Graef M. Introduction to conventional transmission electron microscopy

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148 The transmission electron microscope

equations and they are rather difﬁcult to solve because of the explicit dependence

of the relativistic mass on the velocity components. The non-relativistic version of

these equations is obtained by setting m = m

The second equation can be rewritten by noting that [Rei93]

d(mr

˙ϕ)

d(mr ˙ϕ)

+ m

r ˙ϕ.

This can be veriﬁed easily by explicitly writing out the two derivatives. If we

truncate the expressions for the ﬁeld components B

and B

(equations 3.5 and 3.6)

after the ﬁrst term, then we ﬁnd for equation (3.12):

d(mr

˙ϕ)

|e|

B(z) − 2

rB(z)

which reduces to

d(mr

˙ϕ)

|e|

B(z)

This can be integrated readily to give



m ˙ϕ −

|e|

B(z)



= C, (3.14)

where C is an integration constant.

†

C must be zero for any trajectory that intersects

the optical axis at some point (for such a point r = 0). Such trajectories are known

as meridional trajectories. Trajectories that are not meridional are skew trajectories.

C can be computed for an arbitrary initial point with coordinates (r

,ϕ

, z

3.4.4 The paraxial approximation

It is common practice to transform the time dependence of the equations of motion

into a dependence on one of the coordinates. In most cases the z-coordinate is taken

to be the independent variable, and the goal of the calculation is then to determine

r(z) and ϕ(z) for a charged particle traveling through the ﬁeld described by B(z).

The azimuthal equation can be rewritten using the following relation:

= v

where v is the axial velocity of the electron. It is a good approximation to as-

sume that v is very nearly constant, so that

z = ˙v = 0. In what follows we will

†

If we had chosen to derive the equations of motion from the variational principle, then this equation would have

followed directly from the fact that the coordinate variable ϕ does not explicitly appear in the Lagrangian of the

system, and therefore ˙ϕ must be a constant of motion [Szi88].

3.4 Basic electron optics: round magnetic lenses 149

denote a derivative with respect to z by a prime, i.e. d f /dz ≡ f



. The azimuthal

equation (3.14) for a meridional trajectory (C = 0) becomes

˙ϕ −

|e|

2mv

B(z) = ˙ϕ −

|e|



2em



B(z) = 0.

Using the notation of the previous chapter we ﬁnd

˙ϕ −

ηB(z)

√



= 0.

(3.15)

The total rotation angle of the electron trajectory around the optical axis is derived

by integrating this equation:

ϕ(z) = ϕ

√



B(z)dz. (3.16)

The rotational frequency for a paraxial electron intersecting the optical axis can

then be derived from equation (3.14),

˙ϕ =

B(z),

and this quantity is known as the Larmor frequency.

The radial equation can be simpliﬁed using the so-called paraxial approxima-

tion: we will only consider electrons traveling close to the optical axis (r remains

small), with motion predominantly in the z-direction (both radial and azimuthal

velocities are small compared to the forward velocity, or r



 1; r ϕ



 1). The

radial trajectory equation (3.11) can then be rewritten using the following relation:

r = v





z ≈ v



Since ˙v = 0, we also have

m = 0, and we leave it as an exercise for the reader to

show that this leads to the following radial paraxial equation:



(z)



r = 0.

(3.17)

It is important to note that the prefactor

(z)



in the radial paraxial equation is always positive, and becomes equal to zero only

for a vanishing magnetic ﬁeld strength. This implies that an electron trajectory is

150 The transmission electron microscope

always bent towards the optical axis (for a round lens),

†

regardless of the electron

energy and the details of the magnetic ﬁeld conﬁguration.

3.4.5 Numerical trajectory computation

The radial equation (3.17) can be solved analytically for a few particular choices

of the axial magnetic ﬂux distribution B(z), but in general requires a numerical

approach. We refer to [HK89a, HK89b] for a detailed discussion of the analytical

solutions for Glaser’s bell-shaped ﬁeld, given by

B(z) =

1 +

(

z/a

)

a is a parameter with the dimension of length, which indicates the extent of the ﬁeld:

small values of a correspond to sharply peaked ﬁelds. Although this ﬁeld does not

represent a realistic magnetic ﬁeld distribution (the ﬁeld strength does not drop off

sufﬁciently fast at large distances), it is often employed in theoretical calculations

because it allows closed-form solutions for almost all relevant parameters, including

the lens aberrations.

