The Fresnel effect of a defocused biprism on

the fringes in inelastic holography

Jo Verbeeck a,b Giovanni Bertoni b Peter Schattschneider c,a

aInstitute for Solid State Physics, Vienna University of Technology, A-1040Vienna, Austria.

bEMAT, University of Antwerp, Groenenborgerlaan 171, 2020 Antwerpen,Belgium.

cCEMES CNRS, Rue Jeanne Marvig 29, BP 94347, 31055 Toulouse Cedex 4 ,France.

Abstract

We present energy filtered holography experiments on a thin foil of Al. By prop-agating the reduced density matrix of the probe electron through the microscope,we quantitatively predict the fringe contrast as a function of energy loss. Fringecontrast simulations include the effect of Fresnel fringes created at the edges ofthe defocused biprism, the effect of partial coherence in combination with inelasticscattering, and the effect of a finite energy distribution of the incoming beam.

Key words: holography, EELS, inelastic scattering, coherence, density matrixPACS: 34.80.Pa, 82.80.Pv

1 Introduction

Although the study of partial coherence is a well developed field in optics,relatively few publications deal with its consequences for electron microscopy.This is remarkable since it is well known that a typical electron beam is farfrom fully coherent and it is expected that partial coherence must have alarge influence on the images formed in an electron microscope. Usually itseffect on TEM image formation is approximated by the coherent envelopefunction which dampens the high spatial frequencies of the microscope transferfunction. This approach however fails completely to describe the outcome of

Email address: jo.verbeeck@ua.ac.be (Jo Verbeeck).

Preprint submitted to Elsevier Science 20 June 2007

experiments involving the direct measurement of coherence using an electronbiprism [1,2]. Recently there have been several experiments, mostly based onholography, to test the effects of partial coherence in combination with inelasticscattering [3–8]. The outcome of these experiments have proven to be difficultto understand in simple terms of either coherent or incoherent imaging and itwas shown that partial coherence has to be taken into account properly [9–22],to be able to understand the obtained contrast. Although the theory of partialcoherence can be extremely difficult to solve for an arbitrary setup, we canusually use the symmetry of the system to make suitable simplifications. Inthis contribution, we show that it is currently possible to accurately simulatethe outcome of holography experiments with inelastic scattering and partialcoherence.

2 Experimental

A thin sample of pure Al was obtained by electropolishing a sheet of purealuminum in a mixture of 10% perchloric acid and 90% ethanol. Experimentalenergy filtered holography images are acquired making use of a Philips CM30,300kV microscope with a GIF200 imaging filter. The energy selecting slit ischosen to be 3 eV. A biprism in the selected area aperture plane is usedwith a virtual diameter of 12 nm in the image plane. The specimen plane isshifted approximately z=5 µm above the virtual biprism plane while keepingthe specimen in focus on the viewing screen making use of free lens control.A positive voltage of 70 V is applied to the biprism to create a shear ofapproximately s=8 nm. This creates an overlap between two electron paths,a distance s apart, that went both through the crystal 1 and both have anenergy loss selected by the energy selecting slit. Numerical calculations areperformed using the MatlabTMscripting language.

3 Results and Discussion

The fringe patterns obtained at energy losses of 0, 5, 10, 15 and 20 eV areshown in fig.1. Fringes are clearly present at all energy losses, although theircontrast reduces with energy loss. This effect has been described in detail in[16], and is basically due to the delocalised interaction of the fast electronswith the electrons in the specimen. In this paper we go one step further anddescribe the effect of the defocused biprism on the fringe pattern, an effect thatbecomes more important for small shear values as in the current experiment.

1 as opposed to conventional elastic holography where a specimen wave is made tointerfere with a vacuum wave

2

The necessity for treating this effect properly, is seen in the experimentalimages where Fresnel fringes around the biprism wire are present. They lead tothe problem of unambiguously defining the fringe contrast, since the contrastchanges depending on the position.

