To be published in Ultramicroscopy (Gert Rempfer memorial issue)

TEM-EELS: a personal perspective R.F. Egerton Physics Department, University of Alberta, Edmonton, Canada T6G 2G7 Abstract The development of electron energy-loss spectroscopy in a transmission electron microscope (TEM-EELS) is illustrated through personal anecdote, highlighting the basic principles, instrumentation and personalities involved. The current state of the art is also reviewed, together with some challenges for the future. 1. Introduction The scientific career of Gertrude Rempfer (1912-2011) spans the whole era of the electron microscope, including those early days when it was unclear whether such instruments should be based on electrostatic or electromagnetic lenses. Gert designed and built electron lenses and aberration-correction systems, tested them in the laboratory and constructed entire instruments (transmission and photoelectron microscopes) that incorporated her ideas. Her preference was always for electrostatic lenses, a choice that the TEM community abandoned during its search for higher resolution and higher operating voltage. However 30 kV is sufficient to accelerate the electrons in a photoelectron microscope, where electrostatic lenses readily provide an image with a resolution approaching 7 nm [1]. Gert’s analysis [2] of the hyperbolic electron mirror as a corrector for both spherical and chromatic aberrations eventually led to a PEEM at Portland State University (using a simple UV lamp as the excitation source) with a measured resolution of 5.4 nm [3].

My own contribution to electron microscopy has been more circumscribed: mainly analytical TEM and electron energy-loss spectroscopy. The account that follows stems from an after-dinner talk at the Banff EDGE meeting in 2009, which I had planned to share in written form with Gert on her 100th birthday but must now, with sadness, dedicate to her memory.

I first encountered the transmission electron microscope (a JEOL JEM-7) as a graduate student at Imperial College (London University), during research on vacuum-evaporated thin films of the lead chalcogenides (PbTe, PbSe, PbS). After being shown how to prepare shadowed carbon replicas and remove them from the mica substrate, I was given a supervised TEM session, during which we happened to look at the edge of a film that had been evaporated through an out-of-contact mask, where the thickness decreased to zero over a fraction of a millimeter. That tiny area of specimen revealed the entire growth history of the film: its nucleation, island growth, coalescence and subsequent grain growth; see Fig. 1. The result was an extra chapter in my PhD thesis and a lifelong interest in electron microscopy in general.

Fig. 1. (a) JEOL JEM-7A transmission electron microscope and (b) micrographs of Pt-shadowed carbon replicas, showing the growth sequence of an epitaxial thin film of PbTe [4]. This thickness-gradient trick was later used (with two evaporation sources) to fabricate thin films of binary alloys [5]. After graduating, I joined the UK Research Labs of Zenith Radio Corporation, where PbTe was being investigated as the semiconductor for a thin-film transistor, to be incorporated in a flat-panel TV screen. It took Japanese engineers a further decade to fabricate reliable flat-panel displays, using different materials. In fact, Japanese competition led to the closure of the ZRRC lab in 1973 and then to the demise of Zenith’s TV production in Chicago. 2. Oxford EELS Misreading an advertisement in the New Scientist, I applied in 1972 for a postdoc position in Oxford. Strangely, I read “energy-selecting electron microscope” as “scanning microscope” and it was not until my interview that I found out the energy analysis (now referred to as EELS) measures the various changes in energy of an electrons as they travel though a thin specimen, in order to provide new information about chemical composition for example.

At that time, the Metallurgy Department at Oxford University was a breeding ground for past and future electron microscopists; see Fig. 2. A few years earlier, Mike Whelan and Peter Hirsch had observed the first images of dislocations in metals and, with Archie Howie at Cambridge, had adapted x-ray diffraction theory to provide a detailed description of diffraction-contrast TEM images. Together with miscellaneous practical advice, this theory was documented in the “yellow bible”, so called because the book (as

Fig. 2. Oxford University Metallurgy Department group photograph (1975) with some of the microscopists identified. The Department head (Prof. P.B. Hirsch) is absent from this photo, his place being taken by Prof. Jack Christian.

originally published by Butterworths) had a yellow dust cover [6]. In the Metallurgy annex at 10 Parks Road (Fig. 3), I shared an office with David Cockayne and Ian Ray, who were developing the weak-beam technique for examining the structure of dislocations. Ian’s desk was later taken over by John Spence, who arrived from Australia after completing PhD research that involved the deconvolution of electron energy-loss spectra.

