Real space maps of atomic transitions

ARTICLE IN PRESS Ultramicroscopy 109 (2009) 781–787

Contents lists available at ScienceDirect

Ultramicroscopy journal homepage: www.elsevier.com/locate/ultramic

Real space maps of atomic transitions P. Schattschneider a,c,, J. Verbeeck b, A.L. Hamon c a b c

¨t Wien, A-1040 WIEN, Austria ¨ r Festko ¨rperphysik, Technische Universita Institut fu EMAT, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp, Belgium ´ LMSSMAT, Ecole Centrale Paris, Grande Voie des Vignes, F-92290 Chatenay-Malabry, France

a r t i c l e in fo

abstract

Article history: Received 17 September 2008 Received in revised form 8 January 2009 Accepted 13 January 2009

Considering the rapid technical development of transmission electron microscopes, we investigate the possibility to map electronic transitions in real space on the atomic scale. To this purpose, we analyse the information carried by the scatterer’s initial and final state wave functions and the role of the different atomic transition channels for the inelastic scattering cross section. It is shown that the change in the magnetic quantum number in the transition can be mapped. Two experimental set-ups are proposed, one blocking half the diffraction plane, the other one using a cylinder lens for imaging. Both methods break the conventional circular symmetry in the electron microscope making it possible to detect the handedness of electronic transitions as an asymmetry in the image intensity. This finding is of important for atomic resolution energy-loss magnetic chiral dichroism (EMCD), allowing to obtain the magnetic moments of single atoms. & 2009 Elsevier B.V. All rights reserved.

PACS: 79.20.Uv 68.37.Og 32.10.Dk 34.80.Nz Keywords: Electron energy loss spectrometry Transmission electron microscopy Circular dichroism Coherence

1. Introduction The new generation of transmission electron microscopes (TEMs) , equipped with aberration correctors, energy filters and monochromators provides exciting possibilities. It has been shown recently that the new microscopes bear indeed enormous potential for energy filtered images at atomic resolution (HR EFTEM) [1–3]. In that context, HR EFTEM attract particular interest. A notorious problem has been the combination of elastic and inelastic interaction, i.e. the propagation of the probe electron in the crystal potential before and after the inelastic event. Although theoretically solved [4–11], applications have remained elusive due to the huge computational effort. With a number of insights and proposals, theory has also made progress [12,13], providing advanced methods. It is therefore reasonable to investigate some of those details of electronic transitions that will be important for interpretation of results expected from last generation microscopes. Such considerations may also serve as a guide for future experiments and instrumental development.

 Corresponding author at: Institut fu ¨ r Festko¨rperphysik, Technische Universita¨t Wien, A-1040 WIEN, Austria. E-mail address: [email protected] (P. Schattschneider).

0304-3991/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2009.01.016

In this paper we analyse the possibility to map electronic transitions on an atomic scale, focussing on the information that is contained in the initial and final state wave functions of the scatterer. We concentrate on the inelastic interaction part and calculate the wave function of the probe electron due to single atomic transition channels. In doing so, we derive the inelastic scattering kernel in real space. This treatment clarifies the role of the different channels in the scattering cross section, and makes contact to the density matrix approach [14]. Based on a recent proposal [15] it will be shown that the handedness of electronic transitions (which translates into the change in the magnetic quantum number Dm ¼ 0 or 1) bear unique signatures that can be mapped in real space. This situation occurs in energy-loss magnetic chiral dichroism (EMCD) experiments [16–18]. The particular attractivity of EMCD lies in the possibility to detect atom specific magnetic moments in combination with sum rules for the spin and orbital components [19,20] with nanometer resolution [21]. Whereas the standard EMCD geometry does not allow atomic resolution, it is shown that off-axis HR EFTEM conditions are more favourable. Imaging with a cylinder lens even allows the direct visualisation of the transferred angular momentum and its numerical evaluation on a per atom-basis. Apart from the fundamental interest in the physics of the chiral and non-chiral transitions and its relationship to angular

ARTICLE IN PRESS 782

P. Schattschneider et al. / Ultramicroscopy 109 (2009) 781–787

momenta, the analysis will provide some ideas about unusual scattering geometries.