The image formation in case of partial coherence can be calculated makinguse of the reduced density matrix for the fast electrons [9–22]. The reduceddensity matrix is a quantum mechanical tool to properly describe the inter-action between the fast electrons and the sample. Note that, because of thisinteraction, it is no longer possible to use a wave function as opposed to thesituation of pure elastic scattering. If we describe the total system of fast par-ticles and sample as non-interacting with the rest of the universe we can writethe total wave function as [23]:

Φ(x, y) =∑i,j

ai,jψi(x)Ψj(y), (1)

constructed from a complete orthonormal basis set of wave functions ψi(x)and Ψj(y) for the fast electrons and the sample respectively. Performing thesum in i we get:

Φ(x, y) =∑j

φj(x)Ψj(y), (2)

with:

φj(x) =∑

i

ai,jψi(x). (3)

Note that the φj(x) are in general no longer orthogonal.

In the experiment we only measure the fast electrons which can be described byintegrating out the non observed sample coordinates y leading to the definitionof the reduced density matrix ρ(x, x′) for the fast electrons:

ρ(x, x′) =∫

Φ(x, y)Φ∗(x′, y)dy

=∑i,j

∫φi(x)Ψi(y)φ

∗j(x

′)Ψ∗j(y)dy

=∑

i

φi(x)φ∗i (x

′). (4)

From this equation it is obvious that it is no longer possible to describe thestate of the fast electrons by means of a single wave function, because ofthe sum over the possible states i. If on the other hand we force the system

3

to remain in its ground state (i = 0), the density matrix depends only onone wave function and we recover the usual wave function picture used inelastic scattering 2 . Note that the density matrix contains information aboutthe probability of finding a fast particle in location x along the diagonal ofthe density matrix:

ρ(x, x) =∑

i

|φi(x)|2 (5)

The off-diagonal elements contain information about the mutual coherencebetween two points x and x′.

To apply this approach in electron microscopy we replace the wave functionby the density matrix and the Schrodinger equation by the kinetic equationfor density matrices [21]. The procedure is completely analogously to waveoptical calculations with the notable exception that now one has to treat allcalculations in four dimensional space (x, x′), rather than in two dimensionalspace, which can make numerical computation difficult unless symmetry canbe exploited.

Of special importance for this paper is the Fresnel propagator. In coherentwave optics we know how a wave function propagates from one plane r1 toanother r2 over a distance z along the beam direction (in free space) makinguse of the Fresnel propagator:

φ(r2) = φ(r1) ⊗ pF (z, r), (6)

with

pF (z, r) = −ikeikz

2πzei k

2zr2

, (7)

and k the fast electron wave vector. Completely analogous we can write thefree space propagation over a distance z along the beam direction for densitymatrices as a generalized Fresnel propagator [20]:

ρ(r2, r′2) = ρ(r1, r

′1) ⊗ PF (z, r, r′), (8)

with

PF (z, r, r′) =k2

4π2z2ei k

2z(r2−r′2). (9)

This gives us a tool to calculate the density matrices in different planes of themicroscope, and to treat partial coherence in a formal way. This approach will

2 Note that the elastic signal is in reality influenced by the possibility for inelasticscattering and therefore one should also use the density matrix formalism for elasticscattering [24]. It is common however to approximate the effect of inelastic scatteringas an absorption term

4

now be used to calculate the fringe contrast in the experiment as well as thedetailed fringe shape. The outline of the reasoning is given here.

