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Simulation of TEM and STEM images

277

Scanning Microscopy Vol. 11, 1997 (Pages 277-286) 0891-7035/97$5.00+.25Scanning Microscopy International, Chicago (AMF O’Hare), IL 60666 USA

SIMULATION OF TRANSMISSION AND SCANNING TRANSMISSION ELECTRONMICROSCOPIC IMAGES CONSIDERING ELASTIC AND THERMAL DIFFUSE

SCATTERING

Abstract

A reliable image simulation procedure for transmis-sion (TEM) and scanning transmission (STEM) electronmicroscopic images must take into account plural elasticscattering, inelastic scattering and thermal diffuse scattering.The intensity of the simulated images depends strongly onthe elastic scattering amplitude and the models chosen todescribe inelastic and thermal diffuse scattering. Ourimproved image simulation procedure utilizes the approxima-tion proposed by Weickenmeier and Kohl for the elasticscattering amplitude instead of the Doyle-Turner approx-imation. Thermal diffuse scattering is treated in terms of theEinstein model. In this paper, simulated TEM diffractionpatterns and high-angle annular dark-field (HAADF) STEMimages of thick crystals of silicon are used to demonstratethe influence of plural elastic and thermal diffuse scatteringon the image formation. The comparison of calculated TEMdiffraction patterns using the two different approximationsfor the elastic scattering amplitude shows that the use ofthe Doyle-Turner approximation results in a smaller imageintensity in the range of large scattering angles. SimulatedHAADF STEM images for different thicknesses of a siliconcrystal show strong influence of the dynamical calculationon image formation for thick specimens. In this case a theoryfails which describes the image formation process by asimple convolution of a probe function with an objectfunction.

Key Words: Image simulation, high-angle annular dark-fieldscanning transmission electron microscopic images,transmission electron microscopy diffraction patterns, pluralelastic scattering, thermal diffuse scattering, elasticscattering amplitude, Einstein model, frozen latticeapproximation.

*Address for correspondence:H. RoseInstitute of Applied Physics, TU DarmstadtHochschulstrasse 6, 64289 Darmstadt, Germany

Telephone number: +49 6151 162718FAX number: +49 6151 166053

E-mail: [email protected]

C. Dinges and H. Rose*

Institute of Applied Physics, Darmstadt University of Technology, Darmstadt, Germany

Introduction

In the past, several attempts have been made toinclude thermal diffuse scattering and inelastic scatteringinto the theory of image formation in electron microscopy.The approaches of Rose (1984), Wang (1995) and Dinges etal. (1995) are based on the multislice formalism (Cowley andMoodie, 1957) which allows the calculation of inelasticallyfiltered images for transmission (TEM) and scanningtransmission (STEM) electron microscopy. Allen andRoussow (1993) proposed an ansatz using the Bloch waveformalism.

Thermal diffuse scattering has been included intoan image simulation procedure for the first time by Xu et al.(1990). The authors introduced the so-called frozen latticemodel. This model is based on the approximation that thetime for the electron to pass through the object is muchsmaller than the vibration period of the atoms. Therefore,each electron interacts with another configuration of theatoms. To obtain the resulting intensities one must add upthe image intensity of all partial images resulting from thedifferent configurations of the atoms. The calculation ofsuch an image is very simple: one performs several multislicecalculations for different configurations of the atoms andsums up the resulting images incoherently. The individuallattice configurations can be obtained by assuming simplemodels for the atomic vibrations, e.g., the Einstein model.

In our previous paper (Dinges et al., 1995) we havedescribed another possibility for the calculation of imageswhich considers plural elastic scattering, inelastic scatteringand thermal diffuse scattering. We have shown that theaveraging over the different configurations of atoms can beperformed semi-analytically if the Einstein model is applied.This model assumes that each atom oscillates independentlyalong the three coordinate axes. Our calculations result intransmission functions which can be used for the multisliceformalism.