In this section we opt for a simple numerical approach to determine the paraxial

electron trajectory for an arbitrary axial ﬁeld distribution B(z). From the

website

the reader may download the program lens.f90, which implements a simple paraxial

meridional trajectory computation. The program is written for Glaser’s bell-shaped

ﬁeld, but can be modiﬁed easily for other axial ﬁeld distributions. The reader is

encouraged to modify the ﬁeld parameters to study how a magnetic ﬁeld can be

used to create lenses with varying focal length. It is particularly interesting to

use

lens.f90 to simulate how a two-condensor lens system works. The following

paragraphs describe the program and the PostScript output.

The program

lens.f90 integrates both the azimuthal and radial equations. The

azimuthal equation is straightforward to solve by replacing the integral (3.16) by a

summation. There are many different algorithms which can be used in this case:

trapezoidal integration, Simpson’s rule, Romberg integration and Gaussian quadra-

ture are amongst the more commonly used methods [Szi88]. Since we will be

using fourth-order Runge–Kutta method to solve the radial equation, we will use

Simpson’s rule for the azimuthal equation.

‡

If the integration interval (z

, z

)is

†

A positive second term in the radial equation can only be canceled by a negative ﬁrst term, which means that

the curvature of the trajectory must be negative, i.e. the electron trajectory bends towards the optical axis.

‡

It can be shown that Simpson’s rule is equivalent to a fourth-order Runge–Kutta method.

3.4 Basic electron optics: round magnetic lenses 151

divided into N segments of length z, then the total rotation angle ϕ(z

)isgiven

by [AS77]

ϕ(z

) = ϕ

ηz

√



B(z

) + B

(

)

+ 2

N/2−1



j=1

(

+ 2 jz

)

+ 4

N/2



j=1

(

2 j − 1

)

z

)

+ O

(

z

)

(3.18)

with ϕ

the initial rotation angle at the point z

. The error in the computation

is of order (z)

, as indicated by the last term. This equation is rather simple

to implement in a Fortran routine. In the same computational loop, we can also

calculate an approximate value for the focal length, using Busch’s formula [Bus26]:



(z)dz; (3.19)

z



) + B

) + 2

N/2−1



j=1

+ 2 jz)

N/2



j=1

+ (2 j − 1)z)

+ O

(

z

)

Both numerical values can then be compared

†

with the theoretical values for Glaser’s

bell-shaped ﬁeld [HK89a]:

ϕ =

ηB

√



πa;



πa.

The program

lens.f90 employs fourth-order Runge–Kutta integration with ﬁxed

step size to integrate the radial paraxial equation (the public domain

rksuite90

software package is available from the www.netlib.org website; for a gen-

eral description of Runge–Kutta procedures we refer the interested reader to the

Numerical Recipes book [PFTV89]). The Runge–Kutta procedure operates on a

†

In comparing these equations with others quoted in the literature for Glaser’s bell-shaped ﬁeld one should bear

in mind that there is a difference between the real focal length and the asymptotic focal length. The equations

above all refer to asymptotic quantities. Since Glaser’s ﬁeld has a slow drop off at large distances from the

maximum ﬁeld strength, the numerical procedure will in general underestimate the total rotation angle ϕ and

overestimate the focal length.

152 The transmission electron microscope

system of two coupled ﬁrst-order differential equations which are equivalent to the

radial equations:

= y

; (3.20)

=−

(z)



, (3.21)

from which the solution r(z) = y

follows. The function y

provides the distance

from the optical axis, and the function y

represents the slope of the electron

trajectory with respect to the optical axis. The initial values to be inserted into

this routine are then the initial distance r(z

) and slope r



Pseudo Code PC-11 Simulation of paraxial meridional electron trajectories.

Input: axial ﬂux density B(z), acceleration voltage

Output: PostScript drawing of trajectories

integrate rotation angle ϕ(z) and focal length f {equation 3.18}

use a fourth-order Runge–Kutta procedure {equations 3.20 and 3.21}

(

rksuite.f) to obtain r(z)

generate PostScript output

The source code of the lens.f90 program is rather simple and is represented in

pseudo code

PC-11 ; the radial equation is solved by a driver subroutine rksol2. The

initial values are taken to be r(z

) = 1.0 and r



) = 0.0, i.e. an electron entering

the ﬁeld parallel to the optical axis at unit distance. The initial value for the azimuthal

angle is taken as ϕ(z

) = π/2, i.e. the electron enters the ﬁeld traveling in the plane

(y, z).

