Graef M. Introduction to conventional transmission electron microscopy

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588 Phase contrast microscopy

and we must resort to image simulations to conﬁrm a particular model structure. We

have already seen in Chapters 5–7 how we can compute the exit plane wave for any

given crystal structure or atom arrangement. The multi-beam dynamical scattering

formalism allows us to take a conﬁguration of atoms and an incident beam with

a given energy and orientation, and calculate how the amplitude and phase of this

beam are modiﬁed by the sample. Ultimately, we are interested in ﬁnding out where

all the atoms are, and what types of atoms are present at particular positions. In a

sense, we are fortunate that there are only 92 possibilities for the atom species; in

addition, for perfect crystal structures we can make use of the powerful machinery

of symmetry and group theory. The total number of independent variables may,

therefore, be very small. For instance, for pure copper we would need the lattice

parameter a, the atomic number of Cu (29), the space group number, and the position

of one atom (0, 0, 0), to unambiguously deﬁne the entire structure.

When it comes down to determining a structure (i.e. all atom positions and types

in the asymmetric unit), we must make sure that there is sufﬁcient information

available to determine unambiguously each of the independent parameters. The

following question must therefore be answered: how much information can be

transferred by the microscope? Or, equivalently, how many independent degrees

of freedom can be determined? To answer this question we will follow Van Dyck

and de Jong [VDdJ92].

Since the microscope generates 2D images, we need to determine how many

independent degrees of freedom are transferred per unit area of the image. We

introduce the parameter ρ as the smallest image dimension that can contain mean-

ingful information; we will see later on that ρ is the so-called information limit of

the microscope. For now, it is sufﬁcient to simply assume that this distance can be

calculated in terms of the electron optical parameters of the microscope. The small-

est meaningful area in the image is then equal to ρ

. The distance ρ corresponds to

a distance q = 1/ρ in reciprocal space; q represents the largest spatial frequency

that can meaningfully contribute to the image. All higher frequencies should be

excluded from the image by means of the diffraction aperture. The area covered by

this aperture is πq

= π

. Inside the aperture, we have several reciprocal lattice

points. Each reciprocal lattice point is characterized by the real and imaginary parts

of a complex number, which corresponds to two degrees of freedom for each point.

Because of Friedel’s law the diffraction pattern is centrosymmetric, which means

that there is really only one degree of freedom per reciprocal lattice point. For a

crystal structure with lattice parameter a the area of the reciprocal unit cell is equal

to 1/a

. The total number of degrees of freedom (i.e. reciprocal lattice points) inside

the diffraction aperture is then given by

πa

10.2 The microscope as an information channel 589

The number of degrees of freedom per unit area of the image

is given by n

which means that there are about three degrees of freedom per area ρ

. These

degrees of freedom could represent atom coordinates or atom types.

We will see later on that typically ρ = 0.05–0.15 nm, which is in the range of

typical interatomic distances. If we have a crystal structure which is oriented such

that atom columns line up along the beam direction (i.e. a low-index zone axis

orientation), then the projected crystal structure will contain a small number of

projected atom positions per unit area, and if our microscope has the proper ρ,

then we can, in principle, determine those atom positions. If the crystal is tilted

with respect to the electron beam, so that atoms overlap each other in the projected

image, then the number of atoms per unit area will be too large, and it will be

impossible to determine the coordinates of all the atoms. This means that there is

virtually no chance of determining the atom coordinates of a thin amorphous foil

using transmission electron microscopy methods. For high-symmetry zone axis

orientations, on the other hand, the microscope may transfer sufﬁcient information

for the determination of all independent atom coordinates. It is, therefore, quite

important that we learn how the microscope affects the structural information as

the electron wave travels down the column. In the following sections, we will

describe in great detail how lens aberrations affect the phase and amplitude of the

electron exit wave.

10.2.1 The microscope point spread and transfer functions

In the theoretical Chapters 5–7, we described how the exit wave function can be cal-

culated for a variety of observation conditions: two-beam, systematic row, and zone

axis. In this section, we will analyze how the exit wave propagates to the observation

plane. We already know that for an ideal microscope the intensity distribution I (r)

in the image plane is given by |ψ(r)|

, where ψ(r) is the exit plane wave function.

†

This means that there is a one-to-one correspondence between the exit wave and

the image plane wave; all electrons that leave a single exit plane point converge

to a single image plane point, regardless of their trajectory. For a real lens the

electrons which leave a single exit plane point will not, in general, arrive at the

same image plane point, but instead they will arrive in a neighborhood around

this point. The resulting image will be a “blurred” version of the ideal image. The

illuminated region on the sample typically has an area in the range of several tens

†

Throughout this chapter we will ignore both image magniﬁcation and rotation, since these are simple linear

transformations which do not distort the wave function.

