Limits to the Spatial, Energy and Momentum Resolution of Electron Energy-Loss Spectroscopy

ARTICLE IN PRESS

Ultramicroscopy 107 (2007) 575–586 www.elsevier.com/locate/ultramic

Limits to the spatial, energy and momentum resolution of electron energy-loss spectroscopy

R.F. EgertonÃ

Department of Physics, Room #238 CEB, 11322-89 Avenue, University of Alberta, Edmonton, AB, Canada T6G 2G7

Received 17 August 2006; received in revised form 14 November 2006; accepted 22 November 2006

Abstract

We discuss various factors that determine the performance of electron energy-loss spectroscopy (EELS) and energy-ﬁltered (EFTEM) imaging in a transmission electron microscope. Some of these factors are instrumental and have undergone substantial improvement in recent years, including the development of electron monochromators and aberration correctors. Others, such as radiation damage, delocalization of inelastic scattering and beam broadening in the specimen, derive from basic physics and are likely to remain as limitations. To aid the experimentalist, analytical expressions are given for beam broadening, delocalization length, energy broadening due to core-hole and excited-electron lifetimes, and for the momentum resolution in angle- resolved EELS. r 2007 Elsevier B.V. All rights reserved.

PACS: 79.20.Uv; 68.37.Lp; 61.80.Fe; 42.30.Kq; 34.80.I

Keywords: EELS; TEM; Radiation effects; Fourier optics; Electron scattering

1. Introduction presented here in a form designed to be convenient for the experimentalist. Many considerations affect the capability of electron energy-loss spectroscopy (EELS), in combination with 2. Spatial resolution transmission electron microscopy (TEM), to solve pro- blems in materials or life science. Some of them relate to In EELS, spatial resolution refers to the smallest instrument design, such as the electron-optical design of diameter (lateral dimension within a thin specimen) from the spectrometer and microscope column. Others are partly which spatial information can be obtained. In energy- environmental, such as the electrical and mechanical filtered (EFTEM) imaging, the equivalent quantity is the stability. There are also important human considerations, minimum useful pixel size, below which there is no including the knowledge, skill and patience of the substantial gain in information content. researcher. But over the last few decades, the knowledge base and level of instrument performance have improved to the extent that a third kind of factor becomes important: 2.1. Electron-optical considerations performance limits arising from basic physical principles. This paper reviews all of the relevant factors but gives Because spectroscopy and scanning-transmission emphasis to these fundamental limitations. Although the (STEM) imaging are usually carried out using a tightly physical principles involved are well established, they are focused beam (electron probe), one obvious limit is the smallest beam size that can be produced by a given instrument. In a modern TEM, the minimum probe size is ÃTel.: +1 780 641 1662; fax: +1 780 641 1601. well below 1 nm, thanks to the efficient exploitation of E-mail address: [email protected]. electromagnetic lenses. Use of a Schottky or a cold field-

0304-3991/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2006.11.005 ARTICLE IN PRESS 576 R.F. Egerton / Ultramicroscopy 107 (2007) 575–586 emission (CFE) electron source helps to ensure that there is electrons is: sufficient current (e.g. 1 nA) in such a small probe. d F bt, (2) Correction of the spherical aberration of the probe- n n E E forming lens allows a further increase in probe current where Fn 0.4 for n ¼ 50 and Fn 1.0 for n ¼ 90 [6]. E and can reduce also the full width at half maximum Taking t ¼ 50 nm and b ¼ 10 mrad gives d50 0.2 nm and E (FWHM) of the probe to below 0.1 nm [1–3]. By removing d90 0.5 nm, values considerably less than the total aberration tails that otherwise contain a substantial broadening given by Eq. (1). These estimates may actually proportion of the current, the corrector should provide a be pessimistic since they assume that the scattering ‘‘cleaner’’ current-density profile. (per unit solid angle) is constant up to the angle b.Ifno Although the probe size is determined by the electron- angle-limiting aperture is used, the beam width in EELS source size, lens aberrations and electron diffraction at the should be given in Eq. (1), at least for an amorphous condenser aperture, it can in principle be degraded by specimen. Coulomb interactions between the electrons within the For crystalline specimens, a more correct treatment of TEM illumination system. Fortunately, the TEM uses a beam broadening includes the fact that elastic scattering relatively low beam current, giving conditions that (except in ultra-thin specimens) is dynamical: for depths correspond to the ‘‘pencil-beam regime’’ where the in excess of xg=2, where xg is the extinction distance (in statistical Coulomb broadening depends on the third the range 25–100 nm for 100 keV electrons), many elect- power of the beam current [4]. So for accelerated rons are scattered back towards the optic axis. As a result, electrons, the statistical broadening appears to be negli- the electron beam spreads less than in an amorphous gible, even for aberration-corrected lenses forming a high- material (at least for scattering through less than a typical intensity probe, where the current density can exceed Bragg angle), which benefits XEDS analysis in addition to 1 MA/cm2. EELS [7,8]. In addition, the electron density within the beam becomes distributed non-uniformly: at depths exceeding about 5 nm, electrons are channeled preferen- 2.2. Beam broadening within the specimen tially along the columns of atoms, especially for a crystal oriented with a low-index zone axis parallel to the incident Inside the specimen, the electron beam spreads laterally beam [9]. due to electron scattering. Even without scattering, it would broaden by 2at (of the order of 1 nm for a tightly 2.3. Chromatic aberration focused probe and thin specimen) for a probe of convergence semi-angle a focused onto the top surface of If the electrons transmitted through a specimen are a specimen of thickness t. For an amorphous specimen, subsequently focused to form an energy-filtered (EFTEM) elastic scattering increases the beam broadening by an image, the image resolution is subject to degradation by amount [5]: lens aberrations. Usually chromatic aberration is the most important factor, particularly for core-loss images where 1=2 3=2 b ð625 cmÞðZ=E0Þðr=AÞ ½tðcmÞ the energy and angular widths of the focused electrons can 1=2 3=2 be considerable. Assuming the energy-filtered image is ð0:47 nmÞðrZÞ ð100 keV=E0Þ½t=50 nm , ð1Þ correctly focused for electrons that pass though the center 3 where r is the specimen density in g/cm , E0 the incident- of the energy-selecting slit, the diameter containing 50% of electron energy in keV. For E0 ¼ 100 keV and t ¼ 50 nm, the electrons is increased by an amount Fdc, where b ¼ 1.8 nm for carbon, 2.9 nm for Al, 7.6 nm for Cu and dc ¼ CcbðD=E0Þ, Cc is the chromatic aberration coefficient 17 nm for Au, values that agree rather well with the of the objective lens and D is the energy width of the diameter (containing 90% of the trajectories) deduced from energy-selecting slit. The factor F depends on the angular Monte Carlo calculations [5]. This effect may be added in width of the inelastic scattering: F 0:1 for an energy loss quadrature to the beam-divergence effect (2at) in a first E ¼ 100 eV and E0 ¼ 100 keV, increasing to 0.3 for large E approximation. or a thick specimen [10]. The diameter Fdc is typically Although beam broadening in the specimen largely around 0.2 nm for b ¼ 10 mrad and D ¼ 20 eV. Note that determines the spatial resolution of X-ray energy-dispersive this 50% broadening is substantially less (by factor F) than spectroscopy (XEDS), its effect on transmission EELS the total chromatic width dc, which is often taken as an can be considerably less, provided the spectrum is re- estimate of the chromatic effect. corded through an angle-limiting aperture that removes In principle, the EFTEM resolution also depends on the electrons that deviate from the optic axis by more than spatial resolution of the electron detector (usually a its semiangle b. Considering scattering at a distance z scintillator/CCD arrangement). However, this factor can above the bottom surface of the specimen, electrons be made unimportant by choosing a sufficiently high image entering the spectrometer travel through a cone of radius magnification. bz and volume ðp=3Þb2z3. Averaging over the specimen None of the above factors are fundamental, in the sense thickness t, the diameter containing n% of the detected that they depend on the design of the microscope and ARTICLE IN PRESS R.F. Egerton / Ultramicroscopy 107 (2007) 575–586 577 spectrometer. Even beam broadening can be made in- significant by using a high accelerating voltage or a v/ω sufficiently thin specimen. However two additional factors represent basic limits, imposed by properties of the electron. The first of these is delocalization of the inelastic scattering. b This phenomenon must be distinguished from contrast delocalization, a term used in high-resolution TEM ima- ω ging to describe a lens-aberration effect, and from delocalization of valence or conduction electrons, as used in chemistry and solid-state physics. It is certainly different E0 -E from the nonlocality of entangled quantum states, as impulse region of discussed in quantum mechanics in connection with Bell’s approxn. dynamic screening theorem. E ~ ER electrostatics electrodynamics 2.4. Inelastic delocalization

