![[figure 1]](fig1.jpg)
Quantitative comparisons of experimentally obtained and simulated high resolution electron microscope images have shown that the contrast in experimental images is usually much less than is predicted by simulations. The aim here is to investigate this loss of contrast as a function of image spatial frequency using high resolution images of amorphous carbon. It seems that experimental images of amorphous carbon have an unexpectedly high contrast for the low spatial frequencies but that the loss of contrast is constant for frequencies above 0.5 nm-1.
One of Mike Stobbs' interests in high resolution electron microscopy was whether it was possible to analyse high resolution images quantitatively and thus extract more information from such images [1]. For example high resolution has been used to measure the oxygen content of high Tc superconductors [2-4] by comparing quantitatively the images with simulated YBa2Cu3O7-[delta]. This work showed that when high resolution images are compared quantitatively with simulated images, ie taking into account the intensity (the average brightness of an image) and the contrast (the amplitude of the lattice fringes) as well as the pattern (the shape distribution of bright and dark areas), the experimentally obtained images always had much less contrast than the simulated images. This was true even though it was often possible to obtain good qualitative matches taking into account only the pattern of the image [5, 6]. As a result, in order to obtain a good match for many materials the simulated images that best match a set of experimental high resolution images need to be calculated with unrealistically low specimen thicknesses.
The possible reasons for a lack of agreement between experimental and simulated high resolution images can be considered in a number of stages, the calculation of the crystal potential, the calculation of the exit surface wavefunction from the interaction of an electron wave with the crystal potential and the propagation of this exit surface wave through the electron lenses taking into account practical microscope parameters. In addition, experimental factors have to be taken into account, such as contributions from scattering by the specimen that are not included in simulations (such as phonon scattering), the presence of amorphous layers on the specimen, microscope instabilities and stray scattering in the microscope and loss of contrast in the image recording system [7]. We can to some extent consider the first two stages separately from the propagation of the electron wave through the lens by comparing the intensity of the beams in the diffraction pattern. One way of comparing diffracted beam intensities is to match convergent beam patterns. Considerable success has been achieved in determining the rearrangement of electrons causing chemical bonding by analysing systematic row [8] and zone axis [9, 10] convergent beam patterns. Typically these methods allow the measurement of structure factors to within a few percent. However although the convergent beam patterns are energy filtered, phonon scattering is still present. Because relatively thick specimens are used phonons contribute a background to the convergent beam pattern of typically 5% of the disc intensities, which has to be allowed for by adding a constant background to each disc in simulations. An alternative way of comparing diffracted beam intensities is to match bright field and dark field thickness fringes in a wedge shaped crystal at a systematic row condition [11-13]. It is then found that even with energy filtering it is not possible to obtain a particularly good match at all thicknesses. Typically the first bright fringe in dark field is lower experimentally than is predicted by simulations by around 10 to 20% [13].
Thus from comparing diffracted beam intensities we learn that for large specimen thicknesses simulations and experiment agree fairly well, with the experimental contrast around 5% lower than simulations due to phonon scattering not being taken account of in simulations. For lower thicknesses we can say that diffracted beam intensities may differ from simulations by 10 to 20% but the reasons are less clear. The structure factors and Debye Waller factors must be fairly accurate since extinction distances and thus thickness fringe spacings are matched well and convergent beam patterns from large specimen thicknesses fit well. Absorption parameters model the intensity well for thick specimens both for convergent beam diffraction and for thickness fringes, but clearly work less well at low thicknesses. The discrepancy at low thickness thus could be caused by phonon scattering, although I would expect phonon scattering to cause an error that increased with thickness not decreased, or it could be due to amorphous surface layers or surface damage, or to stray scattering in the microscope.