2. Basic considerations After inelastic interaction, the probe electron is in a mixed state. That means that a density matrix [22,23] describes the system correctly. This concept has been proposed for fast electron scattering by Dudarev et al., [5] and further developed by Schattschneider et al. [14]. One may avoid the use of the density matrix in inelastic electron scattering by propagating the mutually incoherent wave functions of the probe to the detector, and adding the respective intensities. There is a caveat, however: in doing so, one loses information on the coherence between different positions on the detector; secondly, adding intensities instead of amplitudes one must be sure that there is no interference between the different terms. It is convenient to separate the three-dimensional (3D) coordinates into z and x variables where the latter denote a two-dimensional (2D) position vector in a lateral plane. In the single inelastic scattering approximation, the density matrix of the inelastically scattered electron at the exit plane of the specimen (z ¼ d) is Z ro ðro ; r0o Þ ¼ Gdz ðro ; xÞGdz0 ðr0o ; x0 ÞT zz0 ðx; x0 Þ 0

0

 rzz0 ðx; x Þ dSx dSx0 dz dz ,

(1)

where ro and r0o are 2D variables for the positions in the exit planes, Green’s function Gz propagates the electron after inelastic interaction in the crystal potential from (x; z ¼ 0) to (r; z) and T is the inelastic scattering kernel. rzz0 ðx; x0 Þ is the density matrix of the incident electron at positions x; z; x0 ; z0 in the specimen. In single inelastic scattering approximation rzz0 can be written as a product of wave functions: because there was no inelastic interaction before, the electron is still in a pure state 0

rzz0 ðx; x Þ ¼

cz ðxÞcz0 ðx0 Þ

(2)

with

cz ðxÞ:¼cðx; zÞ ¼

Z

Gz ðx; ri Þci ðri Þ dSri

(3)

with ri the lateral coordinate in the entrance plane of the specimen (z ¼ 0), and ci the incident electron wave function. Fig. 1 shows the position of the relevant planes in the specimen. The integral in Eq. (1) is over the whole 3D specimen.

ki

z=0

i ri zz' x, x' O r0, r0'

z=d z post specimen lenses

D s, s'

detector plane

z=D

Fig. 1. Position of relevant planes in an inelastic scattering experiment: ci with wave vector ki at the entrance plane (z ¼ 0) with coordinates ri , rzz0 at any depths z and z0 within the specimen, and ro at the exit plane (z ¼ d). The detector is situated after the post specimen lens system in the plane (z ¼ D) where lateral coordinates are s and s0 .

We should note that we have implicitly fixed the energy loss and omitted this variable for convenience. Eq. (1) is valid in single inelastic scattering approximation, justified for core losses in specimens of usual thickness because the core excitation’s mean-free path (MFP) is several hundred nm. The density matrix rzz0 can be calculated with any dynamical scattering code when the incident wave ci ðri Þ at the entrance surface z ¼ 0 is known. The density matrix ro at the exit plane z ¼ d, Eq. (1), is finally propagated to the detector via GD describing the action of lenses and apertures. The intensity IðsÞ is measured in the detector plane z ¼ D. It is given by the diagonal elements of the density matrix (rD ) Z GD ðs; rÞGD ðs; r0 Þro ðr; r0 Þ dSr dSr0 . (4) IðsÞ ¼ rD ðs; sÞ ¼ To resume, Eqs. (1)–(4) describe the inelastic scattering experiment completely.

