A coherence function approach to image simulation

A coherence function approach to image simulation*

H. MULLER, H. ROSE & P. SCHORSCHInstitut fur Angewandte Physik, TU Darmstadt, Hochschulstraße 6, D-64289 Darmstadt,Germany

Key words. Electron microscopy, image interpretation, image simulation,multislice, partial coherence, phonon scattering, thermal-diffuse scattering,inelastic scattering

Summary

A quantitatively correct theory of the simulation of electronmicrographs is proposed which considers the partiallycoherent process of image formation within the electronmicroscope. The new approach is based on the propagationof the mutual coherence function of the partially coherentelectron wave field. Our method leads to the formulation ofa generalized multislice algorithm. Applications to imagingwith zero-loss electrons and inelastically scattered electronsare presented. In addition the effect of thermal diffusescattering is investigated in detail.

1. Introduction

The simulation of high-resolution electron micrographs is avery suitable tool for determining the atomic structure ofobjects by means of electron microscopical techniques(Reimer, 1991). Most objects encountered in materialsscience have a crystalline structure. Such periodic objectscause strong dynamic scattering of the incident electronwave. As a result, the exit wave is a nonlinear function ofthe potential of the object. Knowledge of the potential givesinformation about the atomic structure.

The total exit wavefunction is the sum of many partialwaves, each of which is connected with a distinct energy losswhich results from a specific excitation of the object. Only inthe case of purely elastic scattering does the object remain inits initial state. With increasing thickness of the object thecontribution of the inelastically scattered electrons to theimage intensity increases, while that of the purely elasticallyscattered electrons decreases. The inelastically scatteredelectrons blur the image formed by the elastically scatteredelectrons owing to the unavoidable chromatic aberration ofconventional electron lenses. In addition, the image isaffected by noise. Owing to these deleterious contributions,the inverse scattering problem cannot be solved in practicewith a sufficient degree of reliability (Newton, 1989).

Image simulation offers a promising alternative forretrieving the structural information about the objectfrom the experimental image. However, to achieve aquantitative agreement between the recorded and thesimulated image intensity, one must consider the influenceof all elements of the image-forming system with anappropriate degree of accuracy.

Unfortunately, present simulation procedures do notsufficiently consider the effect of partially coherent imaging(Self et al., 1983; Kirkland et al., 1987; Stadelmann, 1987).Moreover, inelastic scattering processes are usuallyneglected apart from an absorption potential. Since theimaginary part of this potential removes the inelasticallyscattered electrons from the beam, the use of an absorptionpotential is appropriate only in the case of ideal zero-lossfiltering. In reality, the thermal-diffusely scattered electronscontribute to the image intensity even in zero-loss images.In addition, the incoherent perturbations, such as mechan-ical vibrations, fluctuations of the lens currents and of theaccelerating voltage and alternating external electromag-netic fields suppress primarily the transfer of the high-spatial frequencies and diminish both contrast andresolution. Hence, to achieve reasonable quantitativeagreement between simulation and experiment, the effectof incoherent perturbations must be incorporated in thesimulation. At present, the contrast of the simulated imageis up to three times higher than that obtained in thecorresponding experimental image. To achieve a reasonablequantitative agreement between the intensities of theseimages, one must assume the same information limit for thesimulation as in the experimental situation.

Inelastic scattering can be subdivided into thermal diffusescattering and inelastic electron–electron scattering. Ther-mal diffuse scattering is strongly localized because theincident electron interacts with only a single nucleus. Inthis case the electron penetrates the atomic electron cloudand passes a relatively short distance from the nucleus. As aresult the electron is scattered at a large angle. Owing to therecoil, the atom is generally deflected from its position of

Journal of Microscopy, Vol. 190, Pts 1/2, April/May 1998, pp. 73–88.Received 10 March 1997; accepted 18 August 1997

73q 1998 The Royal Microscopical Society

* Dedicated to Professor Dr Manfred Ruhle on the occasion of his 60th birthday.

rest. Since the transferred energy is in the order of 0·1 eV,this process can be considered as quasielastic. The deflectionof the atom immediately after the collision can be describedby a coherent superposition of many phonons, where eachphonon defines a distinct collective vibration state of thelattice. The interference of the partial waves of the phononwave package produces the deflection of the atom from itsposition of rest. Subsequently this localized wave packagedisperses very rapidly over the entire lattice. Each quasi-elastically scattered electron wave contains high-resolutionspatial information about the object structure. Thisinformation is comparable to that carried by the elasticallyscattered electrons. However, the thermal-diffusely scatteredwave cannot interfere with the unscattered electron wavebecause they belong to different object states which areorthogonal to each other. This behaviour holds under theassumption that the recording time T of the micrograph islarge compared with the coherence time

tc ¼ h=DE: ð1Þ

Here h denotes the Planck constant and DE ¼ Ef ¹ Ei is theenergy difference between the final and the initial state ofthe object. Assuming a typical recording time of about 10 s,Eq. (1) shows that the assumption of incoherent thermaldiffuse scattering is fulfilled if the excitation energy is much

larger than 4 × 10¹16 eV. This condition holds for almost allfinal states excited by the inelastic scattering processes.Accordingly, in the case of thermal diffuse scattering eachatom which has been deflected from its position of rest canbe considered as a self-luminous source. The radius of thissource roughly equals the radius of the atom plus its rootmean square displacement. Since the latter is smallcompared with the size of the atom (less than about0·15 A), the vibration should not appreciably impair theresolution as long as object heating remains sufficiently low.In this case the quasi-elastically scattered electrons form anincoherent high-resolution image of the object structure.This image is superimposed on the coherent imageproduced by the elastically scattered electrons. Since theenergy loss for thermal diffuse scattering is smaller than theenergy width of present electron sources, including fieldemission guns, images which are formed solely by thethermal diffusely scattered electrons cannot be visualized inpractice.

The scattering between the incident electron and anelectron of the object is accompanied by an electronicexcitation or an ionization. Owing to the long range of theCoulomb force, the incident electron interacts simulta-neously with many electrons in the object. As a result thescattered electron transfers little momentum and its

Fig. 1. Schematic overview of the propagation processes within an EFTEM and the parameters affecting the image formation. The bottom rowof the table lists the capabilities of our image simulation software at its present state.

74 H. MULLER E T AL.

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deflection angle is generally small. Nevertheless, the energytransferred is relatively large since the mass of thescattering electron is the same as that of the scatteredparticle.

Plasmons are fluctuations in the spatial density of thenonlocalized electrons in the object, such as the conductionelectrons in metals or the p-electrons in aromatic mole-cules. As a consequence, electrons which have excited suchstates carry only spatial information with a resolutionwhich corresponds to the extension of these collectiveexcitations. On the other hand, electrons which arescattered by an inner-shell electron contain high-spatialinformation because the inner-shell electron is bound to thenucleus of a distinct atom.

A quantitative image simulation procedure is not yet inreach. However, the progress made during the last yearsshows that the application of image simulation based onimproved multislice algorithms is not restricted to imagingwith elastically scattered electrons (Dingles et al., 1995).Preliminary investigations show that a precise imagesimulation procedure must consider the partially coherentnature of the electron wave field within the microscope.Figure 1 gives a schematic overview of the differentpropagation processes encountered in an energy-filteringtransmission electron microscope (EFTEM). Each part of themicroscope can be described by a small set of parameterswhich affect the propagation of the mutual intensity. Toobtain quantitatively correct results, image simulationneeds to account for the influence of each of theseparameters with a sufficient degree of accuracy.

2. The mutual intensity

As stated in the Introduction, quantitative image simulationneeds to account for the partially coherent processes whichaffect the image formation. Partial coherence can bedescribed in terms of the mutual coherence function. Thisfunction is well known from light optics (Born & Wolf,1964) and was introduced into electron optics by Hawkes(1978) and Rose (1976).

