Graef M. Introduction to conventional transmission electron microscopy

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128 Basic quantum mechanics

factor for copper, at temperatures of T = 300 and 1000 K; the scattering factor is

signiﬁcantly reduced for large scattering angles.

2.6.7 Numerical computation of the Fourier coefﬁcients

of the lattice potential

The quantity V

(2.83) is known as the electron structure factor [SZ92]. Some-

times it is convenient to rewrite the summation over all N atoms in the unit cell

as a double summation over all N

atoms in the asymmetric unit and over all

symmetry operators of the space group. If we use the Seitz symbol to denote a sym-

metry element (see equation 1.38 on page 40), then the electron structure factor

becomes





j=1

(s)



(D|t )

−2πig·(D|t )[r

]

The second summation is then essentially a sum over the orbit of each of the atoms

in the asymmetric unit. This expression is particularly useful for implementation

in a computer program, since one would typically ask the user to enter the space

group and the coordinates of the atoms in the asymmetric unit. One must take care

to remove double positions generated by the symmetry operators before computing

the summation.

We can now use equation (2.83) combined with the parameterization of the

electron scattering factors discussed in Section 2.6.1 to compute numerically the

Fourier coefﬁcients of the electrostatic lattice potential for an arbitrary crystal

structure. This computation is implemented in the subroutine

CalcUcg in the module

diffraction.f90 and described in PC-9 ; the routine returns the modulus and phase of

the complex Fourier coefﬁcient (and optionally also the Fourier coefﬁcient of the

absorption potential W

=|V

iθ

; W

=|W

iθ



from which the real and imaginary part can be computed through the use of Euler’s

formula e

= cos x + i sin x. The routine uses either the Doyle–Turner/Smith–

Burge or the Weickenmeier–Kohl atomic scattering and absorptive form factors. A

more complete discussion of the absorptive form factor can be found in Section 6.5.1

on page 371.

The source code for this book incorporates the standard isotropic Debye–Waller

factor in all scattering factor computations. The Debye–Waller factor must be en-

tered in nm

for each atom in the asymmetric unit. The Fourier coefﬁcients V

2.6 The electrostatic lattice potential 129

Pseudo Code PC-9 Compute the Fourier coefﬁcient V

(and W

Input: crystal data ﬁle [*.xtal], Miller indices h, k, l

Output: |V

|, θ

load crystal data { PC-2 , page 63}

generate all atoms in the unit cell {

CalcPositions

}

compute d

hkl

{CalcLength}

convert to scattering parameter s =

hkl

for each atom type do

compute f

(s) (including Debye–Waller factor) {2.75 or 2.76}

for each atom in orbit do

compute χ

= 2πg · r

compute cos χ

and sin χ

end for

add all cos χ

and sin χ

together for the orbit

multiply sum with f

(s)

add to total sum for V

{2.102}

end for

convert V

to modulus and phase

of the electrostatic lattice potential are then given by (using the Doyle–Turner

parameterization):

0.047 878 01





j=1



−B



i=1

−b





(D|t)

−2πig·(D|t)[r

]

. (2.102)

A similar expression is obtained for the Weickenmeier–Kohl version of the elastic

scattering factor ﬁtting function; it is left to the reader to write down the explicit

expression for this case. When  is given in nm

and s in nm

−1

, then V

is expressed

in volts. The factor between large parentheses must be computed only once for

each atom in the asymmetric unit, and for every value of the scattering parameter s;

the orbit (D|t)[r

] of each atom in the asymmetric unit can be precomputed before

the potential Fourier coefﬁcients are determined. This has been implemented in the

routine

CalcPositions in the module symmetry.f90.

As an example of the use of the

CalcUcg routine we will next discuss the com-

putation of the direct space electrostatic lattice potential for an arbitrary crystal

structure. The direct space potential is given by the 3D inverse Fourier transform:

V (r) =



2πig·r

130 Basic quantum mechanics

3000

2000

1000

3000

2000

1000

(a) (b)

3000

2000

1000

3000

2000

1000

3000

2000

1000

(e)

Fig. 2.12. Electrostatic lattice potential in volts in the plane (x, y, 0) for the fcc Al structure.

The number of contributing Fourier coefﬁcients increases from (a) to (d), according to

the entries in Table 2.4. Part (e) includes the room-temperature Debye–Waller factor for

aluminum B(300) = 0.008 33 nm

The potential at any given point r is thus determined by all reciprocal lattice points

g. Since the potential is strongly peaked at the atom positions we expect to need a

large number of Fourier coefﬁcients to represent the potential accurately.