3. Atomic transition channels in real space In the following part, we adapt Eq. (4) to the case of single atom ionisation. The propagator Gz depends on the energy of the probe electron. We can extract a rapidly varying phase factor and obtain Gz ðx; rÞ ¼ G¯ z ðx; rÞeiki z , where ki ¼ 2p=li is the incident electron’s wave number. G¯ z is the propagator normally used in multislice calculations or in Bloch wave methods. When calculating elastic intensities, the phase factor cancels with its complex conjugate, so it is omitted in general. But in treating inelastic interactions we must keep it. The propagator Gdz applies to electrons after energy loss, that is, we can also extract a rapidly oscillating phase factor as previously for Gz : Gdz ðx; rÞ ¼ G¯ dz ðx; rÞeiko ðdzÞ with the outgoing electron’s wave number ko. We can now replace the propagators in Eq. (1) by the normally used ones for Bloch wave propagation Z Z  ro ðro ; r0o Þ ¼ G¯ dz ðro ; xÞG¯ dz ðr0o ; x0 ÞT zz0 ðx; x0 Þ ¯ z0 ðx0 Þ dSx dSx0 eiqe ðzz0 Þ dz dz0 , ¯ z ðxÞc c

(5)

where after extraction of the exponential factors the wave functions c of Eq. (3) are now replaced by Z c¯ z ðxÞ ¼ G¯ z ðx; ri Þci ðri Þ dSri (6) and qE ¼ ko  ki is the minimum wave vector transfer in the inelastic interaction. For energy losses of o1 kV as encountered in EFTEM, the Bloch wave propagators G¯ z for the incident electron and G¯ dz for the inelastically scattered electron can be assumed to be equal, which may simplify the calculations. (Note that this is not the case for G because of the different phase factors.) We have expressed the z dependence explicitly in preparation for the next step. The inelastic scattering kernel T can be written in configuration space as the convolution of the mixed dynamic form factor (MDFF) [4] S with the Coulomb coupling field [24] R1 ¼ 1=jRj where R is the configuration space vector R ¼ ðx; zÞ T zz0 ðx; x0 Þ ¼ SðR; R0 Þ%ðR1 R01 Þ.

(7)

Assuming that we have a plane wave incident parallel to the optical axis, and a single atom in the object plane of a perfect lens, 2 2 2 then GD ðs; ro Þ ¼ d ðs; ro Þ, and Gdz ðro ; xÞ ¼ d ðro ; xÞ, where d is the

ARTICLE IN PRESS P. Schattschneider et al. / Ultramicroscopy 109 (2009) 781–787

783

m m′

2 at.u.

0 0

0 1

0 –1 Fig. 2. Electron probe in the lateral plane, Eq. (13), caused by the various p ! d transition channels for initial magnetic quantum number m ¼ 0. Real part of the wave function (left column), imaginary part of the wave function (middle) and intensity (right). The initial and final state magnetic quantum numbers of the atom are indicated as m; m0 . Incident plane wave and perfect lens assumed. See text for details. Colour coding by rainbow chart shown as insert in the left upper panel (blue: low, red: high). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

2D Dirac distribution. Eq. (4) simplifies to Z 0 0 IðsÞ ¼ dz dz T zz0 ðs; sÞeiqE ðzz Þ .

(8)

Note that the exponential factor comes from the rapidly oscillating axial component of the wave functions, as remarked at the end of the previous section. We evaluate now Eq. (8). This is a Fourier transform with respect to z and z0 . The convolution theorem applied to Eq. (7) gives Z 0 0 (9) IðsÞ ¼ SðR; R0 ÞeiqE ðzz Þ dz dz %x;x0 Fz ½R1 qE Fz0 ½R01 qE , where the Fourier transform can be approximated to good accuracy as [25] 

Fz ½R1 qE ¼ K¯ 0 ðqE xÞ:¼

K 0 ðqE xÞ= lnðqc =qE Þ;

K 0 ðqE xÞo lnðqc =qE Þ

1

else (10)

with the modified Bessel function K 0 . qc is the cut-off wave number, usually taken as the value of the Bethe ridge maximum. The convolution function K¯ 0 broadens the otherwise very narrow factor S which is the Fourier transform of the MDFF. The S factor is the product of density matrices of the target initial and final states [14,26] SðR; R0 Þ ¼ ri ðR; R0 Þrf ðR0 ; RÞ.