In the following, we refer to our image simulationprogram (Dinges et al., 1995). In the second (theoretical)section it will be shown that the use of the Weickenmeier-Kohl approximation (Weickenmeier and Kohl, 1991) for theelastic scattering amplitude allows the calculation ofanalytical expressions which describe thermal diffuse

278

C. Dinges and H. Rose

scattering. In the third section the working scheme of ourimage simulation procedure is explained using a simple exam-ple. In the forth section the new analytical expressions areapplied to the image simulation. First, a simulated TEMdiffraction pattern of a 50 nm thick crystal of silicon iscompared with a previously published image (Dinges andRose, 1996). Then the influence of thermal diffuse scatteringon HAADF STEM images is discussed. The results of ourimage simulation procedure are compared with thoseobtained by Hillyard and Silcox (1995) using the frozen latticeapproximation and the Einstein model. In addition theinfluence of crystal thickness on the image intensity inHAAD STEM images is discussed in detail. The main resultsof the paper are summarized in the fifth section.

Theory Based on Improved Scattering Amplitudes

In our previous work (Dinges et al., 1995; Dingesand Rose, 1996) we employed the Doyle-Turner approx-imation for the electron scattering amplitude. In this casethe scattering amplitude is given as the sum of four Gaussfunctions:

( ) ∑=

−=4

1

2}exp{i

ii KBAKf�

where K denotes the scattering vector and Ai and B

i are

constants tabled by Doyle and Turner (1968) and Smith andBurge (1962). Unfortunately, this fit is only valid in the range

of scattering vectors up to K < 4π⋅2Å-1. The calculation ofdiffraction patterns of thick crystalline specimens whichemploys this approximation underestimates high-anglescattering and, therefore, leads to erroneous intensities forthe diffraction spots (Fox et al., 1989; Bird and King 1990;Weickenmeier and Kohl, 1991).

This approximation has also been used to obtainanalytical expressions for phonon scattering. In this casethe correct structure of the phonon background in thediffraction patterns is obtained because this structure islargely determined by the symmetry of the crystallinespecimen. Nevertheless, the absolute intensity of thephonon background is still underestimated.

In the past several routines have been developed(Fox et al., 1989; Bird and King, 1990; Weickenmeier andKohl, 1991) which approximate the elastic scatteringamplitude in the full range up to K < 4π⋅6Å-1 tabled byDoyle and Turner (1968). Here we will show that theimproved approximation for the elastic scattering amplitude

given by Weickenmeier and Kohl (1991) yields analyticalexpressions for the phonon transmission functions.

An appropriate starting point for the calculation ofthe improved transmission function of a thin slice is themutual dynamic object transparency (MDOT) M( , ′) whichcan be written (Dinges et al., 1995) as

Figure 1. Simulation of images considering single thermal diffuse scattering and plural elastic scattering. Elastic scatteringevents occur in the blank slices. Thermal diffuse scattering events occur in the hatched slices. A detailed descriction is givenin the section on “Modified Multislice Formalism”.

(1)

(2)


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The transmission functions T and T(ab) are defined as

where

is the thermally averaged phase shift,

the phonon term and

the absorption potential. The mean square vibrationamplitude of the i-th atom is denoted by u

i2. The trans-

mission function T(ab)( ) considers the effect of phononscattering only by means of an absorption potential. Thisincomplete approximation merely results in a damping ofthe image intensity. By employing this transmissionfunction for the simulation of TEM diffraction patterns, oneonly obtains a slight change in the intensities of thediffraction spots, yet never a diffuse background. To obtainthe diffuse background the term µ

11(ph) ( , ′) T( )T*( ′) must

be incorporated in the image simulation. It has been shownpreviously (Dinges et al., 1995) that an appropriatefactorization of µ

11(ph) ( , ′) is possible. In this case the well-

known multislice formalism (Cowley and Moodie, 1957) canbe applied for the image simulation. The factorization can

Figure 2. Calculated background diffraction patternof Si in <111>-orientation caused by thermal diffusescattering employing the improved Weickenmeier-Kohl approximation for the scattering amplitude.

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(4)

(5)

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(9)

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be written as a sum of products. Each product consists oftwo factors, one is a function of , the other a function of ′.Reducing the sum of products to a sum of factors multipliedby an additional random phase factor leads to the phonontransmission function. Multiplying this function with itsconjugate complex and averaging over the random phasefactors results in the MDOT for a thin slice.