As far as elastic scattering is concerned, the quantummechanical process of image formation can be fullydescribed by a time-independent complex-valued quantummechanical wave function w(r) depending only on thespatial coordinate vector r. This behaviour follows from thetime-dependent Schrodinger equation since the potentialdoes not vary with time. In this case the time dependence ofthe electron wave function is known to be harmonic

wðr ; tÞ ¼ wðr Þ exp ð¹iðEt="ÞÞ: ð2Þ

The spatial part w(r) of the wave function is governed by thestationary Schrodinger equation.

The situation becomes much more involved if we drop theproposition of energy conservation for the scatteredelectrons. In this case we must employ a time-dependent

formalism. Fortunately, the problem is simplified by the factthat an electron micrograph is not recorded at an instantpoint in time but taken during a time interval T in the orderof seconds. This observation time T is much longer than therelaxation time of almost any excited object state. Also thelifetimes of other incoherent perturbations of the imagesignal are very small on this macroscopic time scale. Inaddition we assume that effects of object heating arenegligibly small. Therefore, the detected signal can beconsidered as stationary because the time of observation islarge compared with the duration of the temporal fluctua-tions of the image intensity caused by inelastic scatteringprocesses and other incoherent perturbations. We use thisstationarity property of the signal to eliminate the timedependence from our further investigations.

To illustrate this procedure (Fertig & Rose, 1977; Kohl &Rose, 1985), we consider a polychromatic electron sourcewith finite size which illuminates an opaque screen withtwo pin holes P and P0 located at the positions r and r0. Thetotal wave at the observation point rO situated in the righthalf space behind the screen is a linear superposition of thepartial waves w and w0 emanating from the pin holes P andP0 respectively. Accordingly, the time-averaged intensity atthe point rO depends not only on the retarded intensities atthe subsequent source points r and r0 but also on thecorrelation between the temporal fluctuation at thesepoints. This correlation is measured by the mutualcoherence function

Gcðr ; r0; tÞ ¼ hwðr ; tÞw¬ðr 0

; t ¹ tÞi ð3Þ

of the wave field. Gc depends on the two space coordinates,yet only on the difference t of the retardation times. Thebrackets h. . .i in Eq. (3) denote a time averaging procedurewhich we use to model the effect of the finite detectiontime T.

In some cases the electron wave field can be considered asquasi-monochromatic. As a result the time dependence ofthe mutual coherence function can be described by anexponential function in t which depends only on the meanenergy E ¼"qa of the wave field. The bar denotes theensemble average taken over the electrons in the beam. Inthis case the mutual coherence function is replaced by themutual intensity function G(r, r0) ¼Gc (r, r0, t ¼ 0). Thisfunction has the property

Gðr ; r 0 ¼ r Þ ¼ Iðr Þ ¼ hwðr ; tÞw¬ðr ; tÞi; ð4Þ

where I(r) is the time-averaged intensity at the point r. Thepropagation properties of the mutual intensity closelyresemble those of the stationary wave function w(r).

3. Propagation of the mutual coherence function

The four-dimensional Fresnel propagator describes theparaxial propagation of the mutual coherence function

q 1998 The Royal Microscopical Society, Journal of Microscopy, 190, 73–88

IMAGE SIMULATION IN EM 75

within the field-free regions of the transfer system shownin Fig. 1. For simplicity we first consider a quasi-monochromatic wave field emanating from an extendedincoherent source and suppose that the correspondingmutual coherence function Gc (rA, r0

A, t) is known for allpairs of points (rA, r0

A) in a plane A perpendicular to theoptical axis. In this case the mutual coherence functionat any arbitrary plane B in the field-free region behindthe plane A can be determined. By employing Huygensprinciple (Born & Wolf, 1964) we eventually find

Gcðr B; r0B; tÞ

¼k

2p

� �2� �Gcðr A; r

0A; t ¹

1v

ðjr B ¹ r Aj ¹ jr 0B ¹ r 0

AjÞÞ

×eikjr B¹r Aj

jr B ¹ r Aj

e¹ikjr 0B¹r 0

Aj

jr 0B ¹ r 0

AjLL0 d2r Ad2r 0

A; ð5Þ

where k denotes the mean wave number and L, L0 aregeometrical factors slightly smaller than unity. The timeincrement (1/v)(jrB ¹ rAj ¹ jr0

B ¹ r0Aj) accounts for the effect

of retardation caused by different geometrical distances.Equation (5) is the fundamental propagation law of themutual coherence function in the field-free region. It showsthat the free-space propagation of the mutual coherencefunction is described by a product, where one factor dependssolely on the primed, the other on the unprimedcoordinates.

In an electron microscope the electron beam is confinedto the region about the optical axis of the system. In thiscase the inclination factors L and L0 in (5) areapproximately unity and the spatial distance

jr B ¹ r Aj ¼

�����������������������������������������������ðrB ¹ rAÞ2 þ ðzB ¹ zAÞ2

q< jzB ¹ zAj 1 þ

12

ð rB ¹ rAÞ2

ðzB ¹ zAÞ2 þ . . .� � ð6Þ

can be expanded in a Taylor series. Since the planes A and Bare parallel to each other, we only need information aboutthe correlation of the disturbance between different lateralpositions. Therefore Gc reduces to a four-dimensionalcorrelation function Gc(r, r0, t; z) ¼G(r, r0, t)jz ¼ z0 with the zcoordinates as a parameter. The same restriction can beapplied to the mutual intensity function G. The dependenceof Gc(r, r0, t; z) on the pair of points (r, r0) accounts for thelateral coherence and the dependence on the time differencet describes the longitudinal or temporal coherence of thewave field.

Considering this definition and the expansion (6), we

obtain from the relation (5) the approximation

Gcðr B; r0B; tÞ ¼ Gcð rB; r

0B; t; zBÞ

¼k

2pd

� �2� �Gcð rA; r

0A; t; zAÞ ei k

2d ðð rB¹rAÞ2¹ð r0B¹r0

AÞ2Þd2rAd2r0A

¼

� �Gcð rA; r

0A; t; zAÞPFð rB ¹ rA; r

0B ¹ r0

AÞ d2rAd2r0A;

ð7Þ

where d ¼ zB ¹ zA defines the distance between the twosuccessive planes. Within the paraxial region retardationeffects can be neglected because Gc varies slowly with t.

Equation (7) describes a linear mapping of the mutualcoherence function from the plane z ¼ zA onto the planez ¼ zB; it can be read as a convolution of a propagatorfunction PF with the mutual coherence function at theinitial plane A. The four-dimensional free space propagator

PFð r; r0Þ ¼k

2pd

� �2

ei k2d ð r2¹r02Þ ð8Þ

decomposes into a product to two two-dimensional Fresnelpropagators.

The Fresnel propagator describes the free space propaga-tion much better than expected from the derivation even forsmall distances d. To understand this behaviour, we considera high-energy electron moving in the direction of k. Itsspatial wave function fulfills the three-dimensional Helm-holtz equation

Dw þ k2w ¼ 0: ð9Þ

The z dependence of w can be expressed as a slightlydistorted plane wave

wð r; zÞ ¼ eikzwð r; zÞ; ð10Þ

Inserting this ansatz into the differential Eq. (9) andconsidering that w( r, z) changes slowly over a distance ofseveral wavelengths, we obtain the approximation

Drw þ 2ik ∂zw ¼ 0: ð11Þ

This differential equation is the high-energy approximationof the Schrodinger equation. It neglects back scattering andcorresponds to the two-dimensional diffusion equation witha complex diffusion coefficient. By employing the Greenfunction technique, we find the solution

wð rB; zBÞ ¼k

2pd

� �2

eikðzB¹zAÞ

�wð rA; zAÞ e¹i k

2d ð rB¹rAÞ2

d2rA:

ð12Þ

The comparison of this result with Eq. (7) shows that theFresnel approximation represents an exact solution of thehigh-energy approximation of the Schrodinger equation.