Figure 2.12 represents the electrostatic potential V (x, y, 0) for a single unit cell

of the face-centered structure of aluminum. The atoms are located at the corners

and at the center of the unit cell face. The computation used the Doyle–Turner

parameterization of the elastic scattering factors. The potential is represented as a

shaded surface and was calculated on an array of 128 × 128 points. The “pixel size”

for this computation is thus 404.9/128 = 3.15 pm. The only difference between

the four shaded surfaces in Fig. 2.12 is the number of Fourier coefﬁcients entering

2.6 The electrostatic lattice potential 131

Table 2.4. Total number of families contributing to the electrostatic lattice

potential of aluminum as a function of the radius |g

max

| (in nm

−1

). The numbers in

this table correspond to the computed potentials in Fig. 2.12.

max

|=10 |g

max

|=20 |g

max

|=40 |g

max

|=80

No. of independent families 7 27 145 559

No. of allowed reﬂections 58 510 4140 19 236

the summation. The details of the computation are tabulated in Table 2.4: the total

number of Fourier coefﬁcients is determined by the truncation radius |g

max

|. The

larger the number of Fourier coefﬁcients, the more accurate the electrostatic lattice

potential. Since V (r) was deﬁned to be a positive potential (see page 94), the

occurrence of negative values is often an indication that an insufﬁcient number of

Fourier coefﬁcients have been used. Oscillations in the potential, in particular in

regions where no atoms are present, are also caused by premature truncation of the

Fourier summation.

In the ﬁrst chapter we deﬁned the concept of a family of lattice planes. All lattice

planes (hkl) related to each other by the point group symmetry operations form

a family {hkl}. It would be tempting, but incorrect, to assume that all members

of a family have the same structure factor F

hkl

(or V

hkl

). The lattice potential has

the symmetry of the complete space group, so that the symmetry of the potential

Fourier coefﬁcients will depend on all the space group symmetry elements and

not just on the point group symmetry. There is no a priori reason why the ﬁnal

summation in equation (2.102),



(D|t)

−2πig·(D|t)[r

]

should be the same for all members of a family {hkl}. The magnitude |V

| is the

same for all members of a given family, but the phase factor e

iθ

is not necessar-

ily the same. For centrosymmetric space groups, the phase factor amounts to a

simple plus or minus sign; for non-centrosymmetric space groups, the relations

between the phases θ

in a given family can be more complicated. Ultimately

the atom positions in the asymmetric unit cell determine what the phase factor

will be.

The computation of V (r) for an arbitrary rectangular section through the unit

cell of an arbitrary crystal structure has been implemented in the Fortran program

vr.f90. The algorithm is explained in PC-10 . To determine the members of each

family, it is convenient to use a 3D array of Booleans to make sure no reciprocal

132 Basic quantum mechanics

Pseudo Code PC-10 Compute the electrostatic lattice potential V (r).

Input: crystal data ﬁle [*.xtal]

Output: N

× N

array of potential values

load crystal data {

PC-2 }

generate all atoms in the unit cell {

CalcPositions}

ask user for |g

max

for each triplet for which |g| < |g

max

| do

if not member of previously considered family then

compute family {hkl} {

CalcFamily}

if reﬂection allowed by Bravais lattice centering then

compute and store V

hkl

for each family member { PC-9 }

end if

mark entire family in Boolean array

end if

end for

ask for section plane (three corner points)

ask for resolution N

× N

for every allowed family {hkl} do

for every member (hkl) of the family do

for each location in section plane do

compute χ = 2πig · r

compute cos χ + i sin χ

multiply by V

hkl

{complex number!}

add to total potential {only real part needed}

end for

store pixel coordinates and potential values

lattice points are either missed or double-counted. The program generates an ASCII

output ﬁle with pixel coordinates and potential values (in volts), which can then

be processed by an appropriate graphics program.

†

The implementation of the

program

vr.f90 does not make use of fast Fourier transforms; while one would

normally use 3D FFTs to speed up the transformation from reciprocal to direct

†

Most rendered drawings and contour plots in this book were generated with the Interactive Data Language

[Res98].

2.7 Recommended additional reading 133

space, the implementation in vr.f90 illustrates in more direct terms how the direct

space potential V (r) is determined by all Fourier coefﬁcients V

. A second program,

vrfft.f90, is provided which makes use of the FFTW library to compute the 3D Fourier

transform.