The fringe contrast can be calculated starting from the kinetic equation forfast electron density matrices [21], neglecting elastic scattering and assumingsingle inelastic scattering to a homogeneous distribution of scattering centers(e.g. a plasmon). This equation gives the exit density matrix, taking into ac-count the inelastic scattering and the fact that the incoming fast electronsare not purely monochromatic. The energy spread of the incoming electronsleads to a mixing of the different energy losses that will be seen at a givenenergy in the exit density matrix. The off diagonal elements of the exit densitymatrix then contain information on the coherence, which is directly related tothe fringe contrast in the experiment. The mathematical details are treated inAppendix 1 and lead to a prediction of the fringe contrast that depends on theenergy spread of the gun (eq.15). The final result deviates from the truncatedK0 function (monochromatic theory) described in [11,16]. Both functions areshown in Fig.2 together with the experimentally observed maximum fringecontrast calculated from a linetrace through the images in Fig.1. The contrastis normalised to the elastic experiment, to cancel out the effect of the par-tial coherence of the incoming beam. Note the good fit with the experimentalvalues and the rather large deviation from the monochromatic theory (espe-cially for the 5 eV result). The observed fringe contrast for 10, 15 and 20 eV isslightly lower then the predicted value, which can be attributed to instabilitiesdue to the longer exposure times (0.5 s for 0 eV, 5 s for 5, 10, 15 eV and 10 sfor 20 eV).

The creation of the Fresnel fringes due to the defocused biprism is treated inappendix 2. The procedure is: one propagates the density matrices for the fastelectrons from the specimen to the biprism plane, then applies the biprismoperator (which cuts a part of the wave according to the size of the biprismwire and applies a linear phase shift according to the distance from the wire)and propagates back to the image plane. The diagonal of the final density ma-trix will be the observed intensity in the experiment. According to Appendix2, this results in a convolution of the elastic fringe contrast (including Fres-nel fringes coming from the cutting of the wave by the biprism wire) with arescaled angular distribution function (coming from the angular distributionof inelastic scattering). This result can intuitively be understood as sketchedin Fig.3 since every inelastic excitation will scatter to a certain angle θ whichleads to a shift over zθ of the intensity distribution in the final image. Sincethe inelastic scattering is assumed angularly incoherent (a consequence of thehomogeneous distribution of scattering centers), we have to sum the intensitiesof these shifted fringe patterns. Effectively this becomes a convolution of theelastic fringe pattern with a scaled version of the angular distribution, whichwill reduce the fringe contrast depending on z. The calculated traces show anextremely good comparison with the experiment (Fig.4) for all energy losses.

5

0 eV 5 eV

10 eV 15 eV

20 eV

−10 0 10 20 30 40

0

2

4

6

8

10

x 104 EELS spectrum

Energy Loss [eV]

coun

ts

Fig. 1. Experimental holography images for different energy losses and an energyselecting slit of 3 eV. The shear is fixed at approximately 8 nm. In the right bottomcorner the low loss EELS spectrum is shown. Line-traces are taken perpendicularto the fringes in a region indicated on the first image in the top left corner.

Both the fringe contrast as well as the Fresnel fringes of the biprism wire arereproduced. An incoming beam coherence of p = 0.6 is used for a fixed shearof s = 7.9 nm and a virtual biprism diameter of b = 12 nm. Slight deviationsbetween theory and experiment can be due e.g. to nonhomogeneous illumina-tion conditions, contamination on the biprism wire and varying thickness ofthe sample. Despite the absence of these imperfections in the simulation, thefit with the experimental line-traces is remarkably good.

6

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1monochromatic theoryfinite energy resolutionexperiment

Fig. 2. Fringe contrast vs. energy loss for s = 7.9 nm, Simulated function takes intoaccount the finite energy resolution and the experimental spectrum. An excellentagreement with experiment is obtained as opposed to the fringe contrast that isobtained from assuming a monochromatic beam.

Fig. 3. Sketch of the formation of the fringe contrast in inelastic image holographyincluding Fresnel fringes from the biprism. A) shows the formation of the imageon both sides of the defocused biprism for a plane wave with the biprism turnedoff. B) shows the effect of an inelastic scattering event with scattering angle θ, thatcauses a shift of the Fresnel shadow of the biprism by approximately zθ if both zand θ are small. C) When the biprism is switched on, a shift of both parts overs/2 creates overlap between waves on the left and right side of the biprism creatingfringes. The total intensity is given by summing incoherently the intensity of everyinelastic scattering event with a distinct scattering angle θ. This can be seen asa convolution of the elastic fringe contrast with a scaled version of the scatteringdistribution depending on z. This will lead to a reduction in the fringe contrastprofile as observed in the experiment.