The concept of delocalization can be introduced in terms 0.9 of the simple experiment performed in 1974 by Isaacson et al. [11], who scanned a 43 keV CFE probe across the 0.8 edge of a thin carbon film and recorded line profiles from 0.7 an annular detector (designed to collect high-angle elastic scattering) and from an energy-loss spectrometer (set to 0.6 all angles detect the total-inelastic component of the lower- angle inelastic scattering). The inelastic signal changed 0.5 100mrad more gradually than the elastic one, showing inelastic 0.4 scattering to be more delocalized than high-angle elastic scattering. 0.3 This basic experiment has been repeated in greater detail 0.2 (measuring the inelastic signal as a function of energy loss) 10 pm E0 = 100 keV by Muller and Silcox [12]. In addition, the spatial 0.1 distribution of inelastic scattering has been discussed in terms of electron-scattering theory in several important 0 publications [12–22] and reviews [6,23]. Here our aim is to 0 0.5 1 1.5 2 2.5 3 3.5 provide an approximate value of the localization distance (3.5e5) (bw/v) x 104 and its dependence on experimental variables, as well as some insight into the cause of delocalization. We Fig. 1. (a) Interaction of a fast-moving incident electron with a bound atomic electron, according to the classical-particle analysis of Bohr [25]. therefore adopt a single-scattering model, assuming that (b) Classical prediction of the fraction of energy loss contained within an the sample is thin enough to make plural inelastic impact parameter less than b, for bo=v in the range 0 to 0.1, E0 ¼ 100 keV, scattering unimportant and that the effect of elastic ho/(2p) ¼ 100 eV and maximum. scattering angles of p rad and 100 mrad. scattering can be dealt with separately as beam broadening (as just discussed). In the case of crystalline specimens and core-loss spectroscopy, these assumptions become ques- linear path with impact parameter b: tionable and a proper treatment requires a simultaneous 2 2 2 treatment of elastic and inelastic scattering, using (for EðbÞ¼ERðbo=vÞ f½K0ðbo=vÞ þ½K1ðbo=vÞ g. (3) example) a combination of multislice and single-inelastic 2 4 2 scattering theory [21,22,24] or Bloch-wave methods Here ER ¼ðkc e =E0Þð1=b Þ is the Rutherford recoil energy [18–20]. that would be imparted to a stationary and free electron, 21 One of the first attempts to describe the problem kc ¼ð4p0Þ being the Coulomb constant. For b5v=o, mathematically was that of Bohr [25], using entirely the energy exchange is approximately ER: the atomic classical physics; this work predates Bohr’s quantum model electron has insufficient time to move during transit of the of the atom [26] by 6 months. The incident electron is incident electron and receives the same momentum as a free represented as a particle that interacts via the Coulomb electron, according to the impulse approximation. For force with a bound atomic electron, represented as a bbv=o, however, the atomic electron moves in response to classical oscillator of angular frequency o; see Fig. 1. the electric field of the incident electron, resulting in very Several pages of non-relativistic mechanics lead to an little energy exchange (adiabatic conditions), represented expression for the energy loss E of the incident electron by the rapidly falling modified-Bessel terms in Eq. (3). when the latter moves at speed v along an approximately Therefore the inelastic scattering is often taken to be ARTICLE IN PRESS 578 R.F. Egerton / Ultramicroscopy 107 (2007) 575–586 confined to impact parameters below v/o and a localiza- inelastically scattered electrons, Eq. (5) becomes tion range taken as L v=o according to these classical- L ð0:52Þl=hyi (6) physics arguments. 50 In fact, E(b) as given by Eq. (3) has a singularity at b ¼ 0 in which /yS could reasonably be taken as the median and a long tail extending to moderate values of ðbo=vÞ, angle of inelastic scattering, containing 50% of the which points to the dangers of representing localization in scattered intensity. Eq. (6) is consistent with the Heisenberg terms of a single number or at least to the need for care- momentum-position uncertainty principle, since if we ful definition. Considering a circular beam of electrons take the median transverse momentum to be Dp ¼ of diameter greater than v/o, E(b) can be multiplied by gm0v sinhyi ð2h=lÞhyi, the associated uncertainty in (2pb) and integrated over b to evaluate the energy loss position is Dx h=Dp ¼ 0:5l=hyi. for impact parameters up to b. A lower limit of integ- Although partly conjecture, Eq. (6) provides a reason- ration bmin must be introduced [27], corresponding to able approximation to most experimental measurements a maximum scattering angle of p radians or to some [12,29–32] and calculations [12,15,21] of the localization smaller value (defined by a spectrometer entrance aper- distance over a wide range of energy loss, as illustrated in ture). Dividing by the total integral (large b) gives the Fig. 2. Exact agreement would hardly be expected: some of fraction of energy loss arising from impact parameters less the data relate to valence-electron excitation at a planar than b, as illustrated in Fig. 1b, which shows that a large interface while others represent excitation of core electrons part of the energy loss occurs at impact parameters from a single atom. Not all data use the 50% definition of considerably less than v/o. localization width and the measurements involve appreci- The classical impact parameter L ¼ o=v can be reframed able experimental error. But in view of the striking in terms of electron wave properties and made semi- correlation with Eq. (6) over a 1000:1 range of energy relativistic by using the de Broglie relation ðp ¼ gm0v ¼ loss, it appears worthwhile to explore the consequences of h=lÞ to replace the fast-electron speed v. Also, o can this relation, treating it as a useful approximation rather be taken as the angular frequency of an electron orbiting than an exact formula. an atomic nucleus, whose binding energy W is (according For single-electron transitions, Bethe theory predicts a to classical electrodynamics) exactly equal to its kinetic Lorentzian ðy2 þ y2 Þ1 angular distribution of inelastic 2 E energy and therefore given by W ¼ðm=2ÞðrnoÞ . For 1=2 scattering, but with a more rapid falloff at yc ðE=E0Þ excitation of this electron to a state just above the 1=2 ionization threshold, the absorbed energy is E W ¼ ð2yEÞ due to decrease of the oscillator strength [6,33]. nðo=2Þðh=2pÞ according to Bohr’s quantum postulate [26]. Approximating this decrease as an abrupt termination of Combined with de Broglie’s relation, these expressions the Lorentzian at y ¼ yc, the median scattering angle is 1=2 1=2 1=4 3=4 give: given [6] by: hyi¼yEðyc=yE 1Þ ðyEycÞ 2 yE , 1 leading to: L ¼ v=w ¼ hðgm0lÞ ðnh=4pEÞ¼nl=ð4pyEÞ, (4) 3=4 2 L ð0:44Þl=y . (7) where yE ¼ E=ðgm0v Þ is the characteristic scattering angle 50 E used in EELS theory [6,28]. According to Eq. (4), L depends on energy loss (through yE) but also on the quantum number n, and would therefore differ for different edges (K, L, etc.) of similar energy loss. Some experimental data have suggested that this might be true [21]. Alternatively, we can simply write E ¼ðh=2pÞo, imply- ing that the impulse field of the transmitted electron is equivalent to a white-noise spectrum of photons, to which each electron responds appropriately, resulting in: 1 L ¼ v=w ¼ hðgm0lÞ ðh=2pEÞ¼l=ð2pyEÞ. (5) A similar result (without the factor of 2p) was deduced by Howie [13], by using the Heisenberg energy-time uncertainty principle. Eq. (5) differs from Eq. (4) by being independent of the quantum number of the excited electron. But in both equations, L is seen to resemble the Rayleigh expression for the total object-plane diameter ð1:22l=aÞ of the Airy disk produced when a point object is imaged using a conical beam of waves of wavelength l and Fig. 2. Measurements and calculations of the localization distance for inelastic scattering, plotted as a function of energy loss. Where necessary, semi-angle a. Here a is several times yE but can be assumed the data has been adjusted to 100 keV incident energy. The dashed curve to be small so that sin aEa. If we define L more represents Eq. (7) combined with Eq. (9), taking b ¼ 10 mrad; the solid conveniently as the diameter L50 containing 50% of the line (for Eo30 eV) represents Eq. (10). ARTICLE IN PRESS R.F. Egerton / Ultramicroscopy 107 (2007) 575–586 579