Fig. 3. Red-brick annex of the Oxford Metallurgy Department and (to the right) the long shed that housed electron microscopes in the 1970’s.

The Oxford electron microscopes were housed in a long concrete shed adjacent to the Parks Road annex. They included a JEM 100B for high-resolution studies and a Philips 300 for analytical work. There was also an energy-analyzing microscope: an old Siemens-1a with a Möllenstedt electrostatic spectrometer added below the TEM screen. In this analyzer [7], electrons enter through a narrow slit and then pass between two rod-shaped electrodes, aligned parallel to the slit and connected to the TEM high-voltage supply. When the slit is moved slightly off-axis, electrons are deflected away from the nearer electrode by an amount dependent on their kinetic energy, equivalent to a lens with high chromatic aberration. The resulting energy-loss spectra were viewed on a second TEM screen below the analyzer, or else recorded on glass plates coated with photographic emulsion (Fig. 4a) before being digitized using a flat-bed scanner. The Möllenstedt analyzer is non-focusing in a direction parallel to the entrance slit, so the photographic plate contained spectral information as a function of position in the specimen or in reciprocal space (allowing the angular dependence of scattering to be measured). At the Hitachi Central Research Laboratory in Tokyo, Hiroshi Watanabe had already used a Möllenstedt analyzer to measure characteristic energy losses [8] and ascribed some of them to a plasma-resonance effect based on their angular dispersion [9].

Fig. 4. (a) Energy-analyzing TEM, showing the entrance slit (B) of a Möllenstedt analyzer (D) and a second viewing chamber (F) with photographic-plate camera (G). The lens controls of the Siemens-1A were moved to a separate rack (T). (b) Energy-filtering TEM at Oxford, with double-prism spectrometer (P) and electrostatic mirror (M). (c) Cavendish-lab TEM with a single-prism spectrometer below the viewing chamber; spectra were displayed using a Tektronix oscilloscope or the pen plotter on the left.

At the far end of the corridor, across the street from Oxford’s Keble College, was an energy-filtering electron microscope that incorporated the prism-mirror system developed by Prof. Raymond Castaing and Lucien Henry at the University of Paris [10]. In the Castaing-Henry filter, the electron beam is bent through 90° by a magnetic field

generated between prism-shaped polepieces, then reflected from an electrostatic mirror: a planar electrode connected to the electron-gun potential. It re-enters the prism and emerges downwards, maintaining the vertical alignment of the TEM column. The Oxford version was based on an EM6 microscope donated by the AEI company, identifiable in Fig. 4b from the analog clock to the right of the column. The manufacturer had taken care to shield this accessory, to prevent its magnetic field from influencing the electrons; before the advent of quartz crystals, electric clocks made use of the ac-mains frequency, accurately maintained at 50 Hz in the UK. The EM6 had circuit diagrams that were easy to follow because its analog electronics was based on vacuum tubes, with no transistors or (heaven forbid!) integrated circuits. The EFTEM had been assembled by Peter Turner in the main Metallurgy building and then moved a few blocks south, its high-voltage cables transported down Parks Road on the shoulders of graduate students. Unfortunately this move resulted in high-voltage instability, which graduate student John Philip and I eventually traced to dirty pressure contacts in the oil-filled high-voltage tank. There was also a HV splitter tank containing dry-cell batteries for biasing the electrostatic mirror, everything connected by long HV cables. Even after repair, the apparatus was unreliable at 100 kV; discharge anywhere in the system caused high-voltage pulses to run along the cables and create sparks between nearby metal objects, even if they were nominally grounded. So for peace of mind, we usually operated at 80 kV, giving the impression to some readers of our publications that 80 kV was somehow special for EELS. Although the EFTEM was built to study plasmon-loss images, I became interested in inner-shell losses and, during our excursion into the oil tank, took the opportunity to install a chain of Zener diodes. Using a motor-driven rotary switch, the gun potential could then be raised in steps corresponding to the ionization thresholds of the light elements (Be – F), a feature appreciated by my successor in Oxford, Richard Leapman, who went on to pioneer the possibilities of core-loss EELS in biology. Inner-shell spectroscopy also benefited from the installation of an electron-counting system based on a plastic scintillator and photomultiplier tube. Photographic plates were still used for recording energy-loss fine structure or, in the diffraction plane, the Bethe ridge [11]. I knew the latter corresponded to close collisions from valence electrons but it took a visit from Bernard Jouffrey to point out Mitio Inokuti’s recent review of Bethe theory, which explained everything [12]. 3. Other-Place EELS Sometime in 1972, I took a day trip to Cambridge University, where Dr. V.E. Cosslett supervised an energy-loss project at the old Cavendish laboratory in Free School Lane. They were also using an AEI-EM6, to which David Wittry had added a single-prism magnetic spectrometer with 45° deflection angle; see Fig.4c. This spectrometer was mounted below the TEM camera chamber and employed the projector-lens crossover as its object point, the arrangement commonly used today. Towards the end of my visit, Dr. Cosslett remarked that direct competition between our two groups would be avoided if they continued work on core losses while we in Oxford concentrated on plasmon