7

−1 0 10

1

2

3

elastic simulation

−1 0 10

1

2

3

elastic experiment

−1 0 10

1

2

3

5 eV simulation

−1 0 10

1

2

3

5 eV experiment

−1 0 10

1

2

3

10 eV simulation

−1 0 10

1

2

3

10 eV experiment

−1 0 10

1

2

3

15 eV simulation

−1 0 10

1

2

3

15 eV experiment

−1 0 10

1

2

3

20 eV simulation

−1 0 10

1

2

3

20 eV experiment

Fig. 4. Simulated and experimental line traces through the fringes as a functionof energy loss. Note the good overall agreement between theory and experiment.Calculation details: z = 5.1 µm, E0 = 300 kV, diameter biprism = 12 nm, shears = 7.9 nm.

4 Conclusion

We are currently able to simulate the contrast of inelastic holography experi-ments involving partial coherence to a very high degree of accuracy. It showsthat the propagation of the density matrices through the microscope is a use-ful and efficient tool for the prediction of inelastic contrast. These ideas will

8

be valuable to develop high resolution EFTEM image simulations includinginelastic scattering and partial coherence. Another area of application mightexist in better understanding the fringe contrast in HRTEM images includinginelastic phonon scattering.

5 Acknowledgments

J.V. and P.S. acknowledge the support of the European Commission, contractnr. 508971 (CHIRALTEM). J.V. and G.B. acknowledge the FWO-Vlaanderenfor financial support under contract nr. G.0147.06 and the financial supportfrom the European Union under the Framework 6 program under a contractfor an Integrated Infrastructure Initiative. Reference 026019 ESTEEM. Allauthors acknowledge stimulating discussions with H. Lichte.

Appendix 1: the effect of a finite energy distribution

Starting from the Kinetic Equation (eq.11 in [21]), assuming no elastic scat-tering and single inelastic scattering we get:

ρexit(r, r′, E) = ρ0(r, r

′, E) +∫T (r, r′, E ′)ρ0(r, r

′, E + E ′)dE ′ (10)

We assume that the incoming density matrix can be written as a product withan energy distribution function f(E):

ρ0(r, r′, E) = ρ0(r, r

′)f(E) (11)

and that the MDOS for inelastic scattering (considering a a homogeneousdistribution of scattering centers) can be written as [16]:

T (r, r′, E) = g(E)K0(qE|r − r′|), (12)

with g(E) the probability for inelastic scattering at a given energy. This equa-tion holds regardless of the type of excitation, as long as it is delocalised enoughso that there is no dependency of the scattering probability with r + r′. Weget:

ρexit(r, r′, E) = ρ0(r, r

′)[δ(E) + g(E)K0(qE|r − r′|)] ⊗ f(E + E ′). (13)

The fringe contrast can easily be obtained from:

9

C(r, r′) =2Re(ρexit(r, r

′, E))

ρexit(r, r, E) + ρexit(r′, r′, E). (14)

Assuming a homogeneous sample we get:

C(r − r′) =Re(ρexit(r, r

′, E))

ρexit(r, r, E)

C(r − r′) =Re(ρ0(r, r

′)[g(E)K0(qE|r − r′|) + δ(E)] ⊗ f(E))

ρ0(r, r)[g(E) + δ(E)] ⊗ f(E)

C(r − r′) = pRe([g(E)K0(qE|r − r′|) + δ(E)] ⊗ f(E))

[g(E) + δ(E)] ⊗ f(E). (15)

The denominator is the experimental spectrum (apart from noise), and thenumerator can be calculated from the experimental spectrum by first deconvo-luting with the zero loss peak f(E), and then multiplying with the truncatedK0 function and reconvoluting with f(E). The reconvolution also takes careof noise amplification problems, similar to the deconvolution with a Gaussianmodifier. In eq. 15 the effect of partial coherence p of the incoming beam isadded.