Rewriting the elastic scattering amplitude(Weickenmeier and Kohl, 1991) as

shows that this approximation can be expressed as a sum ofintegrals over exponential functions with K2 as parameter.Therefore, the formulae given in our first paper (Dinges etal., 1995) can be applied, where the constants A

i must be

replaced by i and the additional integrations over B

i must

be performed in all corresponding formulae.Performing this integration for ⟨χ⟩ results in

The function E1(x) in the last expression denotes the integral

logarithm

The somewhat lengthy calculation of the phonon trans-mission function leads to

and

The details of the calculation and the absorption potentialµ

2(ph) ( ) are given in Appendix A. We obtain two

transmission functions because the last term in Equation(24) (Appendix A) contains a negative sign. To determinethe image including phonon scattered electrons, theimproved multislice formalism must be used three times:First, one has to calculate the image by considering pluralelastic scattering and phonon scattering by means of the

Figure 3. Linescans across the two diffraction patternsshown in Figure 2; the curve (---) corresponds to Figure 2a,the curve (- - -) to Figure 2b.

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(11)

(12)

(13)

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absorption potential. In the next step a partial image iscalculated using the phonon transmission function T(ph1).The third step determines the “negative” partial image bymeans of the second phonon transmission function T(ph2).This image must be subtracted from the other two imagesbecause of the negative sign in Equation (24).

Modified Multislice Formalism

In the following discussion we restrict ourself toplural elastic scattering and thermal diffuse scattering.Inelastic scattering is not taken into account. Nevertheless,for a complete description of the image formation processinelastic scattering should not be neglected.

In the previous section thermal diffuse scatteringhas been considered by applying the Einstein model. Thismodel describes each atom as an independent oscillatoralong the three coordinate axes. Therefore, scatteredelectron waves originating from thermal diffuse scatteringprocesses at different atoms must be superimposedincoherently. In contrast scattered electron waves resultingfrom elastic scattering must be superimposed coherently.In the previous section we have calculated transmissionfunctions which describe elastic scattering and thermaldiffuse scattering. These transmission functions are appliedto simulate images. The calculation has shown that it isnecessary to divide the simulation process into two steps:

In the first step an image calculation is performedwhich takes into account plural elastic scattering and thermaldiffuse scattering by means of an absorption potential. Theabsorption potential describes the influence of an (imaginary)energy filter which removes the thermal diffuse scatteredelectrons from the electron beam. Therefore, the resultingimage intensity is damped in comparison to the intensity ofan image which has been calculated considering only pluralelastic scattering. This image is obtained by performing awell-known multislice calculation.

In the second step an image is calculated with themodified multislice formalism. Every electron wave whichcontributes to this image has performed at least one thermaldiffuse scattering process. In this case an energy filtereliminates all purely elastically scattered electrons from thebeam. As mentioned above the scattered waves stemmingfrom thermal diffuse scattering processes at different atomsmust be superimposed incoherently. Figure 3 shows theworking scheme for the calculation of the “inelastic” image.In this example we consider an object divided in four slices.There should be no empty slice. For a proper calculationtaking into account plural elastic scattering and singlethermal diffuse scattering one has to perform severalmultislice calculations. Thermal diffuse scattering occurswith equal probability in one of the four slices, because themean free path length is large compared to the entire

thickness of the object. If each slice contains only one atomthe calculation of the second step image is finished afterperforming the four cases for scattering events depicted inFigure 3. Normally, there are several atoms in each slice. Inthis case the scattered waves stemming from thermal dif-fuse scattering processes at different atoms must besuperimposed incoherently. To ensure this incoherentsuperposition of the inelastically scattered waves emanatingfrom the atoms of the first slice, a multislice formalism mustbe performed for each atom in the first slice using thephonon transmission function for this slice and the elastictransmission function for the other three slices.Subsequently the images intensities obtained from each ofthese calculations must be superimposed. The sameprocedure occurs for thermal diffuse scattering in the otherthree slices. Therefore, one has to perform Σ

n i

n multislice

calculations to obtain the image. The index n denotes theslice, in which thermal diffuse scattering occurs, the symboli indicates the number of atoms in this slice.

Since one deals with thick objects consisting of alarge number of atoms in a unit cell, one can use a trick toreduce the calculation time. By introducing statistical phasesit is possible to calculate images considering thermal diffusescattering and plural elastic scattering rather fast. Assumingthat thermal diffuse scattering occurs in the first slice onehas to multiply the corresponding scattered waves withrandom phase factors. After performing a few number ofmultislice calculations with different sets of statisticalphases, the interference terms between waves emanatingfrom different atoms cancel out in the averaged image.

The frozen lattice model can also be applied for ourmodel. In this case one has to perform the multislicecalculations by using different configurations of the atomsin each of the four slices where thermal diffuse scatteringoccurs. Therefore the calculation time is comparable for thetwo models.