Figure 2.12(e) represents the same section of the electrostatic lattice potential of

Al, but includes the Debye–Waller factor B(300) = 0.008 33 nm

[GP99]. Since

vibrations cause the atom to appear larger, i.e. to occupy more space with a resulting

smaller electron density than the atom at rest, the elastic scattering factor for large

scattering parameters s is affected more strongly than the small s region. The peaks

in Fig. 2.12(e) are hence broadened and have a lower peak value.

2.7 Recommended additional reading

In addition to the references cited in this chapter, the following books may be of

interest to the reader. Basic texts on quantum mechanics:

Quantum Mechanics: a Modern Introduction, A. Das and A.C. Melissinos, Gordon and

Breach Science Publishers (New York, 1986).

Introduction to Quantum Mechanics, B.H. Bransden and C.J. Joachain, Longman Scien-

tiﬁc & Technical (New York, 1989).

An Introduction

to Theory and Applications of Quantum Mechanics

,A.Yari

v, Wiley

(New York, 1982).

Introductory books on the special theory of relativity:

Introduction to Special Relativity, R. Resnick, Wiley (New York, 1968).

Introduction to Special Relativity, W. Rindler, Oxford University Press (New York,

1982).

Space and Time in Special Relativity, N.D. Mermin, McGraw-Hill (New York, 1968).

A basic introduction to diffraction can be found in:

The Basics of Crystallography and Diffraction, C. Hammond, Oxford University Press

(New York, 1997).

Fourier transforms, convolutions, and their numerical implementation are

discussed in:

The Fourier Transform and its Applications, R.N. Bracewell, 3rd edition, McGraw Hill

(Boston, 2000).

Inside the FFT Black Box: Serial and Parallel Fast Fourier Transform Algorithms,E.Chu

and A. George, CRC Press (Boca Raton, 2000).

Algorithms for Discrete Fourier Transform and Convolution, R. Tolimieri, M. An, and

C. Lu, Springer-Verlag (New York, 1989).

134 Basic quantum mechanics

Additional information on inelastic electron scattering may be found in:

Fundamentals of Inelastic Electron Scattering, P. Schattschneider, Springer-Verlag (Wien,

1986).

Elastic and Inelastic Scattering in Electron Diffraction and Imaging, Z.L. Wang, Plenum

Press (New York, 1995).

Exercises

2.1 Show that the normalization constant C for the momentum plane wave eigenfunctions

(page 84) is given by h

−3/2

2.2 Derive a proof for the multiplication theorem (2.53) and the convolution theorem

(2.54) on page 106.

2.3 Derive an expression, similar to equation (2.102), for the Weickenmeier–Kohl pa-

rameterization of the elastic scattering factor.

2.4 Consider the drawing you have made for Exercise 1.2; determine graphically the

direction of an incident beam (with wave vector in the plane of the drawing) for which

the (201) planes and the (10

1) planes will simultaneously be in Bragg orientation.

What is the angle between the two diffracted beams (both the computed angle and

the angle measured from the drawing)? What is the wavelength of this incident

beam?

2.5 Use the Fortran libraries on the

website to write a program that creates a list of

scattering angles 2θ for an arbitrary crystal structure and a given acceleration voltage.

Each family of reﬂections {hkl} should only occur once

in this list.

2.6 Show that the DFT of a normalized Gaussian function

√

−a

is again a Gaussian function. What is the relation between the widths of this Fourier

transform pair?

2.7 Use the Fortran routines on the

website, in particular the fftw library, to write a

program to compute the convolution product of two one-dimensional functions of

arbitrary length.

2.8 Use the Fortran libraries on the

website to write a program that lists the real

and imaginary Fourier coefﬁcients of the electrostatic lattice potential for an ar-

bitrary crystal structure, using the Weickenmeier–Kohl atomic scattering factor

parameterization.

2.9 Consider a row of reciprocal lattice points ng (Fig. 2.13). If the angle between the

vectors g and k

is given by ω, show that, to a good approximation, the

excitation error of the reﬂection ng is given by

≈−n

|g|



2 cos ω + nλ|g|

1 + nλ|g|cos ω



Exercises 135

Ewald sphere

-2g

-g

-2g

-g

Fig. 2.13. Excitation errors for a row of reﬂections in a random orientation with respect to

the Ewald sphere.

In particular, for ω = π/2 show that the following parabolic approximation is valid:

≈−

λ|g|



≈−

)

1 −



1 − n

|g|



, (E2.1)

where the second expression is the exact solution which will be derived in Sec-

tion 3.9.5.

2.10 Determine the conditions for systematic absences for all different Bravais lattice

centering types.