Given that the mismatch between bright field and dark field images and simulations is worst for the low specimen thicknesses used in high resolution microscopy (in contrast to high resolution simulations, where the mismatch gets increasingly worse with thickness) it is worth investigating high resolution images more quantitatively. Generally the average intensity of an experimental high resolution image matches simulations well since, with a large or no objective aperture, little intensity is lost and what is lost is determined only by V0'. The high resolution contrast, as defined by the standard deviation of the lattice images is, however, much less than predicted by simulations, typically by a factor of three [7, 14]. There is thus a distinct change in the match between predicted and measured image contrast as one goes from low to high spatial frequencies and it is the aim of this paper to investigate this spatial frequency dependence of the contrast. For this purpose the choice of specimen is important. A material with a large until cell is needed whose atom coordinates are well characterised in order to provide many diffracted beams within the information limit of the microscope, and thus many spatial frequencies in the resulting images. Also, the specimen must be free of surface contamination and amorphous layers as are typically found on ion beam thinned specimens. Additionally, there must be some way of accurately determining the specimen thickness, such as specimens in the form of 90deg. cleaved wedges. In practice I have not found a material that satisfies all these conditions at once. In the past I have used 8WO39Nb2O5 which has a unit cell of a=2.625 nm when imaged down [001], but as the sample is prepared by crushing it is also difficult to determine the thickness independently [7]. Here I choose to analyse amorphous carbon, which has the advantage of containing all spatial frequencies (in projection) as well as being easily available in large areas of uniform thickness, although simulations of its structure are an approximation and its thickness is still difficult to determine. It is just as important to know the microscope imaging conditions as accurately as possible and to determine them from a source independent from the image series being analysed. For this reason, a microscope with a field emission gun was used here as the high spatial coherence and low energy spread of the gun mean that these parameters are small and errors in them are of less importance, and defocus and Cs could be determined easily from the rings in the diffractogram of high resolution images of amorphous materials.
Figure 1: Experimental (a to c) and simulated (d to f) high resolution images of an amorphous carbon film for three defoci selected from a series of six, printed on the same intensity scale with black = 0.4 and white = 1.4, where the incident intensity = 1.
Figures 1a to 1c show three images from a focal series taken from amorphous carbon on a Philips CM200 FEG electron microscope at 197 kV using a Gatan imaging filter and figure 2 shows a diffractogram from a similar image taken at a defocus of -441 nm. There are a number of stages to analysing these images and comparing them with simulations. Firstly from the experimental images the microscope contrast transferred as a function of spatial frequency must be derived on an absolute scale. Then simulated images can be calculated, using a simple model for amorphous carbon with matching thickness and microscope parameters and the contrast as a function of spatial frequency derived as for the experimental image. Finally the contrast in the experimental and simulated diffractograms can be compared on an absolute scale as a function of spatial frequency.
![[figure 2]](fig2.jpg)
Figure 2: Computed diffractograms for the image taken at a defocus of -441 nm, a) before and b) after the distortions introduced by the imaging filter had been corrected.
For the experimental images (figure 1a-c) the first step is to remove the artefacts introduced by the imaging filter, ie the CCD camera modulation transfer function and the imaging filter distortions. The ideal order to perform these corrections is to deconvolute the camera modulation transfer function to give an image that is a faithful reproduction of the image incident on the camera, then undistort this image to correct for the image filter distortions. Unfortunately this second step necessarily involves resampling the image, which itself affects the frequency response of the images, particularly at high spatial frequencies (unless very large sampling densities are used) and so must be corrected for as well. In practice, since the intention here is to reduce images of amorphous carbon to one dimensional line traces of the diffractogram intensity, a simplified approach is possible. The imaging filter distortions (measured by analysing a lattice image of a uniformly flat crystal of [0001] oriented sapphire) are corrected first, then diffractograms are calculated as in figure 2, then the various corrections as a function of spatial frequency can be applied to the one dimensional sections of the diffractograms. This approach is computationally much quicker as most of the corrections are applied to the one dimensional sections. The effect of the imaging filter distortions can be seen by comparing diffractograms calculated before (figure 2a) and after (figure 2b) the distortions have been corrected for. Before correcting for the distortions (figure 2a) the rings at the top and bottom of the diffractograms are blurred, because the imaging filter distortions have the effect of stretching the top and bottom of filtered images vertically with respect to the centre of the image.