spectroscopy. When I reported this comment to Mike Whelan, his suggestion was to ignore it, advice that I never regretted. In any event, Dr. Cosslett was very gracious to my wife and myself during summer visits to Cambridge several years later. There were times when progress with the EFTEM seemed painfully slow and at such times I envied David Joy, who was working in another Metallurgy building on “Project 6”, a field-emission STEM inspired by Albert Crewe’s achievments in Chicago. David encouraged me to persevere with EELS and after he left to join Bell Laboratories, I built an electron spectrometer for the STEM and Charlie Lyman installed an EDX system, based on an early Link-systems computer.

After reading pioneering energy-loss publications by French researchers, including Colliex and Jouffrey [13], I wanted to meet some of these messieurs (Fig. 5). So I applied to the Royal Society of London for a travel grant, then booked space for myself and my small Morris Minivan on the SRN4 hovercraft (Fig. 6) that crossed the English Channel (La Manche) in only 30 minutes. This vessel held up to 254 passengers and 30 cars, kept afloat by Rolls-Royce gas–turbine engines that pumped air into a huge rubber skirt. Its operation continued until the year 2000, when it fell victim to competition from ferries and the Channel Tunnel.

Fig. 5. Three French pioneers of EELS: (from left to right) Raymond Castaing, Bernard Jouffrey and Christian Colliex.

Fig. 6. SRN4 Hovercraft at Dover in 1973, just before loading the author’s minivan (extreme right) and departure for Calais.

After stopping to admire the cathedrals at Amiens and Beauvais, I drove to the University of Paris-Sud at Orsay, staying a few nights in a student dormitory. By that time, Bernard Jouffrey had left to become director of the CNRS electron-optics lab in Toulouse but his former student Christian Colliex was continuing energy-loss studies at the CNRS Laboratoire for Solid State Physics, building 510. As a break from science, everyone could enjoy refreshments on the roof of the building, with fine views over the surrounding valley. For his PhD thesis [14], Christian had used a Castaing-Henry spectrometer to produce spectra of various elements, including solidified rare gases. As shown in Fig. 7, the custom in those days was to display spectra with energy loss increasing from right to left. The energy loss was denoted ΔE, rather than E, and the technique was referred to as ELS. Noting that surface scientists seemed to be fond of four-letter words, I mischievously tried EELS [15] and the acronym seems to have stuck.

Fig. 7. One of the first energy-loss spectra to reveal the fine structure of an ionization edge (the K-edge of graphite), recorded with a Castaing-Henry spectrometer [14].

After a pleasant few days in Paris, it was time to drive to Toulouse, taking care not to miss the cathedral at Chartres and various chateaux in the Loire valley. From lectures of Gaston Dupouy, the previous CNRS director, I had the impression that the Toulouse lab sat amid the French countryside, rising majestically from its surroundings like Chartres cathedral. In fact, it is located beside the Canal du Midi and surrounded by houses; see Fig. 8. Even so, the facilities were impressive; the 1MeV microscope was housed in a large spherical building known locally as La Boule. I was allowed to climb to the top of the TEM, using a ladder provided for changing the filament. My CNRS accommodation was in the former director’s apartment above the main entrance, and included maid-service continental breakfast. It is conceivable that I slept in the same bed as some distinguished visitors from earlier years; see Fig. 9.