Image Simulation

In the following simulated TEM diffraction patternsare used to demonstrate the differences in the imageintensity resulting from the chosen approximation for theelastic scattering amplitude.

To investigate the influence of phonon scatteringon HAADF STEM images of thick crystalline objects,linescans through such images for a 20 nm thick crystal ofsilicon in <110>-orientation have been calculated. Byemploying the imaging conditions chosen by Hillyard andSilcox (1995), we can compare the results of our simulationwith their results which are based on the frozen phononapproximation.

By varying the specimen thickness for distinct probepositions, we can roughly determine the fraction of image

282


intensity which stems from thermal diffuse scattering as afunction of thickness.

Simulation of TEM diffraction patterns

Simulated TEM diffraction patterns of a simplecrystalline object are well suited for demonstrating thedifferences in image intensity which result from the use ofdifferent approximations for the scattering amplitude. Thecalculations have been performed for a 50nm thick crystalof silicon in <111>-orientation and a 100kV TEM. A 9 x 9supercell of Si <111> with a = 1.99 nm, b = 1.99 nm, c = 0.939nm, α = β = π/2 and γ = π/3 has been assumed. The multislicecalculations have been performed on a 512 x 512 matrixchoosing a slice thickness c/3 for purely elastic scattering.For slices with single thermally diffuse scattering the thick-ness c was chosen. The diffraction patterns presented inFigure 2 have been obtained after averaging over tendifferent sets of statistical phases.

Figure 2 shows the background caused by thermaldiffuse scattering taking into account only single thermaldiffuse scattering and plural elastic scattering. Figure 2ahas been obtained by applying the Doyle-Turner Gaussianfit for the elastic scattering amplitude, while Figure 2brepresents the image calculated with the scattering amplitudegiven by Weickenmeier and Kohl (1991). The two images,when displayed with the same gray scales, do not differsignificantly from each other. But nevertheless, there aredifferences, as shown in Figure 3. By taking linescans fromthe top left edge to the bottom right edge, the differencesbecome clearly visible. Especially for large scattering anglesthe use of the Doyle-Turner Gaussian fit results in small

image intensities, while those obtained by the Weickenmeier-Kohl approximation are significantly larger. This effect canbe explained by the different asymptotic behaviour of thetwo approximations. The result for the Doyle-Turner fit de-creases faster for large scattering angles than that for theapproximation of Weickenmeier and Kohl which yields thecorrect asymptotic behaviour.

Simulation of HAADF STEM images

Hillyard and Silcox (1995) assumed a 100 kV STEMwith C

s = 0.05 mm, ∆f = 13.6 nm and an aperture angle θ

0 = 23

mrad. These parameters yield a probe size of approximately0.1 nm. Therefore, the atoms of the dumbbells in the <110>-projection of silicon (Fig. 4) are resolved. For our calculationswe assume a 8 x 6 supercell with a = 3.072 nm, b = 3.264 nmand c = 0.384 nm. This cell is sampled on a 512 x 512 matrix.According to the distance between the two planes in whichthe atoms in the supercell are lying, a minimum slicethickness of 0.192 nm has been chosen for the multislicecalculations. All images have been calculated by choosinga slice thickness of 4 x 0.192 nm for the slices in whichthermal diffuse scattering occurs. A thickness of 0.192 nmhas been chosen for slices which contain solely elasticscattering events.

An annular detector with limiting angles θout

= 4θin,

θin = 50 mrad has been assumed. Thermal diffuse scattering

is considered by using the Einstein model with a meanvibration amplitude u

Si = 7.6 pm.

Figure 5a shows linescans taken from images of a19.2 nm thick crystal which are obtained by consideringonly elastic scattering (——), incorporating the effect ofthermal diffuse scattering by means of an absorptionpotential (⋅⋅⋅⋅) and assuming plural elastic scattering plussingle thermal diffuse scattering (----). The calculations haveshown that the average taken over three different sets ofstatistical phases leads already to reliable results. The useof more sets of statistical phases require larger computationtimes yet changes the calculated intensities by no morethan 3%.

Figure 5b shows the scaled image intensities ob-tained by considering only plural elastic scattering (----)and by taking into account, in addition, thermal diffusescattering (——).