Correcting for the camera modulation transfer function is not straightforward. There are three components to the correction applied here. The first component is the modulation transfer function of the camera itself. This was measured using the "noise method" (in which uniformly illuminated images are taken and the modulation transfer derived assuming the individual electrons act as point sources of intensity [15]). However it has subsequently been found that this method over-estimates the transfer at high spatial frequencies and that the "edge method" (in which the modulation transfer function is derived from an image of a sharp edge) is more accurate [16]. Not having an "edge" modulation transfer function for the camera used in these experiments, an "edge" transfer function had to be used from a similar camera at a similar accelerating voltage for which the noise transfer function matched the noise transfer function for the camera used in these experiments. The second component arises because CCD cameras average the electron intensity recorded over the area of each pixel, while image simulations give the intensity sampled at each point, thus a correction to the modulation transfer function has to be made for averaging over the area of each CCD pixel. Finally, the third component, the effect on the spatial frequency response of resampling in correcting for the image filter distortions can be estimated by running the same undistorting transformation on a noisy image with a uniform spread of spatial frequencies so that a correction curve as a function of spatial frequency can be derived. Combining these three corrections together gives an effective modulation transfer function for the camera, including the effects of an "edge" modulation transfer function, the area of the pixels and the distortion correction, and is shown as the dotted line in figure 3.
![[figure 3]](fig3.gif)
Figure 3: Components of the correction applied to the diffractograms plotted as a function of scattering angle, which is equivalent to spatial frequency in the images. ........... Camera modulation transfer function calculated by the "edge" method, - - - - - - Electron scattering factor for carbon, -·-·-·-· Debye Waller factor for carbon (0.005 nm2), _______ Overall correction calculated as the product of the three separate corrections. The correction applied is in fact the square of the overall correction shown here as the modulus squared intensity is plotted in the diffractograms.
The aim in this analysis is to derive the transfer function of the microscope from the amorphous carbon images and thus the scattering factor for carbon (dashed line in figure 3) and the Debye Waller factor, assumed to be 0.005 nm2, (dot-dash line in figure 3) must be corrected for in the experimental diffractograms. Although it would be possible to include the carbon scattering factor in the simulations described subsequently, this factor reduces the intensity of the scattering at high frequencies much more than at low frequencies, so it is thus easier to compare the experimental and simulated diffractograms if the experimental diffractograms are divided by the carbon scattering factor, making them of roughly the same intensity at all frequencies, than if the simulated diffractograms include the carbon scattering factor. The overall correction applied to the experimental diffractograms is shown as the solid line in figure 3d and is the product of the three broken lines in figure 3. After applying the correction in figure 3 the solid lines in figure 4 are obtained, showing the corrected diffractograms for the defoci analysed. These diffractograms are plotted on an absolute scale where the incident intensity is scaled to a value of 1 and as such can be compared directly with simulations.
![[figure 4]](fig4.gif)
Figure 4: Experimental and simulated modulus squared of the diffractograms from amorphous carbon after applying the corrections shown in figure 3 and plotted on an absolute intensity scale. Also shown is the contrast transfer function averaged over the specimen thickness of 40 nm and scaled to the simulated diffractograms.
Accurate simulations of amorphous materials require some knowledge of the arrangement of the carbon atoms and will differ depending on the deposition conditions, and thus whether the carbon is diamond like or graphite like. However high resolution images of amorphous materials do not change much with the structure of the amorphous material and for the purposes of these simulations a random arrangement of carbon atoms provides a good enough approximation. This will model the average scattering of the carbon film as a function of scattering angle well, because this depends mostly on the atomic scattering factor and Debye Waller factor for carbon, but it will not model the diffuse rings in the diffraction pattern as these are determined by the atomic arrangement of the carbon atoms. In practice the diffuse rings in amorphous diffraction patterns are of relatively low amplitude - typically a variation of 10 to 20% on top of the overall decay with scattering angle caused by the atomic scattering factor [17].