The output of the program is a PostScript ﬁle (default name:

lens.ps) which can

be displayed with a PostScript viewer or sent directly to a printer. Typical program

output is shown in Fig. 3.5. The list near the top of the page shows the acceleration

voltage (both the input value and the relativistic value), the electron wavelength, the

maximum axial ﬁeld strength along the trajectory, the total azimuthal rotation angle

ϕ(z

), and the focal length computed using Busch’s formula. The three rectangular

boxes show:

(i) top left: proﬁle of the axial ﬁeld distribution B(z) (arbitrary scale, thinnest line), and

the radial trajectory r(z) (thickest line). When the radial distance becomes larger than

a preset truncation value, the trajectory is clipped;

(ii) top right: projection of the trajectory along the positive z-direction (i.e. looking away

from the electron source). The position of each point is given by (x(z), y(z)) =

(r(z) cos ϕ(z), r(z) sin ϕ(z));

3.4 Basic electron optics: round magnetic lenses 153

Electron Trajectory Computation

Written by MDG, 1998.

Voltage [V] 100000.0000

Relativistic voltage [V] 109784.7500

Wavelength [pm] 3.7014

Total rotation [rad] 2.4584

Focal length [mm] 0.7973

B(z), r(z)

Radial plot

B-max [T] 2.0000

5.0 mm

Azimuthal plot

Perspective plot

5.0 mm

Fig. 3.5. Representative output of the lens.f program, for Glaser’s bell-shaped ﬁeld. The

parameters of the ﬁeld are B

max

= 2.0T,a = 1.0 mm, and the acceleration voltage is

V = 100 kV.

(iii) bottom: perspective representation of four identical trajectories, obtained from the

computed trajectory by rotation by [0, 1, 2, 3] × π/2 in the (x, y) plane. The trajecto-

ries are clipped by the frame of the drawing, to prevent them from overlapping other

parts of the output.

154 The transmission electron microscope

B(z), r(z)

Radial plot

Azimuthal plot

B(z), r(z)

-5.0 mm

100

200

400

5.0 mm

Fig. 3.6. Radial and azimuthal trajectories for the ﬁeld of Fig. 3.5, for acceleration voltages

V = 100, 200 and 400 kV.

This program can be used to study how various parameters affect the focal length of

the lens and the electron trajectory. For instance, by entering a number of different

acceleration voltages for a given lens ﬁeld, we ﬁnd (Fig. 3.6) that a faster electron is

less affected by the ﬁeld or, equivalently, that a higher acceleration voltage requires

more powerful lenses, or lenses that act over a longer distance along the electron

trajectory.

Stationary (i.e. time-independent) magnetic ﬁelds cannot accelerate electrons,

they can only change the travel direction of the electrons. Accordingly, in the

absence of electrostatic ﬁelds, magnetic ﬁelds leave the energy of the electron

unchanged. The action of an axially symmetric magnetic ﬁeld on the trajectory

of an electron can then be understood as follows: for an electron with an initial

axial velocity component v

, the radial component of the magnetic ﬂux density

creates an azimuthal force F

. The particle thus acquires an azimuthal velocity

component v

= r ˙ϕ, which will make the trajectory curve around the optical axis.

This azimuthal component will then interact with the z component of the ﬂux

density to create a radial force component F

, which brings the particle closer to

the optical axis.

If we place two lenses along the optical axis the electron trajectory becomes

more complex. Figure 3.7 shows a conﬁguration of two lenses (B

1,max

= 1.2T,

2,max

=−1.8T,a = 1.0 mm), spaced by 20 mm. The acceleration voltage is

V = 400 kV. This conﬁguration mimics a double-condensor system, where the

ﬁrst lens has a ﬁxed cross-over and the second a variable cross-over; by changing

the current through the second lens the focal length of the system can be changed.

Note that the negative magnitude of the second ﬁeld does not affect the focal length,

but does change the overall rotation angle.

