shape functions in calculations of differential scattering cross-sections

FYSAST

Examensarbete 15 hpJuni 2010

Shape functions in calculations of differential scattering cross-sections

Anders Johansson

Institutionen fr fysik och astronomiDepartment of Physics and Astronomy

AbstractTwo new methods for calculating the double differential scattering cross-section (DDSCS) in electron energy loss spectroscopy (EELS) have beendeveloped, allowing for simulations of sample geometries which have beenunavailable to earlier methods of calculation. The new methods concernsthe calculations of the thickness function of the DDSCS. Earlier programshave used an analytic approximation of a sum over the lattice vectors ofthe sample that is valid for samples with parallel entrance and exit-surfaces.The first of the new methods carries out the sum explicitly, first identifyingthe unit cells illuminated by the electron beam, which are the ones neededto be summed over. The second uses an approach with Fourier transforms,yielding a final expression containing the shape amplitude, the Fourier trans-form of the shape function defining the shape of the electron beam insidethe sample. Approximating the shape with a polyhedron, one can quicklycalculate the shape amplitude as sums over its faces and edges.

The first method gives fast calculations for small samples or beams, whenthe number of illuminated unit cells is small. The second is more efficient inthe case of large beams or samples, as the number of faces and edges of thepolyhedron used in the calculation of the shape amplitude does not need tobe increased much for large beams.

A simulation of the DDSCS for magnetite has been performed, yieldingdiffraction patterns for the L3 edge of the three Fe atoms in its basis.

Supervisor: Jn RuszDepartment: Department of Physics and Astronomy,

Division of Materials TheoryExaminer: Kjell Pernestl

PopulrvetenskapligsammanfattningFr att analysera material anvnder man ofta olika typer av diffraktion, detvill sga man lter en vg av ngot slag spridas mot materialets atomer.Denna vg kan vara en elektromagnetisk vg, vanligtvis rntgenstrlning,eller en elektrons vgfunktion. P andra sidan fngar man upp ett diffrak-tionsmnster, som med rtt tolkningar ger mycket information om vilkenstruktur materialet har.

En annan vanlig metod fr att analysera material r elektronenergifr-lustspektroskopi (EELS). Med den fr man reda p egenskaper hos material,inte bara genom att studera vilka diffraktionsmnster som uppstr, utan ock-s genom att underska hur mycket energi de elektroner som skickas genommaterialet frlorar i de inelastiska processer som sker dr. D man analyse-rar energispektrumet frn experimentet kan man f detaljerad informationom vilka atomer som ingr i materialet. En relativt ny metod inom EELS,som har utvecklats fr att underska materials magnetiska egenskaper, rEMCD (electron magnetic chiral dichroism). Dr jmfr man spektrum frnolika delar av diffraktionsbilden som representerar olika riktningar i vgvek-torverfringen, q, skillnaden mellan elektronens vgvektor fre och efterreaktionen. De mjliga reaktionerna i en magnetiserad atom varierar med qoch genom skillnaden i spektrumen fr olika q kan man utlsa atomspecifikinformation om materialets magnetisering.

Vid diffraktion r det differentiella tvrsnittet r den storhet som beskri-ver den utgende strlens intensitet i en given riktning. Kan man berknadetta kan man frutsga vilket diffraktionsmnster som ett givet experimentkommer att ge. Berkningarna r i de flesta fall omjliga att utfra analy-tiskt och mste gras numeriskt med datorprogram. Tidigare har det barafunnits program som kunnat berkna tvrsnittet fr vissa srskilda typer avexperiment. Dessa program har jag nu utkat fr att mjliggra simulering-ar fr en strre klass av experiment. De formler som har anvnts tidigarekrvde att ingngs- och utgngsytorna fr strlen i det material som skul-le analyseras var parallella och att kristallstrukturen sg ut p ett srskiltstt. Med de tv nya metoder jag implementerat kan dremot berkningarutfras fr godtyckliga kristaller.

Fr att prova dessa nya metoder har jag gjort en simulering av tvrsnit-tet hos magnetit med en viss magnetisering, ngot som inte kunde grastidigare.

Contents

1 Introduction 1

2 Theory 22.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1.1 Scattering experiments in materials science . . . . . . 22.1.2 Electron energy-loss spectroscopy . . . . . . . . . . . . 22.1.3 Dichroic spectroscopy . . . . . . . . . . . . . . . . . . 3

2.2 Cross-section calculations . . . . . . . . . . . . . . . . . . . . 42.3 Thickness functions . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.1 Analytic expression . . . . . . . . . . . . . . . . . . . . 92.3.2 Brute-force summation . . . . . . . . . . . . . . . . . . 102.3.3 Shape amplitude calculations of thickness functions . . 10

2.4 Calculating shape amplitudes . . . . . . . . . . . . . . . . . . 122.4.1 Calculating the shape amplitude . . . . . . . . . . . . 122.4.2 Constructing the polyhedron . . . . . . . . . . . . . . 14

3 Implementation 203.1 Earlier versions and a new structure . . . . . . . . . . . . . . 203.2 Implementing Brute-force calculations . . . . . . . . . . . . . 203.3 Implementing shape amplitude calculations . . . . . . . . . . 20

4 Results and discussion 234.1 Comparisons of calculations . . . . . . . . . . . . . . . . . . . 234.2 Case study: Magnetite in (111)-direction . . . . . . . . . . . . 24

5 Conclusions 27

Acknowledgements 28

References 29

NomenclatureDDSCS Double differential scattering cross section, 2E ,

see equation (2.1)

EELS Electron energy loss spectroscopy

ELNES Electron energy-loss near edge structures

EMCD Electron energy-loss magnetic chiral dichroism

MDFF Mixed dynamic form factor, see equation (2.18)

Shape amplitude The Fourier transform of a shape function, see equa-tion (2.42)

Shape function A function having value 1 when inside a body and0 when outside, see equation (2.40)

TEM Transmission electron microscopy (or microscope)

Thickness function A part of the expression for the DDSCS in dynam-ical diffraction theory, see equation (2.29)

q Wave vector transfer, see equation (2.2)

1 IntroductionThis project is about the calculation of differential scattering cross sectionsin the theory of electron spectroscopic experiments. More specifically, it willexpand upon the theory of calculating the thickness functions being part ofthe expressions for the cross-sections of this theory, as well as expandingearlier programs for calculating the cross-sections.