The simulations were performed using the multislice method with a slice thickness of 1 nm and are shown in figures 1d to 1f on the same intensity scale as the experimental images, figures 1a to 1c. For each slice, the projected potential was obtained by weighting the Fourier transform of Poisson noise, of intensity chosen to match the density of amorphous carbon (1800 kg/m3), by the scattering factor for carbon and this was repeated for each slice so that all the slices were different [18]. The thickness of the carbon film used experimentally was determined from energy loss spectra using the plasmon mean free path for carbon (137 nm) [19] and was found to be 40 nm. The accuracy of the thickness measurement is difficult to determine as it depends on how similar my carbon is to the carbon measured by Egerton [19]. However this must be accurate to within about 10% as there is good agreement between the experimental and simulated dark and bright rings in the diffractograms. The thickness is important because to a first approximation the scattering is proportional to thickness and thus errors in the thickness will be reflected in the ratio between the experimental and simulated scattering intensity.
For the microscope lens part of the multislice calculation the experimental microscope imaging conditions must be known accurately. The defocus is relatively easy to determine from the positions of the rings in the experimental diffractograms. This measured defocus corresponds to the defocus of the middle of the thickness of the carbon film, but multislice simulations measure the defocus from the exit surface of the film, thus the defoci entered into the simulation program have to be corrected by half the thickness of the carbon film, in this case 20 nm. This is a potential cause of error in any high resolution matching experiment that is perhaps not widely appreciated, especially for crystalline materials where there is no easy check that the defocus is correct.
Because of the high coherence of the field emission gun there were sufficient rings in the diffractograms to provide a reasonable estimate of the spherical aberration constant of 3.3 ± 0.1 mm, which was consistent with all of the six diffractograms analysed. This value is higher than the normal value for a Philips CM200 microscope because a non-standard specimen height was used. The astigmatism was found to be negligible but there was a significant horizontal vibration caused by stray fields from a nearby cable that degraded the resolution in the horizontal direction, as can be seen in figure 2. As a result, the experimental diffractograms were averaged only over a 60deg. sector in the vertical direction. The focal spread was determined from the width of the zero loss peak of an energy loss spectrum (1.3 eV for a 0.1 s exposure, this value is relatively high because of the low gun lens setting used to obtain high brightness) and the measured value of the chromatic aberration constant, Cc = 1.45 mm, and found to be 4.7 nm. This covers the energy spread of the filament and assumes any instabilities in the lenses to be negligible. In practice the focal spread is fairly small and has only a small effect on the diffractograms.
The beam divergence is more difficult to determine. In principle it can be measured from the diameter of the 000 beam in a diffraction pattern taken under the same conditions as the high resolution images (this gives a standard deviation of about 0.14 mrad, assuming the 000 beam to be gaussian shaped), but in practice for the low divergence of a field emission microscope this is very inaccurate. A better method is to choose a value that matches diffractograms taken a long way out of focus, such as figure 4c, where beam divergence limits the resolution and this gives a beam divergence of 0.08 mrad. An objective aperture of radius 12.8 nm-1 was used for the experimental images to provide a known resolution cut off that could be included in the simulations and to reduce the amount of any stray scattering in the microscope reaching the image. It is appreciated that such an aperture could be contributing to the somewhat higher than expected value of spherical aberration measured due to charging. However, for any high resolution images it should not matter how good or bad the microscope parameters are, so long as they are measured accurately it should still be possible to simulate the resulting images.
