MICROSCOPY RESEARCH AND TECHNIQUE 46:220–233 (1999) Accurate Structure Refinement and Measurement of Crystal Charge Distribution Using Convergent Beam Electron Diffraction J.M. ZUO* Department of Physics and Astronomy, Arizona State University, Tempe, Arizona 85287 KEY WORDS energy filtering; imaging filter; 2-D detectors; structure factor ABSTRACT The method for accurate structure refinement from energy-filtered convergentbeam electron diffraction (CBED) patterns is described with emphasis on recent progress in using imaging filters and 2-D detectors. Details are given about the underlying theoretical model and the statistical analysis of experimental data. The relationship between crystal potential and charge density is also derived for crystals at thermal equilibrium. The method is applied to the refinement of Si (111) and (222) structure factors using various goodness-of-fit (GOF) criteria. Results show that the refinement method is robust and highly accurate. The importance of the experimentally measured structure factors is illustrated through the study of the charge density in MgO. With the measured structure factors, it is possible to obtain details about the charge redistribution due to crystal bonding. Microsc. Res. Tech. 46:220–233, 1999. r 1999 Wiley-Liss, Inc. INTRODUCTION Refinement is an integral part of the structure determination process. Accurate information about atomic positions, Debye-Waller (D-W) factors, and crystal potentials or charge distributions is obtained through the refinement process of comparing experimental and theoretical intensities, after the initial estimation of atomic positions using direct methods. While considerable difficulties exist for using direct methods with electron diffraction (ED) due to the multiple scattering, refinement of ED data is not only possible, but also can be done with high accuracy. This is because the theory for dynamical ED is well developed, and, once the structure model is known, theoretical intensities can be calculated relatively quickly with today’s fast computers. In addition to X-ray or neutron diffraction with their established methods and sophisticated software packages, the ED method is needed for the following reasons: (1) Electron beam can be focused down to a probe of sub-nanometer size. Such a small probe can be used to characterize materials of similar dimensions in combination with other electron microscopic methods. (2) Electron beam also interacts with crystals differently from X rays or neutrons; with its Coulomb interaction the electron probes both electrons and nuclei in crystals. (3) The interaction strength with matter is ⬃104 times stronger for 100 kV electrons than for X rays. Recent developments show that it is possible to obtain accurate measurements of unit cell parameters (Tomokiyo et al., 1993; Zuo, 1992; Zuo et al., 1997a), structure factors (Holmestad et al., 1995; Saunders and Bird 1995; Zuo and Spence 1991), atomic positions (Tsuda and Tanaka 1995), and D-W factors (Nüchter et al., 1995) using quantitative convergent beam electron diffraction (CBED). This is due to the fact that most applications in materials science involve the characterr 1999 WILEY-LISS, INC. ization of structural modifications, for which refinement methods can be readily applied. It is also significant because there is a genuine need of methods for accurate structural parameter measurement; e.g., to determine the charge distribution due to crystal bonding. Accurate structure parameters, such as the distribution of charge, provide the indispensable test of the theoretical approximations (Zuo et al., 1997b). Traditional X-ray methods have been less successful in measuring charge distributions of crystals, especially the charge redistribution from bonding. A very small fraction of charge redistributes when atoms form crystals; this can only be measured with highly accurate structure factors. The accuracy of X-ray structure factors obtained using the