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4. Properties of the source

Each electron source has an extended region of emissionand a finite energy width. The energy distribution isgenerally characterized by the full width of half maximum(fwhm) dE. According to the definition of the mutualcoherence function (3), its spectral representation J(r, r0, q)fulfills the relations

Gcðr ; r0; tÞ ¼

12p

�∞

¹∞Jðr ; r 0

;qÞ e¹iqtdq;

Jðr ; r 0;qÞ ¼

wðr ;qÞw¬ðr 0;qÞ

T; ð13Þ

where w(r, q) denotes Fourier transform of the wavefunction with respect to the time t; T is the observationtime. To obtain the second relation in (13), we must performthe time-average in the definition of Gc (3) by using therepresentation

Gcðr ; r0; tÞ ¼ hwðr ; tÞw¬ðr 0

; t ¹ tÞi

:¼1T

�∞

¹∞wðr ; tÞw¬ðr 0

; t ¹ tÞ dt: ð14Þ

Here we have assumed that the wave function w(r, t)vanishes outside the time interval [¹T/2, T/2] of observa-tion. The mutual spectral density J is connected with themutual intensity via the relation

Gðr ; r 0Þ ¼1h

�∞

¹∞Jðr ; r 0

;EÞ dE; E ¼ "q: ð15Þ

Each real electron source emits electrons from anextended region of its surface. Since the wave of anoutgoing electron extends over a finite region of this surface,we cannot consider the individual points of the real surfaceof the cathode as incoherent point sources. Instead of thephysical source the virtual or real crossover is considered asthe effective source. In the case of a field-emission source theeffective source is the smallest spot formed by theasymptotes of the trajectories which the electrons have infront of the anode. The effective source is roughly located atthe centre of curvature of the tip. The individual points rS ofthis effective source can be considered as incoherent withrespect to each other. Moreover, we assume that the energydistribution of the emitted electrons does not depend on theposition rS of the source point. Accordingly, the correlationfunction JS can be written as

JSðr S; r0S; qÞ ¼ Gðr S; r

0SÞ pðqÞ: ð16Þ

From Eq. (13) we find that under this assumption alsoGc(r, r0, t) ¼G(r, r0) g(t) factorizes, where the first factordepends on the spatial coordinates and the second solelyon t. The function g(t) is the Fourier-transform of thespectral density p(q) with respect to q. The spectral densityfunction p(q) of most cathodes can be approximated with a

sufficient degree of accuracy by a Gaussian distribution

pðqÞ ¼1������������

2pDqp exp ¹

ðq ¹ qaÞ2

2Dq2

� �; ð17Þ

where "Dq ¼ dE/(2√

(2 ln 2)). The mean energy "qa ¼ eU isdefined by the nominal acceleration voltage U.

Since partial waves emanating from different points of theeffective source are considered as incoherent with respect toeach other, the mutual intensity function on the surface ofthe effective source is given by the relation

Gðr S; r0SÞ ¼ ISðr SÞ dðr S ¹ r 0

SÞ: ð18Þ

The function IS(r) is the normalized intensity of the effectivesource. By employing formula (5) with L¼L0 ¼ 1 for thepropagation of the mutual intensity and neglectingretardation effects, we obtain for the mutual intensity atan arbitrary plane behind the effective source the expression

Gðr ; r 0Þ ¼k

2p

� �2�S

Iðr SÞexp ikðjr ¹ r Sj ¹ jr 0 ¹ r SjÞ

jr ¹ r Sjjr 0 ¹ r Sjd2r S;

ð19Þ

The light-optical analogue of this formula is known as theVan–Cittert–Zernike theorem.

For most applications it suffices to assume a planeeffective source with a Gaussian spatial distribution

Iðr SÞ ¼ Ið rSÞ ¼I0

2pj2 exp ¹r2

S

2j2

� �: ð20Þ

The variance j2 defines the mean area of emission of the effec-tive source. Using this assumption together with the Fresnelapproximation, we eventually find from (19) for the mutualintensity at the plane z ¼ zs þ d the analytical expression

Gðr; r0Þ¼ I0k

2pd

� �2

exp ¹k2j2

2d2 ð r¹r0Þ2� �

exp ik

2d

� �ð r2¹r02Þ

� �:

ð21Þ

In this approximation the intensity I(r) ¼G(r, r) is constantwithin the paraxial domain about the optical axis anddecreases quadratically with the axial distance d. From Eq.(21) we find that the mutual intensity of an incoherentsource is sharply peaked for r¼ r0. Only in the limiting caseof a very small effective source with kjp1 is the wave fieldcoherent in the paraxial region.

5. Propagation through the object

The propagation of the electron wave through thin objectscan be described with a sufficient degree of accuracy bymeans of the Glauber high-energy approximation (Glauber,1959). However, this formalism must be generalized inorder to include inelastic scattering events in our calcula-tions. In this section we shall outline a generalization of theGlauber method for objects with internal degrees offreedom.

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IMAGE SIMULATION IN EM 77

For simplicity we assume a plane incident wave whichmoves in the direction of the optical axis:

w0ðr ; tÞ ¼ ei ðkiz¹qtÞ; q ¼

k2i "

2mg: ð22Þ

The wave number ki is connected with the accelerationvoltage U by the relation

"ki ¼ mgu ¼

���������������������������������������2meU 1 þ

eU2mc2

� �s; g ¼ 1 þ

eUmc2 ; ð23Þ

where m denotes the mass at rest of the electron.The total wave function Q at the exit plane of the object

depends not only on the spatial position r of the scatteredelectron but also on all internal degrees of freedom of thescatterers. The positions of the nuclei and electrons of theobject are defined by the vectors Ri, i ¼ 1, . . . l.

The Hamiltonian operator H of the time-dependentSchrodinger equation

HQ ¼ i" ∂tQ; Q ¼ Qðr ;R1; . . . ;Rl; tÞ ¼ Qðr ; t;RÞ ð24Þ

can be written as

H ¼ HO þ He þ V: ð25Þ

The Hamiltonian HO(R) defines the unperturbed object,He(r) is the Hamiltonian of the incident electron and

Vðr ;RÞ ¼Xl

i¼1

Viðr ;RiÞ: ð26Þ

describes the interaction potential between the incidentelectron and the object.

We assume that all energy eigenvectors jmi andeigenvalues Em of the operator HO are known. Byintroducing the transformed wave function

W ¼ Wðr ; t;RÞ ¼ expi"

HOt� �

Qðr ; t;RÞ; ð27Þ

and defining the time-dependent position operators

RiðtÞ ¼ expi"

HOt� �

Ri expi"

HOt� �

; i ¼ 1; . . . ; l; ð28Þ

we eventually arrive at the transformed Schrodingerequation with relativistic kinematics

"2

2mg=2 þ i" ∂t

� �W ¼ VW; ð29Þ

where

V ¼ Vðr ; RðtÞÞ ¼Xl

i¼1

Við r þ zez ¹ RiðtÞÞ ð30Þ

denotes the interaction operator.Considering that the energy of the incident particle is

much higher than the interaction energy and that the wavelength of the incident electron is much smaller than therange of the interaction potentials Vi, we can solve the

differential Eq. (29) by the ansatz

Wmðr ; tÞ ¼ w0ðr ; tÞJðr ; tÞjmi ð31Þ

for any initial energy eigenstate jmi of the object. Theoperator-valued function J(r, t) describes the interactionbetween the incident electron and the object. This functionvaries slowly within the range of the potential V anddepends implicitly on the internal degrees of freedom of theobject. By considering these assumptions the differential Eq.(29) adopts the form

1u

∂t þ ∂z

� �Jðr ; tÞ ¼ ¹

i"u

VJ; u ¼"ki

gm; ð32Þ

where we have neglected all second-order partial derivativesof J with respect to x, y and z. Therefore, this equation canonly be used in the case of very thin objects.

To extend our approach to thick objects, we employ thewell-known multislice formalism introduced by Cowley &Moodie (1957). In this case the potential V represents onlythe contribution of a distinct slice of the object.

