Graef M. Introduction to conventional transmission electron microscopy

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88 Basic quantum mechanics

superposition of two waves, e.g. surface waves on a liquid, can give rise to interfer-

ence patterns: at positions where the waves are in-phase, constructive interference

will give rise to a displacement equal to the sum of the two wave amplitudes,

whereas partial or complete destructive interference occurs when the waves are

out-of-phase. A crucial requirement for this type of classical wave interference to

occur is that both waves must be present simultaneously. In quantum mechanics it

is possible for a single particle (or wave) to give rise to an interference pattern.

The double-slit experiment provides a standard illustration of this phenomenon

[FLS63]. A screen with two identical slits A and B is placed between a particle

source S and a detection screen D. If one of the slits, say A, is covered, then the

particles which pass through the other slit, B, will arrive at the detector with a certain

spatial distribution, described by a probability function |ψ

(indicated in Fig. 2.1

by a thin solid line). The same happens when slit B is covered. When both slits

are uncovered, the detector cannot determine through which of the slits the particle

has traveled, and, hence, the intensity distribution on the detector is given by the

function |ψ

+ ψ

(the oscillating thicker line in Fig. 2.1), and not by the sum

|ψ

+|ψ

. If, on the other hand, the experiment is modiﬁed such that it can be

determined through which of the two slits the particle traveled, then the intensity

distribution is given by the sum of the individual distributions |ψ

+|ψ

(the

dashed line in Fig. 2.1). The difference between the two patterns is the interference

term ψ

∗

+ ψ

∗

Fig. 2.1. Schematic illustration of the double-slit interference experiment (details given in

text).

2.3 Elements of the special theory of relativity 89

It is important to emphasize that this type of quantum mechanical interference

is entirely due to the way the experiment is carried out. This is an example of

how the observer is intricately linked to the experiment. In the standard double-slit

experiment one cannot distinguish electrons that passed through slit A from those

which passed through slit B, and therefore interference will occur. This interference

will occur even when the time between subsequent particles is large, say one particle

per minute. Even then interference will occur because we still cannot distinguish

between the two slits. We can say that the particle interferes with itself, in a way

determined by the layout of the experiment. This is precisely what happens in a

TEM. At any given moment in time, only one single beam electron is present in

the sample (see Section 3.7.6). The crystalline specimen acts as a “screen” with

multiple slits. Since we do not attempt to determine where exactly the electron

traveled through the sample (i.e. by which atom(s) it was scattered), the wave

function of the electron will interfere with itself, and form an interference pattern

on the detector. We will see later on that the microscope operator plays an important

role in determining the type of interference pattern that will be formed, and which

component of the pattern is used to obtain an image.

2.3 Elements of the special theory of relativity

2.3.1 Introduction

In modern transmission electron microscopes, the electrons travel at substantial

fractions of c, the velocity of light. Thanks to the efforts of Lorentz, Einstein, and

many others, we now know that such fast motion obeys the laws of the special

theory of relativity; the classical Newtonian laws of motion must be modiﬁed to

take into account the fact that c is the maximum velocity for any type of motion.

In our daily lives this is not an important fact, since we “know” from personal

(i.e. classical) experience that velocities are additive quantities. When observer

B moves with respect to observer A with a velocity v along the x-direction, and

observer C moves with respect to B with velocity u in the same direction, then

the relative velocity between C and A is simply the sum w = u + v. This is the

principle of relativity, ﬁrst formulated by Newton. The coordinate transformation

associated with such relative velocities is known as a Galilean transformation, and

is mathematically described by













= x − vt;



= y;



= z;



= t.

90 Basic quantum mechanics

All laws of classical physics are invariant under a Galilean transformation. It was

noted by various researchers around the end of the nineteenth century, that the

Maxwell equations of electrodynamics are not invariant under such a simple trans-

formation. In 1887 Voigt [Voi87] and subsequently Lorentz in 1904 [Lor04] dis-

covered that the Maxwell equations are invariant under the following remarkable

transformation:













= γ (x − vt);



= y;



= z;



= γ (t − vx/c

(2.25)

where

γ =



1 − β

and β =

. (2.26)

It is easy to see that for small velocities v, these equations reduce to those for

the Galilean transformation, since lim

c→∞

γ = 1. Equations (2.25) are now known

as a Lorentz transformation, sometimes referred to as a boost, and the modiﬁed

principle of relativity states that all laws of physics must be invariant under such a

transformation.