applicable to most of the data presented. In the case of STEM or small-probe spectroscopy, the reciprocity principle can be applied to electron intensities [23], in which case b in Eq. (9) is the incident-probe semi-angle, assuming a small-angle axial detector. For energy losses below 30 eV, it may be more appropriate to consider a plasmon model for the inelastic scattering. The angular distribution is again Lorentzian but its cutoff occurs when the magnitude of the plasmon phase velocity ðo=qÞ approaches the speed vF of electrons at the Fermi surface and the cutoff angle is:

yc o=ðk0vFÞ¼2pE=ðhk0vFÞ. (10) This approximation is shown as the solid line in Fig. 2, based on values of Ep and vF for the elements K, Li, Be, Al and Si [6], and perhaps gives better agreement with the experimental data [12,32]. In fact, plasmon excitation provides an alternative way to visualize the delocalization effect, in terms of the variations in potential and charge density induced in the specimen by the transmitted electron. Any charged particle Fig. 3. Localization length (relative to the value at E0 ¼ 100 keV) as a moving through solid matter produces an oscillation in function of incident energy E0, according to Eq. (8). potential (of angular frequency op and wavelength 2pv=op) in the wake trailing the particle [36,37]. The wake angle is aE(vF/v)E10 mrad, so the wavelength measured perpen- 2 2 2 Taking yE ¼ E/(gm0v ) in Eq. (7), with g ¼ð1 v =c Þ, dicular to the particle trajectory is að2pv=opÞ L50 should depend on the incident-electron speed v as 2pvF=op 0:5 nm. The fact that this transverse periodicity follows: is independent of v is consistent with Eq. (8). Underlying Eqs. (6), (7) and (9) is the wave-optical 3=4 1 1 3=4 3=2 1=2 2 2 1=8 L50 / l=ðyEÞ / g v =ðg v Þ¼v ð1 v =c Þ . theory of imaging and Fraunhofer diffraction [38–41].If (8) the angular dependence of the scattering amplitude A (square root of the intensity dI/dO) is used as an aperture For E04100 keV, the right-hand side of Eq. (8) varies function, a point-spread function (PSF) can be calculated: only weakly with the kinetic energy E0 of the incident electron; L50 is predicted to increase by only about 10% PSF ¼ FT2ðAÞ FT2 ðAÞ, (11) from 100 to 300 keV, as shown in Fig. 3. In terms of where FT2 denotes a two-dimensional Fourier transform Eq. (6), the decrease in electron wavelength almost cancels and * represents a complex conjugate. the fall in median scattering angle above 100 keV. This As an example, Fig. 4 shows the PSF evaluated using a limited E0-dependence appears compatible with the results hydrogenic K-shell angular distribution for dI/dO, together of hydrogenic calculations of the root-mean-square impact with the fraction of intensity F(r) contained within an parameter for K-shell ionization [34]. object radius r. The diameter L50 containing 50% of the At higher energy loss, the median scattering angle can scattered intensity is about twice the full-width at half become comparable to the collection semi-angle used in maximum of the PSF, indicating that the PSF has an EFTEM imaging or the probe semi-angle used in STEM, extended tail. so that the angular distribution of scattering is partly In a Fourier-optics treatment, it is straightforward to determined by this aperture. One advantage of using the include the effect of an angle-limiting aperture or to 50% criterion is that it allows optical blurrings to be simulate the effect of an annular detector. In the latter combined with reasonable accuracy by power-law addition case, the width L50 of the PSF falls as the inner diameter [35]. Accordingly, with nearly parallel illumination and a of the detector is increased (see Fig. 5), readily exp- spectrometer collection angle b (the EFTEM case), the lained in terms of the higher spatial frequencies present in localization distance at higher E should be given approxi- the signal. A similar effect is expected on the basis of mately by classical theory but would be interpreted in terms of the ðd Þ2 ðL Þ2 þð0:52l=bÞ2 (9) reduced impact parameter associated with higher scattering 50 50 angles [13]. in which the last term represents the diffractive effect (Airy- One assumption behind Eqs. (6) and (11) is that the disk diameter) of an on-axis aperture. The dashed curve in electron waves reaching the detector are incoherent. For Fig. 2 represents Eq. (9) with b ¼ 10 mrad, which is coherent radiation, the factor in the Rayleigh formula (for ARTICLE IN PRESS 580 R.F. Egerton / Ultramicroscopy 107 (2007) 575–586

F(r) 0.8

0.6 Z=6 E = 300 eV E0 = 200 keV r.PSF hydrogenic 0.4

PSF 0.2

0 0 10 20 30 40 50 60 radial distance in units of 2 pm

Fig. 4. Point-spread function for inelastic scattering (PSF) and the scattered intensity per unit radius (dI/dr), both with arbitary vertical scale, Fig. 5. Localization distance as a function of the internal diameter of an together with the fraction F(r) of intensity within a radius r (the horizontal annular detector, for an external diameter corresponding to 45 mrad axis, in which each numerical unit represents 2 pm). scattering angle. The calculation assumes parallel incident illumination and incoherent inelastic scattering.

PSF radius) changes from 0.61 to 0.82 if certain assump- inelastic scattering is well below the Bragg angle), and at tions are made [40] or becomes 0.5 or 1.0 under particular high energy loss where it becomes a fraction of the atomic conditions [41]. In the case of inelastic (plasmon) scattering diameter. This latter case corresponds to the use of a of electrons, the degree of coherence of the scattered waves delocalization distance in the analysis of channeling has been measured as 0.3 [42,43], meaning 70% incoher- experiments [46–49]. ence. Therefore an assumption of incoherence in the scattered waves is unlikely to lead to a large error in the estimate of delocalization. 2.5. Radiation damage A related concept is the lateral coherence length associated with inelastic scattering. Holographic measure- Another fundamental limit to spatial resolution is set by ments suggest that this quantity is also given by Eq. (7) for radiation damage to the specimen. In order to obtain a energy losses in the range 50–400 eV [44]. In a recent statistically meaningful spectrum or image, a large enough theoretical treatment of bulk plasmon losses [45], number of electrons must be recorded. But as the electron the coherence length appears closely linked to the dose D (usually measured in C/cm2) approaches some angular distribution of plasmon scattering. If so, all critical value Dc, the specimen undergoes damage and the information relating to delocalization and lateral coherence measurement no longer reﬂects the original properties of is contained within the angular distribution of inelastic the specimen. scattering. The value of Dc depends on specimen composition, The beam-broadening effect (discussed earlier) differs specimen temperature and the incident-electron energy. from inelastic delocalization in that it increases with Organic specimens are the most beam-sensitive and they specimen thickness. For an amorphous specimen, the two damage mainly by radiolysis, caused by electronic excita- broadening effects could be combined (for example) in tions induced by the inelastic scattering. This damage is quadrature. In a crystalline specimen that is not extremely manifest as mass loss (specimen thinning or hole forma- thin, the elastic and inelastic scattering are more closely tion) and/or structural change, including loss of crystal- coupled and require a more rigorous mathematical linity. Radiolysis also occurs in most inorganic compounds treatment [21,22]. Yet even here, there appear to be but the value of Dc is usually much larger. Metallic and situations in which the concept of a thickness-independent semi-metallic specimens may be immune to radiolysis but delocalization distance remains useful: for low-loss spectro- are expected to undergo mass loss (for higher incident scopy or imaging, where the delocalization length is large energies and sufﬁciently high electron dose) as a result of compared to interatomic distances (the angular width of electron-beam sputtering. ARTICLE IN PRESS R.F. Egerton / Ultramicroscopy 107 (2007) 575–586 581