Employing the method of successive approximation, thesolution of Eq. (32) can be written as a power series

Jð r; z; tÞ ¼ 1 ¹i

"u

�z

¹∞V z0; t ¹

z ¹ z0

u

� �dz0

þi

"u

� �2�z

¹∞V z0; t ¹

z ¹ z0

u

� � �z0

¹∞V z1; t ¹

z ¹ z0

u

� �dz1dz0

¹ . . . : ð33Þ

This expression is also known as Dyson series (Sakurai,1985). The function Vacts as an operator. Therefore, its valuestaken at different positions z do not commute. Nevertheless, byusing the time-ordering brackets [. . .]þ with the time replacedby the z coordinate, we can sum up this power series:

Jðr ; tÞ

¼

�exp�

¹i

"u

�z

¹∞

Xl

i¼1

Vi

�r þ z0ez ¹ Ri t ¹

z ¹ z0

u

� ��dz0

��þ

;

ð34Þ

where we have already inserted the definition of theoperator-valued potential (30).

The use of the time-ordering brackets accounts for thefact that the potential of the object changes during thepassage of the scattered electron. The time retardationoriginates from the same effect because the potential at adistinct point of the trajectory z0 must be evaluated at thetime of passage t ¹(z ¹z0)/u. The time dependence can beneglected for phonon scattering because the transition timeof the electron is much shorter than the time of vibration ofa displaced nucleus. In this case the application of the timeordering brackets is obsolete. The retardation of time canalways be neglected if the extension of V(t, z) in the z-direction is sufficiently small.

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The expression (34) yields the amplitude of the scatteredelectron at an arbitrary point r behind the object at a time tafter the scattering event. The probability that the objectstate has been changed from the initial state jmi to the finalstate jni is determined by the function

wmnðr ; tÞ ¼ hnjJðr ; tÞjmi

¼ w0ðr ; tÞ hnj½expfixð rÞgÿþjmi exp iEm ¹ En

"uðz ¹ utÞ

� �;

ð35Þ

where the observation point r is located outside the range ofthe potential V. In Eq. (35) x(r) denotes the formal definitionof an operator-valued projected potential

xð rÞ ¼ ¹1"u

�∞

¹∞

Xl

i¼1

Vi

�r þ z0 ez ¹ Ri

z0

u

� ��dz0

: ð36Þ

In the time-dependent case this expression is only mean-ingful in the context of an exponential function enclosed bytime-ordering brackets, as in Eq. (34).

The mutual product of the wave function at an arbitraryplane behind the object for two distinct points r and r0 isgiven as a superposition of all final energy eigenstates of theobject

Wmð r; z; tÞW¬mð r0

; z; t0Þ ¼X

n

wmnjni

! Xn0

wmn0 jn0i

!¬

¼ w0w¬0

Xn

exp ¹iEm ¹ En

"t

� �hnj½expfixð rÞgÿþjmi

×hmj½expf¹ixð r0Þgÿþjni: ð37Þ

During the time of exposure an incident electron meets theobject in the initial states jmi with the probability Pm. Sincethe product (37) depends only on the difference t ¼ t ¹ t0 thetime average in the definition of the mutual coherencefunction (3) can be replaced by an average taken over allinitial states jmi of the object. Accordingly, the mutualcoherence function behind the object adopts the stationaryform

Gð r; r0; tÞ ¼ w0ð r; z; tÞw¬

0ð r0; z; t0Þ

Xm;n

Pm exp ¹iEm ¹ En

"t

� �×hnj½expfixð rÞgÿþjmi hmj½expf¹ixð r0Þgÿþjni: ð38Þ

This relation does not depend on a specific form of theincident wave. Therefore, we can replace the incident planewave w0 by a somewhat distorted plane wave. Since we haveneglected backscattering, we can consistently assume thatthe fluctuation of the initial wave and the object potentialare stochastically uncorrelated when performing the timeaverage.

The mutual dynamic object transparency (Rose 1984) of

a thin object is defined as

Mð r; r0; tÞ ¼

Xm;n

Pm exp ¹iEm ¹ En

"t

� �×hnj½expfixð rÞgÿþjmi hmj½expf¹ixð r0Þgÿþjni: ð39Þ

This function connects the mutual coherence functionG( f )

c at the exit plane with G(i)c at the entrance plane via the

relation

Gð f Þc ð r; r0

; tÞ ¼ Mð r; r0; tÞGðiÞ

c ð r; r0; tÞ: ð40Þ

The sum in (39) can be rewritten as the time-average of thetime-dependent operator-valued projected potential

Mð r; r0; tÞ ¼ h½expfixð r; tÞgÿþ½expf¹ixð r0

; tÞgÿþi: ð41Þ

The time-dependent operator-valued projected potential isdefined similarly to Eq. (36) as

xð r; tÞ ¼ ¹1"u

�∞

¹∞V�

r; z; R�

t þzu

��dz ð42Þ

The formula (40) allows one to calculate the mutualcoherence function for subsequent slices. The mutualdynamic object transparency (MDOT) represents a general-ization of the well-known transmission function used in thetheory of purely elastic scattering. In the case of elasticscattering the double sum in (39) reduces to a single sumsince the transition matrix hnj[exp{ix(r)}]þjmi is diagonal.Usually one assumes that the object is in its ground state. Weobtain this special case from the definition (39) by choosingPm ¼ dm0, where dmn denotes the Kronecker symbol. In thiscase the mutual dynamic object transparency M(r, r0, t)reduces to the product Mel(r, r0) ¼ T(r)T*(r0), where T(-) ¼ exp{ix(r)} is the complex transmission function for astatic potential.

For thick objects the MDOT behind the last object slice is ahighly non-linear function of the object potential. Thisbehaviour results from plural scattering processes withinthe object.

In the conventional multislice algorithm the interactionof the object with the wavefunction is described by theelastic transmission function. In order to extend theconventional multislice formalism to inelastic scattering,we assume that (a) each object state jni is sufficientlylocalized within a distinct slice of the object, (b) the rangesof the interaction potentials are not larger than the slicethickness and (c) the object is at thermal equilibrium. In thiscase one can replace the wave function in front of each sliceby the mutual coherence function Gc and describe theinteraction of the object with the mutual coherencefunction by the MDOT.

To obtain the mutual coherence function Gc(rN, r0N, t; zN)

at the exit plane of the object, we subdivide the object into N

q 1998 The Royal Microscopical Society, Journal of Microscopy, 190, 73–88

IMAGE SIMULATION IN EM 79

thin slices. The propagation of the mutual coherencefunction between any two successive intermediate planesz ¼ zi and z ¼ ziþ1 is described by multiplying the mutualcoherence function at the plane z ¼ zi with the MDOTfunction Mi of the ith slice and performing a four-dimensional free space propagation to the planeziþ1 ¼ zi þ d. By iteration we find

Gcð rN ; r0N ; tÞ ¼

�. . .�

Gcð r0; r00; tÞ

×YN¹1

i¼0

Mið ri; r0i; tÞPFð riþ1 ¹ ri; r

0iþ1 ¹ r0

iÞ d2rid2r0

i: ð43Þ

This generalized multislice formalism represents a five-dimensional description of the propagation of an electronthrough the object. Unfortunately, it is not possible toevaluate Eq. (43) numerically, because present workstationscannot perform the calculations within a reasonableamount of time. In the next section we will show how Eq.(43) can be solved approximately with moderate numericaleffort by employing a two-dimensional formalism.

