International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. C. ch. 4.3, pp. 416419

4.3.7. Measurement of structure factors and determination of crystal thickness by electron diffraction
Current advances in quantitative electron diffraction are connected with improved experimental facilities, notably the combination of convergentbeam electron diffraction (CBED) with new detection systems. This is reflected in extended applications of electron diffraction intensities to problems in crystallography, ranging from valenceelectron distributions in crystals with small unit cells to structure determination of biological molecules in membranes. The experimental procedures can be seen in relation to the two main principles for measurement of diffracted intensities from crystals:

Integrated intensities are not easily defined in the most common type of electrondiffraction pattern, viz the selectedarea (SAD) spot pattern. This is due to the combination of dynamical scattering and the orientation and thickness variations usually present within the typically micrometresize illuminated area. This combination leads to spot pattern intensities that are poorly defined averages over complicated scattering functions of many structure factors. Convergentbeam electron diffraction is a better alternative for intensity measurements, especially for inorganic structures with smalltomoderate unit cells. In CBED, a fine beam is focused within an area of a few hundred ångströms, with a divergence of the order of a tenth of a degree. The diffraction pattern then appears in the form of discs, which are essentially twodimensional rocking curves from a small illuminated area, within which thickness and orientation can be regarded as constant. These intensity distributions are obtained under well defined conditions and are well suited for comparison with theoretical calculations. The intensity can be recorded either photographically, or with other parallel recording systems, viz YAG screen/CCD camera (Krivanek, Mooney, Fan, Leber & Meyer, 1991) or image plates (Mori, Oikawa & Harada, 1990) – or sequentially by a scanning system. The inelastic background can be removed by an energy filter (Krahl, Pätzold & Swoboda, 1990; Krivanek, Gubbens, Dellby & Meyer, 1991). Detailed intensity profiles in one or two dimensions can then be measured with high precision for loworder reflections from simple structures. But there are limitations also with the CBED technique: the crystal should be fairly perfect within the illuminated area and the unit cell relatively small, so that overlap between discs can be avoided. The current development of electron diffraction is therefore characterized by a wide range of techniques, which extend from the traditional spot pattern to twodimensional, filtered rocking curves, adapted to the structure problems under study and the specimens that are available.
Spotpattern intensities are best for thin samples of crystals with light atoms, especially organic and biological materials. Dorset and coworkers (Dorset, Jap, Ho & Glaeser, 1979; Dorset, 1991) have shown how conventional crystallographic techniques (`direct phasing') can be applied in ab initio structure determination of thin organic crystals from spot intensities in projections. Two main complications were treated by them: bending of the crystal and dynamical scattering. Thin crystals will frequently be bent; this will give some integration of the reflection, but may also produce a slight distortion of the structure, as pointed out by Cowley (1961), who proposed a correction formula. The thickness range for which a kinematical approach to intensities is valid was estimated theoretically by Dorset et al. (1979). For organic crystals, they quoted a few hundred ångströms as a limit for kinematical scattering in dense projections at 100 kV.
Radiation damage is a problem, but with lowdose and cryotechniques, electronmicroscopy methods can be applied to many organic crystals, as shown by several recent investigations. VoigtMartin, Yan, Gilmore, Shankland & Bricogne (1994) collected electrondiffraction intensities from a beamsensitive dione and constructed a 1.4 Å Fourier map by a direct method based on maximum entropy. Large numbers of electrondiffraction intensities have been collected from biological molecules crystallized in membranes. The structure amplitudes can be combined with phases extracted from highresolution micrographs, following Henderson Unwin's (1975) early work. Kühlbrandt, Wang & Fujiyoshi (1994) collected about 18 000 amplitudes and 15 000 phases for a protein complex in an electron cryomicroscope operating at 4.2 K (Fujiyoshi et al., 1991). Using these data, they determined the structure from a threedimensional Fourier map calculated to 3.4 Å resolution. The assumption of kinematical scattering in such studies has been investigated by Spargo (1994), who found the amplitudes to be kinematic within 4% but with somewhat larger deviations for phases.