2.11 Building on the previous exercise, write the structure factor for all seven non-primitive

Bravais lattices as the product of a structure factor due to lattice centering and a term

containing the atoms in the asymmetric unit. As an example, here is the expression

for the face-centered cubic lattice:

fcc

hkl

1 + (−1)

h+k

+ (−1)

h+l

+ (−1)

k+l

N/4



j=1

2πig·r

. (E2.2)

Under which conditions can the second term become equal to zero?

The transmission electron microscope

3.1 Introduction

In this chapter, we introduce the components of the transmission electron micro-

scope. After a brief review of the history of the electron and the electron micro-

scope, we will consider a cross-section of a real microscope (JEOL 120CX) and

identify the various stages and components inside the column. We will discuss

peripheral support systems (vacuum, compressed air, water cooling, etc.) without

which the microscope could not possibly function. Then we focus our attention on

the general properties of round magnetic lenses and discuss basic charged particle

dynamics in magnetic ﬁelds. We will illustrate the behavior of electrons in a mag-

netic ﬁeld by means of simpliﬁed paraxial trajectory simulations, and introduce the

concept of cardinal elements of a lens. Then we will consider deviations from the

paraxial behavior and introduce various lens aberrations of importance in electron

microscopy. We conclude the section on magnetic lenses with a brief overview of

multipole lenses, used as deﬂectors and correctors.

The electron gun is next, and we will introduce the basic gun types and their oper-

ational parameters, along with a discussion of beam properties (current, brightness,

coherence, etc.). In the remainder of the chapter, we break the microscope column

down into four regions: the illumination system, the goniometer stage and objective

lens, the imaging stage, and the detector stage. For each stage, we will describe the

common components, why they are present, and how the microscope operator can

affect their behavior. In particular, we will relate the geometry of the goniometer

stage to reciprocal space and Bragg’s Law. We will conclude this chapter with a

brief review of electron detection systems.

136

3.2 A brief historical overview 137

3.2 A brief historical overview

When Max Knoll and Ernst Ruska built the ﬁrst transmission electron micro-

scope

†

in Berlin in 1931–1934 [KR32a, KR32b, Rus34a, Rus34b], it took only

a few years before the ﬁrst commercial machines became available (Metropolitan

Vickers, 1936; Siemens, 1939; RCA, 1941; Hitachi, 1941; Philips, 1949; JEOL,

1949; etc.) [Hal53, Fuj86, Haw85]. Ruska’s 1933 microscope had three lenses, a

magniﬁcation of 12 000×, and a resolution of 50 nm, signiﬁcantly better than the

best optical microscope of that time.

The basic components of the TEM have not changed substantially since the late

1940s. The electronic circuitry, however, has changed signiﬁcantly from the pre-

transistor electronics of the ﬁrst microscopes to the sophisticated modern machines

that permit remote operation of nearly all microscope functions. There has been a

steady improvement of the quality of the main lenses in the microscope, the stabil-

ity of high voltage and lens currents, and an increased theoretical understanding of

lens aberrations and how they can be corrected. This evolution continues to date,

and in this chapter we will on occasion cite current research in various areas of

electron optics.

It is fascinating to consider the history of the discovery of the electron, which

preceded the electron microscope by about 35 years. Indirect evidence for the

existence of a fundamental charged particle was only slowly obtained, starting

with the extensive electrolysis experiments by Faraday in the 1830s [Far39]. The

name electron was ﬁrst used by Stoney in 1891, when he named the fundamental

unit of charge [Sto91]. It was only after the actual discovery of the electron itself

in 1897

‡

that the name was also used to denote the particle. For an overview of

selected important events around the turn of the nineteenth century leading up to

the construction of the ﬁrst electron microscope we refer to Table 3.1.

The currently accepted value for the unit of electric charge is e = 1.602 177 ×

−19

C, and the rest mass of the electron is m

= 9.109 389 × 10

−31

kg [CT95].

Additional physical constants associated with the electron may be found in

Table A2.1 on page 665 (Appendix A2).

For a detailed account of the early days of the electron we refer to the wonderful

book by A. Pais: Inward Bound: of Matter and Forces in the Physical World [Pai86],

which provides a wealth of historical information and anecdotes about the main

players in the ﬁeld. The ﬁrst chapter of the book Electron: a centenary volume

discusses the role of J.J. Thomson in the discovery of the electron [Spr97]. For a de-

tailed account of the history of electron microscopy in various countries worldwide

†

An earlier two-lens version of an electron microscope with a magniﬁcation of 17× was built in 1931, but its

resolution was worse than that of an optical microscope.

‡

One could debate about the actual date of the discovery. Some authors prefer the year 1899 as the year of

discovery, since J.J. Thomson determined e in that year.