The approximate evaluation of the MDOT leads toanalytical expressions for the absorption potential and forthe terms describing inelastic and quasi-elastic scattering.Using the fact that the projected potential can be consideredas a small quantity if the object slices are sufficiently thin,we can approximate the MDOT (41) by an expression whichclosely resembles the transmission function with absorptionpotential Tab(r) ¼ exp (ix(r) ¹ (1/2)m2(r)). This expression isused in conventional image simulation where it is known asoptical potential. We expect additional terms in theexponent of the new expression if we incorporate theinelastically scattered electrons into the calculation. For thispurpose we expand the logarithm of M(r, r0, t) in a powerseries. Using the Taylor series ln (1 þ x) < x ¹ (1/2)x2 þ . . .and considering only terms up to second order in x, weobtain

ln Mð r; r0; tÞ < ln

�1 þ ixð r; tÞ ¹ ixð r0

; t0Þ þ xð r; tÞxð r0; t0Þ

¹1"u

� �2�∞

¹∞Vð r; z; tÞ

�z

¹∞Vð r; z0

; tÞ dz0 dz

¹1"u

� �2�∞

¹∞Vð r0

; z; t0Þ�z

¹∞Vð r0

; z0; t0Þ dz0 dz

�< iðhxð r; tÞi ¹ hxð r0

; t0ÞiÞ þ hxð r; tÞxð r0; t0Þi

þ 12ðhxð r; tÞi ¹ hxð r0

; t0ÞiÞ2

¹1"u

� �2��∞

¹∞Vð r; z; tÞ

�z

¹∞Vð r; z0

; tÞ dz0 dz�

¹1"u

� �2��∞

¹∞Vð r0

; z; t0Þ�z

¹∞Vð r0

; z0; t0Þdz0 dz

�¼ iðhxiðrÞ ¹ hxiðr0ÞÞ ¹ 1

2ðm2ðrÞ þ m2ðr0ÞÞ þ m11ðr; r0; tÞ: ð44Þ

Here we have used the definitions

hxið rÞ ¼ hxð r; tÞi; ð45Þ

m2ð rÞ ¼2

ð"uÞ2

��∞

¹∞Vð r; z; tÞ

�z

¹∞Vð r; z0

; tÞ dz0 dz�

¹ hxi2ð rÞ

<hx2ð r; tÞi ¹ hxi2ð rÞ; ð46Þ

m11ð r; r0; tÞ ¼ hxð r; tÞxð r0

; t0Þi ¹ hxið rÞhxið r0Þ: ð47Þ

The second relation in Eq. (46) is obtained by consideringthat the slices are very thin. In this case the double integralapproximately represents (("u)2/2)hx2i. With this result wefind for the mutual dynamic object transparency of a thinslice the approximation

Mð r; r0; tÞ < exp ðiðhxið rÞ ¹ hxið r0ÞÞ

¹12ðm2ð rÞ þ m2ð r0ÞÞ þ m11ð r; r0

; tÞÞ: ð48Þ

Elastic scattering is described by the first two terms of theexponent. Each of these terms represents a phase shift of theelectron wave which is proportional to the projectedpotential of the slice. The terms m2(r) and m2(r0) representan absorption potential. The term m11(r, r0, t) describesthe contribution of the inelastically scattered electrons tothe image intensity. This term is a direct consequence of theoptical theorem which ensures the conservation of thenumber of particles:

Mð r; r0 ¼ r; t ¼ 0Þ ¼ 1;

m2ð rÞ ¼ m11ð r; r0 ¼ r; t ¼ 0Þ: ð49Þ

The MDOT (48) factorizes if m11 ¼ 0. Each of the twofactors represents the transmission function with absorptionpotential for the standard multislice algorithm. Ourapproach is, therefore, a true generalization of the conven-tional multislice theory.

6. The generalized multislice formalism

The generalized multislice formula (43) describes thepropagation of the mutual coherence function Gc(r, r0, t; z)through the object. It correctly accounts for elastic andinelastic plural scattering processes within the object. Inmost situations of practical importance it suffices to considerplural elastic scattering in combination with single inelasticscattering. This approximation holds even for moderatelythick specimens because partial waves originating fromdifferent inelastic scattering events are incoherent witheach other and hence do not interfere. Neglecting inelasticplural scattering, the image intensity I in the detector planez ¼ zD is given as the sum over all intensities Ij, j ¼ 1, . . . , N,with the inelastic scattering process confined to the jth sliceof the object:

Ið rÞ ¼XN

j¼1

Ijð rÞ: ð50Þ

80 H. MULLER E T AL.

q 1998 The Royal Microscopical Society, Journal of Microscopy, 190, 73–88

The mutual coherence function depends on the spatialcoordinates r, r0 and the time increment t. This parameterdistinguishes between the different energy losses within theobject. Owing to the chromatic aberration of the objectivelens, partial waves belonging to different energy losses aretransferred differently by the microscope. Therefore, it isadvantageous to discuss the generalized multislice formal-ism in terms of the mutual spectral density J(r, r0, q; z), asdefined in Eq. (13). If we assume a discrete set of possibleexcitations of the object with excitation energiesqj, j ¼ 0, 1, . . . , the Fourier transform of the MDOT withrespect to t is given by�

Mð r; r0; tÞ eiqt dt ¼ 2p

Xj

Mð r; r0;qjÞ dðq þ qjÞ: ð51Þ

Therefore, the relation between the mutual spectral densityat a plane z ¼ z0 in front of the nth object slice and at theplane z ¼ zn þ e directly behind this slice adopts the form

Jð r; r0;q; zn þ eÞ ¼

Xj

Mnð r; r0;qjÞJð r; r0

;q þ qj; znÞ: ð52Þ

This result has been obtained by inserting the approxima-tion (51) into the Fourier transform of Eq. (40). Equation(52) shows that the image intensities belonging to differentenergy losses can be calculated separately and added upsubsequently. Nevertheless, partial waves belonging todifferent energy losses must be propagated separatelythrough the optical system up to the recording plane wherethe incoherent superposition is performed. This necessityresults from the chromatic aberration which causes anenergy-dependent transfer of the spatial frequencies. Con-sidering the relation (52) and assuming single inelasticscattering, we derive from the Fourier transform of formula(43) the following expression for the mutual spectral densityat the exit plane z ¼ zZ:

Jð rN ; r0N ;qÞ ¼

Xj

XN¹1

k¼0

�. . .�

Jð r0; r00;q þ qjÞ

×YN¹1

i¼0

Mið ri; r0i;q ¼ dikqjÞPFð riþ1 ¹ ri; r

0iþ1 ¹ r0

iÞ d2ri d2r0i:

ð53Þ

The sum over the index j comprises all possible energylosses. The MDOT Mi reduces to the elastic objecttransparency Mi(r,r0, q ¼ 0) ¼ T(r)T*(r0) if no inelasticscattering event occurs within the ith slice. In the followingwe restrict our discussion without loss of generality to asingle excitation energy qex.

The convolution with the four-dimensional free-spacepropagator in Eq. (53) can be computed efficiently byemploying the Fourier convolution theorem and the mutual

Fourier transform of the Fresnel propagator

FF 0½PFð r; r0; dÞÿ ¼

� �PFð r; dÞP¬

Fð r0; dÞ eirqe¹ir0q0

d2rd2r0

¼ PFðq; dÞPFð¹q0; dÞ: ð54Þ

Here q defines the component of the scattering vectorperpendicular to the optical axis and F denotes the two-dimensional Fourier transform.

In front of the object the energy dependence of themutual spectral density

Jð r; r0;q; z0Þ ¼ Gð r; r0; z0ÞpðqÞ ð55Þ

is determined by the energy width of the source. The spatialand the energy dependence are separated as stated in Eq.(16). In the case of elastic scattering the mutual intensityG(r, r 0, z0) ¼w(r, z0)w*(r0, z0) factorizes into a product of thestationary wave function w evaluated at the lateral positionsr and r0, respectively. Unfortunately, this relation does nothold true in the general case of partially coherentillumination. Nevertheless, the mutual intensity in front ofthe object can still be expanded into a series of products

Gð r; r0; z0Þ ¼X

l

Flð r; z0ÞF¬l ð r

0; z0Þ; ð56Þ

where each factor depends solely on the primed orunprimed coordinates. For partially coherent illuminationthe expansion functions Fl have no physical meaning. Onlyin the special case of an incident plane wave w0(z, t) doesthe sum in (56) reduce to a single term l ¼ 0 withF0(r; z0) ¼w0(z0, t ¼ 0). Without loss of generality we restrictour further investigations to a single term of the sum in Eq. (56),bearing in mind the representation (56) and the summationover the index l. Then the mutual spectral density at theplane z ¼ z0 in front of the object has the form J(r, -0, q; z0) ¼F(r; z0)F*(r0; z0)p(q). In this case the mutualspectral density at the exit plane zN ¼ z0 þ N d is given by

Jð rN ; r0N ;q; zNÞ ¼

XN¹1

j¼0

�. . .�

Fð r0; z0ÞF¬ð r00; z0Þpðq þ qexÞ

×YN¹1

i¼0

PFð riþ1 ¹ ri; r0iþ1 ¹ r0

i; dÞMið ri; r0i;q ¼ djiqexÞ d2rid

2r0i:

ð57Þ

In accordance with the representation (56) we alsoassume a series expansion for the MDOT

Mið r; r0;qexÞ ¼

Xm

TðmÞi ð r;qexÞT

ðmÞ¬i ð r0

;qexÞ ð58Þ

of the ith slice with energy loss qex. The modifiedtransmission functions T (m)

i , m ¼ 0, 1, . . . have a physicalmeaning only if we neglect the mutual term m11(r, r0, t) inthe MDOT (48). In this case the sum in (58) reduces to asingle term m ¼ 0. We identify T (0)

i with the ordinary elastictransmission function with absorption potential T (ab)

i .

q 1998 The Royal Microscopical Society, Journal of Microscopy, 190, 73–88

IMAGE SIMULATION IN EM 81

If we insert the expansion (58) into the generalizedmultislice formula (57), use the factorization of the incidentmutual intensity, and exchange the order of summation andintegration, we find that the propagation of each term in theexpansion of the final mutual intensity can be computed byemploying a conventional two-dimensional multislice form-alism. Since the four-dimensional Fresnel propagator PF(r, -0; d) ¼ PF(r; d) P*

F(r0; d) decomposes into a product of two two-dimensional Fresnel propagators, the propagation of thefunction F(r; z0) through the object is governed by themodified transmission functions T (m)

i , m ¼ 0, 1, . . . . At theplane z ¼ zN behind the last slice we find

Fðm;jÞð rN ; zNÞ ¼

�. . .�

Fð r0; z0ÞYN¹1

i¼0

PFð riþ1 ¹ ri; dÞ

× TðmÞi ð ri;q ¼ djiqexÞ d2ri: ð59Þ

This equation closely resembles the conventional two-dimensional multislice formula with the stationary wavefunction substituted by F(m) and the ordinary transmissionfunction of each slice replaced by T (m)

i . For the energy lossqex the mutual intensity at the plane z ¼ zN is given by

Gð r; r0; zNÞ ¼XN¹1

j¼0

Xm

Fðm;jÞð r; zNÞFðm;jÞ¬ð r0; zNÞ: ð60Þ

The five-dimensional MDOT of a thin object sliceconsisting of M atoms has the form

Mð r; r0; tÞ ¼ exp

�Xj

ifhxjið r ¹ rjÞ ¹ hx0jið r

0 ¹ rjÞg

¹12 fmðjÞ

2 ð r ¹ rjÞ þ mðjÞ2 ð r0 ¹ rjÞg þ mðjÞ

11ð r ¹ rj; r0 ¹ rj; tÞ

�;

ð61Þ

where the atoms are situated at the lateral positions rj,j ¼ 0, . . . , M ¹ 1. This formula follows from Eq. (48) with theassumption that the temporal fluctuations of the projectedpotentials of different atoms are stochastically uncorrelatedhxi x

0ji¼ hxii hx0

ji for i Þ j. The functions m( j)2 and m( j)

11 dependon the atomic number of the atom located at the position rj.

The terms m( j)2 and m( j)

11 in the exponent of expression (61)are small compared with unity. Therefore, we can expandthe exponential function with respect to these quantities.Keeping only the first-order terms, we obtain

Mð r; r0; tÞ ¼

exp�X

j

ifhxji ¹ hx0jig ¹ 1

2 fmðjÞ2 ð r ¹ rjÞ þ mðjÞ

2 ð r0 ¹ rjÞg

�

þX

j

mðjÞ11ð r ¹ rj; r

0 ¹ rj; tÞ exp�X

j

ifhxji ¹ hx0jig

�¼ TðabÞð rÞTðabÞ¬ð r0Þ þ

Xj

mðjÞ11ð r ¹ rj; r

0¹ rj; tÞTðelÞð rÞTðelÞ¬ð r0Þ:

ð62Þ

The first term is a product of two transmission functionswith absorption potential; each factor depends solely on theprimed or unprimed coordinates, respectively. The secondpart on the right-hand side of (62) does not decompose inthis simple manner because m11(r, r0,t) depends mutuallyon the primed and unprimed coordinates. Employing theFourier convolution theorem, we can conveniently calculatethe spatially shifted functions for each atom

mðjÞ11ð r ¹ rj; r

0 ¹ rj; tÞ ¼

F¹1F 0¹1½mðjÞ11ðq;q0

; tÞFF 0½dð r ¹ rjÞdð r0 ¹ rjÞÿÿ; ð63Þ

where m( j)11(q, q0, t) denotes the Fourier transform of

m( j)11(r, r0t) with respect to r and r0. Therefore, it suffices to

find a decomposition of m( j)11 in Fourier space.

It is evident from the definition of the MDOT (39) that itcan always be decomposed in the form (58). In the followingwe shall derive such representations for the special cases ofthermal diffuse (phonon) scattering and incoherent (elec-tronic excitation) scattering, respectively. For simplicity weemploy the Einstein model (Einstein, 1907) for phononscattering and the Raman–Compton approximation (Comp-ton, 1930; Lenz, 1954) for inelastic scattering. Theapplicability of these approximations for image simulationhas been shown previously (Dinges et al., 1995).

Phonon scattering

Phonon scattering is described with a sufficient degree ofaccuracy by the Einstein model because the resulting imageis affected only by the time average of the lattice dynamicsover the time of exposure. For our purpose it suffices toconsider the individual atoms of the object as independenttwo-dimensional harmonic oscillators. The mean squareelongation u2

j /2 depends on the atomic number of the jthatom. The time-average of the oscillating projected objectpotential yields the expressions

hxið rÞ ¼2p

kF ¹1

�felðqÞ exp

�¹

u2

4q2��

;

hxx0i ð r; r0Þ ¼2p

k

� �2

×F¹1F 0¹1�

felðqÞ exp�

¹u2

4ðq ¹ q0Þ2

�felðq

0Þ

�: ð64Þ

Here fel(q) denotes the elastic scattering amplitude of asingle atom. To retain the analytical character of the Eqs.(64) it is advantageous to use the Kohl–Weickenmeierapproximation (Weickenmeier & Kohl, 1991) for the elasticscattering amplitudes (Dinges & Rose, 1997).

The mixed term in the second expression of (64) can bewritten as a sum of products by means of the generatingfunction of the modified Bessel function In (Abramowitz &

82 H. MULLER E T AL.

q 1998 The Royal Microscopical Society, Journal of Microscopy, 190, 73–88

Stegun, 1970):

e2ðu2Þ

2qq0

¼Xjnj

In

�2�

u2

�2

qq0

�einðf¹f0 Þ

¼Xjnj;k

ððu2Þ

2qq0Þnþ2k

k!ðk þ jnjÞ!einðf¹f0Þ

: ð65Þ

Inserting this expansion into Eq. (64) yields thedecomposition

hxx0iðq; q0Þ ¼2p

k

� �2Xjnj;k

�felðqÞ

expf¹ðu2Þ

2q2gðuq2 Þjnjþ2k����������������������

k!ðk þ jnjÞ!p einf

�×fq0

;f0g¬; ð66Þ

where {q0, f0}* denotes the complex-conjugated of the firstfactor with the unprimed coordinates replaced by theprimed ones. In most cases only a few terms of the sum inEq. (66) must be taken into account.

Inelastic scattering

Inelastic scattering results in electronic excitations of theatoms. To describe this effect, we use the modified Raman–Compton approximation (Rose, 1976) for the inelastic partS(in)

j (K, K0, q) of the mixed dynamic form factor Sj (K, K0, q)of the jth atom with atomic number Zj. Assuming anaverage excitation energy q, the expression

SðinÞj ðK ;K 0

;qÞ ¼ dðq ¹ qÞ FjðK ¹ K 0Þ ¹FjðK ÞFjðK

0Þ

Zj

��ð67Þ

holds, where K, K0 denote the three-component scatteringvectors. Within the frame of validity of the small angleapproximation the scattering vector can be written as

K < q ¹ kQE ez; ð68Þ

where QE < "q/2eU denotes the characteristic meanscattering angle of the inelastic scattering process. There-fore, the minimum momentum transfer from the incidentelectron to the scattered is given by "kQE.