For inorganic structures, spotpattern intensities are less useful because of the stronger dynamical interactions, especially in dense zones. Nevertheless, it may be possible to derive a structure and refine parameters from spotpattern intensities. Andersson (1975) used experimental intensities from selected projections for comparison with dynamical calculations, including an empirical correction factor for orientation spread, in a structure determination of V_{14}O_{8}. Recently, Zou, Sukharev & Hovmöller (1993) combined spotpattern intensities read from film by the program ELD with image processing of highresolution micrographs for structure determination of a complex perovskite.
A considerable improvement over the spot pattern has been obtained by the elegant doubleprecession technique devised by Vincent & Midgley (1994). They programmed scanning coils above and below the specimen in the electron microscope so as to achieve simultaneous precession of the focused incident beam and the diffraction pattern around the optical axis. The net effect is equivalent to a precession of the specimen with a stationary incident beam. Integrated intensities can be obtained from reflections out to a Bragg angle equal to the precession angle for the zeroth Laue zone. In addition, reflections in the first and second Laue zones appear as broad concentric rings. Dynamical effects are reduced appreciably by this procedure, especially in the nonzero Laue zones. The experimental integrated intensities, I_{g}, must be multiplied with a geometrical factor analogous to the Lorentz factor in Xray diffraction, viz where nh is the reciprocal spacing between the zeroth and nth layers. The intensities can be used for structure determination by procedures taken over from Xray crystallography, e.g. the conditional Patterson projections that are used by the Bristol group (Vincent, Bird & Steeds, 1984). The precession method may be seen as intermediate between the spot pattern and the CBED technique. Another intermediate approach was proposed by Goodman (1976) and used later by Olsen, Goodman & Whitfield (1985) in the structure determination of a series of selenides. CBED patterns from thin crystals were taken in dense zones; intensities were measured at corresponding points in the discs, e.g. at the zoneaxis position. Structure parameters were determined by fitting the observed intensities to dynamical calculations.
Higher precision and more direct comparisons with dynamical scattering calculations are achieved by measurements of intensity distributions within the CBED discs, i.e. one or twodimensional rocking curves. An uptodate review of these techniques is found in the recent book by Spence & Zuo (1992), where all aspects of the CBED technique, theory and applications are covered, including determination of lattice constants and strains, crystal symmetry, and fault vectors of defects. Refinement of structure factors in crystals with small unit cells are treated in detail. For determination of bond charges, the structure factors (Fourier potentials) should be determined to an accuracy of a few tenths of a percent; calculations must then be based on manybeam dynamical scattering theory, see Chapter 8.8 . Removal of the inelastic background by an energy filter will improve the data considerably; analytical expressions for the inelastic background including multiplescattering contributions may be an alternative (Marthinsen, Holmestad & Høier, 1994).
Early CBED applications to the determination of structure factors were based on features that can be related to dynamical effects in the twobeam case. Although insufficient for most accurate analyses, the twobeam expression for the intensity profile may be a useful guide. In its standard form, where U_{g} and s_{g} are Fourier potential and excitation error for the reflection g, k wave number and t thickness. The expression can be rewritten in terms of the eigenvalues γ^{(i, j)} that correspond to the two Blochwave branches, i, j: where Note that the minimum separation between the branches i, j or the gap at the dispersion surface is where _{g} is an extinction distance. The twobeam form is often found to be a good approximation to an intensity profile I_{g}(s_{g}) even when other beams are excited, provided an effective potential , which corresponds to the gap at the dispersion surface, is substituted for U_{g}. This is suggested by many features in CBED and Kikuchi patterns and borne out by detailed calculations, see e.g. Høier (1972). Approximate expressions for have been developed along different lines; the best known is the Bethe potential Other perturbation approaches are based on scattering between Bloch waves, in analogy with the `interband scattering' introduced by Howie (1963) for diffuse scattering; the term `Blochwave hybridization' was introduced by Buxton (1976). Exact treatment of symmetrical fewbeam cases is possible (see Fukuhara, 1966; Kogiso & Takahashi, 1977). The threebeam case (Kambe, 1957; Gjønnes & Høier, 1971) is described in detail in the book by Spence & Zuo (1992).