(11)

The product of density matrices of the target atom is given by the sum over respective (initial or final) wave functions f: X ra ðR; R0 Þ ¼ fa ðRÞfa ðR0 Þ. a

where a ¼ i or f. The integral over z; z0 in Eq. (9) can be evaluated term-wise for the single wave functions, and we obtain for the

energy filtered image of a single atom X IðxÞ ¼ jci;f ðxÞj2 ,

(12)

i;f

where

ci;f ðxÞ ¼ K¯ 0 ðqE xÞ%x;x0

Z



dzfi ðRÞff ðRÞeiqE z

(13)

is the wave function of the electron probe after having excited a transition from state i to f in the target atom. This shows again, by the way, that the probe electron is in a mixed state. A similar formalism has been used by Weickenmeier and Kohl [27] without explicit reference to the density matrix. In order to study the details of the different transition channels, we present results for a simple model system. As in a previous paper [15] we choose Si. It is a standard testing material in electron microscopy, the appearance of Si high resolution images are familiar to many TEM users. To make connection to experiment one can imagine that the energy filter is ideal, capable of selecting a particular transition when a Zeeman splitting is induced by the magnetic field of the objective lens.1 Atomic wave functions were used for calculation. All channels contributing to the Si L23 edge were numerically evaluated. The p ! s transitions are more than an order of magnitude smaller than the p ! d transitions and can be neglected—see Fig. 4. In Fig. 2 and 3 we show the real and imaginary parts of the wave functions ci;f from Eq. (12) and the intensity, corresponding to different transition channels. The initial and final magnetic quantum numbers are given for each row in the figure. 1 In the usual 2 Tesla field of the objective lens, the Zeeman splitting is of the order of 104 eV. This is far from the resolving power of existing spectrometers. However, as successful EMCD experiments in the TEM show the strong spin–orbit coupling in the ferromagnets leads to an energy split of the order of 10 eV between final states with different orbital polarisation [16].

ARTICLE IN PRESS 784

P. Schattschneider et al. / Ultramicroscopy 109 (2009) 781–787

m m′

10

2 at.u.

11

12

The initial states m ¼ 0 in Fig. 2 are more localized than the m ¼ 1 states in Fig. 3. We also see that the intensities are ringshaped when the magnetic quantum number changes. This is indicative for a parity change in the probe electron in the image plane. This effect is even better visible in the wave functions. Whereas the initial probe has even parity (it is radially symmetric), the channels withDma0 reveal odd parity in the real and imaginary part of the wave function. The parity change is a necessary consequence of the dipole transitions forced by the selection rules. It should be noted however that Eq. (11) contains all multipole orders and is therefore exact when all energetically allowed final states are included. Note that the image has a volcano-like structure typical for p-waves. The interaction has changed the symmetry of the incident electron wave from s-type to p-type. The maximum of the p lobe is found more than one atomic unit from the centre. Modern microscopes with C s corrector are capable of resolving such structure [12,3,28]. The channels Dm ¼ 0 seem to have conserved the even symmetry of the incident plane wave, in contradiction to the above statement. At second thought it becomes clear that only the parity with respect to vertical mirrors is even. The probe electron has changed its symmetry with respect to a mirror in the drawing plane. This change is not visible because of the integral over the axial coordinates z; z0 in Eq. (8).

4. Chiral transitions Intuitively one would think that at least the m ¼ 0 ! m ¼ 1 and the m ¼ 0 ! m ¼ 1 would show some asymmetry that cancels when adding the contributions in the density matrix. This is true for the wave function—compare the second and third rows for m0  1 in Fig. 2. But this is not true for the intensity. The characteristic phase factor eiDmf of chiral transitions (Dma0) shows up in the scattering amplitude but disappears of course in the intensity. It is therefore not visible in the image. Chiral

arb.u.