Fig. 8. French housewives enjoying their proximity to La Boule. Photograph by Jean Dieuzaide, thanks to Virginie Serin.

Returning from France, I found that I had missed an interview for a faculty position at London University, where Professor Ron Burge had just acquired the prototype of the HB5 field-emission STEM manufactured by the Vacuum Generators company. Production models of the HB5 and its successors were later shipped to Cambridge and Glasgow, where Mick Brown, Archie Howie and Alan Craven (among others) developed EELS in combination with high spatial resolution, complementing the work of John Cowley and colleagues in Arizona.

Fig. 9. Famous visitors to the CNRS labs in Toulouse. (a) President Charles de Gaulle being shown how to evaporate a carbon film by laboratory director Gaston Dupouy in 1958. (b) Prof. Sir Neville Mott at the controls of a high-voltage TEM in 1973.

4. New-World EELS The HB5 was based on the work of Albert Crewe at the University of Chicago in the 1960’s, where several STEMs operated at around 30 kV and produced the first images of single atoms; see Fig. 10. One of these microscopes was fitted with an electron spectrometer, enabling Mike Isaacson, Dale Johnson and Joe Wall to produce the first spectra of biological molecules such as nucleic acids [16] and to advocate the use of core-loss EELS for quantitative microanalysis [17]. After a careers officer at Oxford University advised me to get a permanent job, I applied to several universities around the world. The first offer came from the University of Alberta in Edmonton, Canada. On the way to the interview, I broke the journey in Toronto to visit Peter Ottensmeyer, who at short notice came into his downtown lab on a Sunday to show me his Zeiss TEM fitted with a Castaing-Henry filter. He was producing some of the first elemental maps of biological specimens, such as mineralizing cartilage, using photographic subtraction to remove the non-characteristic background beneath ionization edges [18]. When I took the position in Edmonton, vacated by Peter Turner when he returned to Australia, I found myself following in his footsteps for a second time. On the way to Canada, I stopped at Cornell University where John Silcox and colleagues had installed a Wien-filter spectrometer below the screen of an old Hitachi TEM [19]. They were using it to study plasmon dispersion and Cerenkov modes in thin films [20], work continued at Cornell by Phil Batson and more recently by David Muller.

Fig. 10. The original field-emission STEM at the University of Chicago (photo courtesy of Prof. Albert Crewe). Images were produced on the circular display tube on the left. The inset shows images of uranium atoms, which appear circular in outline since they were used to adjust the lens stigmators.

The Cornell system used the microscope high voltage to slow down electrons during their passage through the spectrometer; see Fig. 11. This retardation principle, which can be applied to other kinds of spectrometer, has several advantages: it increases the energy dispersion while making the mechanical accuracy and alignment less critical. Even more important, it renders the position of the exit beam insensitive to changes in high voltage: any increase in accelerating voltage is compensated by an equal increase in retarding potential, allowing very high stability to be achieved. After joining the IBM Watson Laboratories, Phil Batson added a retarding Wien filter to a Vacuum Generators STEM, resulting in an energy resolution of the order of 0.1 eV at 100 kV [21]. The retardation method becomes awkward and possibly unsafe at higher voltages, which is probably why it has not been applied more widely. It could be reconsidered for new instruments that use accelerating voltages below 40 kV to examine very thin specimens with a minimum of radiation damage. The Physics Department in Edmonton had recently purchased a JEOL JEM-100B, fitted with an early-version STEM attachment. Influenced by spectrometers in Arizona [22] and Bell Laboratories [23], I persuaded the Physics machine shop to make a below-column magnetic spectrometer with 90° deflection angle and a bend radius of 20 cm [24]. Large

Fig. 11. Hitachi TEM at Cornell University, with a retarding Wien filter below the camera chamber [19].

bend radius maximizes the energy dispersion but results in a spectrometer that has to be tailored to the space available. Unlike Wittry’s design, my spectrometer projected backwards, into space below the diffusion pump of the JEM-100B. Weighing 17 kg, it had to be aligned with a hammer. It took the insight of Ondrej Krivanek (at the University of California, then at Gatan) to realize that by reducing the bend radius to 10 cm, the weight could be reduced by a factor of 8 and the same spectrometer would fit as an attachment to virtually any TEM [25].