590 Phase contrast microscopy

of square nanometers to several hundreds of square micrometers, which is small

compared with the physical dimensions of the lens. We do not expect the imaging

properties of the lens to vary much across this region of interest, and this allows us

to introduce the so-called isoplanatic approximation: the imaging properties of the

objective lens are the same across the entire region of interest.

When we discussed the double-slit experiment in Chapter 2, only two beams

originating from a single source were considered. In the present case, every exit

plane point is effectively a point source from which spherical waves originate. The

amplitude and phase of these waves is described by the amplitude and phase of

ψ(r). The electrons which leave a given exit plane point will travel in discrete di-

rections described by the Bragg equation, and, similar to the double-slit experiment,

electrons in different scattered beams can interfere with each other. This means that

the mathematical formalism describing the image formation process must allow for

such an interference between all pairs of beams. We will postpone a more detailed

discussion of this formalism until Section 10.2.2. For now, we will assume that the

transmitted beam is signiﬁcantly stronger than any of the scattered beams, so that

only interference effects between a scattered beam and the transmitted beam are

taken into account. This is known as linear image formation.

The two approximations in the preceding paragraphs allow us to write the imaging

properties of the lens as a simple convolution product of the exit plane wave function

with the function T (r), the point spread function. The image plane wave function

is then given by

(r) = ψ(r) ⊗ T (r). (10.1)

For an ideal lens, the point spread function reduces to the Dirac delta-function, and

= ψ. For a real lens, the point spread function has a ﬁnite width, and it should

be obvious to the reader that the narrower this width, the closer the image wave will

approximate the exit plane wave. However, intuition is not going to take us very

far because the point spread function is a complex function which will affect the

amplitude and phase of the exit wave in different ways. In particular, the amplitude

and phase of the exit plane wave will become mixed in the image plane, which

means that image interpretation will not be straightforward.

The Fourier transform of equation (10.1) is given by

(q) = F[ψ(r) ⊗ T (r)] = ψ(q)T (q), (10.2)

where q is a vector in the lens back focal plane. We see that the effect of the ob-

jective lens under isoplanatic, linear imaging conditions is simply a multiplication

by the function T (q) in the back focal plane of the lens. T (q) is known as the co-

herent transfer function of the lens, or, more commonly, as the microscope transfer

function. For the ideal lens, we have T (q) = 1, since the Fourier transform of a

10.2 The microscope as an information channel 591

delta-function is a constant. In what follows, we will describe the components of

the transfer function in detail.

10.2.1.1 The aperture function

We can eliminate all but one or a few scattered beams by means of a physical

aperture in the back focal plane (BFP) of the objective lens, so we will need to have

a mathematical description in the form of an aperture function. We introduce the

diffraction aperture function A(q), which is deﬁned by

A(q − q

) =

1 |q − q

| < q

0 |q − q

|≥q

(10.3)

where q

is the aperture radius in reciprocal length units, and q is a 2D vector in

the BFP (i.e. a reciprocal space vector). The center of the aperture is determined

by the position vector q

; for most observations the aperture is centered around

the optical axis and therefore q

= 0. We will see in Section 10.4 on Lorentz mi-

croscopy methods that the aperture need not always be centered on the optical axis.

In mathematical terms, the aperture provides a truncation point for the reciprocal

space Fourier expansion of the lattice potential. Only lattice periodicities with spa-

tial frequencies smaller than the reciprocal aperture radius can contribute to the

image.

†

It is a trivial matter to measure the reciprocal radius of the diffraction aperture.

The “true” (i.e. direct space) radius of the aperture is usually known, since the

microscope manufacturer speciﬁes the diameters of all the apertures in the column.

Even when custom aperture strips are used, it is still a trivial matter to calibrate

the reciprocal aperture radius. All that is needed is a material with a known lattice

parameter, say Ti with a = 0.255 and c = 0.468 nm. Insert the sample in the column

and obtain a zone axis orientation, say [11.0]. Since the lengths of the reciprocal

lattice vectors are known (1/d

1.0

= 3.914 nm

−1

and 1/d

00.2

= 4.272 nm

−1

), a

simple multiple exposure of the zone axis pattern with and without the aperture

provides an immediate value of the aperture radius, as shown schematically in

Fig. 10.2. The aperture radii are R

= 7.94 nm

−1

= 19.9 mrad, R

= 4.45 nm

−1

11.2 mrad, and R

= 1.83 nm

−1

= 4.58 mrad. The location of the aperture center

can also be measured easily and is typically expressed in the same way as the Laue

center; i.e. as a fractional coordinate with respect to two g-vectors.