Appendix 2: Fresnel fringes and partial coherence

To obtain all the details of the fringes including Fresnel diffraction around thebiprism, we have to go one step further. The image of the specimen is at adistance |z| below the SA plane (see fig.3) containing the biprism. We look ata plane focused on the specimen. Starting from eq.13, we can apply a Fresnelpropagator for density matrices [20], to obtain the density matrix in the SAplane:

ρSA(r, r′, E) = ρexit ⊗ Pf (z, r, r′) (16)

with Pf (z, r, r′) = k2

4π2z2 ei k2z

(r2−r′2) and z < 0 (going up). Assuming an incomingplane wave:

ρ0(r, r′, E) = I0f(E) (17)

with I0 the intensity, we get that:

ρSA(r, r′, E) = ρexit(r, r′, E). (18)

10

Since ρexit only depends on r− r′ and therefore the Fresnel propagator has noeffect. In the SA plane we apply the biprism operator [20] which gives a phaseramp according to the position of both r and r′ with respect to the biprismwire:

TB(r, r′) =H(r · eb − b/2)H(r′ · eb − b/2)e−iα(r−r′)·eb

+H(r · eb − b/2)H(−r′ · eb − b/2)e−iα(r+r′)·eb

+H(−r · eb − b/2)H(r′ · eb + b/2)e−iα(−r−r′)·eb

+H(−r · eb − b/2)H(−r′ · eb − b/2)e−iα(−r+r′)·eb , (19)

with eb a unit vector perpendicular to the wire, b the diameter of the wire, andα is a constant related to biprism voltage. The density matrix in the imageplane after the biprism then becomes:

ρB(r, r′, E) = ρSA(r, r′, E)TB(r, r′). (20)

We image the specimen plane, so we have to apply another Fresnel propagatorto reach the final image density matrix:

ρv(r, r′, E) = ρB(r, r′, E) ⊗ Pf (−z, r, r′). (21)

Simplifying the notation and leaving the convolution in energy till the end weget:

hv1(r, r′, E) = h(r, r′, E)TB(r, r′) ⊗ Pf (−z, r, r′) (22)

with:

h(r, r′, E) = δ(E) + g(E)K0(qE|r − r′|). (23)

Writing this out we get 4 terms (due to the four terms in the biprism operator).The first one is:

hv1(r, r′, E) =

∫∫h(r − ξ, r′ − ξ′, E)H(r.eb − ξ − b

2)H(r′.eb − ξ′ − b

2)

e−iα(r−ξ−r′+ξ′).ebe−i k2z

(ξ2−ξ′2)d2ξd2ξ′

hv1(r, r′, E) = e−iα(r−r′).eb

∫ r.eb−b/2

−∞

∫ r′.eb−b/2

−∞h(r − ξ, r′ − ξ′, E)

e−i k2z

((ξ− zαk

eb)2−(ξ′− zα

keb)

2)d2ξd2ξ′

hv1(r, r′, E) = e−iα(r−r′).eb

∫ r.eb−b/2− zαk

eb

−∞

∫ r′.eb−b/2− zαk

eb

−∞h(r − ξ, r′ − ξ′, E)

e−i k2z

(ξ2−ξ′2)d2ξd2ξ′. (24)

11

We are finally interested in the diagonal elements only, because they willdetermine the observed image:

Hv1(r, E) = ρv1(r, r) =∫∫ r.eb−b/2− zα

keb

−∞h(r − ξ, r − ξ′, E)

e−i k2z

(ξ2−ξ′2)d2ξd2ξ′. (25)