Before 1975, EELS employed either photographic parallel recording or serial recording with the spectrum scanned across a slit in front of a single-channel detector, typically a scintillator-photomultiplier combination connected to a pen recorder. But it was clear from other forms of spectroscopy that digital acquisition of the spectrum leads to much easier data processing. In Oxford, I had found that the background of an ionization edge approximates to AE-r, where E represents energy loss and the exponent r depends on the edge energy and collection angle. For background subtraction, these two parameters need to be found by curve fitting to the pre-edge intensity. At the Bell Telephone Labs, David Joy and Dennis Maher adapted an EDAX x-ray spectrometer system to acquire energy-loss spectra via multichannel scaling, then process and display them [26]. My solution was to use one of the first general-purpose microcomputers (Texas Instruments model 990) for acquisition, storage and processing, with the JEOL STEM attachment for scanning and display (Fig. 12). A graduate student (Dan Kenway) wrote an assembly-language that took about 10 minutes to load to the TI-990 from audio-cassette tape [27].

Fig. 12. (a) Texas Instruments 990 micro-computer with (on left) its mains-voltage stabilizer and (on top) one of its printed circuit boards. Buttons on the front were used to make small program corrections, by changing the content of selected memory locations. (b) Carbon, nitrogen and oxygen K-ionization edges recorded from a specimen of nitrocellulose; the bright curve is an extrapolated power-law background fitted to data preceding the carbon edge [27].

After a year, I realized the futility of this go-it-alone approach and purchased a Tracor Northern 1710 multichannel analyzer,whose 4096 memory channels could be store typically four energy-loss spectra (Fig.13). Inexpensive plug-in modules were available for multichannel scaling, spectrum calibration, computing fast-Fourier transforms, and so on. Best of all, recorded data could be manipulated by means of short programs written in a simple high-level language (FLEXTRAN, a close cousin of BASIC). At the Gatan company, Macintosh computers were used for EELS acquisition and processing but abandoned some years later when Apple no longer supported plug-in boards. During this time, many electron microscopists became interested in core-loss EELS for light–element analysis, encouraged by the creation of databases of ionization edges and low-loss spectra at Argonne Laboratories and Arizona State University [28,29].

These early databases were compiled using serial-EELS acquisition, with advantages in terms of simplicity the ability to cope with a large dynamic range of intensities. But core-loss data is inherently noisy, reflecting the limited number of scattered electrons that pass through the detection slit and resulting in unwanted electron-beam shot noise. The situation is improved by recording an extended range of energy loss in parallel, employing some form of position-sensitive detector. One of the options available for my TN-1710 analyzer was a parallel-recording system that used an image intensifier and silicon photodiode array. It was designed for optical spectroscopy but easily adapted for EELS by interposing a thin plastic scintillator that converted the 100keV electrons into visible photons. To achieve 3eV energy resolution, I tilted the detector through 70°, thereby increasing the energy dispersion along the array (Fig. 14a).

Fig. 13. Tracor Northern TN-1710 multichannel analyzer with Texas Instruments keyboard (and thermal printer) and dual 8”-diskette storage system, connected to the energy-loss spectrometer below the JEM-100B TEM. The TN-1710 replaced the mechanical pen plotter visible in the background.

Fig. 14. Three generations of parallel-EELS system at the University of Alberta. (a) Indirect exposure using an intensified photodiode array and plastic-film scintillator [54]. (b) Direct exposure of a cooled Reticon RL-1024S photodiode array [38]. (c) Indirect exposure, using a YAG scintillator preceded by magnifying quadrupole lenses [30].