10.2.1.2 The phase transfer function

We have already seen one way in which the objective lens can change the phase

of the exit plane wave. In Section 5.6.2, we derived that the Fresnel propagator

†

Assuming linear imaging conditions hold, non-linear contributions may have higher spatial frequencies than

those admitted by the aperture.

592 Phase contrast microscopy

00.2

11.0

Fig. 10.2. Calibration of the reciprocal diffraction aperture radius using the [11.0] zone

axis pattern of Ti as an internal standard.

expresses how a wave travels over a distance in free space. When we change the

current through the objective lens, we effectively change the location of the image

plane along the optical axis by an amount  f , which we will call the image defocus.

In the back focal plane, the Fresnel propagator can be written as

 f

(q) = e

−πiλ fq

, (10.4)

and this phase factor expresses how the phase of the exit plane wave is af-

fected by a deviation from the perfect (Gaussian) focus. We have for the phase

shift

χ

 f

(q) =−πλfq

. (10.5)

A magnetic lens will impart other phase changes to the back focal plane wave.

We have seen in Chapter 3 that the main aberration of the objective lens is spherical

aberration, expressed by the coefﬁcient C

. Spherical aberration results in a shorter

effective focal length for electrons which travel at an angle β with respect to the

optical axis. Following Reimer [Rei93, page 82], we will compute the path length

difference between a trajectory at an angle β with respect to the optical axis and a

trajectory along the optical axis.

Consider the trajectories in Fig. 10.3. The trajectory at an angle β would, in

the absence of spherical aberration, intersect the optical axis in the image plane.

Spherical aberration causes an additional bending towards the optical axis by an

angle ω. The trajectory intersects the image plane at a distance C

M, with M the

lens magniﬁcation. This causes a path length difference given by the distance ds,

10.2 The microscope as an information channel 593

ds = ωR

Fig. 10.3. Schematic of the path length difference computation for spherical aberration.

which is equal to ωR. The path length difference increases with increasing distance

R, so the total path length difference s is given by the integral

s =

ds =

ω dR.

The angle ω is given by

ω =

where we have used the fact that v = fM and R = fβ. This is a trivial integral

and we ﬁnd

s =

ω dR =

. (10.6)

If we convert from the angle β to the back focal plane variable q, using β = λq,

we ﬁnd for the path length difference

s =

. (10.7)

This can be converted into a phase difference by multiplication with the factor 2π/λ

to result in

χ

(q) =

. (10.8)

Combining equations (10.5) and (10.8), we ﬁnd for the total phase shift due to

defocus and spherical aberration:

χ(q) = π



− λ fq



. (10.9)

594 Phase contrast microscopy

Aberration function χ(Q)

-10

-20

Dimensionless spatial frequency Q

0123

D=0.0

0.5

1.0

1.5

2.0

2.5

3.0

Fig. 10.4. Aberration function (10.12) for several values of the dimensionless defocus

value D.

Positive values of  f are known as underfocus values, whereas negative values are

overfocus values.

†

It is convenient to use dimensionless variables in equation (10.9),

so that the equation becomes independent of the electron wavelength. Following

Kirkland [Kir98, page 14], we deﬁne the following variables:

Q = q(C

)

1/4

; (10.10)

D =

 f

√

. (10.11)

The quantity

√

λ is known as 1 Scherzer or 1 Sch; (C

)

1/4

is 1 Glaser or 1 Gl.

It is easy to show that the total phase shift χ(Q) becomes

χ(Q) = π



− DQ



. (10.12)

This function is known as the aberration function

‡

and is represented in Fig. 10.4

for several values of the dimensionless defocus D. This function is valid for

†

It is possible to use an alternate sign convention for which underfocus is negative. In that case the second term

in equation (10.9) would be preceded by a + sign. The reader should always check which sign convention is

being used, in particular when entering parameters into a simulation program.

‡

Some authors do not include the factor 2π in the deﬁnition of the aberration function.

10.2 The microscope as an information channel 595

every microscope, regardless of electron wavelength and spherical aberration

constant.

Example 10.1 Consider a microscope operated at 300 kV, with a spherical aber-

ration of C

= 1.2 mm and a defocus value of  f = 45 nm. What are the dimen-

sionless parameters D and Q for the (111) planes of aluminum? What would be

the phase shift for this situation?

Answer: Aluminum has a lattice parameter of 0.4049 nm, so the q value for the

(111) planes is equal to q = 1/d

111

= 4.277 nm

−1

. The electron wavelength λ =

1.969 pm, so we ﬁnd

Q = q





1/4

= 4.277[1.2 × 10

× (1.969 × 10

−3

)

]

1/4

= 1.323.