Table 1 shows approximate values of Dc for the three ionization damage (radiolysis) should occur largely types of specimen. The radiolysis doses are based on through the excitation of inner shells, with small cross experimental data published in several review papers section but large excitation energy, so that the deposited [50–52]. The dose range for hole formation due to energy can be larger than for the valence electrons. As a sputtering (in a 20 nm specimen) is based on calculations result, the damage should be mainly associated with small for elements of atomic number below 40, assuming a impact parameters and in the aloof mode of EELS [54] a planar surface potential [53]. highly focused electron beam placed very close to but just The electron dose D needed for EELS depends greatly on outside the specimen should cause less damage than a beam the type of measurement involved. Valence-loss spectro- that penetrates the specimen [55]. Unfortunately this scopy or imaging require relatively low dose, the intensity argument fails for light elements such as carbon, where being relatively high in the low-loss region of the spectrum. the K-shell stopping power is only a few percent of that of Probably the application that demands the least dose is a the valence electrons. measurement of local thickness via the formula t=l ¼ lnðI t=I 0Þ, which uses only the integrated total and zero-loss intensities It and I0. For t=l51, this formula 3. Energy resolution becomes t=lp I p=I 0 (where Ip and lp are the plasmon intensity and mean free path) and the largest uncertainty is The energy resolution of an EEL spectrum or energy- in the measurement of Ip. For 10% accuracy, 0:1 ¼ Dt=t ¼ filtered image is influenced by several instrumental factors: 1=2 DI p=I p and taking the statistical error in Ip as DI p ¼ðI pÞ energy width of the electron source, energy broadening gives IpE100 electrons and I0E1000 electrons for t=lp ¼ (Boersch effect) arising from electron interaction in the 0:1 (e.g., tE20 nm, lpE200 nm). Ignoring elastic scatter- illumination system, electron-optical design of the electron ing, the resolution limit dmin imposed by radiation damage spectrometer, and stability of the high-voltage and spectro- 2 is given by I 0 DcðdminÞ , resulting in the dmin values given meter power supplies. There are also environmental in column 4 of Table 1. These estimates indicate that factors: level of AC magnetic fields and mechanical radiation damage need not be a limitation for valence-loss vibrations, although their effects can be minimized by spectroscopy, although measurement of low-loss fine good spectrometer design. structure would imply higher dose and worse resolution Whereas thermionic (LaB6 and W-filament) sources than given in column 4. provide an energy spread in the range 1–2 eV, a Schott- Because of the relatively low inner-shell ionization ky-emission source gives typically 0.7 eV and a cold field- cross sections, core-loss spectroscopy and elemental map- emission (CFEG) source can give 0.5 eV, or even as low as ping demand higher radiation dose. The resolution 0.27 eV with fast data recording [56]. The use of a gun estimates in the final column of Table 1 relate to a monochromator (a spectrometer with a narrow energy- measurement (with signal/noise ¼ 5) of 10% of calcium in selecting slit) in the illumination system can reduce the a 20 nm specimen, via the L-edge at 350 eV [6, p. 355]. energy spread to below 0.2 eV, with some loss of intensity Radiolysis is seen to be the main factor limiting the spatial [57,58]. This type of monochromator operates at close to resolution of core-loss analysis of organic (and some the gun potential and disperses the electrons before they inorganic) specimens. are accelerated, resulting in relatively high dispersion. A From Table 1, electron-beam sputtering appears to be disadvantage of the gun monochromator is that its unimportant; the dose involved is large, allowing resolution monochromation is relative to the gun potential: any down to atomic dimensions. However, the situation would change DV in the high voltage supply results in an equal be less favorable for ionization-edge energies above 350 eV, change (DV eV) in kinetic energy of the accelerated so in some circumstances sputtering could set a practical electrons. limit to the spatial resolution. The highest energy resolutions (in the range 2–100 meV) As a final comment on radiation damage, it has been have been obtained by combining these relative mono- noted that for atoms of medium or high atomic number, chromators with relative spectrometer systems; the same high voltage is used to retard the electrons when they are analyzed, so that high-voltage fluctuations are compen- Table 1 Limits to spatial resolution set by radiation damage, for t ¼ 20 nm and sated [59–62]. Such an arrangement has the disadvantage E ¼ 200 keV of requiring high potential to be applied to the post- specimen portion of the TEM column. However, it would Damage Type of Damage Resolution Resolution greatly relax the stability needed for the spectrometer and mechanism specimen dose Dc (C/ (nm) for (nm) for cm2) valence- 350 eV- microscope power supplies and might make possible the EELS EELS analysis of vibrational and phonon-mode features (which mostly occur at energy losses below 0.1 eV) combined with Radiolysis Organic 103–1 5 104–0.5 1–300 6 the high spatial resolution. The deployment of aberration Radiolysis Inorganic 0.1–10 o0.05 0.01–30 correctors could make the use of lower acceleration Sputtering Metallic 2000–40 000 o106 o0.2 voltages more attractive. ARTICLE IN PRESS 582 R.F. Egerton / Ultramicroscopy 107 (2007) 575–586

Energy broadening due to electron interaction within the TEM column (e.g. at beam crossovers) appears to be negligible at typical beam currents. The only signiﬁcant effect may be in the electron gun itself, where the electrons move slowly, which presumably explains why the energy width of an electron source increases with emission current. As an alternative (or in addition) to the use of a monochromator, the resolution in an energy-loss spectrum can be improved by a useful factor through deconvolution, using either Fourier [6,63,94] or Bayesian techniques [64–66]. In both cases, the maximum resolution enhance- ment is determined by spectral noise, which (in the absence of instrumental instability) is limited by radiation damage to the specimen. In theory, maximum-entropy deconvolution offers a resolution improvement several times larger than Fourier processing, although it has to be used with care to avoid introducing spectral artifacts [66]. Fine structure present in an energy-loss spectrum is closely related to the energy dependence of a density of states (DOS) of the material being investigated. For valence-electron losses, this implies a convolution of the Fig. 6. Width of the K, L3 and L2 levels according to Krause and Oliver [67]; the solid lines show the parameterization represented by conduction- and valence-band DOS, including gap states if Eqs. (11)–(13). Triangles are data provided by Brown [70]. the energy resolution is good enough. In the case of an ionization edge, the background-subtracted spectrum reﬂects the density of unoccupied states above the Fermi level, multiplied by an atomic matrix element (assumed to vary slowly with energy) and convolved with a function that represents the energy width of the core level. The width of this function (essentially the natural linewidth) has been calculated for K and L shells [67] and the results can be parameterized as follows:

0:472 DEK ¼ 0:0216ðEÞ 0:285, (12)