Fig. 3. Same as in Fig. 2, for initial magnetic quantum number m ¼ 1. See text for details. Colour coding by rainbow chart (blue: low, red: high). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 -4

-2

0

2

4

r [a.u.] Fig. 4. Intensity trace across an EFTEM image of Si L23 energy loss in a free atom. Full line: p ! d. The p ! s transition (dashed) is negligible. (Incident plane wave and perfect lens assumed.)

transitions create radially symmetric intensity distributions as non-chiral channels do. The sum over all channels gives the diagonal terms of the density matrix ro . This image is also radially symmetric, as has been shown previously [29]. A radial intensity trace across the atom image, due to p ! d transitions in Si is shown in Fig. 4, together with the faint p ! s contribution. The reason for the smallness is the small overlap integral between initial and final states. For the present purpose we can safely neglect p ! s transitions. It seems there is no way of fingerprinting chirality of transitions in the standard HR EFTEM set-up. How can we render it visible? Recall that the reason for the non-appearance of chirality in the image is that the phase eiDmf characteristic for those transitions (where f is the azimuthal angle with respect to the centre of the projected atom) cancels in the diagonal element of the kernel T. A clue to a possible mapping of chirality comes from the following argument. Using Cartesian components of the

ARTICLE IN PRESS P. Schattschneider et al. / Ultramicroscopy 109 (2009) 781–787

785

py × sx = 0 py py

2 at.u.

py × sx = 0

py sx

sx py

py × sx > 0

sx

py × sx > 0

py

sx py

py × sx < 0

py × sx < 0

Fig. 5. Images (intensity maps) of the m ¼ 0 ! m0 ¼ 0 (upper row), m0 ¼ 1 (middle), m0 ¼ 1 (lower row) transitions, with the lower (left) or upper (right) half of the diffraction pattern blocked. The dashed rectangle symbolises the contrast aperture. The py and sx vectors are also shown, demonstrating the sign of the vector product which is classically given by Lz ¼ sx py . The dashed lines are an aid to the eye for finding the centre of the atom. Colour coding by rainbow chart (blue: low, red: high). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

lateral coordinates (x,y), at fixed x, say at 1 a.u from the centre, a vertical trace (coordinate y) would carry a phase eiDmf . For small y one has approximately the amplitude g  ðyÞ ¼ f x ðyÞeiy=ao . In the intensity map, the phase factor disappears; but we can retain the phase information when considering that the Fourier transform of g  is f~ x ðqy  a1 0 Þ by virtue of the shift theorem. The diffracted amplitude will have its centre of gravity shifted by a1 0 above or below the original symmetry line qy ¼ 0. It is reasonable to exploit this difference by posing an off-axis aperture in the diffraction plane (DP). In order to keep it simple we block the upper (lower) half of the DP. The resulting images are given in Fig. 5. Blocking the upper half means that we restrict the linear momentum py to negative values. This action forces the probe electron (which has hLz i40) to xo0. The opposite is true when the lower half of the DP is blocked. With C s corrected microscopes it should be possible in principle to detect this faint shift of the maximum. The question is then: is it technically possible to separate the different mrelated channels in an energy filter? Chiral transitions have been detected with electron energy-loss spectroscopy (EELS) in the ferromagnets [16,30–33]. So in principle it is possible to map the chirality. This would be a unique method for probing magnetism in HR TEM. It must however be cautioned that the signal that may create a chiral map is of the order of 5–10% of the signal in the ionisation cross section of the L23 edges, and is taken at high scattering angle. For the moment noise and stability problems are definitely an obstacle to such an experiment.

k0 atom,  (r, r’) objective lens py px

diffraction panel,  (p, p’)

cylinder lens

x py

image plane, D (x,py, x’, p’y)

Fig. 6. A cylinder objective lens forms an ðx; py Þ-basis for the representation of the density matrix r. The detector measures the diagonal elements rD ðx; py ; x; py Þ.