From these equations, we can derive the Einstein addition law for velocities

(referring to the same reference frames A, B and C as before):

w =

u + v

1 + uv/c

. (2.27)

Einstein was the ﬁrst to fully understand the implications of the Lorentz trans-

formation on the concepts of space and time. He showed that Newton’s equations

of motion could be made invariant by making the mass of a particle dependent on

its velocity with respect to a reference frame. In particular, Newton’s equation of

motion can be made invariant under a Lorentz transformation by the substitution:

m = γ m



1 − v

. (2.28)

Newton’s equation of motion under an applied force F then reads as

F =

(mv) = m



1 − v

= m

(γ v). (2.29)

This will be the starting equation for the analysis of electron trajectories in an

electromagnetic ﬁeld; the force F is then the Lorentz force, as we will see in

Chapter 3.

2.3 Elements of the special theory of relativity 91

For an extensive in-depth discussions of the special theory of relativity we refer

the reader to the following books: [Wey22], [FLS63], and [MTW73]. For the pur-

poses of this book it is sufﬁcient to list a few consequences of the theory that are

of importance for electron microscopy.

(i) Equivalence of mass and energy: the kinetic energy T of a particle traveling with a

velocity v is given by

T = c

(m − m

) = m

(γ − 1).

In the non-relati

vistic limit when

β → 0, one can

readily show that

γ → 1 +

+···

and substitution in the above expression for the kinetic energy T results in the standard

expression T =

. The energy relation is often rewritten as

= m

+ T, (2.30)

which states that the total energy equals the sum of the rest energy and the kinetic

energy. This relation expresses the equivalence of mass and energy.

(ii) Lorentz–Fitzgerald contraction of length scales: The Lorentz–Fitzgerald contraction

refers to the fact that an object of length l = x

− x

to one observer acquires a length



− x



= l/γ to an observer in the primed coordinate system. The contraction occurs

along the direction of motion. In the TEM this means that when an electron traveling

at a velocity v = βc encounters a crystal

of thickness

(measured in the

reference

frame of the microscope), then the electron will “see” a crystal of thickness z

/γ in

its own reference frame. As we will see in the next section, a 400 kV electron travels

at 83% of the velocity of light, which means that a 100 nm thick crystal appears only

55.7 nm thick to the electron.

(iii) Einstein dilatation of time scales: the duration of an interval in the unprimed reference

system, t

− t

(for a given location in space) becomes t



− t



= γ (t

− t

) in the

primed reference system. This is often restated as “stationary clocks run slow”.

In the following subsection, we will rederive the expression for the wavelength of

the electron, this time making use of the relativistic relations for mass and energy.

2.3.2 The electron wavelength (relativistic)

The total energy of the relativistic electron is given by equation (2.30):

= mc

When the electron is accelerated by a potential drop E, it acquires a potential energy

eE, which is completely converted into kinetic energy T when the particle leaves

92 Basic quantum mechanics

the electrostatic ﬁeld. The potential energy is then given by

eE = (m − m

. (2.31)

The wavelength according to de Broglie is given by

λ =

. (2.32)

The mass of a relativistic particle increases with increasing velocity according to

relation (2.28) which, using (2.31), can be rewritten as (following [HK89a])

γ = 1 +

= 1 + ω,

where ω is introduced as a shorthand notation. From the deﬁnition of γ we can

derive the electron velocity:

v = c

1 −

1 + ω



(1 + ω)

− 1.

The momentum p is then given by

p = mv = m

(1 + ω)v = m



2ω + ω



1 +



Substitution into the de Broglie expression for the wavelength yields:

λ =

(

)

1 +

Substitution of the physical constants listed in Appendix A2 then leads to

λ =

1 226.39

√

E + 0.978 45 × 10

−6

, (2.33)

where again the wavelength is given in picometers when the voltage E is ex-

pressed in volts. Table 2.2 lists the relativistic wavelengths for a few commonly

used voltages. Note that the relativistic correction increases with increasing voltage;

at E = 400 kV, the electrons travel at 83% of the speed of light and have a mass

78% heavier than the rest mass, while at E = 3 MeV the electron mass increases

by nearly 700%.

2.3 Elements of the special theory of relativity 93

Table 2.2. Relativistic acceleration potential

, electron wavelength λ,

wavenumber K

= 1/λ, mass ratio γ = m/m

, relative velocity β = v/c, and

interaction constant σ for various acceleration voltages E.