4 2 DEL3 ¼ð6:9 10 ÞE 0:14 exp½ðE 680Þ=400 0:05 exp½ðE 30Þ=902, ð13Þ

4 2 DEL2 ¼ð7:4 10 ÞE 0:16 exp½ðE 690Þ=400 0:05 exp½ðE 30Þ=902, ð14Þ where E is the appropriate threshold energy in eV. The L-widths are roughly proportional to threshold energy, the Fig. 7. Final-state broadening for the Al L-edge, from measurements [71], theoretical predictions [76,77] and as given in Eqs. (15) and (16) with L2 width being slightly higher than that of the L3 level because Coster–Kronig transitions provide an addi- m ¼ m0 ¼ free-electron mass (solid line) and with m ¼ effective mass of Al ¼ 1.096m (dashed line). tional decay channel, decreasing the core-hole lifetime 0 in a solid. However, some data [68–70] differ substantially from the Krause and Oliver values [67], as illustrated in and up to an energy loss E ¼ [71–75]. This life- Fig. 6. time applies to an energy loss e above the ionization edge This core-level width can be viewed as lifetime broad- and the corresponding energy broadening DE can be ening of the initial state, but there is also ﬁnal-state estimated from the uncertainty principle: DE:t h=ð2pÞ. broadening, representing the fact that the excited electron Recent measurements for aluminum [71,72] are shown in has a ﬁnite lifetime. This electron is subject to the same Fig. 7. inelastic scattering processes as a high-energy transmitted Alternatively, the lifetime of the excited electron (energy electron, so its lifetime t(e) and inelastic mean free path l(e) e and speed v) can be taken as t l=v, where v ¼ð2 mÞ1=2 (for a given starting energy e) can be estimated by according to a free-electron approximation, m being the integrating the low-loss spectrum over all scattering angles electron mass, giving: ARTICLE IN PRESS R.F. Egerton / Ultramicroscopy 107 (2007) 575–586 583

exploited as a means of exploring the mechanisms of electron scattering in solids [79–88]. In the low-loss region, surface plasmons, bulk plasmons and radiation-loss processes are found to have distinctly different dispersion ðo qÞ characteristics, so dispersion (or lack of it) can help to identify spectral peaks, even in amorphous materials [89]. The q-dependence can also be used to investigate the band structure of crystalline materials, especially when combined with measurements made at different sample orientations [86,88]. For this purpose, the q-resolution should correspond to some small fraction of the Brillouin zone. In the case of anisotropic materials, measurement of the angular dependence of scattering can provide useful information; a good example is provided by energy-filtered diffraction patterns of graphite, taken above and below the carbon K-edge, which show up the directionality of bonding [90]. To provide a standard ‘‘fingerprint’’, a collection aperture can be chosen such that the spectrum is independent of orientation, although the ‘‘magic angle’’ (angular range) required is rather small. A recent discovery, following up on earlier work [91], is that the Fig. 8. Final-state broadening for ionization edges of Al, Ag, Au and Cu, according to Eqs. (15) and (16). relativistic effects that occur at higher beam energies must be reformulated to correctly account for electron scattering in anisotropic materials [28,92]. DE ðh=2pÞ=t ¼ hð2plÞ1ð2=mÞ1=2. (15) It is easy to see that there is a basic-physics limit to the angular or momentum resolution and that this is linked The slow-electron mean free path lðÞ has been measured to the spatial resolution of the measurement. High from photoelectron spectroscopy and the results for several 4 spatial resolution can be achieved by using a focused elements have been parameterized (for 1 eVoeo10 eV) in probe (Fig. 9a) but a fundamental limit to probe size is set the form [78]: by diffraction at the probe-lens aperture and is given by the l ðnmÞ¼538a2 þ 0:41a3=21=2, (16) Rayleigh criterion: Dx ð0:5Þl=a. Increasing the aperture semi-angle a should improve the resolution, although in where e is in eV and a is the atomic diameter (thickness 3 most instruments spherical aberration makes things worse of a monolayer) in nm, given by: a ¼ A=ð602rÞ, where A for a greater than about 10 mrad. In an aberration- is the atomic mass number and r the specific gravity of 3 corrected instrument, a can increased beyond this value the specimen (the density in g/cm ). The result of but any appreciable spread in incident-electron direction applying Eqs. (15) and (16) to aluminum is shown in introduces a spread Dq ¼ð2p=lÞð2DaÞ in the scattering Fig. (7) and is seen to be in reasonable agreement with vector q. For band-structure studies, Dq needs to be a small the experimental data. For e460 eV, the second term in fraction of the wavevector qB that corresponds to the first Eq. (16) is dominant and exactly cancels the e-dependence Brillouin-zone boundary, equivalent to diffraction in Eq. (14), giving: (through an angle 2yB) by atomic planes of spacing dhkl; 3=2 see Fig. 9a. Using the Rayleigh criterion and Bragg’s law DEmax ð0:93 eVÞa . (17) ðl ¼ 2dhklyBÞ, the fractional momentum resolution can be In other words, the final-state broadening is predicted to expressed as: reach a limit beyond about 60 eV from the edge threshold. Using data from several elements, this limit appears to be Dq=qB ¼ð2aÞ=ð2yBÞ¼ðl=DxÞðdhkl=lÞ¼dhkl=Dx. (18) between 7 and 9 eV, independent of atomic number; see For 5% momentum resolution, the spatial resolution is Fig. 8. therefore limited to Dx ¼ 20dhkl, amounting to 2 nm or more in a typical material. These considerations become 4. Momentum resolution limiting when determining the electronic structure of nanowires [89]. Note that Eq. (18), which is equivalent to To extract all of the information available from EELS, use of the Heisenberg uncertainty principle ðDpDx hÞ in the spectrum must be recorded as a function of scattering the specimen plane, is independent of source brightness and angle y, yielding the dependence of the scattering cross accelerating voltage. 2 2 1=2 section on scattering vector q k0ðy þ yEÞ or trans- Midgley [93] has described a method for obtaining ferred momentum ðh=2pÞq. This capability has been angular resolution down to a few microradians by ARTICLE IN PRESS 584 R.F. Egerton / Ultramicroscopy 107 (2007) 575–586

a operating the TEM in image mode. The electron probe is focused at the usual eucentric plane but the specimen is raised by a distance z,asinFig. 9b. Because the probe is out of focus at the specimen, the spatial resolution is no better than 2az; see Fig. 9b. From geometry at the eucentric plane, the angular resolution is limited to Dy ¼ð2yBÞ d ½Dx=ðz2yBÞ ¼ Dx=z ¼ð0:5Þl=ðazÞ¼dhklyB=ðazÞ and the fractional angular or momentum resolution is: Dy=2yB ¼ specimen dhkl=Dx, in agreement with Eq. (18).