providing a possibility to map angular momenta. We shall shortly repeat the arguments given in [15]. The expectation value of the angular momentum operator of the probe electron is ^ Lz  ¼ Tr½r ^ x^ p^ y   Tr½r ^ y^ p^ x , hLz i ¼ Tr½r

5. Mapping angular momenta It has been shown [15] that the angular momentum of the probe electron, and thus the transfer of magnetic quantum numbers in an inelastic interaction can be represented in a particular basis. Here, we extend on this idea, proposing an appropriate scattering geometry to realise this basis, thus

(14)

^ is the electron’s density operator. The density matrix where r used above is the density operator in a real space basis

rðx; x0 Þ ¼ hxjr^ jx0 i. Instead of this basis, we choose for the first term on the right hand side the basis jx; py i which is complete and orthonormal.

ARTICLE IN PRESS 786

P. Schattschneider et al. / Ultramicroscopy 109 (2009) 781–787

5

qy [mrad]

2.5

x [a.u.]

mrad 10

20

-2.5 0

0 -5 -4

-2

2

-10

4

-20

x [a.u.] Fig. 7. Left: EFTEM image of a single Si atom after a transition m ¼ 0 ! m0 ¼ 1, obtained with a perfect cylinder lens as sketched in Fig. 6. Parameters as before. Abscissa: atomic units; ordinate: scattering angle in mrad (200 kV incident electrons). Colour coding by rainbow chart (blue: low, red: high). Right: Perspective view with wider xrange. The dashed line indicates the centre of the atom. The broad tail of the profile is caused by the long range Coulomb coupling between probe and target. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

This gives ^ x^ p^ y  ¼ Tr½r

Z

dx dpy rðx; py ; x; py Þxpy .

(15)

It can be shown that the second term is the negative of the first one [15]. If we were able to map the ðx; py Þ plane we could experimentally determine the angular momentum from an ðx; py Þ—or equivalently from a ðy; px Þ representation. This can be accomplished by a cylinder lens that projects the x coordinate of the exit surface as well as the focal line px ¼ 0 (coordinate perpendicular to x) onto the detector plane. The set-up is sketched in Fig. 6. Green’s function of the cylinder lens is

x ¼ ðx; yÞ, s ¼ ðsx ; py Þ. Eq. (1) is then Z Z 0 0 dz dz Iðsx ; py Þ ¼ dy dy T zz0 ðsx ; y; sx ; y0 Þ 0

0

Acknowledgements J.V. acknowledges the FWO-Vlaanderen for support (contract no. G.0147.06) and the European Union under the Framework 6 program under a contract for an Integrated Infrastructure Initiative. Reference 026019 ESTEEM.

GD ¼ dðx  sx Þeiysy

eiqE ðzz Þþpy ðyy Þ;

real space wave functions of the scatterer. We proposed two experimental set-ups that allow to distinguish between chiral and non-chiral transitions, and even to detect their handedness. Numerical simulations show that a change in angular momentum of the probe due to a chiral transition translates into a shift of the image intensity that should be detectable with modern instruments. This opens the way for atomic resolution EMCD, and possibly for other experiments involving broken rotational symmetry.

(16)

which is the double Fourier transform of the density matrix Iðs; s0 Þ with respect to the y coordinates of the real space representation, taking sx ¼ s0x . (The diagonal elements of Iðs; s0 Þ are given in Eq. (8).) Fig. 7 shows the intensity distribution equation (16) for a chiral transition with Dm ¼ 1 in the ðsx ; py Þ-plane. The asymmetry in the figure is significant for the transfer of angular momentum to the electronic system. In fact, the probability to find a probe electron with x40 and py 40, or xo0 and py o0 is high, i.e. the probe electron has angular momentum hLz i40. Numerical evaluation via Eq. (14), (15) gives hLz i ¼ 0:97_. The nominal value is _, corresponding to a change Dm ¼ 1 of the magnetic quantum number. The deviation from the exact value is due to numerical integration errors and to the cutoff at x ¼ 22 a.u., qy ¼ 6:25 a.u.2

6. Conclusion We have applied the density matrix approach to electronic transitions in a simple atomic model for the calculation of HR EFTEM images. The inelastic scattering kernel was derived from 2 The numerical value depends strongly on the integration range. For a cutoff at 5 a.u. the value is 0.93 [15].