Simulations of the cross-sections for various different samples and geo-metries are important to provide guidance to experimentalists doing elec-tron energy loss spectroscopy (EELS) and more specifically EMCD, elec-tron magnetic chiral dichroism. The simulations give information on howthe diffraction patterns should look for specific samples, and also on thedichroic signals, the differences in intensities between different directionscorresponding to different wave-vector transfers, which are important forEMCD experiments.

The earlier program packages have implemented a method which re-stricts the calculations to samples with parallel surfaces and with two of thelattice vectors parallel to these surfaces and the third perpendicular. Thenew methods will allow for an arbitrary orientation of sample surfaces andlattice vectors but be more computationally demanding.

The report is structured into several parts: First comes theory, with ageneral introduction to the experiments and theory as well as details regard-ing dynamical diffraction theory and my own derivations aiming at the newmethods for calculating the cross-section. The second part is about imple-mentation, describing briefly what was done in order to make programs withthe new methods. Following this is results and discussion, containing testsand comparisons of the methods as well as the results from a simulation ofthe cross-section of magnetite. Last comes conclusions.

1

2 Theory2.1 GeneralIn this section a general introduction to scattering experiments and theorywill be given ending in the subject of Electron energy-loss magnetic chiraldichroism.

2.1.1 Scattering experiments in materials science

Determining the structure and properties of solids have been a major con-cern in materials science aiming towards a better understanding of materialproperties and the ability to control them.

One specific and widely used method of determining crystal structure isdiffraction, which can, in a simplified way, be described as directing waveswith a sufficiently short wavelength of some kind into a crystal and seewhat comes out on the other side. What comes out is a so called diffractionpattern resulting from the scattering of the wave on the atoms of the crystal.The most common diffraction experiments use X-rays.

Another way of determining material properties is through spectroscopy.In spectroscopic experiments one illuminates a sample with waves of varyingwavelength and, looking at the spectrum coming out, one can deduce whichwavelengths have been absorbed and from that obtain information aboutthe properties of the material.

In scattering experiments, the fundamental quantity of interest to boththeorists and experimentalists is the differential cross section, , whichmeasures in what solid angle a given cross-sectional area of an incidentbeam is spread. This determines the observed intensity at various positionsof the detector. The cross section can be calculated from theory. [1]

2.1.2 Electron energy-loss spectroscopy

A specific spectroscopic method for determining material properties is Elec-tron energy loss spectroscopy (EELS), which deals with the inelastic scat-tering of an electron beam with on a studied material. These experimentsare often carried out in a transmission electron microscope (TEM) wherethe intensity of electrons after leaving the sample is measured as a func-tion of energy by an electron spectrometer. The inelastic events can be ofmany types, as explained in figure 2.1, and sorting those out enables oneto deduce different properties of the sample. The excitations to valencebands of inner shell electrons (atom core-loss) give rise to ionization edgesin the spectrum. Since every element have different binding energies for the

2

2.1. General Theory

electrons in the inner shells, examining these edges is a way of determiningwhat elements constitute the sample. The spectrum near these edges givesdetailed information and is referred to as ELNES (Electron energy-loss nearedge structures) [2]. [3]

Rel

ativ

efr

equen

cy

Zero loss peak

Atom core-loss peak

Fine structure

Plasmon resonance

Electron energy loss (eV)

Figure 2.1: Idealized version of an EELS spectrum. The zero-loss peakcorresponds to electrons passing through the material without takingpart in any inelastic events. The plasmon resonance corresponds toelectrons interacting with electrons in the valence band of the material.The atom core-loss and its fine structure corresponds to events wherean electron of some inner shell of the materials atoms are excited bythe incident electron. (Source: [4])

When doing experiments with an energy resolution the differential crosssection is replaced by the double differential scattering cross-section (DDSCS),in a general form described by

2

E I,F

|I|V |F |2 (E EF EI), (2.1)

where I and F refers to possible initial and final states of the combinedsystem of the target and probe electron.

2.1.3 Dichroic spectroscopy

For determining the magnetic properties of materials on the microscopicscale several techniques have been developed. A material is said to exhibit

3

2.2. Cross-section calculations Theory

dichroism if the absorption of light (or electron waves for that matter) de-pends on its polarization. This often depends on the magnetisation of thematerial and can thus be used as a quantitative measure of magnetisation.

The method of using circularly polarized x-rays in x-ray absorption spec-troscopy (XAS) to discover dichroism in materials has been around since1987 [5] and is commonly referred to as XMCD, x-ray magnetic circular di-chroism. Starting in 2006 a new technique called electron energy-loss mag-netic chiral dichroism (EMCD) have been developed and used for measuringdichroism with the ordinary methods of ELNES spectroscopy[6]. Using elec-trons instead of x-rays allows for a higher spatial resolution. In the theoryof EMCD the polarisation vector of the x-rays is replaced by the wavevector transfer q, the difference between final and initial wave vector of theincident electron:

q = kf ki. (2.2)

Instead of having the waves polarized in a specified way, something whichis harder to do with electrons than with x-rays, one can use an unpolarizedbeam and place the detector at two different positions representing differ-ent wave vector transfers that are orthogonal and phase shifted by 14 of awavelength [6]. This corresponds formally to circular polarization. The di-chroic signal is then obtained as the difference between the near-edge spectrafor these detector positions, displaying the difference in excitations allowedfor different atomic orbitals in a magnetized sample.

This is just a schematic way of describing the interaction of the electronbeam with the atoms of a sample. In a crystal, the wave functions of theprobe electrons cant be described as single plane waves, which is impliedin the description above, but rather has to be written as a superpositionof Bloch waves, the solutions to the Schrdinger equation for electrons ina periodic potential. Describing this theory will be the subject of the nextsection.