The programs discussed in this section are greatly simpliﬁed versions of the

types of computations a lens designer would have to carry out to create an entire

3.4 Basic electron optics: round magnetic lenses 155

Electron Trajectory Computation

Written by MDG, 1998.

Voltage [V] 400000.0000

Relativistic voltage [V] 556556.0625

Wavelength [pm] 1.6439

Total rotation [rad] −0.3657

B(z), r(z)

Radial plot

B-max [T] 1.7970

-30 mm

Azimuthal plot

Perspective plot

30 mm

Fig. 3.7. Double-lens conﬁguration; the centers of the lenses are 20 mm apart. The pa-

rameters of the ﬁeld are B

max

= 1.2, −1.8T,a = 1.0 mm, and the acceleration voltage is

V = 400 kV.

microscope column, or any other lens conﬁguration. Professional designers would

typically employ variable step-size Runge–Kutta integration algorithms or the

Burlisch–Stoer algorithm [PFTV89] to integrate the equations of motion accu-

rately for realistic ﬁeld distributions. Although most microscopists will rarely need

156 The transmission electron microscope

to worry about the physics of a magnetic lens, it is useful to play with simple

programs such as

lens.f90 to acquire some understanding of how a magnetic lens

works.

3.4.6 General properties of round magnetic lenses

The general solution to any second-order differential equation (and therefore also

equation 3.17) can be written as a linear combination of two independent solutions.

We select two solutions with special properties:

†

the ﬁrst solution, r

(z), satisﬁes

the condition

lim

z→−∞

(z) = 1, (3.22)

or, the incident ray is asymptotically parallel to the optical axis, at unit distance

from the axis. The second solution, r

(z), is asymptotically parallel to the outgoing

optical axis, i.e.

lim

z→+∞

(z) = 1. (3.23)

A general solution of equation (3.17) thus has the form:

r(z) = Ar

(z) + Br

(z). (3.24)

We will refer to each solution as a ray. Since magnetic ﬁelds have inﬁnite range, a

ray can only become asymptotically identical to a straight line.

The rays r

(z) and r

(z) become asymptotically straight lines far from the lens

center, and hence we have (using the equation for a straight line intersecting the

optical axis):

lim

z→+∞

(z) = (z − z



;

lim

z→−∞

(z) = (z − z



We will call z

the asymptotic image focus (see Fig. 3.8) and z

the asymptotic

object focus. As before, a prime denotes differentiation with respect to z.

A ray parallel to r

(z), but at a different distance from the optical axis, can be

written as λr

(z), where λ is a real number; the asymptotic outgoing ray belonging

to λr

(z)isλ(z − z



, according to the relations above. This ray intersects the

optical axis in the same point z

, hence the name image focus.

We can now determine the intersection of the incident asymptote to the ray

(z) and its emergent asymptote; the intersection point lies in the plane z = z

†

We will follow closely Section 16.2 in [HK89a] and Chapter 4 in [Szi88], but with a slightly different notation.

3.4 Basic electron optics: round magnetic lenses 157

r (z) r (z)

slope r'

Fig. 3.8. Deﬁnition of the asymptotic ray parameters and location of the general points z

and z

(see Fig. 3.8), which is called the asymptotic principal image plane:

incident asymptote → 1 = (z

− z



← emerging asymptote, (3.25)

from which

= z



= z

− f

, (3.26)

where f

is the asymptotic focal length. Similarly, the asymptotes to r

(z) intersect

in the asymptotic principal object plane z = z

, with associated asymptotic focal

length f

= 1/r



. The equality f

= f

is valid, provided the electron does not

experience any change in its kinetic energy while traveling through the lens. Insert-

ing the limit values for r

(z) and r

(z) into equation (3.24), we obtain the following

equations for the asymptotic general ray:

lim

z→−∞

r(z) = A + B

z − z

;

lim

z→+∞

r(z) =−A

z − z

+ B.

There are two components to each ray: the distance to the optical axis, and the angle

of the trajectory with respect to the optical axis. It is then convenient to introduce

a two-component ray-vector:

U(z) =



r(z)



(z)



. (3.27)

We leave it up to the reader to eliminate A and B from equation (3.27), for two

positions z

and z

(see Fig. 3.8), z

somewhere on the incident asymptote and

somewhere on the emergent asymptote of the general solution. We obtain the