Many intensity features can be related to the structure of the dispersion surface, as represented by the function γ(k_{x}, k_{y}). The gap [equation (4.3.7.4)] is an important parameter, as in the fourbeam symmetrical case in Fig. 4.3.7.1 . Intensity measurements along one dimension can then be referred to three groups, according to the width of the gap, viz:
A small gap at the dispersion surface implies that the twobeamlike rocking curve above approaches a kinematical form and can be represented by an integrated intensity. Within a certain thickness range, this intensity may be proportional to , with an angular width inversely proportional to gt. Several schemes have been proposed for measurement of relative integrated intensities for reflections in the outer, highangle region, where the lines are narrow and can be easily separated from the background. Steeds (1984) proposed use of the HOLZ (highorder Lauezone) lines, which appear in CBED patterns taken with the central disc at the zoneaxis position. Along a ring that defines the firstorder Laue zone (FOLZ), reflections appear as segments that can be associated with scattering from strongly excited Bloch waves in the central ZOLZ part into the FOLZ reflections. Vincent, Bird & Steeds (1984) proposed an intensity expression for integrated intensity for a line segment associated with scattering from (or into) the ZOLZ Bloch wave j. is here the excitation coefficient and β^{(j)} the matrix element for scattering between the Bloch wave j and the plane wave g. μ^{(j)} and μ are absorption coefficients for the Bloch wave and plane wave, respectively; t is the thickness. From measurements of a number of such FOLZ (or SOLZ) reflections, they were able to carry out ab initio structure determinations using socalled conditional Patterson projections and coordinate refinement. Tanaka & Tsuda (1990) have refined atomic positions from zoneaxis HOLZ intensities. Ratios between HOLZ intensities have been used for determination of the Debye–Waller factor (Holmestad, Weickenmeier, Zuo, Spence & Horita, 1993).
Another CBED approach to integrated intensities is due to Taftø & Metzger (1985). They measured a set of highorder reflections along a systematic row with a wideaperture CBED tilted off symmetrical incidence. A number of highorder reflections are then simultaneously excited in a range where the reflections are narrow and do not overlap. Gjønnes & Bøe (1994) and Ma, Rømming, Lebech & Gjønnes (1992) applied the technique to the refinement of coordinates and thermal parameters in highT_{c} superconductors and intermetallic compounds. The validity and limitation of the kinematical approximation and dynamic potentials in this case has been discussed by Gjønnes & Bøe (1994).
Zero gap at the dispersion surface corresponds to zero effective Fourier potential or, to be more exact, an accidental degeneracy, γ^{(i)} = γ^{(j)}, in the Blochwave solution. This is the basis for the criticalvoltage method first shown by Watanabe, Uyeda & Fukuhara (1969). From vanishing contrast of the Kikuchi line corresponding to a secondorder reflection 2g, they determined a relation between the structure factors U_{g} and U_{2g}. Gjønnes & Høier (1971) derived the condition for the accidental degeneracy in the general centrosymmetrical threebeam case 0,g,h, expressed in terms of the excitation errors s_{g,h}_{} and Fourier potentials U_{g,h,g−h}, viz where m and m_{0} are the relativistic and rest mass of the incident electron. Experimentally, this condition is obtained at a particular voltage and diffraction condition as vanishing line contrast of a Kikuchi or Kossel line – or as a reversal of a contrast feature. The secondorder criticalvoltage effect is then obtained as a special case, e.g. by the mass ratio: Measurements have been carried out for a number of elements and alloy phases; see the review by Fox & Fisher (1988) and later work on alloys by Fox & Tabbernor (1991). Zoneaxis critical voltages have been used by Matsuhata & Steeds (1987). For analytical expressions and experimental determination of nonsystematic critical voltages, see Matsuhata & Gjønnes (1994).