2.2 Cross-section calculationsHere I will briefly present the theory of dynamical diffraction. It is the theoryused to calculate the DDSCS of ELNES and thus the theory employed inthe programs simulating those experiments.

The DDSCS in ELNES, within the first-order Born approximation andfor an incident electron represented by a plane wave, can be written as [7]

2

E =42

a20

kfki

S(q, E)q4

, (2.3)

withS(q, E) =

i,f

|i|eiqr|f|2(Ef Ei E), (2.4)

4


where a0 is the Bohr radius, = 1/

1 v2/c2 a relativistic factor, (Ei, |i),(Ef , |f) the initial and final energies and electron states of the ionised atomand E the observed energy loss of the incident electron. S(q, E) is knownas the dynamical form factor (DFF).

As mentioned in the previous section, approximating the probe electronas one plane wave is in general not applicable for scattering inside a crystal.The wave functions of the incoming and outgoing electron should instead bewritten as superpositions of Bloch waves with different wave vectors (indexedj, l below), in r-representation this is

in(r) =jg

(j)C(j)g ei(k(j)+g)r, (2.5)

out(r) =lh

(l)D(l)h e

i(k(l)+h)r, (2.6)

where (j), (l) are the excitation factors of the Bloch waves labelled j, l andC

(j)g , D

(l)h are the so called Bloch coefficients. Finding these coefficients is

the objective of the theory of dynamical diffraction.To calculate the DDSCS one has to solve the Schrdinger equation in

the crystal potential using the proposed forms of the wave-functions given inEq. (2.5) and (2.6). In doing this one has to apply continuity conditions ofthe wave-functions (which are modelled as plane waves outside the crystal)at the crystal surfaces. We have:

k(j) = in + (j)(n1), (2.7)k(l) = out + (l)n2, (2.8)

where (in),(out) are the respective wave-vectors of the incoming and out-going electron outside the sample, n1, n2 are the normal vectors of thecrystal faces where the electrons enter and leave the sample as pictured inFig. 2.2 and (j), (l) are small parameters giving the change in the normalcomponents (the only ones allowed to change) of the wave vectors.

The details of solving the Schrdinger equations of incoming and outgo-ing waves are given in ref. [2, 7]. The solution is, for high energies, in the formof a linear eigenvalue problem with the s as eigenvalues and vectors formedof the Bloch coefficients as eigenvectors. Solving this, in principle infinitedimensional, eigenvalue problem thus gives the s and Bloch coefficients.Defining the planes containing the crystal faces by

n1 r = t1, (2.9)n2 r = t2, (2.10)

for the face where the beam enters and leaves the sample respectively, and

5


n1

n2

x

yz

Figure 2.2: The conventions for coordinate systems and surface nor-mals used throughout.The xyz-coordinates are defined by the latticeof the crystal. One usually chooses z to be parallel to the third latticevector and x parallel to the first reciprocal lattice vecto. The order ofthe lattice vectors are choosen after conventions for differnt lattices.

imposing the continuity conditions of Eq. (2.7) and (2.8) one gets[7]:

(j) = C(j)0?ei(j)t1 , (2.11)

(l) = D(l)0?ei(l)t2 , (2.12)

where ? here and in the following denotes complex conjugation. We willhowever always use an origin in the plane of the entrance face which givest1 = 0 and hence (j) = C(j)0

?.

Adding an absorptive part to the crystal potential and treating it as afirst order perturbation, the Bloch coefficients will not change but the sobtain an imaginary part:

(j) (j) + i(j), (2.13)

where the real part, (j), will equal the (j) of the non-absorptive calculation.For the outgoing wave we will, for reasons explained further by Nelhiebel[2], need to do some complex conjugation, which here will be equivalent topostulating

(l) (l) i(l). (2.14)

This follows the convention (l) > 0 and the physical requirement that theoutgoing wave is attenuated more if the inelastic event happened furtheraway from the exit surface.

6


This allows us to write the wave functions as:

in(r) =jg

C(j)0

?C(j)g e(i

(j)+(j))n1rei(in+g)r, (2.15)

(out)(r) =lh

D(l)0?D

(l)h e

(i(l)+(l))(n2rt)ei(out+h)r, (2.16)

where t hereafter will be used instead of t2. This t is what will be referredto as the thickness of a sample.

The DDSCS, which was already given in Eq. (2.3), should actually bewritten as being proportional to the square of the transition matrix element[2], i. e.

2

E |out|f |V |i|in|2 . (2.17)

This means it will obtain phase factors composed of the terms in the expan-sions of the wave functions in Eq. (2.15) and (2.16). In squaring the sums wewill have to sum over each index twice. This means that we must introduceprimed indices. Furthermore we will have interference terms so that theDFF will be generalized to the MDFF (mixed dynamical form-factor)[8]:

S(q, q, E) =i,f

i|eiqr|ff |eiqr|i(Ef Ei E), (2.18)

where the wave vector transfers q, q are defined as1

q = out + ((l) + i(l))n2 in ((j) + i(j))(n1) + h g, (2.19)q = out + ((l

) + i(l))n2 in ((j) + i(j))(n1) + h g. (2.20)

All this will yield the final DDSCS:

2

E =42

a20

outin

1Na

aghgh

jljl

Xjljl

ghgh(a)Sa(q, q, E)

q2q2, (2.21)

where the phase factor Xjljl

ghgh is given by

Xjljl

ghgh(a) = C(j)0

?C(j)g D

(l)0 D

(l)h

?C

(j)0 C

(j)g

?D

(l)0

?D

(l)h

=Y jljl

ghgh

e[i((l)(l))((l)+(l))]t ei(qq

?)a,

(2.22)

and where a denotes the positions of all atoms in the sample. The MDFF canbe calculated in various ways but that will not be covered here and insteadI will refer to [7, 9, 10]. The summation in Eq. (2.21) can be simplified indifferent manners and that will be the focus of the next section.

1 Here it should be noted that Nelhiebel [2] uses the opposite definition, Q = kinitial kfinal, of wave vector transfer.

7

2.3. Thickness functions Theory

2.3 Thickness functionsThis section will focus on different ways of simplifying the calculation of theDDSCS of Eq. (2.21). First the thickness function, being a part of the totalDDSCS, will be defined and then three different methods of calculating itwill be presented.

