coil

winding has an areal density of nearly 300 copper wires per cm

. The pole pieces, visible

through the slot on the left, are made of a high-permeability Fe–Co alloy.

take a closer look at the magnetic ﬁeld components inside a round magnetic lens.

3.4.2 Magnetic ﬁeld components for a round lens

Let us consider a magnetic ﬁeld with circular symmetry around its central axis.

It is convenient to employ cylindrical coordinates (r,ϕ,z), related to the standard

Cartesian coordinates (x, y, z) by the relations:







x = r cos ϕ;

y = r sin ϕ;

z = z.

(3.1)

The z-axis is taken to be the symmetry axis of the ﬁeld. In the absence of ﬁeld-

producing currents, we can use a scalar magnetic potential ω(r, z), satisfying the

homogeneous Laplace equation ω = 0, to derive the magnetic ﬁeld components

B =−µ∇ω [Jac75] (µ is the permeability). This potential does not depend on the

azimuthal angle ϕ because of the rotational symmetry.

The Laplace equation in cylindrical coordinates is given by (e.g. [Jac75])

∂

∂r



∂ω

∂r



∂

∂ϕ

∂

∂z

= 0,

and the magnetic ﬁeld components are given by

=−µ

∂ω

∂r

and B

=−µ

∂ω

∂z

3.4 Basic electron optics: round magnetic lenses 145

Following Szilagyi [Szi88], we will express the potential ω(r, z) as a radial series,

as follows:

ω(r, z) =

∞



i=0

(z)r

. (3.2)

After substitution into the Laplace equation we ﬁnd

∞



i=−2

(

i + 2

)

i+2

(z)r

∞



i=0

(2)

(z)r

= 0,

where the superscript (2) denotes the second-order derivative with respect to z.For

i =−1 we ﬁnd a

(z) = 0 and therefore all odd functions a

2i+1

(z) vanish. Denoting

(z)byU (z) we also ﬁnd

(z) =−

(2)

(z)

;

(z) =−

(2)

(z)

(4)

(z)

;

(z) =···

from which we ﬁnd the general relation

(z) =

(−1)

(i!)

(2i)

(z). (3.3)

This leads to the following expression for the scalar magnetic potential:

ω(r, z) =

∞



i=0

(−1)

(2i)

(z)

(

)

. (3.4)

If we deﬁne the axial ﬂux density B(z)by

B(z) =−µ

∂ω

∂z

r=0

then we ﬁnd for the ﬁeld components

∞



i=1

(−1)

(2i−1)

(z)

i!(i − 1)!

)

2i−1

=−B



(z)

+ B

(3)

(z)

−···; (3.5)

∞



i=0

(−1)

(2i)

(z)

(i!)

)

= B(z) − B



(z)

+ B

(4)

(z)

+···. (3.6)

In other words, the magnetic ﬁeld components for a round lens at any point in space

are completely determined by the ﬂux density B(z) along the optical axis and all of

146 The transmission electron microscope

its derivatives. We will use the series expansions above to derive the paraxial ray

equation for a round magnetic lens in the next sections.

There are many different techniques for computing electrostatic and magnetic

ﬁelds, many of them relying on some form of Laplace’s equation for a domain with a

boundary of complicated shape. Finite element methods are among the more popular

methods for the calculation of ﬁelds. For an extensive review of these methods and

experimental measurements of ﬁeld conﬁgurations we refer to [HK89a, Chapters

6–13] and [Szi88, Chapter 3].

3.4.3 The equation of motion for a charged particle in a magnetic ﬁeld

We have seen in Chapter 2 that electrons in a TEM must be treated as relativistic

charged particles. Newton’s equation of motion,

= F,

with p = mv the so-called kinetic momentum and F the total external force, can

be rewritten for a particle with charge −|e| in an arbitrary, time-dependent electro-

magnetic ﬁeld:

d(mv)

=−|e|[E(r, t) + v × B(r, t)]. (3.7)

The right-hand side is the Lorentz force, a velocity-dependent force. If the mass m

is written in the relativistic form γ m

, then equation (3.7) describes the relativistic

motion of an electron. We recall that the magnetic ﬁeld B can be written as the curl

of the magnetic vector potential A (B =∇×A) or as the gradient of the magnetic

scalar potential ω in the absence of currents in the ﬁeld region, and the electric ﬁeld

E is equal to −∇ with  the electrostatic potential [Jac75]. For a conventional

transmission electron microscope, the electrostatic potential is zero along most of

the electron trajectory, except for the initial acceleration by the voltage E, and the

electrostatic lattice potential V (r) inside the crystal.

Equation (3.7), with B = 0, is used to describe the electron optical characteristics

of electron guns and electrostatic deﬂectors. For E = 0, the equation describes a

purely magnetic ﬁeld, and it is the starting point of trajectory analysis procedures

for magnetic lenses. The design of an electromagnetic device thus consists of two

separate problems:

(i) for a given electrode or lens geometry, the ﬁeld must be computed;

(ii) then, with this ﬁeld, the trajectory equation must be solved for particles traveling at

various distances from the electrode surfaces or the optical axis of the lens.

For a very general derivation of the relativistic trajectory equations for a combined

electrostatic and magnetic ﬁeld distribution in arbitrary curvilinear coordinates

3.4 Basic electron optics: round magnetic lenses 147

we refer to Szilagyi [Szi88, Chapter 2], whose derivation starts from the Euler–

Lagrange equations of motion. The equations of motion derived in this way are

valid for a general time-dependent magnetic ﬁeld, and are thus applicable to a wide

range of ﬁeld geometries, including synchrotrons, particle accelerators, electron

microscopes, cathode ray tubes, and so on. It should come as no surprise that the

literature on the subject of charged particle dynamics is vast, going back to the

second half of the nineteenth century. We refer the interested reader to the volumes

on Principles of Electron Optics [HK89a, HK89b] for an extensive list of papers

relevant to electron microscopy.

For the purposes of this textbook, it will be sufﬁcient to derive the relevant

equations from the Lorentz equation of motion (3.7) in cylindrical coordinates.

The basis vectors of the cylindrical coordinate system are dependent upon the

particular point were they are considered:

= cos ϕ e

+ sin ϕ e

;

=−sin ϕ e

+ cos ϕ e

;

= e







(3.8)

The time derivatives are given by (denoting a derivative with respect to time by a

dot above the variable):

= ˙ϕ e

;

=−˙ϕ e

;

= 0.







(3.9)

The velocity vector is given by

v =

r e

+r ˙ϕ e

z e

. (3.10)

We now have all the tools we need to express the Lorentz equation in cylindrical

coordinates, and after a simple computation (taking E = 0) we ﬁnd the following

three component equations:

d(m

− mr ˙ϕ

=−|e|r ˙ϕ B

; (3.11)

d(mr ˙ϕ)

+ m

r ˙ϕ =−|e|[

−

]; (3.12)

d(m

=|e|r ˙ϕ B

;

where m =



1 −

(

˙ϕ

)

. (3.13)

In this derivation we have used the fact that B

= 0, as it must be for an axially sym-

metric ﬁeld distribution. These equations are non-linear, coupled partial differential