References [1] M. Bosman, V.J. Keast, J.L. Garcia-Munoz, A.J. D’Alfonso, S.D. Findlay, L.J. Allen, Two-dimensional mapping of chemical information at atomic resolution, Physical Review Letters 99 (8) (2007). [2] D.A. Muller, L. Fitting Kourkoutis, M. Murfitt, J.H. Song, H.Y. Hwang, J. Silcox, N. Dellby, O.L. Krivanek, Atomic-scale chemical imaging of composition and bonding by aberration-corrected microscopy, Science 319 (5866) (2008) 1073–1076. [3] U. Dahmen, R. Erni, C. Kisielowski, An update of the TEAM project, Proceedings of the EMC2008 1 (2008) 3–4. [4] H. Kohl, H. Rose, Theory of image formation by inelastically scattered electrons in the electron microscope, Advances in Electronics and Electron Optics 65 (1985) 173–227. [5] S.L. Dudarev, L.M. Peng, M.J. Whelan, Correlations in space and time and dynamical diffraction of high-energy electrons by crystals, Physical Review B 48 (1993) 13408–13429. [6] L.J. Allen, Electron energy loss spectroscopy in a crystalline environment using inner-shell ionization, Ultramicroscopy 48 (1993) 97–106. [7] L.J. Allen, T.W. Josefsson, Inelastic scattering of fast electrons by crystals, Physical Review B 52 (1995) 3184–3198. [8] D.A. Muller, J. Silcox, Delocalization in inelastic scattering, Ultramicroscopy 59 (1995) 195–213. [9] L.J. Allen, D.C. Bell, T.W. Josefsson, A.E.C. Spargo, S.L. Dudarev, Inner-shell ionization cross sections and aperture size in electron energy-loss spectrometry, Physical Review B 56 (1997) 9–11. [10] M. Nelhiebel. Effects of crystal orientation and interferometry in electron energy loss spectroscopy. Ph.D. Thesis, Ecole Centrale Paris, ChaˆtenayMalabry, 1999. [11] M. Nelhiebel, P.H. Louf, P. Schattschneider, P. Blaha, K. Schwarz, B. Jouffrey, Theory of orientation-sensitive near-edge fine-structure core-level spectroscopy, Physical Review B 59 (1999) 12807–12814. [12] A.J. D’Alfonso, S.D. Findlay, M.P. Oxley, L.J. Allen, Volcano structure in atomic resolution core-loss images, Ultramicroscopy 108 (7) (2008) 677–687.