First defining

(ll) = (l) (l

),

(ll)+ = (l) + (l

),(2.23)

with similar definitions for j, j, we can write the DDSCS of Eq. (2.21) in aperhaps more suggestive form:

2

E =42

a20

outin

ghgh

jljl

Y jljl

ghghe(i(ll

)

(ll)+ )t

1Na

a

ei(qq?)aSa(q, q, E)

q2q2.

(2.24)

This can be simplified if we write a as composed of a lattice vector, R, anda basis vector, u,

a = R+ u. (2.25)

The differences in q and q comes mostly from the h,h, g, g-vectors sincethe and are small [7]. This allows us to take the MDFF out of the sumover j, l, j, l. The MDFF does not depend on what unit cell we are in, R,but only on the basis, u, enabling us to only sum it over the basis vectors.We can note that the combination hg (hg) = G, a reciprocal latticevector.2

All this enables us to simplify things. We write

1Na

a

ei(qq)a = 1NR

R

e[(i(ll)

(ll)+ )n2+(i

(jj)

(jj)+ )n1]R eGR

=1

1Nu

u

ei(qq?)u.

(2.27)2Writing things out explicitly we have:

q q? = ((l) + i(l))n2 ((j) + i(j))(n1)

((l) i(l

))n2 ((j) i(j

))(n1) +G

= ((ll)

+ i(ll)+ )n2 + (

(jj) + i

(jj)+ )n1 +G

(2.26)

8


This allows us to write the DDSCS as

42

a20

outin

ghgh

1Nu

u

ei(qq?)uSu(q, q, E)

q2q2

jljl

Y jljl

ghghTjljl(t), (2.28)

where Tjljl(t) is the so called thickness function defined by:

Tjljl(t) = e(i(ll)

(ll)+ )t 1

NR

R

e[(i(ll)

(ll)+ )n2+(i

(jj)

(jj)+ )n1]R. (2.29)

This function can be evaluated by different means for different samplegeometries. These are explored in the following sections.

2.3.1 Analytic expression

For specific samples the summation in Eq. (2.29) can, with some approx-imations, be carried out analytically and brought to a closed form. Theconditions are as follows:The samples surfaces are parallel and orthogonal to the z-axis. The sampleslattice have one lattice vector c along z and the other two orthogonal to z.This could for instance be a cubic or hexagonal lattice oriented accordingto above mentioned conditions and with 001-surfaces. The lattice vectorparallel to z are denoted c and the other two a and b. This means that wehave

n1 = z n2 = z. (2.30)

The summation over R can now be split up into three sums over a, b and cwhere the sum over a, b will yield the total number of atoms in each xy-planeof the lattice, denoted Nxy. The total number of planes (lattice points) inz-direction is denoted Nz.

We introduce the notation,

= (jj)

(ll) =

(jj)+

(ll)+ , (2.31)

enabling us to write Tjljl(t) as

Tjljl(t) =NxyNa

e(i(ll)

(ll)+ )t

Nzn=0

e[(i(ll)

(ll)+ )(z)+(i

(jj)

(jj)+ )z](z)nc

= NxyNa

e(i(ll)

(ll)+ )t

Nzn=0

e(i)nc.

(2.32)

The sum can be simplified further with some approximations. If we assumethe thickness of the sample to be much larger than the lattice constants,

9


t c, or equivalently Nz 1, c(i + ) to be a small number andwrite NzNxy = NR we get (in accordance with Nelhiebel [2]) 3

Tjljl(t) = e[i((jj) +

(ll) )(

(jj)+

(ll)+ )]

t2

cosh t2 sin

t2 + i sinh

t2 cos

t2

( + i) t2,

(2.33)

where the factors NR and 1/NR have cancelled. In the case of no absorptionthis simplifies to

Tjljl(t) = ei((jj) +

(ll) )

t2

sin t2 t2

. (2.34)

2.3.2 Brute-force summation

For more general lattices, or nonparallel surfaces, one has to evaluate Tjljl(t)of Eq. (2.29) in other ways. One such way is determining what lattice pointsare illuminated by the electron beam and then carry out the summation inEq. (2.29) for all those lattice vectors. For reference I state the generalthickness function assuming no absorption:

Tjljl(t) =ei

(ll) t

NR

R

ei((ll) n2+

(jj) n1)R. (2.35)

2.3.3 Shape amplitude calculations of thickness functions

A different approach to calculating the thickness function is using a shapefunction to define the illuminated volume and after some manipulationsarrive at an expression containing the shape amplitude, i. e. the Fouriertransform of the shape function. This section will explore that possibility.The formalism below is limited to the case with no absorption4.

The summation over lattice vectors of Eq. (2.35) is to be carried out overall lattice vectors inside a body (t):

Tjljl(t) =ei

(ll) t

NR

R(t)

ei((ll) n2+

(jj) n1)R. (2.36)

Definingp = ((ll

) n2 +

(jj) n1), (2.37)

3 To be explicit: we evaluate the sum as a geometric sum, using Nz + 1 Nz in thedenominator, replacing the resulting e(i)c in the numerator with the first two termsin the Taylor series and then expand the exponentials in trigonometric and hyperbolicfunctions and collect terms.

4The kernel, eipr, of the Fourier transform used below requires p to be a real vector

10


this can be rewritten as:

Tjljl(t) =ei

(ll) t

NR

R(t)

eipR. (2.38)

The sum can be rewritten as an integral with a Dirac-comb:R(t)

eipR =

eipr

R(t)(r R) dr

=

eiprR

(r R)(r) dr,(2.39)

where (r) is the shape function of , defined by:

(r) ={

1 if r (t),0 if r / (t).