E (kV)

 (V) λ (pm) K

(nm

−1

) m/m

β = v/c σ (V

−1

)

100 109 784 3.701 270.165 1.196 0.548 0.009 244

120 134 090 3.349 298.577 1.235 0.587 0.008 638

200 239 139 2.508 398.734 1.391 0.695 0.007 288

300 388 062 1.969 507.937 1.587 0.777 0.006 526

400 556 556 1.644 608.293 1.783 0.828 0.006 121

800

1 426 224 1.027 973.761 2.566 0.921 0.005 503

1000 1 978 475 0.872 1146.895 2.957 0.941 0.005 385

1250 2 778 867 0.736 1359.228 3.446 0.957 0.005 296

2000 5 913 900 0.504 1982.876 4.914 0.979

0.005 176

3000 11 806 277 0.357 2801.657 6.871 0.989 0.005 122

It is convenient to introduce a new notation for the relativistically corrected

acceleration potential [HK89a]:

 ≡ E



1 +



= E +  E

, (2.34)

where  = 0.978 45 × 10

−6

−1

. The quantity

 is known as the relativistic ac-

celeration potential. At an acceleration voltage of 200 kV, the electron experiences

a relativistic acceleration potential of 239.1kV.

We also introduce a new parameter σ , deﬁned by

σ ≡

2πγm

eλ

, (2.35)

which is known as the interaction constant. It has dimensions of V

−1

when

lengths are expressed in nanometers. The interaction constant will become useful

later on in this book when we introduce the dynamical diffraction equations. One

can surmise from the last column in Table 2.2 that σ approaches a constant value

with increasing acceleration voltage E. Formally this limit is computed as follows:

lim

E→∞

σ =

2πm

lim

E→∞

γλ =

2πm

lim

ω→∞

(1 + ω)

2ω + ω

0 12 3

lim=1

The numerical value of the limit is given by 0.005 067 73 V

−1

. The ratio

h/m

c is known as the Compton wavelength λ

= 2.4263 pm. The Compton

94 Basic quantum mechanics

wavelength is the non-relativistic wavelength of an electron traveling at the ve-

locity of light (from the de Broglie equation 2.21), i.e. an electron with momentum

p = m

Starting from the acceleration voltage E, the Fortran function

CalcWaveLength

in the module diffraction.f90, available from the website, computes all the entries on

a single line of Table 2.2.

2.3.3 Relativistic correction to the governing equation

Dirac was among the ﬁrst to note that the Schr¨odinger equation is not invariant under

a Lorentz transformation, and he then proceeded to derive the equation which now

bears his name: the Dirac equation [Dir28a, Dir28b]. While relativistic quantum

theory is far beyond what we need for this textbook, it is interesting to note that

both the spin of the electron and the existence of the positron (and anti-matter in

general) are direct consequences of this theory. We refer to [Tha92] and [Gre90]

for an in-depth introduction to relativistic quantum theory.

One can explicitly work out the Dirac equation for a relativistic plane wave

incident upon a crystal, but the mathematics is fairly involved and beyond the scope

of this book. Fujiwara [Fuj61] has shown in a fundamental paper that the Dirac

equation can be simpliﬁed for scattering of high-energy electrons, and that spin

can be neglected entirely. He concluded that the stationary Schr¨odinger equation

(2.20) can be used to describe high-energy electron scattering, provided the electron

wavelength and mass are replaced by their relativistically corrected values. For a

critical analysis of this derivation we refer to [FHS86].

In this section, we will follow an alternative derivation, which starts from the gen-

eral expression for a time-independent wave equation of a wave with wavenumber

(known as the Helmholtz equation):

 + 4π

 = 0. (2.36)

This equation is valid for electromagnetic radiation with wavenumber K

, but it

can also be used to derive the relativistically corrected Schr¨odinger equation for

the electron scattering process [Rei93].

We deﬁne the electron-optical refractive index n as the ratio of the wavelength of

the electron outside and inside the crystal. If the crystal is represented by a positive

potential energy term V , then the energy of the electron inside the crystal is given

by E + V . The wavelength of the electron can be written in terms of the relativistic

acceleration potential as

λ =





and λ





2.3 Elements of the special theory of relativity 95

where



is the relativistic acceleration potential inside the crystal. Explicit com-

putation of



yields



= E + V + (E + V )

;

= E +  E

+ V (1 + 2 E) + V

;

∼

 + γ V ,

where we have used the fact that V

is much smaller than the other terms in

the equation. Since V depends on position in the crystal,



does so as well. The

electron-optical refractive index is then computed as follows:

n =



(

1 +

γ V



In the presence of refraction (i.e. when n is not equal to 1) we must replace the

factor K

in equation (2.36) by n

, which yields

 + 4π

 =  + 4π



1 +

γ V





 = 0,

from which we ﬁnd, after rearranging terms and substituting m for γ m

 + 4π

 =−4π

U (r). (2.37)

= 1/λ is the relativistic wavenumber and

U (r) ≡

2me

V (r) =

πλ

V (r), (2.38)

where we have used the deﬁnition of the interaction constant σ in equation (2.35).