The resulting simulated diffractograms from 40 nm of amorphous carbon after correcting for the carbon scattering factor and Debye Waller factor are shown as the dotted lines in figure 4. They are plotted on the same absolute intensity scale as the experimental diffractograms (solid lines in figure 4) and can be clearly seen to be much more intense at most frequencies, although the positions and relative sizes of the features match the experimental diffractograms well. At this point, to find the ratio between the experimental and simulated diffractograms I should divide the corresponding lines in figure 4 by each other. However both are very noisy and the noise in the simulated diffractogram line can be eliminated by using the contrast transfer function of the microscope instead, as shown by the dashed line in figure 4. Since there is essentially no multiple scattering in this thickness of amorphous carbon then the scattering intensity as a function of scattering angle, after compensating for the scattering factor and Debye Waller factor for carbon, should match the microscope weak phase contrast transfer function. In practice allowance has to be made for the thickness of the carbon film, whose effect is to make the top and bottom of the carbon film out of focus with respect to the middle. This thickness effect can be modelled by averaging together contrast transfer functions over a defocus range equal to the thickness of the carbon film, in this case 40 nm, and the result, scaled to match the simulated carbon diffractograms is shown as the dashed line in figure 4. Dividing the simulated diffractograms by the corresponding contrast transfer functions gives a line that is close to 1 everywhere, apart from the noise, confirming that the contrast transfer function provides a good model of simulated diffractograms.
![[figure 5]](fig5.gif)
Figure 5: Experimental diffractograms divided by the contrast transfer functions in figure 4 and smoothed. The average ratio is unsmoothed and is derived from all six defoci analysed by weighting each ratio by its contrast transfer function so as to minimise the contributions from areas where the contrast transfer is near zero.
Figure 5, however, shows the result of dividing the experimental diffractograms from figure 4 by their corresponding contrast transfer functions. Due to the amount of noise present, especially from the spatial frequencies corresponding to zeros in the contrast transfer function, the lines from the individual defoci (shown as broken lines in figures 5) have been smoothed. An average of the six defoci analysed, weighted by each contrast transfer function and not smoothed, is shown as the solid line in figure 5. There are a number of ways in which this average line differs from a straight line at a value of 1. Firstly, at high scattering angle, ie above about 4 nm-1, the ratio of experimental to simulated diffractograms becomes high. This is because this region is close to the microscope information transfer limit and the corrections in figure 3d have amplified the noise in the original experimental diffractograms. I will therefore not concern myself with the scattering above about 5 nm-1. The broad peaks present at about 4 nm-1 and 2 nm-1 correspond to structure in the amorphous carbon that is not predicted by these simplified simulations, ie the broad rings in the diffraction pattern of amorphous carbon. It can be seen from the dotted lines in figure 5 that these broad peaks are present at all defoci (in fact they are easily visible in all the six defoci analysed) demonstrating that they are a feature of the amorphous carbon and not artefacts of poorly fitting contrast transfer functions. The most important feature of figure 5 is that when the above two areas are ignored, the average value of the ratio of experimental to simulated amorphous carbon diffractograms is about 0.3 over the range of scattering angles from 0.5 nm-1 to about 5 nm-1. Below 0.5 nm-1 the ratio increases dramatically suggesting there is a large amount of long wavelength contrast present in the experimental images that is not there in the simulations. This extra low frequency contrast does not arise from streaks in the diffractograms associated with discontinuities at the edges of the original experimental images, as the images were high pass filtered with a large kernel size filter to remove any very long range intensity variations or mismatches in the intensity at the edges of the images and no such streaks are visible in the diffractograms in figure 2. This high pass filter will have the effect of reducing the very long wavelength contrast in the original experimental images and will thus reduce the contrast in the experimental diffractograms for scattering angles below about 0.1 nm-1.