The scaling factor has been chosen as the ratiobetween the mean image intensities of the two images. Thecomparison reveals that for the given position of the probeand detector geometry the image intensity decreases ifthermal diffuse scattering is considered. This result is ingood agreement with the result obtained by Hillyard andSilcox (1995). Their calculations are based on the frozenphonon approximation. The comparison between theabsolute values of the intensity plots of Figure 10d in thepaper of Hillyard and Silcox (1995) with those of Figure 5bleads to an additional scaling factor.

Figure 4. Si in <ll0>-orientation. The atoms located in theplane c = 0.192 nm are drawn black, the atoms marked whiteare positioned in the plane c = 0 nm. The dashed line marksthe linescans depicted in Figure 5. The two crosses indicatethe probe positions for the thickness series presented inFigure 6.


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Calculations based on the frozen phonon approxi-mation are not restricted to single thermal diffuse scatter-ing. In contrast our calculations are based on the assumptionof single thermal diffuse scattering. Since the two methodsproduce identical results, the assumption of only singlethermal diffuse scattering processes is justified for mostobject thicknesses used in practice.

The fact that we need two partial images to obtain

the total HAADF STEM image allows us to estimate thefraction of the image intensity which stems from thermaldiffuse scattering. Figure 6 shows calculated thicknessseries for the two positions marked in Figure 4. Thecalculations have been performed for the imaging conditionsdescribed above. The figure clearly demonstrates that thefraction of the image intensity that results from plural elas-tic scattering rapidly decreases with increasing thickness,

Figure 5. Linescans through HAADF STEM images of a 19.2 nm thick crystal of silicon in <110>-orientation. (a) unscaledimage intensities obtained by considering only plural elastic scattering (____), thermal diffuse scattering by means of anabsorption potential (⋅⋅⋅⋅) and single thermal diffuse scattering and plural elastic scattering (----), (b) scaled image intensitiesobtained by considering only plural elastic scattering (----) and single thermal diffuse scattering plus plural elastic scattering(____).

Figure 6. (a) Relative fraction of image intensity resulting from single thermal diffuse scattering and (b) from plural elasticscattering for the case where the probe is centered on the atom column (----) and where it is located midway between twoadjacent columns (____). Each slice has a thickness of 0.192 nm.

284


while the corresponding fraction of image intensity causedby thermal diffuse scattering shows the opposite behaviour.At a crystal thickness of 19.2 nm only 10% of the imageintensity results from purely elastic scattering. The curvesfor the two different probe positions do not differsignificantly from each other.

The curves depicted in Figure 6 can also be used toevaluate the column contrast as a function of specimenthickness. The column contrast is defined as

where Itop

denotes the intensity when the probe is centeredon top of an atom row and I

between is the intensity when the

probe is located at the center of hexagon shown in Figure 4.The column contrast C resulting from Figure 6 is depictedin Figure 7. The solid curve describes the correctconsideration of thermal diffuse scattering, while the dashedcurve considers this effect approximately by means of anabsorption potential. In both cases the contrast C decreasesoscillatorily with increasing thickness. Nevertheless, evenfor thicker specimens the contrast is large enough to makethe atom columns clearly visible in the image. Forthicknesses larger than about 8 slices the column contrastC, obtained by including correctly the effect of thermaldiffuse scattering, is always larger than that obtained by

employing merely an absorption potential. The oscillationsare present over the entire range of thickness. They areproduced by interference effects and may serve as a measurefor the “degree of coherence” of the detected signal. This“degree of coherence” strongly depends on the chosendetector geometry since it determines the fraction of theelastically scattered electrons which contributes to thesignal. Experimentally the “degree of coherence” of thesignal can be obtained relatively easily by means of a wedge-shaped Si crystal.

Conclusion

The use of the Doyle-Turner Gaussian fit for theelastic scattering amplitude underestimates scattering inhigh angles and predicts, therefore, smaller intensities inthe outer regions of diffraction patterns. The implementationof the modified formulae in our program allows a moreaccurate simulation of filtered and unfiltered diffractionpatterns.

Furthermore, we have shown that the results of ourHAADF STEM image simulations which consider onlysingle thermal diffuse scattering are in good agreement withcalculations of Hillyard and Silcox (1995) who employ thefrozen phonon approximation.

Our calculations of thickness series for special probepositions on a Si crystal in <ll0>-orientation demonstratethat approximately 90% of the intensity in HAADF STEMresults from thermal diffuse scattering if the object thicknessis larger than about 10 nm.