ARTICLE IN PRESS P. Schattschneider et al. / Ultramicroscopy 109 (2009) 781–787

[13] C. Dwyer, S.D. Findlay, L.J. Allen, Multiple elastic scattering of core-loss electrons in atomic resolution imaging, Physical Review B 77 (18) (2008) 184107. [14] P. Schattschneider, M. Nelhiebel, B. Jouffrey, The density matrix of inelastically scattered fast electrons, Physical Review B 59 (1999) 10959–10969. [15] P. Schattschneider, Exchange of angular momentum in EMCD experiments, Ultramicroscopy 109 (2008) 91–95. [16] P. Schattschneider, S. Rubino, C. He´bert, J. Rusz, J. Kunesˇ, P. Nova´k, E. Carlino, M. Fabrizioli, G. Panaccione, G. Rossi, Experimental proof of circular magnetic dichroism in the electron microscope, Nature 441 (2006) 486–488. [17] C. He´bert, P. Schattschneider, S. Rubino. Circular dichroism in the transmission electron microscope, Encyclopedia of Materials: Science and Technology (2007) 1–11. [18] P. Schattschneider, S. Rubino, M. Sto¨ger-Pollach, C. He´bert, J. Rusz, L. Calmels, E. Snoeck, Energy loss magnetic chiral dichroism: a new technique for the study of magnetic properties in the electron microscope, Journal of Applied Physics 103 (2008) 07D9311–6. [19] L. Calmels, F. Houdellier, B. Warot-Fonrose, C. Gatel, M.J. Hy¨tch, V. Serin, E. Snoeck, P. Schattschneider, Experimental application of sum rules for electron energy loss magnetic chiral dichroism, Physical Review B 76 (6) (2007). [20] J. Rusz, O. Eriksson, P. Nova´k, P.M. Oppeneer, Sum-rules for electron energyloss near-edge spectra, Physical Review B 76 (2007) 060408. [21] P. Schattschneider, M. Sto¨ger-Pollach, S. Rubino, M. Sperl, C. Hurm, J. Zweck, and J. Rusz. Detection of magnetic circular dichroism on the two-nanometer scale. Physical Review B 78 (2008) 104413. [22] L.D. Landau, Zeitschrift fur Physik 45 (1927) 430. [23] J. von Neumann. Go¨ttinger Nachr. 246 (1927). [24] P. Schattschneider, J. Verbeeck, Fringe contrast in inelastic LACBED holography, Ultramicroscopy 108 (5) (2008) 407–414.

787

[25] J. Verbeeck, D. Van Dyck, H. Lichte, P. Potapov, P. Schattschneider, Plasmon holographic experiments: theoretical framework, Ultramicroscopy 102 (3) (2005) 239–255. [26] M. Gusso, Theoretical study of near edge electron energy loss spectroscopy of metal nanoclusters, Journal of Physics—Condensed Matter 18 (2006) 1211–1226. [27] A. Weickenmeier, H. Kohl, Computation of the atomic inner-shell excitation cross section for fast electrons in crystals, Philosophical Magazine B 60 (1989) 467–479. [28] B. Freitag, G. Knippels, S. Kujawa, P.C. Tiemeijer, M.V.D. Stam, D. Hubert, C. Kisielowski, P. Denes, A. Minor, U. Dahmen, First performance measurements and application results of a new high brightness Schottky field emitter for HR-S/TEM at 80–300 kV acceleration voltage, Microscopy and Microanalysis 14 (Suppl. 2) (2008) 1370–1371. [29] P. Schattschneider, M. Nelhiebel, H. Souchay, B. Jouffrey, The physical significance of the mixed dynamic form factor, Micron 31 (2000) 333–345. [30] P. van Aken, L. Gu, D. Goll, G. Schu¨tz, Electron magnetic linear dichroism (EMLD) and electron magnetic circular dichroism (EMCD) in electron energyloss spectroscopy, Microscopy and Microanalysis 13 (S03) (2007) 426–427. [31] B. Warot-Fonrose, F. Houdellier, M.J. Hy¨tch, L. Calmels, V. Serin, E. Snoeck, Mapping inelastic intensities in diffraction patterns of magnetic samples using the energy spectrum imaging technique, Ultramicroscopy 108 (5) (2008) 393–398. [32] J. Verbeeck, C. He´bert, S. Rubino, P. Nova´k, J. Rusz, F. Houdellier, C. Gatel, P. Schattschneider, Optimal aperture sizes and positions for EMCD experiments, Ultramicroscopy 108 (9) (2008) 865–872. [33] H. Lidbaum, J. Rusz, A. Liebig, B. Hjorvarsson, P.M. Oppeneer, E. Coronel, O. Eriksson, K. Leifer, EMCD in the TEM—optimization of signal acquisition and data evaluation, Microscopy and Microanalysis 14 (Suppl. 2) (2008) 1148–1149.