(2.40)

Equation (2.39) can be interpreted as a Fourier transform using the kerneleipr. The conventions for the Fourier transform used here will be:

f(p) = F [f ] (p) =f(r)eiprdr, (2.41a)

f(r) = F1[f]

(r) = 1(2)3f(p)eiprdp. (2.41b)

The Fourier transform of a product of two functions is, according to theconvolution theorem of Fourier analysis, equal to the convolution of theFourier transform of each function. We can hence rewrite the last expressionof Eq. (2.39) as

R(t)

eipR = F[R

(r R)]

(p)F [(r)] (p)

= 1Vu

G

(pG)(p),(2.42)

recognising that the Fourier transform of a Dirac comb is a Dirac combin reciprocal space divided by the unit cell volume Vu [11] and denotingthe Fourier transform of the shape function, called the shape amplitude, by(p). If we note that that NR, being the number of unit cells inside (t),roughly (but close enough for large NR) equals V/Vu and finally evaluatethe convolution of Eq. (2.42) we can write the thickness function as:

Tjljl(t) =ei

(ll) t

V

G

(pG), (2.43)

11

2.4. Calculating shape amplitudes Theory

where the sum over G in practical cases only has to be computed for a fewG within some radius since the shape amplitude decays rapidly with largerG-vectors.

To use the formula Eq. (2.43) in calculations one needs a convenient wayto calculate the shape amplitude. The shape of the beam can for a generalsample be modelled by a cone cut off by the entrance and exit surfaces aspictured in Fig. 2.2. How to calculate the shape amplitude of such shapeswill be explored in the next section.

2.4 Calculating shape amplitudesIn this section I present a method for calculating the shape amplitude ofa general cut-off cone, a body which approximately describes the electronbeam through some sample with arbitrarily oriented top and bottom sur-faces. This shape amplitude can easily be calculated by approximating thecone with a polyhedron since, as described by Komrska [11], the shape amp-litude of all polyhedrons can be calculated as a finite sum. The first of thefollowing sections will present the formulae for calculating the shape amp-litude of a general polyhedron while the second will describe how to constructa suitable polyhedron from the parameters describing the cone.

2.4.1 Calculating the shape amplitude

The trivial case for p = 0 gives

(0) =(r) dr =

V

1 dr = V. (2.44)

The equations below deals with the cases p 6= 0.As mentioned earlier, the shape amplitude, (p), of any polyhedron

can be expressed analytically as finite sums, allowing for easy computation.The formalism below follows Komrska [11] but is corrected to correspondto the convention of the Fourier transform mentioned above in Eq. (2.41).Following Komrska [11] Eq. (3.3) we have:

(p) =Ff=1

f (p), (2.45)

where F is the number of faces of the polyhedron and f (p) is the two-dimensional shape amplitude calculated for each face indexed f . f (p), inturn, is given by:

f (p) = ip Nfp2

eip(Of ) i

[p2 (p Nf )2]

Ee=1

fe(p), (2.46)

12


where Nf defines the unit outward normal of each face, (Of ) is a vectorlying in each face serving as an origin for that face, E the number of edgesbounding each face and fe(p) the contribution of each edge, e, belongingto the face f . The expression for fe(p) is:

fe(p) = ip neeip(Ve)Of

Le0

eiptel dl

=

ip neeip(Ve)Of Le p te = 0,

pnepte

(eip(Ve)Of eip

(Ve+1)Of ) p te 6= 0,

(2.47)

where ne is the unit outward normal of each edge (lying in the plane of theface), (Ve)Of defines the vertex at the beginning of the edge (relative to

(Of ))and (Ve+1)Of the one at the end, Le is the length of the edge and te is theunit vector pointing along the edge, i. e. from (Ve)Of towards

(Ve+1)Of .

If p is perpendicular to the face indexed f , f (p) has to be calculatedin another way ([11] Eq. 3.11):

f (p) = ipPfeipdf , if p = pNf . (2.48)

Here Pf denotes the area of the face and df denotes the orthogonal distancefrom the plane the face is part of to the origin.

All these cases can be brought together and rewritten in the slightlymore handy form:

(p) ={V if p = 0,Ff=1 f (p) otherwise,

(2.49)

where f (p) is given by

f (p) =

ipPfe

ipdf if p = pNf ,ip2

pNf[p2(pNf )2]

Ee=1 fe(p) if p 6= pNf ,

(2.50)

where fe(p) is given by

fe(p) =

ip ne eip(Ve)Le if p te = 0,

pnepte

(eip(Ve) eip(Ve+1)) if p te 6= 0.(2.51)

Here (eip(Ve) eip(Ve+1)) could equally well be written

eip(Ve)(1 eipteLe). (2.52)

This could suit some implementations more.

13


2.4.2 Constructing the polyhedron

The quantities defining the shape are the following (see also Fig. 2.3):

k The normalised direction of the beam.

r0 The radius of the beam before entering the sample.

The spreading angle of the beam once inside the sample.

u0 The point where the centre of the beam enters the sample. This is canmostly be set to zero, using the beam impact point as the origin. Theorigin will, as noted after Eq. (2.9), always be placed in the upperplane.

n1 The upper surface normal.

n2 The lower surface normal.

u0

k

r0

n1

n2

Figure 2.3: The parameters defining the cut-off cone. The vector u0defining the central impact point of the beam is drawn from someorigin in the upper plane.

The first task is tracing out the circular cross-section of the beam outsidethe sample. One can start by finding any normal vector not parallel to k.Then xk as depicted in Fig. 2.4 can be constructed as

xk = k , (2.53)

allowing us to defineyk = k xk, (2.54)

14


yielding a right handed coordinate system with basis xk, yk, k where xk, ykis a basis for the cross-sectional plane of the cylinder representing the beam.

n1

n2

x

yz

k

xk

ykrn

Figure 2.4: The basis of the cross-sectional plane of the cylinder rep-resenting the beam outside the sample, xk, yk. The rn-vector is oneof the evenly distributed vectors which projected on the upper planewill give the upper vertices of the polyhedron.

Now a set ofN points evenly distributed around the circular cross-sectionof the beam can be constructed as

rn = r0[cos

(n

N2)xk + sin

(n

N2)yk

], (2.55)

where n runs from 1 to N .Constructing the vertices of the polyhedron is now a task of projecting

these points firstly onto the upper surface along k and secondly from thereto the lower surface with the correct slope away from k.