The dimensionless defocus is given by

D =

 f

√

1.2 × 10

× 1.969 × 10

−3

= 0.926.

The phase shift is given by equation (10.12):

χ(Q) = π



− DQ



= 3.1416 ×



0.5 × 1.323

− 0.926 × 1.323



=−0.28 rad.

Example 10.2 For the parameters of the previous example, what would be the

spherical aberration and the defocus if a 400 kV microscope were used with the

calculated values of Q and D?

Answer: The electron wavelength is λ = 1.644 pm, so the spherical aberration is

given by

1.323



1.644 × 10

−3



× 4.277

= 2.06 mm.

The defocus is then given by

 f = D



λ = 0.926



2.06 × 10

× 1.644 × 10

−3

= 53.89 nm.

In Chapter 3, we introduced the ﬁve primary or Seidel aberrations: spherical aber-

ration, coma, distortion, ﬁeld curvature, and astigmatism. Each of these aberrations

can be expressed as an equivalent phase shift. It is easy to see that astigmatism is

essentially equivalent to an azimuthally varying defocus. We leave it to the reader

to verify that the phase shift due to astigmatism with strength C

and phase angle

596 Phase contrast microscopy

is given by

χ

(q) =−πC

λq

cos[2(φ − φ

)], (10.13)

where φ

is a reference angle in the back focal plane. The wavelength and q de-

pendence of this phase shift are identical to those of the defocus phase shift, so that

we can combine the two:

χ(q) =−πλq

[ f + C

cos 2(φ − φ

)].

The dimensionless form of C

scales in the same way as f . The two-fold astig-

matism is illustrated in Fig. 10.5(a); the height of the solid line above or below the

dashed circle is equal to the astigmatism contribution. While spherical aberration

is usually in the range 0.3–3.0 mm, astigmatism is measured in micrometers; if the

stigmator coils are turned off completely, then the intrinsic astigmatism is usually in

the range C

= 1–2 µ

m. Two-fold astigmatism can be corrected completely using

the objective lens stigmator coils.

Astigmatism is caused primarily by imperfections in the lens pole piece and

there is no real reason why these imperfections should have two-fold symmetry

with respect to the lens axis. In fact, the defocus variation around the unit cir-

cle in Fig. 10.5(a) would probably look more like that shown in Fig. 10.5(d).

2-fold 3-fold

4-fold

(a) (b)

combination

Fig. 10.5. Illustration of two-fold, three-fold,

and four-fold astigmatism as a defocus change

around a unit circle. A more general case consisting of random contributions of all three

astigmatism orders is shown in (d).

10.2 The microscope as an information channel 597

Mathematically, this can be described by an expansion in terms of two-fold, three-

fold, four-fold terms, and so on. The phase shift corresponding to the three-fold

astigmatism is given by [Kri94]:

χ

=−

2π

Gλ

cos 3(φ − φ

), (10.14)

where G is the magnitude and φ

is the phase angle of the three-fold astigmatism.

This is a third-order contribution in q and the typical magnitude of G is around

µm for modern high-resolution instruments. Another third-order contribution is

given by the axial coma K :

χ

=−2π K λ

cos(φ − φ

), (10.15)

where K is the magnitude and φ

is the phase angle of the axial coma.

We can, of course, continue on with higher-order aberrations, such as ﬁfth-

order spherical aberration C

. Four-fold astigmatism contributes a fourth-order

term in q, which is the same order as the spherical aberration term C

. For a

long time, equation (10.9) has been used as the phase shift needed to understand

image formation in a high-resolution microscope. As the quality of the lenses

improved, the importance of higher-order terms in the phase shift has increased.

For microscopes with spherical aberration correctors, it is important to have a clear

understanding of the contributions of all third-order aberrations and several of the

ﬁfth-order aberrations. The correction of these aberrations requires progressively

more complex optical elements: e.g. three-fold astigmatism can be corrected with

two sextupole coils rotated 30

◦

with respect to each other. The more aberrations

one wishes to correct, the more complex the microscope becomes.

10.2.1.3 The weak phase object approximation

We have seen in Section 5.6.1 that the exit wave function may be calculated by

repeated operation of the phase grating function and the Fresnel propagator, either

in the multi-slice formalism or the equivalent real-space formalism. If the object

is very thin, then it is a good approximation to represent the exit wave function

by the ﬁrst phase grating factor, and ignore the Fresnel propagator altogether; i.e.

the entire foil is considered as a single slice. In that case the exit wave function is

given by

φ(r) = e

iσ V

(r)

. (10.16)

The object is therefore a pure phase object, and this approximation is known as the

strong phase object approximation or SPOA.

If the thin foil consists of low atomic number elements, then the projected poten-

tial will be small and the argument of the phase grating exponential will be small