The X-ray scattering amplitude Fj in Eq. (67) depends onthe electronic charge distribution of the jth atom. It isadvantageous for further calculations to use the Doyle–Turner Gaussian fit

FðK Þ ¼X4

i¼0

ai e¹bi K 2

ð69Þ

for the atomic form factor F. The constants ai, bi, i ¼ 0, 1, 2,3, 4 are tabulated for most elements (Doyle & Turner, 1968).

The factorizsation of the analytical expression for

m(in,j)11 follows from the representation

mðin; jÞ11 ð r; r0

; tÞ ¼4pa

b

� �2

F¹1F 0¹1

�

SðinÞj ðq ¹ kQE ez;q

0 ¹ kQEez;qÞ

ðq2 þ k2Q2EÞðq02 þ k2Q2

EÞe¹iqt dq

" #; ð70Þ

where a . 1/137 is the fine structure constant and b¼ v/cdenotes the normalized velocity of the incident electron.Inserting the approximation (67) into Eq. (70), weeventually obtain

mðjÞ11ðq; q0

; tÞ ¼4pa

b

� �2 1K 2K 02

�

FjðK ¹ K 0Þ ¹1Zj

FjðK ÞFjðK0Þ

�e¹iqext

: ð71Þ

To factorize this expression, we need only to decomposethe first term inside the bracket on the right-hand side.Employing the formula (65), we find

FðK ¹ K 0Þ

K 2K 02 ¼Fðq ¹ q0Þ

ðq2 þ kQ2EÞðq02 þ k2Q2

EÞ

¼Xi;jnj;k

� ����ai

pe¹biq

2

ðq2 þ k2Q2EÞ

��bi

pqÞjnjþ2k����������������������

k!ðk þ jnjÞ!p einf

�× fq0

;f0g¬; ð72Þ

in close analogy to the phonon case.The result (72) has been obtained with the assumption of

an average energy loss q for all atomic excitations. Thisexcitation energy depends on the atomic number of theatom. To derive a more accurate approximation, we mustconsider that the electron energy loss spectrum (EELS) ofeach atom is a continuous function of the excitation energy.In a first approximation we describe this spectrum by adelta function d(q ¹qex), where the specific excitationenergy qex ¼ q is chosen as the average energy loss of thescatterer. The parameter qex enters the expression (70)through the definition of the characteristic scattering angleQE. The smaller the mean energy loss, the more delocalizedthe scattering process. Therefore, we can drop the assign-ment qex ¼ q and reinterpret qex as a formal parameterquantifying the degree of localization of the scatteringprocess.

This interpretation allows us to obtain an improvedapproximation for the term m11 of each atom. For thispurpose we use the energy-loss spectrum fi(qex) of the jthatom which can for example be taken from the EELSAtlas (Ahn & Krivanek, 1983). Moreover, we replace inthe expressions (67) and (68) the average energy loss q

by the energy loss qex. In addition we substitute fi(qex) ford(q ¹q) in relation (67). By taking the Fourier transformof the resulting expression, we obtain the improved

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IMAGE SIMULATION IN EM 83

approximation

min; j11 ðq; q0

; tÞ <4pa

b

� �2 �fjðqexÞ

e¹iqext

K 2K 02

�

FjðK ¹ K 0Þ ¹1Zj

FjðK ÞFjðK0Þ

�dqex: ð73Þ

This formula should be especially useful for the simulationof energy-filtered images obtained with an energy windowof a particular finite width.

Statistical phases

To achieve an efficient implementation of the generalizedmultislice procedure, the method of statistical phases can beused to compute sums like

IðqÞ ¼XM

j¼1

FjðqÞF¬j ðqÞ; ð74Þ

where Fj, j ¼ 1, . . . , M are arbitrary complex-valued func-tions. We assume a set ajk, j ¼ 1, . . . , M and k ¼ 1, . . . , N ofequally distributed random phases within the range0 # ajk<2p. For each k we define the expression

IkðqÞ ¼

�XM

i¼1

FiðqÞ eiaik

��XM

j¼1

FjðqÞ eiaik

�¬

: ð75Þ

The complex exponential factors eiaik are called statisticalphase factors. The value of ik(q) differs from I(q) only by thecontribution of the mixed factors originating from theproduct in (75):

IkðqÞ ¹ IðqÞ ¼XM

i¼1

Xi¹1

j¼1

2 RefFiðqÞF¬j ðqÞ eiðaik¹ajkÞg: ð76Þ

The terms on the right-hand side cancel out if we average Ik

over a series of different sets of statistical phasesajk, k ¼1, . . . , N. As a result we find

IðqÞ <1N

XN

k¼1

IkðqÞ: ð77Þ

In the case of the generalized multislice formalism thefunctions Fj represent the partial waves emanating fromdifferent inelastic scattering events. Because we employ alinear algorithm to calculate these functions from the inputdata, we can calculate any linear combination of thesefunctions by applying the generalized multislice formalismto an appropriate linear combination of the input data.Therefore, we need only one multislice calculation toevaluate Eq. (75) for a fixed k. In this case the numericaleffort for calculating I(q) is proportional to N, the number ofsets of statistical phases, and not proportional to M, thenumber of different inelastic scattering events. Therefore themethod of the statistical phases is especially useful if M islarge and the calculation of the functions Fj in (74) is very

time consuming. In the current implementation of ourimage simulation tool the incoherent superposition of theinelastically scattered partial waves originating fromdifferent atoms is realized in this manner. The number Nof sets of statistical phases required for obtaining a sufficientdegree of accuracy must be determined by numericalexperiments.

7. Propagation through the microscope

Apart from the different scattering processes within theobject a precise image simulation must also consider theeffect of the individual elements of the electron microscopeon the image formation. High-resolution images can beobtained either with a fixed-beam (CTEM) or with ascanning transmission (STEM) electron microscope. Fortu-nately, the framework of the theory of image simulation isessentially the same for both instruments, because thegeneralized reciprocity theorem (Kohl & Rose, 1985) closelyrelates image formation in CTEM with that in STEM. In thispaper we will concentrate on the fixed-beam case.Nevertheless, only minor modifications are needed to applythe method to STEM imaging (Dinges & Rose, 1995; Hartelet al., 1996).

It is convenient to separate the electron microscope intotwo parts: the illumination system in front of the object andthe imaging system between the object and the recordingplane. The influence of the imaging system must be highlycoherent in order to obtain a high-resolution image(Scherzer, 1949). Without this coherence no contrastwould arise in the detection plane.

In the case of an aberration-free imaging system, thedegree of coherence of the illumination does not appreciablyaffect the resolution of the image. However, this behaviourdoes not generally hold true for the contrast. In thepresence of chromatic lens defects the longitudinal coher-ence strongly limits the attainable resolution. Since theenergy width of the cathode determines the degree oflongitudinal coherence, the effect of the energy spread mustbe considered in the simulation. The finite size of theeffective source determines the lateral coherence of theillumination system. This lateral coherence primarily affectsthe phase contrast and the resolution of the incoherent partof the image intensity.

The incoherent parasitic perturbations shorten the rangeof the spatial frequencies transferred by the optical systemand hence determine the information limit. Especially thehigh spatial frequencies of the image signal are dampedduring their passage through the microscope. Consequently,the resolution decreases even if it is not limited by the lensaberrations.

The degree of lateral coherence at the object plane can beadjusted by an appropriate condenser system. A simple

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condenser system consists of a field and an illuminationaperture and of two lenses. By varying of the focal length ofthe condenser lenses, the mode of illumination can bechanged from Kohler illumination to critical illumination.

In the limiting case of Kohler illumination the wavefunction in the object plane is a superposition of planewaves with different directions of incidence. These partialwaves are incoherent with each other because eachdirection corresponds to a distinct point of the effectivesource. Accordingly, the degree of lateral coherence at theobject plane can be varied by changing the diameter of theopening of the illumination aperture.