The general problem of projecting a vector, r, unto a plane, alongsome unit vector, m can be solved in the following way: The projectedvector rm is

rm = r +mm, (2.56)where m is the unknown distance between the plane and r along m. Theplane itself is defined by its normal vector, n and the normal distance tothe origin, t, i. e.

n = t (2.57)defines all points . Since rm we have the following equation fordetermining m:

(r +mm) n = t, (2.58)

15


yieldingm = t r n

m n . (2.59)

Returning to the initial problem of projecting the circle outlining thebeam onto the first surface, hereafter denoted , the vertices of the poly-hedron lying in can be written:

(Vn) = rn rn n1k n1

k, (2.60)

where, as mentioned earlier, the origin is in , meaning t = 0.

rn

ln

(Vn)

(Vn)

n1

n2

Figure 2.5: The vertices of the polyhedron lying in the upper andlower planes. The ones with the same index are connected via an edgealong ln. The upper plane is denoted and the lower .

To get the vertices lying in the lower plane, , we have to constructvectors ln pointing out from k with the correct angle. ln is

ln = cos()k + sin()rnrn, (2.61)

as can be seen from Fig. 2.6. The vertices in the lower plane can now bewritten

(Vn) = (Vn) + lnln, (2.62)

where ln is defined as in Eq. (2.61) and ln is

ln =t (Vn) n2

ln n2, (2.63)

as follows from Eq. (2.59) with t the normal distance from u0 to .

16


rn

rn

ln

ln

ln

(Vn)

(Vn)

(Vn)

k

k

ln

Figure 2.6: The vector ln pointing from upper to lower vertex can becalculated as shown in this figure. The length along the nth edge isalso shown. The inset shows the vectors in the grey plane of the largepicture in more detail.

Knowing that the edges of the polyhedron are those connecting (Vn)and (Vn) as well as those connecting (,Vn) and (,Vn+1) the polyhed-ron is completely specified.

Further specifications of the polyhedron

To use the polyhedron defined above for calculating the shape amplitude asin section 2.4.1 the directions and normals of all edges and faces as well asthe lengths of all edges and areas of and distances to all faces have to befound. How to find these is explored here.

Using the conventions depicted in Fig. 2.7 all the necessary vectors andlengths for each face are obtained by specifying:

t(n)1 =

(Vn+1) (Vn)(Vn+1) (Vn) L(n)1 =(Vn+1) (Vn) , (2.64a)

t(n)2 = ln (of Eq. (2.61)) L

(n)2 = ln (of Eq. (2.63)), (2.64b)

t(n)3 =

(Vn+1) (Vn)(Vn+1) (Vn) L(n)3 =(Vn+1) (Vn) , (2.64c)

N (n) = t(n)1 t

(n)2t(n)1 t(n)2 , (2.64d)

n(n)1 =

n1 t(n)1n1 t(n)1 n(n)2 =

t(n)1 N (n)t(n)1 N (n) , (2.64e)

17


n1

n2

L(n)1

L(n)2

L(n)3

n(n)1

n(n)2

n(n)3 n

(n)4

n(n)5

n(n)6

t(n)1 t

(n)2

t(n)3

N (n)

(Vn)

(Vn)

(Vn+1)

(Vn+1)

Figure 2.7: The conventions used for all normal vectors, edge vectorsand edge lengths defined by Eq. (2.64).

n(n)3 =

t(n)3 n2t(n)3 n2 n

(n)4 =

N (n) t(n)3N (n) t(n)3 , (2.64f)n

(n)5 =

t(n)2 N (n1)t(n)2 N (n1) n

(n)6 =

N (n) t(n)2N (n) t(n)2 , (2.64g)for all n. For the calculation of the shape amplitude the correct vectorshave to be picked out and sometimes reversed to keep the correct orientationaround the faces.

The area of a convex polygon (which all faces will be) can be calculatedby choosing one vertex and drawing the vectors pointing to all other vertices,calculating the sum of the absolute value of the cross-product of all pairsand dividing by 2 (i. e. dividing the polygon into triangles and using A4 =ab sin

2 ).The distance df from the origin to each plane can be calculated as df =

rf Nf where rf is any vector in the plane, e. g. one of the vertices.To calculate the volume of a general polyhedron one can utilise the local

topology formula stated by Franklin [12]:

V = 16nij

t(n)i

(Vn) n(n)ij

(Vn) N(n)ij

(Vn), (2.65)

where the sum is taken over all quadruples ((Vn), t(n)i , n(n)ij , N

(n)ij ) with (Vn)

denoting the vertices, t(n)i the unit vectors along all edges i radiating from

18


x1

x2

x3

x4

x5

Figure 2.8: The area of any convex polygon can be calculated by themethod which in this picture gives: A = |x1 x2 + x2 x3 + x3 x4 + x4 x5|/2

the vertex n, n(n)ij the unit normal of the edge i pointing into the adjacentface j and N (n)ij the unit normal of the face j pointing into the polyhedron.These vectors are not exactly the ones defined by the conventions of Eq.(2.64) but are nevertheless easy to retrieve from those vectors, if care istaken in introducing minus signs where it is appropriate.

19

3 ImplementationIn this section I will present the steps needed to implement the ways ofcalculating the thickness function described above into an existing programpackage.

3.1 Earlier versions and a new structure

This project has been built upon programs developed by Rusz et al. [7]. Theearlier version of the program, implemented in FORTRAN90, contained onepart for solving the problems of dynamical diffraction, called dyndif, andanother separate part for calculating the MDFF, given the necessary data(pairs of q, q-vectors) from dyndif. The MDFF can be calculated withvarious methods [7, 9, 10]. The version that have been used in the testing ofthis project is mdff_dip using a dipole approximation and sum rules [9] tocalculate the MDFF in a simple way. In the earlier versions, the analyticalthickness function, Eq. (2.33), was implemented in dyndif and the programflow was as described in Fig. 3.1.