θ 2 B 5. Conclusions

The spatial resolution of EELS or energy-filtered imaging depends on the mechanical and electric stability of the instrumentation, and on the design of the electron optics and electron source. But as these aspects are improved, more fundamental limits become significant. Elastic scattering in the specimen causes beam spreading, especially in thicker specimens, degrading the spatial resolution of the inelastic signal. Inelastic scattering itself is somewhat delocalized, especially at lower energy loss. On a classical (particle) picture of scattering, this delocalization reflects the long-range nature of the Coulomb potential. Using a wave-optical description, the degree of delocalization depends on the angular distribution of the electrons reaching the detector. In beam-sensitive speci- diffraction mens, radiation damage imposes a more serious practical plane limit to resolution, especially at high-energy loss where the 2α scattered signal is weak. Energy resolution depends on the design and stability of the microscope and spectrometer, the type of electron source and whether a monochromator is used. However, qBZ the resolution of core-loss fine structure is considerably affected by the core-level width (initial-state broadening) b and scattering of the excited electron (final-state broadening). To provide additional information about the specimen, 2zα specimen the momentum resolution of EELS can be increased, but at the expense of spatial resolution. Again this limit follows from the wavelike nature of the electron, as embodied in the Heisenberg uncertainty relation or the Rayleigh criterion for resolution.

Acknowledgments

z The author appreciates discussions with Archie Howie, John Spence, Chris Allen, Steve Pennycook, M. Terauchi, Qian Li, Paul Midgley and Marek Malac. Funding from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.

object References d plane Fig. 9. Momentum resolution achieved (a) with an aperture in a [1] M. Haider, S. Uhlemann, E. Schwan, H. Rose, B. Kabius, K. Urban, diffraction plane and an electron probe (diameter d) focused on the Nature 392 (1998) 768. specimen, and (b) with lenses focused on an object plane and the TEM [2] O.L. Krivanek, N. Dellby, A.R. Lupini, Ultramicroscopy 78 (1999) 1. specimen raised a distance z above this plane [93]. [3] P.E. Batson, N. Dellby, O.L. Krivanek, Nature 418 (2002) 617. ARTICLE IN PRESS R.F. Egerton / Ultramicroscopy 107 (2007) 575–586 585