Using the Fourier transform of h(r, r′, E) wrt. ξ and ξ′ and noting that h(r−ξ, r − ξ′, E) only depends on ξ − ξ′:

h(r − ξ, r − ξ′, E) =∫∫

h(q, q′, E)ei(qξ−q′ξ′)d2qd2q′

h(r − ξ, r − ξ′, E) =∫∫

[δ(E)δ(q)δ(q′) + g(E)δ(q − q′)1

q2 + q2E

]

ei(qξ−q′ξ′)d2qd2q′

h(r − ξ, r − ξ′, E) =∫

[δ(E)δ(q) + g(E)1

q2 + q2E

]eiq(ξ−ξ′)d2q

h(r − ξ, r − ξ′, E) =∫h(q, E)eiq(ξ−ξ′)d2q, (26)

with h(q, E) the angular distribution of scattering as a function of energyloss. This consists of two parts: an elastic part which has a delta function inq and an inelastic part which has a Lorentzian scattering distribution with aweighting factor g(E), the energy loss spectrum. Filling this in we have:

Hv1(r, E) =∫∫ r.eb−b/2− zα

keb

−∞[∫h(q, E)eiq(ξ−ξ′)d2q]e−i k

2z(ξ2−ξ′2)d2ξd2ξ′

Hv1(r, E) =∫h(q, E)

[ ∫ ∫ r.eb−b/2− zαk

eb− zqk

−∞e−i k

2z(ξ2−ξ′2)d2ξd2ξ′

]d2q. (27)

The integral in square brackets can be solved using Fresnel integrals:

F (u) =∫ u

0ei π

2x2

dx. (28)

We get assuming eb = ex:

12

∫ r.eb−b/2− zαk

eb− zqk

−∞e−i k

2zξ2

d2ξ =

=∫ rx−b/2− zα

k− zqx

k

−∞e−i k

2zξ2xdξx

∫ ∞

−∞e−i k

2zξ2ydξy

=πz

k[F ∗(

√(k

πz)(rx − b/2 − zα

k− zqx

k)) − F ∗(−∞)]

[F ∗(∞) − F ∗(−∞)]

=πz

k[F ∗(

√(k

πz)(u− z

kqx)) +

1

2(1 − i)]

= f ∗(u− z

kqx), (29)

with u = rx − b/2 − zαk

. The total intensity for the first term then becomes:

Hv1(r, E) =∫h(q, E)

[f ∗(u− z

kqx)f(u− z

kqx)

]d2q, (30)

which can be seen as a convolution of the elastic intensity with h(q kz, E),

describing the angular distribution of scattering scaled with the z-shift:

Hv1(r, E) =∫h(q

k

z, E)

[f ∗(u− qx)f(u− qx)

]d2q. (31)

This is nothing but a 2D convolution of the intensity that we would have forelastic scattering with h(q k

z, E). The other three terms are analogous:

Hv1(r, E) =∫h(q

k

z, E)

[f ∗(u− qx)f(u− qx)

]d2q (32)

Hv2(r, E) =∫h(q

k

z, E)

[e−iα[(u−qx)+(v−qx)]f ∗(u− qx)f(v − qx)

]d2q (33)

Hv3(r, E) =∫h(q

k

z, E)

[e+iα[(u−qx)+(v−qx)]f ∗(v − qx)f(u− qx)

]d2q (34)

Hv4(r, E) =∫h(q

k

z, E)

[f ∗(v − qx)f(v − qx)

]d2q, (35)

with

v = rx + b/2 +zα

k. (36)

The total intensity is therefore:

13

Hv(r, E) =∫h(q

k

z, E)

[f ∗(u− qx)f(u− qx)

]

+2Re[e−iα[(u−qx)+(v−qx)]f ∗(u− qx)f(v − qx)

]

+[f ∗(v − qx)f(v − qx)

]d2q. (37)

The function between square brackets does not depend on qy so we can performthe qy integral and write the result as a convolution:

Hv(r, E) =[|f(u)|2 + 2Re

[e−iα(u+v)f ∗(u)f(v)

]+ |f(v)|2

]