Upon extending the earlier program with different ways of calculatingthe thickness function I choose to split the thickness function into a separateprogram, taking as input from dyndif the coefficients Y jlj

l

ghgh of Eq. (2.28)paired with all |+(ll

)|(jj) and outputting the products Y jljl

ghghTjljl(t)to be used in the final summation done in dyndif. This flow of the pro-grams is depicted in Fig. 3.2 When doing calculations of the cross-sectionsof materials one now has the choice between three different methods of cal-culating the thickness function: Analytic, Brute-force (called Flood), andShape amplitude (called Shape).

3.2 Implementing Brute-force calculationsThe brute-force summation was implemented with the help of a programdeveloped earlier by J. Rusz. This program, utilising the flood-fill algorithmfor determining what lattice vectors are inside the beam, was incorporatedinto thick_func. A list of illuminated unit cells is generated and used forthe explicit summation performed according to Eq. (2.29) (or Eq. (2.35) inthe case of no absorption).

3.3 Implementing shape amplitude calculationsThe implementation of the shape amplitude calculation of the thicknessfunction was performed along the lines presented in section 2.3.3 and 2.4.

20

3.3. Implementing shape amplitude calculations Implementation

dyndif

mdff

dyndif (with analyticthickness function)

Figure 3.1: Program flowbefore implementation ofdifferent thickness func-tions.

dyndif

mdff thick_func

dyndif

Figure 3.2: Program flowafter the implementationof different thicknessfunctions.

Figure 3.3: An example of illuminated unit cells, here displayed assingle spheres, identified by the flood-fill algorithm.

21

3.3. Implementing shape amplitude calculations Implementation

The polyhedron approximating the beam was constructed as described insection 2.4.2 along with all edge vectors, normal vectors, surface areas etc. Asphere of a few reciprocal lattice vectors is also calculated for the summationin Eq. (2.43), where only a few vectors around 0 will give non-negligiblecontributions. Then, using the data from dyndif, the thickness functionis calculated as in Eq. (2.43) with the expressions for the shape amplitudegiven in Eq. (2.49) and the following equations.

22

4 Results and discussionFirst, this section will present the results of some comparisons done betweenthe old and new ways of calculating the DDSCS. Second, a simulation ofthe DDSCS of magnetite in orientation with zone axis (111), a system whichcould not be simulated earlier, will be described.

4.1 Comparisons of calculationsTo see whether the new implementations of Flood and Shape were correctthey were tested against the old implementation of the analytic calculationand each other. All these tests were done using a Fe3O4 FCC (magnetite)crystal. Figure 4.1 shows a comparison of the Analytic and Flood-fill cal-culation with an increasing tilt of the upper surface. There is a perfectmatch for a tilt angle of 0 but the difference increases with larger angles,as the approximations for the analytic formula become less valid. A similar

0

5

10

15

20

25

30

35

0 10 20 30 40 50 60 70 80 90 100

DDSC

S(arb.un

its)

t/nm

051015202530

AnalyticFlood-fill

Figure 4.1: Comparison of DDSCS between analytic calculation andflood-fill calculation with increased tilt of the upper surface. The linesfor each tilt angle are shifted upwards, hence the scale on the y-axisshould not be interpreted as an absolute scale.

comparison between Flood and Shape, which should give the same results,does show a large difference with the current implementation of the shapeamplitude method. There are probably still some bugs to be sorted out inthe program. However, for samples with parallel surfaces orthogonal to z,

23

4.2. Case study: Magnetite in (111)-direction Results and discussion

all methods seem to work as indicated by Fig. 4.2, which shows a perfectcorrespondence for different beam directions for all three methods.

0

5

10

15

20

25

30

35

0 10 20 30 40 50 60 70 80 90 100

DDSC

S(arb.un

its)

t/nm

048

1216

20 AnalyticFloodShape

Figure 4.2: Comparison of all three methods for a sample satisfyingthe conditions for the analytic approximations to be valid. The plotsare for some different tilts of the incoming electron beam relative tothe sample surface and the values are both shifted and scaled with afactor 10 in all cases but the 0-beam.

The problems with Shape seems to lie in how small but non-zero p-vectors are handled and this seems to be a larger problem for tilted surfaces.Figure 4.3 shows plots of some tests in calculating the shape amplitude withsmall p-vectors. The values around the origin get very high, although theyshould not exceed 1. At p = 0 the shape function is explicitly set to 1.

4.2 Case study: Magnetite in (111)-directionTo test the new programs on a system that could not be simulated with theearlier versions, a simulation of the DDSCS of magnetite was done. Thecrystal structure of magnetite is shown in Fig. 4.4. The incident beam aswell as the magnetisation were set to the (111)-direction. Then calculationsof the DDSCS were performed for the L3 edge for all three Fe atoms inthe basis, separating out the real and imaginary parts of the MDFF beforedoing the last summation in dyndif. Calculations were made over a grid ofdirections around the transmitted beam with (220) along the x-axis. Thisgave the diffraction patterns in Fig. 4.5. One can clearly see the pendellsungoscillations, the variations in the intensity with thickness, and some smalldifferences for the different atoms.

24


1000

100200300400

0.2 0.1 0 0.1 0.2

Im

(p)/V

|p| = px

1000

100200300400500

0.2 0.1 0 0.1 0.2

Im

(p)/V

|p| = py

1000

100200300400500600700

0.2 0.1 0 0.1 0.2

Im

(p)/V

|p| = pz

0.50

0.51

1.52

2.53

0.2 0.1 0 0.1 0.2

Re

(p)/V

|p| = px

0.50

0.51

1.52

2.53

0.2 0.1 0 0.1 0.2

Re

(p)/V

|p| = py

0.50

0.51

1.52

2.53

0.2 0.1 0 0.1 0.2

Re

(p)/V

|p| = pz

Figure 4.3: The real and imaginary parts of the shape amplitude cal-culated for some small p-vectors in x, y and z direction. This wasdone with a beam in z-direction and a slightly tilted exit surface. Thesample thickness was set to 50 nm and the surface was tilted with 6along x.

Figure 4.4: Magnetite structure, two unit cells shown. The red atomsare Fe and the brown O.