[4] P. Kruit, G.H. Jansen, Space charge and statistical Coulomb effects, [47] J. Tafto, O.L. Krivanek, Nucl. Instrum. Methods 194 (1982) 153. in: J. Orloff (Ed.), Handbook of Charged Particle Optics, CRC Press, [48] S.J. Pennycook, Ultramicrosopy 26 (1988) 239. Boca Raton, 1997, pp. 275–318 (Chapter 7). [49] W. Qian, B. Totdal, R. Hoier, J.C.H. Spence, Ultramicroscopy 41 [5] J.I. Goldstein, in: J.J. Hren, J.I. Goldstein, D.C. Joy (Eds.), (1992) 147. Introduction to Analytical Electron Microscopy, Plenum, New York, [50] M.S. Isaacson, in: Principles and Techniques of Electron Microscopy, 1979 p. 101. vol. 7, Van Nostrand, New York, 1977 p. 1. [6] R.F. Egerton, Electron Energy-Loss Spectroscopy in the Electron [51] L.W. Hobbs, in: J.J. Hren, J.I. Goldstein, D.C. Joy (Eds.), Microscope, second ed, Plenum, New York, 1996. Introduction to Analytical Electron Microscopy, Plenum, New York, [7] W.A. Furdanowicz, A.J. Garratt-Reed, J.B. Vander Sande, in: 1987 p. 399. Proceedings of EMAG 1991 (IOP Conference Series No. 119, [52] R.F. Egerton, P. Li, M. Malac, Micron 35 (2004) 399. Institute of Physics, Bristol), p. 437. [53] R.F. Egerton, F. Wang, P. Crozier, Microscopy and Microanalysis 12 [8] Y. Sato, M. Ichihashi, Y. Ueki, Microbeam Anal. 3 (1994) 293. (2006) 65. [9] R.F. Loane, E.J. Kirkland, J. Silcox, Acta. Cryst. A 44 (1988) 912. [54] L.D. Marks, Solid State Communications 43 (1982) 727. [10] R.F. Egerton, P.A. Crozier, Micron 28 (1997) 117. [55] M.G. Walls, A. Howie, Ultramicroscopy 28 (1989) 40. [11] M. Isaacson, J.P. Langmore, H. Rose, Optik 41 (1974) 92. [56] K. Kimoto, G. Kothleitner, W. Grogger, Y. Matsui, F. Hofer, [12] D.A. Muller, J. Silcox, Ultramicroscopy 59 (1995) 195. Micron 36 (2005) 185. [13] A. Howie, J. Microsc. 117 (1979) 11. [57] M. Mukai, T. Kaneyama, T. Tomita, K. Tsuno, M. Terauchi, K. [14] R.H. Ritchie, Philos. Mag. A 44 (1981) 931. Tsuda, M. Naruse, T. Honda, M. Tanaka, Microsc. Microanal. 11 [15] H. Kohl, H. Rose, Adv. Electron. Electron. Phys. 65 (1985) 175. (Suppl. 2) (2005) 2134. [16] P.E. Batson, Ultramicroscopy 47 (1992) 133. [58] P.C. Tiemeijer, Ultramicroscopy 78 (1999) 53. [17] M. Schenner, P. Schattschneider, Ultramicroscopy 55 (1994) 31. [59] H. Boersch, J. Geiger, W. Stickel, Zeitschrift fur Physik 180 (1964) [18] B. Rafferty, S.J. Pennycook, Ultramicroscopy 78 (1999) 141. 415. [19] D.W. Essex, P.D. Nellist, C.T. Whelan, Ultramicroscopy 80 (1999) [60] W.H.J. Anderson, J.B. LePoole, J. Phys. E 3 (1970) 121. 183. [61] M. Terauchi, M. Tanaka, K. Tsuno, M. Ishida, J. Microsc. 194 [20] P. Schattschneider, B. Jouffrey, Ultramicroscopy 96 (2003) (1999) 203. 453. [62] P.E. Batson, Rev. Sci. Instrum. 57 (1985) 43. [21] E.C. Cosgriff, M.P. Oxley, L.J. Allen, S.J. Pennycook, Ultramicro- [63] R.F. Egerton, P.A. Crozier, Scanning Microscopy Supplement 2, in: scopy 102 (2005) 317. J. Fairing (Ed.), Proceedings of the Sixth Pfefferkorn Conference on [22] M.P. Oxley, E.C. Cosgriff, L.J. Allen, Phys. Rev. Lett. 94 (2005) Image and Signal Processing, Scanning Microscopy International, 203906. AMF O’Hare, Chicago, pp. 245–254. [23] J.C.H. Spence, Rep. Prog. Phys. 69 (2006) 725. [64] M.H.F. Overwijk, D. Reefman, Micron 31 (2000) 325. [24] J.C.H. Spence, J. Lynch, Ultramicroscopy 9 (1982) 267. [65] A. Gloter, A. Douri, M. Tence, C. Colliex, Ultramicroscopy 96 [25] N. Bohr, Philos. Mag. 25 (1913) 10. (2003) 385. [26] N. Bohr, Philos. Mag. 25 (1913) 1. [66] R.F. Egerton, H. Qian, M. Malac, Micron 37 (2006) 310. [27] J.D. Jackson, Classical Electrodynamics, second ed, Wiley, [67] M.O. Krause, J.H. Oliver, J. Phys. Chem. Ref. Data 8 (1979) 329. New York, 1975 (Chapter 13). [68] C. Mitterbauer, G. Kothleitner, W. Grogger, H. Zandbergen, B. [28] P. Schattschneider, C. Hebert, H. Franco, B. Jouffrey, Phys. Rev. B Freitag, P. Tiemeijer, F. Hofer, Ultramicroscopy 96 (2003) 469. 