It might be expected that experimental and simulated image contrast would agree for low spatial frequencies (as is found in comparing bright field images, such as thickness fringes, with simulations [13]) and that the ratio of experimental to simulated image contrast would decrease smoothly with image spatial frequency to the value of around 0.3 found for high resolution images [7]. It is thus surprising that this ratio is found to be higher than 1 for spatial frequencies below 0.5 nm-1 but remains approximately constant at a value of 0.3 for higher spatial frequencies. It is not the simplicity of the random atoms model for amorphous carbon that is at fault here, this model if anything should have too much low frequency contrast as it will tend to produce random regions of unrealistically high and low density. Slightly more realistic models limit the maximum atom density when determining the coordinates for the carbon atoms [20]. One possible explanation for the extra low frequency contrast is that the carbon film is not of constant thickness. That this is true is demonstrated in figure 6, which shows a map of the carbon film thickness ("t/[lambda] map"), from which it can be seen that the carbon film thickness varies by about +/-3%. In addition the microscope transfer function is very low at low frequencies, like at high frequencies, and thus the Poisson noise present in the experimental diffractograms will be amplified by the division in calculating the ratios in figure 5. However a comparison of the relative heights of the first, second and third bright rings in the diffractogram furthest from focus with those from simulations (fig 4c) show that there is a real excess of contrast below 0.5 nm-1 experimentally.
![[figure 6]](fig6.jpg)
Figure 6: Thickness of the carbon film ("t/[lambda] map") derived from -ln(1 + I 0/I1), where I 0 is the zero loss image and I 1 is the first plasmon loss image of the same area and printed with black = 0.965 t, white = 1.035 t where t is the mean thickness of the image.
The unexplained low frequency contrast does not affect the main conclusion that for the whole spatial frequency range where the measurement is reliable, ie from 0.5 to 5 nm-1, the experimental contrast is about a third of the simulated contrast. It is worth considering some of the errors that could affect this analysis. The random atoms model for amorphous carbon does not contain any bond length or atom position information that is present in amorphous carbon. Thus the diffraction pattern calculated by this model is a uniformly decaying function whose envelope as a function of scattering angle depends on the scattering and Debye Waller factors for carbon, while an experimental diffraction pattern for carbon contains two broad rings in addition. However in practice the amplitude of these broad rings in the diffraction pattern is small, and they represent only small oscillations either side of a general trend based on the atomic scattering and Debye Waller factors. It is certainly not possible for the structure of the carbon film to change the diffracted intensity over a broad range of scattering angles by as much as a factor of three.
Perhaps the most important source of error is the measurement of the specimen thickness, since the final ratio of experimental to simulated contrast depends directly on this. The film thickness was initially derived from energy loss spectra using a plasmon mean free path of 137 nm [19] and it is unlikely that this mean free path could be wrong by a factor of three even allowing for any differences in the preparation method and thus densities of the films. In addition the simulated diffractograms for a thickness of 40 nm match the experimental diffractograms well, in particular the number of rings visible as a function of defocus is well matched. If the thickness in the simulations was reduced by a factor of three then many more rings would be expected in the diffractograms. Although the number of rings can be reduced in the simulations by increasing the vibration or beam convergence or energy spread above the values measured, all these parameters have the effect of reducing the amount of intensity in the diffractograms, ie reducing the intensity of both the bright and the dark rings by the same factor, so as to reduce the information transferred at high frequencies. Only an increase in specimen thickness will cause the observed effect where the amplitude of the rings is reduced (ie the dark rings get brighter and the bright rings get darker) but information is still transferred at the higher frequencies. This is particularly noticeable at a scattering angle of around 3 nm-1 and at the higher defoci in figure 4 where it can be seen that there is intensity in the experimental diffractogram even though no rings are present, as in the simulated diffractograms. It should be pointed out that if the carbon film is tilted this will have the same effect on the diffractograms as an increase in thickness by causing an average over a larger range of defocus, and that there was some evidence that the film was not quite flat from a slight change in defocus from the top to the bottom of the image taken at -61 nm defocus. However for this to explain the factor of three difference in contrast would require much more specimen tilt than was observed, in addition to a similar error in the thickness determination from energy loss spectra, which would not be affected by a specimen tilt.