25


1.0

0.5

0.0

0.5

1.0 4 nm 14 nm 24 nm 34 nm 44 nm

1.0

0.5

0.0

0.5

1.0

1.0

0.5

0.0

0.5

1.0

1.00.5 0.0 0.5 1.0 1.00.5 0.0 0.5 1.0 1.00.5 0.0 0.5 1.0 1.00.5 0.0 0.5 1.0 1.00.5 0.0 0.5 1.0

1.0

0.5

0.0

0.5

1.0 54 nm 64 nm 74 nm 84 nm 94 nm

1.0

0.5

0.0

0.5

1.0

1.0

0.5

0.0

0.5

1.0

1.00.5 0.0 0.5 1.0 1.00.5 0.0 0.5 1.0 1.00.5 0.0 0.5 1.0 1.00.5 0.0 0.5 1.0 1.00.5 0.0 0.5 1.0

Figure 4.5: DDSCS (diffraction patterns) for magnetite around thetransmitted beam with (220) along the x-axis. The rows correspondto the three different Fe atoms in the basis and the columns to differentthicknesses. The tick labels are multiples of the Bragg vector G220 andthe intensities of the maps are relative to the maximum and minimumDDSCS in the calculation.

26

5 ConclusionsThe earlier software package dyndif used to calculate the DDSCS in ELNEShas been extended with new methods of calculating the thickness function.This allows for simulations with arbitrarily oriented sample-surfaces andlattice vectors. One method, called Flood since it uses an implementationof the flood-fill algorithm, has been implemented identifying the illuminatedunit cells of the sample and then explicitly summing over them. This is veryfast for small samples or beams, where the number of illuminated cells issmall.

Another method, called Shape calculates the thickness function by usingthe shape amplitude of the body representing the beam in the sample. Thismethod scales very well since the number of summations depends on thenumber of faces used in the polyhedron representing the beam, a numberthat does not have to be increased much as the beam or the sample isenlarged.

The DDSCS of magnetite with magnetisation along (111)-direction andan incoming beam in (111)-direction have been calculated for a grid of dir-ections around the transmitted beam, yielding the first (111) zone axis sim-ulation of the energy-filtered diffraction pattoern of this system and beingthe first in a now much wider class of possible simulations to be used inguiding the experiments in e. g. EMCD.

27

AcknowledgementsThanks to my supervisor Jn Rusz, who has provided excellent guidance,encouragement and proofreading. Thanks also to Stina and my other friendsfor encouragement and some proofreading.

28

References[1] David B. Williams and C. Barry Carter. Transmission Electron Mi-

croscopy: A Textbook for Materials Science. 2nd ed. Springer, 2009.[2] Michael Nelhiebel. Effects of crystal orientation and interferometry in

electron energy loss spectroscopy. PhD thesis. cole Centrale Paris,1999.

[3] R F Egerton. Electron energy-loss spectroscopy in the TEM. Reportson Progress in Physics 72.1 (Jan. 2009), p. 016502. issn: 0034-4885.doi: 10.1088/0034-4885/72/1/016502.

[4] Wikipedia User HatnCoat. EELS_Idealised (graphic). Licensed un-der the Creative Commons Attribution-ShareAlike 3.0 License. 2009.url: http://en.wikipedia.org/wiki/File:EELS\_Idealised.svg(visited on 13/05/2010).

[5] G. Schtz et al. Absorption of circularly polarized x rays in iron.Physical Review Letters 58.7 (Feb. 1987), pp. 737740. issn: 0031-9007. doi: 10.1103/PhysRevLett.58.737.

[6] P. Schattschneider et al. Detection of magnetic circular dichroism us-ing a transmission electron microscope. Nature 441.7092 (May 2006),pp. 4868. issn: 1476-4687. doi: 10.1038/nature04778.

[7] Jan Rusz et al. First-principles theory of chiral dichroism in electronmicroscopy applied to 3d ferromagnets. Physical Review B 75.21 (Apr.2007), pp. 214425214434. issn: 1098-0121. doi: 10.1103/PhysRevB.75.214425.

[8] H. Kohl and H. Rose. Theory of Image Formation by InelasticallyScattered Electrons in the Electron Microscope. Advances in Elec-tronics and Electron Physics 65 (1985), pp. 173227.

[9] Jn Rusz et al. Sum rules for electron energy loss near edge spectra.Physical Review B 76.6 (June 2007), p. 4. issn: 1098-0121. doi: 10.1103/PhysRevB.76.060408.

[10] L Calmels and J Rusz. Momentum-resolved EELS and EMCD spec-tra from the atomic multiplet theory: Application to magnetite. Ul-tramicroscopy In print (Apr. 2010). issn: 1879-2723. doi: 10.1016/j.ultramic.2010.04.015.

[11] Ji Komrska. Algebraic expressions of shape amplitudes of polygonsand polyhedra. Optik 80.4 (1988), pp. 171183.

[12] W. Randolph Franklin. Local Topological Properties of Polyhedra.Unpublished talk. 1992. url: http : / / wrfranklin . org / wiki /Research/localtopo.pdf (visited on 12/05/2010).

29

http://en.wikipedia.org/wiki/File:EELS\_Idealised.svghttp://wrfranklin.org/wiki/Research/localtopo.pdfhttp://wrfranklin.org/wiki/Research/localtopo.pdf

AbstractPopulrvetenskaplig sammanfattningContentsNomenclature1 Introduction2 Theory2.1 General2.1.1 Scattering experiments in materials science2.1.2 Electron energy-loss spectroscopy2.1.3 Dichroic spectroscopy

2.2 Cross-section calculations2.3 Thickness functions2.3.1 Analytic expression2.3.2 Brute-force summation2.3.3 Shape amplitude calculations of thickness functions

2.4 Calculating shape amplitudes2.4.1 Calculating the shape amplitude2.4.2 Constructing the polyhedron

3 Implementation3.1 Earlier versions and a new structure3.2 Implementing Brute-force calculations3.3 Implementing shape amplitude calculations

4 Results and discussion4.1 Comparisons of calculations4.2 Case study: Magnetite in (111)-direction

5 ConclusionsAcknowledgementsReferences

shape functions in calculations of differential scattering cross-sections

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