72 (2005) 045142. [69] J.C. Fuggle, S.F. Alvarado, Phys. Rev. A 22 (1980) 1615. [29] K.M. Adamson-Sharpe, F.P. Ottensmeyer, J. Microsc. 122 (1981) [70] F.C. Brown, Solid State Phys. 29 (1974) 1. 309. [71] C. Hebert, Microscopy and Microanalysis 12 (Suppl. 2) (2006) [30] H. Shuman, C.-F. Chang, A.P. Somlyo, Ultramicroscopy 19 (1986) 1168. 121. [72] C. Hebert, Micron 38 (2007) 12. [31] C. Mory, H. Kohl, M. Tence, C. Colliex, Ultramicroscopy 37 (1991) [73] S. Luo, X. Zhang, D.C. Joy, Radiat. Eff. Def. Solids 117 (1991) 191. 235. [32] K. van Benthem, R.H. French, W. Sigle, C. Elsasser, M. Ruhle, [74] D.C. Joy, Microsc. Microanal. 7 (2001) 159. Ultramicroscopy 86 (2001) 303. [75] D.C. Joy, S. Luo, R. Gauvin, P. Hovington, N. Evans, Scanning [33] M. Inokuti, Rev. Mod. Phys. 43 (1971) 297. Microsc. 10 (1996) 653. [34] L.J. Allen, C.J. Rossouw, Phys. Rev. B 42 (1990) 11644. [76] D.A. Muller, D.J. Singh, J. Silcox, Phys. Rev. B 57 (1998) 8181. [35] J.R. Barth, P. Kruit, Optik 101 (1996) 101. [77] P.J.W. Weijs, M.T. Czyzyk, J.F. van Acker, W. Speier, G. van der [36] P.M. Echenique, R.H. Ritchie, W. Brandt, Phys. Rev. B 20 (1979) Laan, K.H.J. Buschow, G. Wiech, J.C. Fuggle, Phys. Rev. B 41 2567. (1990) 11899. [37] P.E. Batson, J. Bruley, Phys. Rev. Lett. 67 (1991) 350. [78] M.P. Seah, W.A. Dench, Surf. Interface Anal. 1 (1979) 2. [38] J.W. Goodman, Introduction to Fourier Optics, McGaw-Hill, [79] H. Raether, Excitation of Plasmons and Interband Transitions by New York, 1968. Electrons. Springer Tracts in Modern Physics, vol. 88, Springer, [39] R.G. Wilson, Fourier Series and Optical Transform Techniques in Berlin, 1980. Contemporary Optics, Wiley, New York, 1995. [80] P.E. Batson, J. Silcox, Phys. Rev. 27 (1983) 5224. [40] M. Born, E. Wolf, Principles of Optics, seventh ed, Cambridge [81] G.A. Botton, C.B. Boothroyd, W.M. Stobbs, Ultramicroscopy 59 University Press, Cambridge, 2001, p. 471. (1995) 93. [41] E.G. Steward, Fourier Optics: An Introduction, Wiley, Chichester, [82] Y.Y. Yang, S.C. Cheng, V.P. Dravid, Micron 30 (1999) 379. 1983, p. 93. [83] R.D. Leapman, P.L. Fejes, J. Silcox, Phys. Rev. B 28 (1983) 2361. [42] R.A. Herring, Ultramicroscopy 104 (2005) 261. [84] R.F. Klie, Y. Zhu, G. Schneider, J. Tafto, Appl. Phys. Lett. 82 (2003) [43] R.A. Herring, Ultramicroscopy 106 (2006) 960. 4316. [44] K. Kimoto, Y. Matsui, Ultramicroscopy 96 (2005) 335 (see [85] P. Schattschneider, J. Fidler, V. Chopov, J. Electron Spectrosc. Rel. Fig. 8). Phen. 31 (1983) 25. [45] J. Verbeek, D. van Dyck, H. Lichte, P. Potapov, P. Schattschneider, [86] B. Rafferty, L.M. Brown, Phys. Rev. B 58 (1998) 10326. Ultramicroscopy 102 (2005) 239. [87] J. Fink, Adv. Electron. Electron Phys. 75 (1989) 121. [46] A.J. Bourdillon, P.G. Self, W.M. Stobbs, Philos. Mag. A 44 (1981) [88] N.D. Browning, J. Yuan, L.M. Brown, Ultramicroscopy 38 (1991) 1335. 291. ARTICLE IN PRESS 586 R.F. Egerton / Ultramicroscopy 107 (2007) 575–586

[89] Juan Wang, Quan Li, R.F. Egerton. Probing the electronic structure [91] C.H. Chen, J. Silcox, Phys. Rev. B 20 (1979) 3605. of ZnO nanowires by valence electron energy loss spectroscopy, [92] Y. Sun, J. Yuan, Phys. Rev. B 71 (2005) 125109. Micron (2006), in press, doi:10.1016/j.micron.2006.06.003. [93] P.A. Midgley, Ultramicroscopy 76 (1999) 91. [90] G.A. Botton, J. Electron Spectrosc. Related Phenomena 143 (2005) [94] P.E. Batson, D.W. Johnson, J.C.H. Spence, Ultramicroscopy 41 129. (1992) 137.