It thus must be concluded that, within the frequency range measured here, high resolution images have about three times less contrast than simulated images over all spatial frequencies. This suggests that the cause of this contrast reduction is the addition of a fairly uniform constant background to the images, such as might arise from phonon scattering (as intensity between the spots in an energy filtered diffraction pattern from a crystalline material) or from stray scattering within the microscope. Looking at this optimistically it means that the traditional approach of stretching the contrast in experimental high resolution images to match the simulations (in effect subtracting a constant background) gives the correct answer even if it is not very quantitative.
I would like to thank Dr CJD Hetherington and the Lawrence Berkeley National Laboratory for the use of their facilities, Dr RE Dunin-Borkowski and Dr WO Saxton for useful discussions and the late Dr WM Stobbs for the inspiration behind this work.
[1] M.J. Hÿtch and W.M. Stobbs, Microsc. Microanal. Microstruct. 5 (1994) 133.
[2] M.J. Hÿtch and W.M. Stobbs, in: Proceedings of the 46th annual meeting of the electron microscopical society of America, Ed G.W. Bailey (San Francisco Press, San Francisco, 1988) 958.
[3] R.A. Camps, S.B. Newcomb, W.O. Saxton and W.M. Stobbs, in: EMAG 87, IOP conf. ser. no. 90, Ed L.M. Brown (IOP, Bristol, 1987) 299.
[4] N.P. Huxford, D.J. Eaglesham and C.J. Humphreys, Nature 329 (1987) 812.
[5] D. Cherns, G.R. Anstis, J.L. Hutchinson and J.C.H. Spence, Philos. Mag. A46 (1982) 849.
[6] B.J. Inkson and C.J. Humphreys, Philos. Mag. Lett. 71 (1995) 307.
[7] C.B. Boothroyd, J. Microsc. 190 (1998) 99.
[8] J.M. Zuo and J.C.H. Spence, Ultramicroscopy 35 (1991) 185.
[9] D.M. Bird and M. Saunders, Ultramicroscopy 45 (1992) 241.
[10] P.A. Midgley and M. Saunders, Contemp. Phys. 37 (1996) 441.
[11] H. Watanabe, A. Fukuhara and K. Kohra, J. Phys. Soc. Japan, 17 (1962) 195.
[12] A.J.F. Metherell, Philos. Mag. 15 (1967) 755.
[13] R.E. Dunin-Borkowski, R.E. Schaüblin, T. Walther, C.B. Boothroyd, A.R. Preston and W.M. Stobbs, in: Electron microscopy and analysis 1995, IOP Conf. Ser. no. 147, Ed D. Cherns (Institute of Physics Publishing, Bristol, 1995) 179.
[14] C.B. Boothroyd, R.E. Dunin-Borkowski, W.M. Stobbs and C.J. Humphreys, in: Beam-solid interactions for materials synthesis and characterization, Eds D.C. Jacobson, D.E. Luzzi, T.F. Heinz and M. Iwaki, MRS symp. proc. 359 (MRS, Pittsburgh, 1995) 495.
[15] J.M. Zuo, Ultramicroscopy 66 (1996) 21.
[16] R.R. Meyer and A. Kirkland, Ultramicroscopy 75 (1998) 23.
[17] David J. Smith, W.O. Saxton, J.R.A. Cleaver and C.J.D. Catto, J. Microsc. 119 (1980) 19.
[18] J.P. Chevalier and M.J. Hytch, Ultramicroscopy 52 (1993) 253.
[19] R.F. Egerton (Plenum Press, New York, 1996) 305.
[20] A. Pitt, in: EMAG 1979, Ed T. Mulvey, IOP conf. Ser. no. 52 (IOP, Bristol, 1980) 269.