Acknowledgments

Financial support by the Volkswagen-Stiftung isgratefully acknowledged. We thank Dr. Earl Kirkland fromCornell University (USA) for valuable discussions. Thanksare also due to Professors J. Silcox and M. Isaacson, CornellUniversity, for giving one of the authors (H.R.) theopportunity to spend part of his sabbatical at the Instituteof Applied and Engineering Physics.

Appendix A

The evaluation of µ11

(Ph) ( , ′)i by means of the

Doyle-Turner approximation has been outlined (Dinges et

al., 1995). Replacing the constants Aµ by the constants µand adding the two integrations over Bµ and B

v leads to

µ11

(Ph)( , ′) in terms of the elastic scattering amplitude givenby Weickenmeier and Kohl (1991):

Figure 7. Column contrast C as a function of specimenthickness. The solid curve has been calculated byconsidering correctly single thermal diffuse scattering plusplural elastic scattering. The dashed curve has been obtainedfrom images which have been calculated assuming pluralelastic scattering and considering thermal diffuse scatteringonly by means of an absorption potential. Each slice has athickness of 0.192 nm.

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With the abbreviations Gµ = Bµ+ui2/4, G

v = B

v + u

i2/4 and the

subsitutions -i0→ , ′-

i0 → ′, u

i2/4 = x, Equation (16) is

rewritten as

In the following only the first term of Equation 17

is evaluated, because this term is not given as a sum ofproducts, where the factors are only functions of or ′.Since x is small compared to unity, f(x) can be expanded ina Taylor series

where f(i) (0) denotes the i-th derivative at x = 0 and R(x) isthe residual term.

The evaluation of the first two derivatives at x = 0yields the expressions

By inserting these relations into Equation (20) andconsidering dB

v = dG

v, we can approximately rewrite

Equation (17) as follows:

The integration over Gv can readily be performed.

The subsequent integration over the variable Gµ yields theanalytical expressions

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(20)

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by means of which the function (24) can be expressed. Theresult leads to the two phonon transmission functions givenin Equations (13) and (14). The necessity for twotransmission functions is a consequence of the negativesign of the last term in Equation (24).

In the special case = ′ we obtain the absorption

potential µ2(ph) ( ) = µ

11(ph) ( ′ = ).

By neglecting the term f(2)(0)x2/2, we derive the dipoleapproximation of phonon scattering. In this case only asingle transmission function exists.

References

Allen LJ, Rossouw CJ (1993) Delocalization inelectron-impact ionization in a crystalline environment. PhysRev B47: 2446-2452.

Bird DM, King QA (1990) Absorptive form factorsfor high-energy electron diffraction. Acta Cryst A46: 202-last page.

Cowley JM, Moodie AF (1957) The scattering ofelectrons by atoms and crystals. I. A new theoreticalapproach. Acta Cryst 10: 609-613.

Dinges C, Berger A, Rose H (1995) Simulation ofTEM and STEM images considering phonon and electronicexcitations. Ultramicroscopy 60: 49-70.

Dinges C, Rose H (1996) Simulation of filtered andunfiltered TEM images and diffraction patterns. Phys statsol (a) 150: 23-30.

Doyle PA, Turner PS (1968) Relativistic Hartree-Fockx-ray and electron scattering factors. Acta Cryst A24: 390-397.

Fox AG, O’Keefe MA, Tabbernor MA (1989)Relativistic Hartree-Fock x-ray and electron atomic scatteringfactors at high angles. Acta Cryst A45: 786-793.

Hillyard S, Silcox J (1995) Detector geometry, thermaldiffuse scattering and strain effects in ADF STEM imaging.Ultramicroscopy 58: 6-17.

Rose H (1984) Information transfer in electronmicroscopy. Ultramircoscopy 15: 173-192.

Smith GH, Burge RE (1962) The analytical repre-sentation of atomic scattering amplitudes for electrons. ActaCryst A15: 182-last page.

Wang ZL (1995) Title of chapter. In: Elastic andInelastic Scattering in Electron Diffraction and Imaging.

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Plenum, New York. Chapter 11.Weickenmeier A, Kohl H (1991) Computation of

absorptive form factors for high-energy electron diffrac-tion. Acta Cryst A47: 590-603.

Xu P, Loane RF, Silcox J (1990) Energy-filteredconvergent-bean electron diffraction in STEM. Ultrami-croscopy 38: 127-133.

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