Large gaps at the dispersion surface are associated with strong inner reflections – and a strong dynamical effect of twobeamlike character. The absolute magnitude of the gap – or its inverse, the extinction distance – can be obtained in different ways. Early measurements were based on the split of diffraction spots from a wedge, see Lehmpfuhl (1974), or the corresponding fringe periods measured in bright and darkfield micrographs (Ando, Ichimiya & Uyeda, 1974). The most precise and applicable largegap methods are based on the refinement of the fringe pattern in CBED discs from strong reflections, as developed by Goodman & Lehmpfuhl (1967) and Voss, Lehmpfuhl & Smith (1980). In recent years, this technique has been developed to high perfection by means of filtered CBED patterns, see Spence & Zuo (1992) and papers referred to therein. See also Chapter 8.8 .
The gap at the dispersion surface can also be obtained directly from the split observed at the crossing of a weak Kikuchi line with a strong band. Gjønnes & Høier (1971) showed how this can be used to determine strong loworder reflections. High voltage may improve the accuracy (Terasaki, Watanabe & Gjønnes, 1979). The sensitivity of the intersecting Kikuchiline (IKL) method was further increased by the use of CBED instead of Kikuchi patterns (Matsuhata, Tomokiyo, Watanabe & Eguchi, 1984; Taftø & Gjønnes, 1985). In a recent development, Høier, Bakken, Marthinsen & Holmestad (1993) have measured the intensity distribution in the CBED discs around such intersections and have refined the main structure factors involved.
Twodimensional rocking curves collected by CBED patterns around the axis of a dense zone are complicated by extensive manybeam dynamical interactions. The Bristol–Bath group (Saunders, Bird, Midgley & Vincent, 1994) claim that the strong dynamic effects can be exploited to yield high sensitivity in refinement of loworder structure factors. They have also developed procedures for ab initio structure determination based on zoneaxis patterns (Bird & Saunders, 1992), see Chapter 8.8 .
Determination of phase invariants. It has been known for some time (e.g. Kambe, 1957) that the dynamical threebeam case contains information about phase. As in the Xray case, measurement of dynamical effects can be used to determine the value of triplets (Zuo, Høier & Spence, 1989) and to determine phase angles to better than one tenth of a degree (Zuo, Spence, Downs & Mayer, 1993) which is far better than any Xray method. Bird (1990) has pointed out that the phase of the absorption potential may differ from the phase of the real potential.
Thickness is an important parameter in electrondiffraction experiments. In structurefactor determination based on CBED patterns, thickness is often included in the refinement. Thickness can also be determined directly from profiles connected with large gaps at the dispersion surface (Goodman & Lehmpfuhl, 1967; Blake, Jostsons, Kelly & Napier, 1978; Glazer, Ramesh, Hilton, & Sarikaya, 1985). The method is based on the outer part of the fringe profile, which is not so sensitive to the structure factor. The intensity minimum of the ith fringe in the diffracted disc occurs at a position corresponding to the excitation error s_{i} and expressed as where n_{i} is a small integer describing the order of the minimum. This equation can be arranged in two ways for graphic determination of thickness. The commonest method appears to be to plot (s_{i}/n_{i})^{2} against 1/n_{i}^{2}and then determine the thickness from the intersection with the ordinate axis (Kelly, Jostsons, Blake & Napier, 1975). Glazer et al. (1985) claim that the method originally proposed by Ackermann (1948), where is plotted against n_{i} and the thickness is taken from the slope, is more accurate. In both cases, the outer part of the rocking curve is emphasized; exact knowledge of the gap is not necessary for a good determination of thickness, provided the assumption of a twobeamlike rocking curve is valid.