A better solution is to magnify the dispersion using one or more quadrupole lenses in front of the detector (Fig. 14c). In their Model 666 parallel-recording spectrometer, introduced in 1987 [31], the Gatan company used four quadrupoles, single-crystal yttrium aluminum garnet (YAG) as the conversion screen and a 1024-element photodiode array as the position-sensitive detector. However, two-dimensional CCD arrays were shown to have advantages [32] and Gatan later developed their GIF system, in which an elaborate system of quadrupoles and sextupoles could be used to form an energy-filtered image on the CCD array [33]. A recent version, the GIF Quantum [34], incorporates an electrostatic beam switch [35] that allows low-loss and core-loss spectra to be recorded almost simultaneously. Nowadays, EELS instrumentation is highly sophisticated and relies heavily on computer control and the development of the appropriate software.

The parallel-EELS systems currently in use employ indirect recording: a thin scintillator preceding the diode array converts the electrons to visible photons. A simpler alternative is to expose the array directly; a single 100keV electron produces about 27,000 electron-hole pairs in silicon, offering the possibility of electron counting at low incident flux. The challenge is then to increase the dynamic range sufficiently to allow higher intensities to be recorded, achievable in principle by sufficiently fast readout [36]. The necessary instrumentation is being developed for image recording and may perhaps be eventually useful for the energy-loss spectrum. A potential problem is radiation damage to the detector, which can be minimized in various ways, including cooling the diode array [37,38]. 5. Present and Future EELS After several decades of development, TEM-EELS now offers a spatial resolution down to 0.1 nm (taking advantage of electron-lens aberration correctors) and an energy resolution down to 0.1 eV (thanks to electron monochromators). These advances in electron optics became practical only after painstaking improvement in the stability of the microscope and the spectrometer power supplies. As a result, the spatial and energy resolution of the analysis is, for many types of specimen, determined by physical principles rather than by instrumental factors. One basic limit to spatial resolution is the delocalization of inelastic scattering, a consequence of the fact that the scattering involves long-range electrostatic forces. In principle, an object or point-spread function for delocalization can be calculated, and might be used as a correction. More accurate measurements of this effect, as a function of energy loss and incident energy, would be valuable. Delocalization becomes worse and more complicated as the specimen thickness increases, due to simultaneous elastic scattering. Conversely, the inelastic signal should be more localized if only higher-angle scattering is collected [39]. A second fundamental limit to spatial resolution is radiation damage to the specimen, predominantly radiolysis in poorly conducting specimens and atomic displacement (within the bulk or at the surface) in metals. There is current interest in using accelerating voltages that are below the threshold for displacement damage. The result might be an instrument that resembles Albert Crewe’s STEM but with lens-aberration correction, something that Albert himself worked on with limited success. To avoid excessive scattering, the specimen must be extremely thin, but the lower incident energy then provides high contrast from certain types of specimen, such as nanotubes. Figure 15 is a slide I prepared to stimulate discussion at the Banff EDGE-2009 meeting. Some aspects, such as atomic resolution at low voltage, have since been realized [40, 41]. When I suggested that lower accelerating voltage should make the equipment less expensive, the idea that seemed to evoke amused skepticism. Certainly it should be easier to achieve good energy resolution in EELS and good mechanical stability in a more compact instrument. The ability to maintain or produce a clean surface in the microscope

would allow the possibility of Auger spectroscopy, or of atomic-scale imaging from the secondary-electron image. With efficient secondary-electron detectors, atomic-resolution images might even be obtainable from bulk specimens [42]. Single-atom sensitivity could also be achievable in the image formed from emitted x-rays, especially if more than 50% of them can be collected using windowless silicon-drift detectors above and below the specimen [43].

Fig. 15. Possible low-voltage STEM, incorporating in-situ specimen cleaning, x-ray and secondary-electron detectors above/below the specimen, and an energy-stabilized EELS system with high dynamic range.

The energy resolution in core-loss spectroscopy is now often limited by core-hole and excited-electron lifetimes. If an energy resolution of the order of 10 meV can be achieved [44], EELS would compete with optical spectroscopy for examining electron states in semiconductors and might resolve vibrational and phonon modes in nanostructures. Slit-free monochromators, as used in reflection-mode high-resolution EELS (HREELS) at low primary energy, would make more efficient use of the electrons but it seems hard to combine this dispersion-compensation principle with sub-nm spatial resolution. There is some indication that electron sources based on field emission from a carbon nanotube might achieve high stability and 0.1 eV energy resolution without the need for a monochromator [45,46]. Pulsed photoemission sources [47,48] offer sub-ps time resolution, making possible dynamic studies of chemical reactions [49] and perhaps eventual freedom from radiation damage, although Coulomb-repulsion effects make this goal harder to achieve than in x-ray diffractive imaging [50].