Standard image simulation assumes an incident planewave which propagates in the direction of the optical axis.This illumination mode corresponds to Kohler illuminationfor a point source situated on the optical axis. Such a sourcecan be realized by a point-like opening of the illuminationaperture. To account correctly for arbitrary illuminationmodes, one must incorporate tilted partial waves into thecalculation. For this purpose we assume a small tilt anglev ¼ q/jqj with jvjp1, where q denotes the projection of theincident wave vector k onto a plane perpendicular to theoptical axis. For a tilted plane wave of incidence

w0ðr Þ ¼ eikr < eikðvrþzÞ ð78Þ

the modified four-dimensional Fresnel propagator has theform

PFðr; r0; v; dÞ ¼2p

k

� �2

ei k2d ð r2¹r02Þ eik ðvr¹vr0Þ

: ð79Þ

The definition of the projected potential of each slice of theobject remains unchanged because the slices are assumed tobe very thin.

The propagation of the mutual intensity through an ideallens with focal length f is described in thin lens approxima-tion by the mutual transparency function on the lens

MLð r; r0Þ ¼ e¹i k

2f ð r2¹r02Þ: ð80Þ

The effect of the axial aberrations of the objective lens and ofthe beam limiting aperture can be introduced abruptly inthe back focal plane of the lens, as is known from thestandard theory of image formation. The correspondinggeneralized mutual aperture function has the form

MAð r; r0Þ ¼ Að rÞAð r0Þ eiðgð r;EÞ¹gð r0;EÞÞ

: ð81Þ

Here A( r) describes the transmission of the apertureopening. The phase shift

gð r;EÞ ¼ kCs

4v4 þ

Cc

2E ¹ E

Ev2 ¹

Df2

v2� �

; v ¼ jrj=f ð82Þ

results from the spherical and the chromatic aberrationwith constants Cs and Cc, respectively; Df denotes the

defocus. The phase shift (82) depends on the differencebetween the average beam energy E and the energyE ¼ "q ¼ Ei ¹ DE of a particular electron. This energy isthe difference between the initial energy of the electron Ei

and its energy loss DE suffered by an inelastic scatteringprocess within the object. To conveniently account for theenergy dependence of the chromatic aberration in formula(82), we discuss the propagation through the lens system bymeans of the mutual spectral density J(r, r0, E) instead of themutual coherence function Gc(r, r0, t).

Using Eqs. (80) and (81) together with the four-dimensional free-space propagator (8), the propagation ofthe mutual spectral density through the microscope canbe separated into a free-space propagation from the exitplane z ¼ zO of the object to the midplane zL of theobjective lens, a multiplication with the mutual transpar-ency function of the ideal lens ML, a Fresnel propagationfrom the lens to the back focal plane zA of the lens, amultiplication with the generalized mutual aperturefunction MA and finally a Fresnel propagation to theimage plane zI. In mathematical terms the image intensityin the image plane has the form

Ið rÞ ¼1"

�Jð r; r;E; zIÞWðEÞ dE

¼1

l6g2f 4M2

�. . .�

Jð rO; r0O;E; zOÞWðEÞ

× exp�

ik

2g½ð rL ¹ rOÞ2 ¹ ð r0

L ¹ r0OÞ2ÿ

�d2rOd2r0

O|�������������������������������������������{z�������������������������������������������}prop: from object to lens

× exp�

¹ ik

2f½ð r2

L ¹ r02L ÿ

�|���������������������{z���������������������}

ideal lens

× exp�

ik

2f½ð rA ¹ rLÞ

2 ¹ ð r0A ¹ r0

LÞ2ÿ

�|���������������������������������{z���������������������������������}

prop: from lens to backfocal plane

d2rLd2r0L

× Að rAÞAð r0AÞ expfigð rA;EÞ ¹ igð r0

A;EÞg|�����������������������������������{z�����������������������������������}generalized mutual aperture function

× exp�

ik

2ðb ¹ f Þ½ð r ¹ rAÞ2 ¹ ð r0 ¹ r0

AÞ2ÿ

�|��������������������������������������{z��������������������������������������}

prop: from backfocal plane to image plane

d2rAd2r0AdE;

ð83Þ

where M denotes the magnification of the image. The two-dimensional integration with respect to rL can be performedanalytically. The integration with respect to the energy Eextends over the spectral range of the image-formingelectrons. The transmitted energies are selected by theenergy window function W(E). In the case of an EFTEM thisrange is determined by the position and width of the energywindow of the energy filter.

q 1998 The Royal Microscopical Society, Journal of Microscopy, 190, 73–88

IMAGE SIMULATION IN EM 85

8. Results and conclusion

The generalized multislice procedure outlined in thepreceding sections is not yet fully implemented in ourcomputer program. Nevertheless, we present first numericalexamples which demonstrate the possibility of incorporatingquasi-elastic and inelastic scattering events into the multi-slice calculation. As an example we choose the well-knowncrystalline object Si in h111i orientation to facilitate theinterpretation of our results. The thickness of the crystal is37·55 mm.

A 9 × 9 supercell with dimensions a ¼ 1·99 nm,

b ¼ 1·99 nm, c ¼ 0·94 nm is taken into account. Thespecimen is subdivided into 120 slices of equal thickness.For the computation of the inelastic contribution thethickness of the slice is three times larger. This choicedoes not significantly affect the result. All calculations areperformed using 512 × 512 matrices. An ideal EFTEMoperating at an acceleration voltage U ¼ 120 kV and anaperture angle v0 ¼ 25 mrad is assumed. To preventaliasing effects, the spatial frequencies are band widthlimited in the calculation to 2/3 of the maximumfrequency. To efficiently simulate the superposition of theincoherent partial waves, ten different sets of statistical

Fig. 2. (a) Unfiltered image of Si in h111i orientation. (b) Thermal diffuse (phonon) image.

Fig. 3. (a) Corresponding elastic diffraction pattern, without absorption. (b) Unfiltered diffraction pattern.

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phases suffice for the computation of the inelastic andquasi-elastic contributions.

The resulting images and diffraction patterns are depictedin Figs. 2–4. In the case of thermal diffuse scattering wehave assumed a vibration amplitude uSi ¼ 10·95 pm andin the case of inelastic scattering a mean energy loss"q ¼ 6Z eV ¼ 84 eV with Z ¼ 14 for Si.

Figure 2a shows an unfiltered image of Si in h111i

orientation. It represents the superposition of the elastic,quasi-elastic and inelastic image. The contrast of this imageis 2·6 times lower than the contrast of the purely elasticimage. Figure 2b shows solely the phonon backgroundimage. Although such an image cannot be realized inpractice, it demonstrates clearly that thermal diffusescattering produces a high-resolution image.

Figure 3a shows the corresponding elastic diffractionpattern without absorption and Fig. 3b represents theunfiltered diffraction pattern. It results from a superpositionof the thermal diffuse (Fig. 4a) and the inelastic diffractionpattern (Fig. 4b) with the ideal zero-loss filtered diffractionpattern calculated using an absorption potential. Thediffraction background caused by thermal diffuse scatteringis illustrated in Fig. 4a. Kikuchi lines are visible. Theyoriginate from phonon scattering in combination withelastic scattering.

In Figs. 3b and 4b black pixels correspond to maximumintensity. For Figs. 2a, 2b, 3a and 4a the coding scheme isreversed.

The coherence function approach allows one to includeelastic, quasi-elastic and inelastic scattering processes andthe influence of the microscope in the simulation. Thenumerical results presented within this section demonstrate

that our approach is even well suited for efficient numericalcomputations. In future work we will concentrate on theincorporation of models describing the properties of themicroscope into our software implementation. This task iscrucial to successfully take the step from qualitative toquantitative image simulation.

Acknowledgments

Financial support by the Deutsche Forschungsge-meinschaft and Volkswagen Stiftung is gratefully acknowl-edged. The software implementation of the generalizedmultislice alogrithm is partly based on a FORTRAN codewritten by Dr C. Dinges. We would like to thank Dr G.Hoffstatter for valuable discussions and critical reading ofthe manuscript.

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IMAGE SIMULATION IN EM 87

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