References
Ackermann, I. (1948). Observations on the dynamical interference phenomena in convergent electron beams. II. Ann. Phys. (Leipzig), 2, 41–54.Google ScholarAndersson, B. (1975). Structure analysis of the γphase in the vanadium oxide system by electron diffraction studies. Acta Cryst. A31, 63–70.Google Scholar
Ando, Y., Ichimiya, A. & Uyeda, R. (1974). A determination of values and signs of the 111 and 222 structure factors of silicon. Acta Cryst. A30, 600–601.Google Scholar
Bird, D. M. (1990). Absorption in highenergy electron diffraction from noncentrosymmetrical crystals. Acta Cryst. A46, 208–214.Google Scholar
Bird, D. M. & Saunders, M. (1992). Inversion of convergentbeam electron diffraction patterns. Acta Cryst. A48, 555–562.Google Scholar
Blake, R. G., Jostsons, A., Kelly, P. M. & Napier, J. G. (1978). The determination of extinction distances and anomalous absorption coefficients by scanning electron microscopy. Philos. Mag. A37, 1–16.Google Scholar
Buxton, B. F. (1976). Bloch waves in high order Laue zone effects in high energy electron diffraction. Proc. R. Soc. London Ser. A, 300, 335–361.Google Scholar
Cowley, J. M. (1961). Diffraction intensities from bent crystals. Acta Cryst. 14, 920–926.Google Scholar
Dorset, D. L. (1991). Is electron crystallography possible? The direct determination of organic crystal structures. Ultramicroscopy, 38, 23–40.Google Scholar
Dorset, D. L., Jap, B. K., Ho, M. M. & Glaeser, R. M. (1979). Direct phasing of electron diffraction data from organic crystals: the effect of nbeam dynamical scattering. Acta Cryst. A35, 1001–1009.Google Scholar
Fox, A. G. & Fisher, R. M. (1988). A summary of lowangle Xray atomic scattering factors measured by the critical voltage effect in high energy electron diffraction. Aust. J. Phys. 41, 461–468.Google Scholar
Fox, A. G. & Tabbernor, M. A. (1991). The bonding charge density of ′NiAl′′. Acta Metall. 39, 669–678.Google Scholar
Fujiyoshi, Y., Mizusaki, T., Morikawa, K., Yamagishi, H., Aoki, Y., Kihara, H. & Harada, Y. (1991). Development of a superfluid helium stage for highresolution electron microscopy. Ultramicroscopy, 38, 241–251.Google Scholar
Fukuhara, A. (1966). Manyray approximation in the dynamical theory of electron diffraction. J. Phys. Soc. Jpn, 21, 2645–2662.Google Scholar
Gjønnes, J. & Høier, R. (1971). The application of nonsystematic manybeam dynamic effects to structurefactor determination. Acta Cryst. A27, 313–316.Google Scholar
Gjønnes, K. & Bøe, N. (1994). Refinement of temperature factors and charge distributions in YBa_{2}Cu_{3}O_{7} and YBa_{2}(Cu,Co)_{3}O_{7} from CBED intensities. Micron Microsc. Acta, 25, 29–44.Google Scholar
Glazer, J., Ramesh, R., Hilton, M. R. & Sarikaya, M. (1985). Comparison of convergent beam electron diffraction methods for determination of foil thickness. Philos. Mag. A52, 59–63.Google Scholar
Goodman, P. (1976). Examination of the graphite structure using convergentbeam electron diffraction. Acta Cryst. A32, 793–798.Google Scholar
Goodman, P. & Lehmpfuhl, G. (1967). Electron diffraction study of MgO h00 systematic interactions. Acta Cryst. 22, 14–24.Google Scholar
Henderson, R. & Unwin, P. N. T. (1975). Threedimensional model of purple membrane obtained by electron microscopy. Nature, 257, 28–32.Google Scholar
Høier, R. (1972). Displaced lines in Kikuchi patterns. Phys. Status Solidi A, 11, 597–610.Google Scholar