6. Two puzzles Returning to Professor Rempfer: during my year at Portland State University, I observed both her skill at handling high voltages and her mathematical alertness. One day, a question arose about the temperature at the free end of a cylindrical nanotube when its fixed end is held at a known (elevated) temperature. I guessed an answer that turned out to be good enough but Gert derived an exact formula within a day or two, probably while commuting by train from Forest Grove to Portland. I had wanted to conclude this article by suggesting a couple of mathematical puzzles to be solved by Gert in her 100th year. Now I present them as a challenge to anyone who might find them of interest.

The first one concerns the angular distribution of plural inelastic scattering, something of practical interest to EELS, where it is usually assumed that the number of electrons that were inelastically scattered n times is described by Poisson statistics. In fact, procedures such as Fourier-log deconvolution depend on this assumption. However, energy-loss spectra are often acquired with an angle-limiting aperture, so only a fraction F1 of the single-scattering intensity is recorded. The angular distribution of double scattering (n=2) is a two-dimensional self-convolution of the single-scattering distribution and is therefore broader in angle, so an even smaller fraction F2 gets measured. It is easy to show that Poisson statistics is no longer valid, unless the fraction Fn of n-fold scattering passing through the aperture happens to satisfy the relation:

Fn = (F1)n (1)

Jouffrey and colleagues [51] came across this surprising possibility while using Monte Carlo procedures to calculate plural-scattering intensities, and they used it to simplify their calculations. Taking the angular distribution of inelastic scattering to be Lorentzian with a halfwidth θE, numerical evaluation of the plural-scattering angular distributions shows that Eq. (1) is indeed valid, to within typically 3% [52, 53]. Alternatively, some cumbersome algebraic analysis reveals that Eq. (1) holds for n = 2 if the aperture collection angle is large compared to θE, so Eq.(1) seems to be a property of a θ-2 angular distribution. But why? A simple formula invites a simple explanation, in this case probably a geometrical one. The second puzzle arose from background fitting to energy-loss ionization edges. Seeking a faster alternative to least-squares fitting, I found that the parameter r in the AE-r dependence can be obtained [54] by defining a pre-edge region (energy loss from E1 to E2) and dividing it into two halves of equal width, then measuring the corresponding background-intensity integrals I1 and I2 and applying the formula:

r = 2 log(I1/I2)/log(E2/E1) (2)

Note the mysterious factor of 2, which arises because Eq.(2) assumes that the area beneath a power-law function AE-r is equal to the horizontal width (E2-E1) multiplied by a geometric mean of maximum and minimum intensities: [(AE1

-r)(AE2-r)]1/2. Simple algebra

confirms that this assumption is correct for r = 2, yet Eq. (2) gives background integrals to better than 1% accuracy for r up to 5. If the power law is replaced by a linear function, the right answer requires taking an arithmetic mean, so why is a geometric average appropriate to a power law? Is there a geometric explanation?

7. Conclusions Like electron microscopy in general, EELS is a technically demanding technique, offering many challenges in terms of instrumentation. Over the last 40 years, these problems have been largely overcome through improved hardware and software design, thanks to continuous effort in Japan, Europe and North America. With suitable equipment and certain types of specimen, core-loss EELS is now possible at a spatial resolution down to atomic dimensions. Assuming continuing human ingenuity and longevity, future improvements in energy and time resolution can be expected. Acknowledgments Several colleagues helped with the compilation of this article by supplying photographs or giving permission for their use; I am particularly grateful to Alan Craven, Christian Colliex, Sir Peter Hirsch, Bernard Jouffrey, Virginie Serin, John Silcox and John Spence. I would also like to acknowledge the Natural Sciences and Engineering Research Council of Canada for support over the past 35 years, appreciative of the fact that NSERC has funded basic research projects lasting 3 to 5 years without requiring a progress report until the next application. Finally, I remain indebted to the late Professor Gert Rempfer for her friendship, patience and generosity of spirit. References

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