kinematical approximation is limited by extinction and absorption effects. Although quantitative analysis of ED is a more recent development, the history of electron crystallography dates to the early stages of the development of electron microscopy. The Bloch wave theory for electron multiple scattering was originally formulated by H. Bethe (1928) in his thesis. The first CBED pattern was obtained by Möllenstedt as early as 1939 (Möllenstedt, 1939) from mica. MacGillavry (1940) first attempted structure-factor measurement from an experimental CBED pattern by using the two-beam theory of Blackman (1939). During the 1970s, there were a number of attempts to measure structure factors from CBED patterns using more sophisticated methods, most noticeably by Cowley, Goodman, Moodie, Lemphful, and Ichimiya (for a review, see Spence, 1993). Recently, the dramatic increase in computer speed and the development of new detectors have brought significant progress Contract grant sponsor: NSF; Contract grant number DMR 9412146. *Correspondence to: J.M. Zuo, Dept. of Physics and Astronomy, Arizona State University, Tempe, AZ 85287. E-mail address: firstname.lastname@example.org Received 6 October 1997; accepted in revised form 5 April 1999 221 REFINEMENT USING CBED in this field. The fast computer makes it possible to do a full dynamical calculation of CBED intensities. The combination of the energy-filter (Krivanek et al., 1995; Lanio, 1986; Rose and Krahl, 1995; Tsuno, 1997) and two-dimensional (2-D) digital detectors—slow-scan CCD cameras (SSC) (Spence and Zuo, 1988) and imaging plates (IP) (Ogura et al., 1994)—provides accurate experimental intensities with a speed that was not possible before. The principle of structure-factor measurement from CBED patterns was outlined in the study of GaAs for structure-factor amplitudes (Zuo et al., 1988) and CdS for structure-factor phases (Zuo et al., 1989). Spence (1993) gave a detailed review and a survey of other ED techniques. Since then, the method has been improved significantly. Especially, by the combined use of an imaging energy filter and 2-D detector, 2-D diffraction patterns can be routinely recorded and used in fitting. A 2-D diffraction pattern recorded in regular arrays gives significantly more experimental information than a few line scans. To take advantage of the 2-D information available from the new detectors, we have developed a new algorithm (Zuo, 1997a). However, these new technologies have also introduced a new set of problems. Especially, it is necessary to correct for the nonuniformity and the limited resolution in the filter and/or detector. It is also necessary to estimate the amount of noise introduced by the detector. We will address these problems in this paper. This paper concentrates on the refinement method and the measurement of individual structure factors. It is also possible to refine structure parameters directly from CBED patterns. Many works in this area have been published recently. Specifically, for the measurement of atomic positions see Tsuda and Tanaka (1995) and Tomokiyo et al. (1996); for the measurement of the D-W factor see Nüchter et al. (1995); and for the measurement of lattice parameters using the program described here see Zuo et al. (1997a). An attempt has also been made to measure atomic positions using the method described here; for an initial report see Holmestad et al. (1997). For the application of quantitative CBED to other areas of crystallography and materials problems, see Holmestad et al. (1999). The method discussed here is general and can be applied to both systematic and zone-axis cases. However, in a zone-axis orientation, additional benefits can be obtained using symmetry. For refinement using zone-axis patterns, see Saunders and Bird (1995). independent electron in a potential field: [K 2 ⫺ (k ⫹ g)2] Cg ⫹ 兺U ⫽ 0. (1) Here we have K 2 ⫽ k 2o ⫹ Uo, (2) Ug ⫽ U gC ⫹ U 9g ⫹ iU8g, (3) Ug ⫽ 2m 0 e 0 Vg /h2 (4) and where