Høier, R., Bakken, L. N., Marthinsen, K. & Holmestad, R. (1993). Structure factor determination in noncentrosymmetrical crystals by a twodimensional CBEDbased multiparameter refinement method. Ultramicroscopy, 49, 159–170.Google Scholar
Holmestad, R., Weickenmeier, A. L., Zuo, J. M., Spence, J. C. H. & Horita, Z. (1993). Debye–Waller factor measurement in TiAl from HOLZ reflections. Electron microscopy and analysis 1993, pp. 141–144. Bristol: IOP Publishing.Google Scholar
Howie, A. (1963). Inelastic scattering of electrons by crystals. I. The theory of small angle inelastic scattering. Proc. R. Soc. London Ser. A, 271, 268–287.Google Scholar
Kambe, K. (1957). Study of simultaneous reflections in electron diffraction by crystals. J. Phys. Soc. Jpn, 12, 13–36.Google Scholar
Kelly, P. M., Jostsons, A., Blake, R. G. & Napier, J. G. (1975). The determination of foil thickness by scanning transmission electron microscopy. Phys. Status Solidi A, 31, 771–780.Google Scholar
Kogiso, M. & Takahashi, H. (1977). Grouptheoretical method in the manybeam theory in electron diffraction. J. Phys. Soc. Jpn, 42, 223–229.Google Scholar
Krahl, D., Pätzold, H. & Swoboda, M. (1990). An aberrationminimized imaging energy filter of simple design. Proceedings of 12th International Conference on Electron Microscopy 1990, Vol. 2, pp. 60–61.Google Scholar
Krivanek, O. L., Gubbens, A. J., Dellby, N. & Meyer, C. E. (1991). Design and first applications of a postcolumn imaging filter. Microsc. Microanal. Microstruct. (France), 3, 187–199.Google Scholar
Krivanek, O. L., Mooney, P. E., Fan, G. Y. Leber, M. L. & Meyer, C. E. (1991). Slowscan CCD cameras for transmission electron microscopy. Electron microscopy and analysis 1991, pp. 523–526. Bristol: IOP Publishing.Google Scholar
Kühlbrandt, W., Wang, D. N. & Fujiyoshi, Y. (1994). Atomic model of plant lightharvesting complex by electron crystallography. Nature (London), 367, 614–621.Google Scholar
Lehmpfuhl, G. (1974). Dynamical interaction of electron waves in a perfect single crystal. Z. Naturforsch. Teil A, 27, 424–433.Google Scholar
Ma, Y., Rømming, C., Lebech, B. & Gjønnes, J. (1992). Structure refinement of Al_{3}Zr using singlecrystal Xray diffraction, powder neutron diffraction and CBED. Acta Cryst. B48, 11–16.Google Scholar
Marthinsen, K., Holmestad, R. & Høier, R. (1994). Analytical filtering of lowangle inelastic scattering contributions to CBED contrast. Ultramicroscopy, 55, 268–275.Google Scholar
Matsuhata, H. & Gjønnes, J. (1994). Blochwave degeneracies and nonsystematic critical voltage: a method for structurefactor determination. Acta Cryst. A50, 107–115.Google Scholar
Matsuhata, H. & Steeds, J. W. (1987). Observation of accidental Blochwave degeneracies of zoneaxis critical voltage. Philos. Mag. B55, 39–54.Google Scholar
Matsuhata, H., Tomokiyo, Y., Watanabe, H. & Eguchi, T. (1984). Determination of the structure factors of Cu and Cu_{3}Au by the intersecting Kikuchiline method. Acta Cryst. B40, 544–549.Google Scholar
Mori, N., Oikawa, T & Harada, Y. (1990). Development of the imaging plate for the transmission electron microscope and its characteristics. J. Electron Microsc. (Japan), 39, 433–436.Google Scholar
Olsen, A., Goodman, P. & Whitfield, H. (1985). Tl_{3}SbS_{3}, Tl_{3}SbSe_{3}, Tl_{3}Sb_{3−x}Se_{x} and Tl_{3}Sb_{y}As_{1−y}Se_{3}. J. Solid State Chem. 60, 305–315.Google Scholar