⊗hx(qxk

z, E), (38)

with:

hx(qxk

z, E) =

∫h(q

k

z, E)dqy

= δ(E) +g(E)√

r2x + ( z

kqE)2

. (39)

The total intensity then becomes:

Iv(rx, E) = I0

([|f(u)|2 + 2pRe

[e−iα(u+v)f ∗(u)f(v)

]+

|f(v)|2]⊗ hx(qx

k

z, E)

)⊗ f(E). (40)

where the effect of partial coherence of the incoming beam is added throughp, the amount of spatial coherence for a given shear value. We see that, for amonochromatic system f(E) = δ(E), the resulting fringe pattern is a convo-lution of the elastic fringe pattern with a projected Lorentzian function. Thisleads to a reduction of the fringe contrast for increasing energy loss. The effectof a finite energy distribution in the incoming density matrix is a convolutionwith approximately the same projected Lorentzian, but with a δ(q) addedto it, coming from the tail of the elastic scattering. This increases the fringecontrast with respect to the monochromatic result, especially at low energylosses, where f(E) �= 0.

14

References

[1] D. Gabor, Nature 161 (1948) 777-778

[2] G. Mollenstedt, H. Duker, Naturwiss. 42 (1954) 41

[3] H. Hashimoto, A. Kumao, phys. stat. sol. (a) 107 (1988) 611-618

[4] F. Zhou, E. Plies, Proc. of the 11th European Congress on Electron Microscopy(EUREM), Dublin (1996), Vol. 2, pp. 555-556

[5] A. Harscher, H. Lichte, J. Mayer, Ultramicroscopy 69-3 (1997) 201-209

[6] H. Lichte, B. Freitag, Ultramicroscopy 81 (2000) 177-186

[7] R. A. Herring Ultramicroscopy 104 (2005) 261

[8] P.L. Potapov, H. Lichte, J. Verbeeck and D. van Dyck, Ultramicroscopy 106-11-12 (2006)1012-1018

[9] H. Rose, Optik 45 (1976) 139-187

[10] H. Rose, Ultramicroscopy 15 (1984) 173-192

[11] P. Schattschneider, H. Lichte, Phys. Rev. B 71 (2005) 045130

[12] M. Nelhiebel, Effects of crystal orientation and interferometry in electron energyloss spectroscopy, Ph. D. thesis, Ecole Centrale Paris (1999)

[13] P. Schorsch, Simulation energiegefilterter elektronenmikroskopischer Bilderunter Berucksichtigung inelastischer Streuprozesse, Ph. D. thesis, Universityof Darmstadt (2001)

[14] M. Nelhiebel, N. Luchier, P. Schorsch, P. Schattschneider, B. Jouffrey,Philosophical Magazine B 79-6 (1999) 941-953

[15] P. Schattschneider, M. Nelhiebel, B. Jouffrey, Physical Review B, 59-16 (1999)10959

[16] J. Verbeeck, D. van Dyck, H. Lichte, P. Potapov, P. Schattschneider,Ultramicroscopy 102 (2005) 239-255

[17] P. Schattschneider, B. Jouffrey, Ultramicroscopy 96 (2003) 453-462

[18] P. Schattschneider, M. Nelhiebel, H. Souchay, B. Jouffrey, Micron 31 (2000)333-345

[19] M. Nelhiebel, P. Schattschneider, B. Jouffrey, Physical Review Letters 85-9(2000) 1847

[20] P. Schattschneider, J. Verbeeck, Ultramicroscopy (2007),doi:10.1016/j.ultramic.2007.05.011

[21] S.L. Dudarev, L.-M. Peng, M.J. Whelan, Phys. Rev. B 48-18 (1993) 13408-13429

15

[22] J. Verbeeck, Ultramicroscopy 106 (2006) 461-465

[23] R. P. Feynman, Statistical Mechanics (a set of lectures), Perseus Books Group,p. 368, United States, ISBN : 0201360764

[24] H. Rose, personal communication

16