Equation (2.37) will form the starting point for all theoretical derivations in the

remainder of this book. For a much more detailed and mathematically more rigor-

ous approach to the derivation of the relativistic wave equation we refer to [Fuj61],

[FHS86] and to Chapters 55 and 56 in [HK94]. Note that some authors (e.g. [Rei93])

deﬁne the electrostatic potential to be a negative potential, so that the minus sign in

equation (2.37) changes to a plus sign. The differences between the standard non-

relativistic Schr¨odinger equation (2.20) and equation (2.37) are the relativistic val-

ues for both the electron mass (m = m

γ ) and wavelength (λ from equation 2.33).

Technically, equation (2.37) is not the Schr¨odinger equation, but a scalar form of

the Dirac equation (known as the Klein–Gordon equation), which looks just like the

standard Schr¨odinger equation. Equation (2.37) correctly includes all relativistic

effects relevant to the scattering of high-energy electrons, and we will return to it

in Chapter 5.

96 Basic quantum mechanics

(hkl)

(a) (b)

Fig. 2.2. (a) Geometrical construction leading to the direct space Bragg equation; (b) the

incident and diffracted directions and the plane normal must lie in a planar

section through

a conical surface with its top in the plane.

2.4 The Bragg equation in direct and reciprocal space

2.4.1 The Bragg equation in direct space

Consider the drawing in Fig. 2.2(a). A plane wave with wavelength λ is incident

upon a set of parallel planes with Miller indices (hkl). The incidence angle is θ.

If we regard each plane as a semi-transparent mirror, then part of the incident

intensity will pass through the ﬁrst plane, and part will be reﬂected. For an ideal

mirror we can apply Snell’s law, which states that the incident and reﬂected angles

must be equal, and both directions must be coplanar with the normal n to the (hkl)

plane.

The same process occurs on the second plane. This wave has traveled a bit further

than the ﬁrst wave, so for an observer far away from the crystal the two reﬂected

waves may arrive somewhat out of phase. The condition for in-phase arrival is

derived easily as follows: the path length difference  between waves 1 and 2 is

given by the sum of the distances PO



and O



Q. These distances can be expressed

in terms of the interplanar spacing d

hkl

and the angle θ as follows:

 = PO



+ O



= d

hkl

sin θ + d

hkl

sin θ,

= 2d

hkl

sin θ.

For in-phase arrival at the observation point, also known as constructive interfer-

ence, this path length difference must be equal to an integral number of wavelengths

nλ,or

hkl

sin θ = nλ. (2.39)

2.4 The Bragg equation in direct and reciprocal space 97

This is the direct space Bragg equation, ﬁrst derived in 1915 by W.H. Bragg [Bra15].

The integer n labels the various diffraction orders for a given set of planes.

In the electron microscopy world it is common practice to move the integer n to

the left-hand side of equation (2.39),

hkl

sin θ = λ.

We recall from the discussion of Miller indices in the previous chapter that the

planes with Miller indices (nh nk nl) are parallel to the planes (hkl), but with an

interplanar spacing equal to

nh nk nl

hkl

So instead of talking about nth-order diffraction from the planes (hkl) we may

equivalently talk about ﬁrst-order diffraction from the planes (nh nk nl), even when

those planes do not physically exist in the crystal lattice (i.e. they do not have any

atoms located on them). We thus arrive at the working version of the Bragg equation:

hkl

sin θ = λ. (2.40)

The angle θ is known as the Bragg angle. This equation should be used with

the understanding that, from a diffraction point of view, the planes (nh nk nl) are

not equivalent to the planes (hkl), and that only ﬁrst-order diffraction should be

considered.

The Bragg equation describes the geometric condition for diffraction to occur. It

does not say anything about the intensity of the diffracted wave. The Bragg equation

is derived with respect to a speciﬁc plane, and states that the incident and diffracted

waves must travel in directions which lie on a conical surface with top in the

diffracting plane and opening angle π/2 − θ , as illustrated in Fig. 2.2(b). In other

words, for each plane (hkl) in a crystal there is a conical surface centered around

the plane normal with opening angle determined by the interplanar spacing d

hkl

and the radiation wavelength λ. While the Bragg equation in direct space is rather

simple and elegant, it is not very useful when one wishes to determine the absolute

direction of a diffracted wave for a given crystal structure, since both the incident

wave direction and the crystal orientation must be speciﬁed with respect to some

reference frame. The direct space version of the Bragg equation is mostly used to

compute the diffraction angle 2θ for a given wavelength and crystal structure; this

has been implemented in the function

CalcDiffAngle in the module diffraction.f90,

available from the

website. To incorporate absolute wave directions it is common

practice to convert the Bragg equation to the reciprocal reference frame, which

leads to the Ewald sphere construction, as described in the next section.