Saunders, M., Bird, D. M., Midgley, P. A. & Vincent, R. (1994). Structure factor refinement by zoneaxis CBED pattern matching. Proceedings of 13th International Congress on Electron Microscopy, Paris, France, 17–22 July 1994, Vol. 1, pp. 847–848.Google Scholar
Spargo, A. E. C. (1994). Electron crystallography and crystal structure. Proceedings of 13th International Congress on Electron Microscopy, Paris, France 17–22 July 1994, Vol. 1, pp. 959–960.Google Scholar
Spence, J. C. H. & Zuo, J. M. (1992). Electron microdiffraction. New York: Plenum.Google Scholar
Steeds, J. W. (1984). Further development in the analysis of convergent beam electron diffraction (CBED) data. EMAG 1983. Inst. Phys. Conf. Ser. No. 69, pp. 31–36.Google Scholar
Taftø, J. & Gjønnes, J. (1985). The intersecting Kikuchi line technique: critical voltage at any voltage. Ultramicroscopy, 17, 329–334.Google Scholar
Taftø, J. & Metzger, T. H. (1985). Largeangle convergentbeam electron diffraction; a simple technique for the study of modulated structures with application to V_{2}D. J. Appl. Cryst. 18, 110–113.Google Scholar
Tanaka, M. & Tsuda, K. (1990). Determination of positional parameters by convergentbeam electron diffraction. Proceedings of 12th International Congress on Electron Microscopy 1990, Vol. 2, pp. 518–519.Google Scholar
Terasaki, O., Watanabe, D. & Gjønnes, J. (1979). Determination of crystal structure factor of Si by the intersectingKikuchiline method. Acta Cryst. A35, 895–900.Google Scholar
Vincent, R., Bird, D. M. & Steeds, J. W. (1984). Structure of AuGeAs determined by convergent beam electron diffraction. II. Refinement of structural parameters. Philos. Mag. A50, 765–786.Google Scholar
Vincent, R. & Midgley, P. A. (1994). Double conical beamrocking system for measurement of integrated electron diffraction intensities. Ultramicroscopy, 53, 271–284.Google Scholar
VoigtMartin, I. G., Yan, D. H., Gilmore, C. J., Shankland, K. & Bricogne, G. (1994). The use of maximum entropy and likelihood ranking to determine the crystal structure of 4[4′(Ndimethylamino)benzylidene]pyrazolidine3,5dione at 1.4 Å resolution from electron diffraction and highresolution electron microscopy image data. Ultramicroscopy, 56, 271–288.Google Scholar
Voss, R., Lehmpfuhl, G. & Smith, D. J. (1980). Influence of doping on the crystal potential of silicon investigated by the convergent beam electron diffraction technique. Z. Naturforsch. Teil A, 35, 973–984.Google Scholar
Watanabe, D., Uyeda, R. & Fukuhara, A. (1969). Determination of the atomic form factor by highvoltage electron diffraction. Acta Cryst. A25, 138–140.Google Scholar
Zou, X. D., Sukharev, Y. & Hovmöller, S. (1993). ELD – a computer program for extracting intensities from electron diffraction patterns. Ultramicroscopy, 49, 147–158.Google Scholar
Zuo, J. M., Høier, R. & Spence, J. C. H. (1989). Threebeam and manybeam theory in electron diffraction and its use for structurefactor phase determination in noncentrosymmetrical crystal structures. Acta Cryst. A45, 839–851.Google Scholar
Zuo, J. M., Spence, J. C. H., Downs, J. & Mayer, J. (1993). Measurement of individual structurefactor phases with tenthdegree accuracy: the 00.2 reflection in BeO studied with electron and Xray diffraction. Acta Cryst. A49, 422–429.Google Scholar