is the optical potential for beam electrons, which consists of the crystal potential V, absorption V8, and a correction to the crystal potential due to virtual inelastic scattering V9. The theory of optical potentials for an electron beam was developed by Yoshioka (1957). The most important contribution to U8 and U9 for g ⫽ 0 comes from inelastic phonon scattering. Both U’ and U9 can be calculated, given the phonon distribution in the crystal. Details on the evaluation of the absorption potential are given by Bird and King (1990), by Weickenmeier and Kohl (1991) for atoms with isotropic D-W factors, and by Weickenmeier and Kohl (1998) and Peng (1997) for anisotropic D-W factors. The virtual scattering term U9 has been estimated by Rez (1978b) for limited cases, and more recently by Anstis (1996). The U9 term is significantly smaller than U’ for silicon according to Rez. However, for heavy elements the U9 may become important (Anstis, 1996). For an electron beam with a sufficiently small diameter, the diffraction geometry can be approximated by a crystal slab with surface normal n. Letting k ⫽ K ⫹ ␥n, (5) where ␥ is the dispersion of the wavevector inside the crystal, and inserting equation 5 into 1 leads to K 2 ⫺ (K ⫹ g)2 ⫺ 2Kn␥ ⫺ ␥2 ⫽ 2KSg ⫺ 2Kn␥ ⫺ ␥2, inside the square bracket. Neglecting the backscattering term of ␥2, we obtain from equation 1 2KSgCg ⫹ 兺U ghCh 1 ⫽ 2Kn 1 ⫹ h THEORY The Bloch wave method is used to calculate theoretical intensities. For ED from perfect crystals with relatively small unit cells, the Bloch wave method is preferable because of its flexibility and accuracy. The multi-slice method or other similar methods are best in the case of diffraction from strained crystals or crystals containing defects. A more detailed description of the Bloch wave method can be found in Spence and Zuo (1992). For high-energy electrons, the exchange and correlation between the beam and crystals electrons can be neglected (Rez, 1978a), and the problem of ED reduces to solving the Schrödinger equation for an ghCh h gn 2 ␥C . Kn g (6) Here Kn ⫽ K · n and gn ⫽ g · n. Equation 6 reduces to an eigen equation by renormalizing the eigenvector (Metherell, 1975): 1 Bg ⫽ 1 ⫹ K2 gn 1/2 Cg. n To simulate a CBED pattern, a coordinate system is needed to specify each beam direction, K. We use the zone-axis coordinate system, in which the incident beam direction is specified by its tangential component (Kx, Ky) in the closest zone axis. For each point inside 222 J.M. ZUO the CBED disk of g, the intensity is given by: Ig(Kx, Ky) ⫽ 0 g(Kx, Ky) 02 ⫽ 0兺 and 0 Vg ⫽ 2 ci(Kx, Ky) C gi(Kx, Ky) exp [2i␥i(Kx, Ky) t] , i U geff ⫽ Ug ⫺ 兺 2KSh h , 3 (7) ⫽ with eigenvalues ␥i and eigenvector C gi from diagonalizing equation 6. Here ci is the first column of the inverse of the eigenvector matrix or the excitation coefficient as determined by the incident-beam boundary condition (Spence and Zuo, 1992). The calculations converge with increasing number of beams included. Straightforward use of equation 6 with a large number of beams is often impractical since the matrix diagonalization is very time-consuming. The computer time needed to diagonalize an N ⫻ N matrix is proportional to N3. A solution to this is to use the Bethe potential: Ug⫺hUh 兰 d r exp (2ig · r)7V(r)8 (8) 0e0 4⑀o · ⫽ 兰d 3 r8 exp (2ig · r8) 57Z(r8)8 ⫺ 7(r8)86 (12) exp [2ig · (r ⫺ r8)] 兰d r 3 0 r ⫺ r8 0 0e0 4 ⑀o g2 2 (Zg ⫺ Fg). (13) Here we have Fq ⫽ ⫽ 1 N 1 兰 d r exp (2iq · r) 7(r)8 3 N exp (2iq · R ) 兺 f (q) exp (2iq · r ) N 兺 l l⫽1 i i i ⫻ 7exp (2iq · ui)8 and ⫽ ␦(q ⫺ g) (14) 兺 f (q) exp (2iq · r ) i i i 2KS geff ⫽ 2KSg ⫺ 兺 Ug⫺hUh⫺g (9) 2KSh h ⫻ 7exp (2iq · ui)8, and similarly, for weak beams. Here the summation is overall weak beams h. The Bethe potential accounts for the perturbation effects of weak beams. A weak beam is defined when 0U 0ⱖ KSg max. (10) g The max is selected for the best theoretical convergence (Birkeland et al., 1996; Zuo and Weickenmeier, 1995). Beams are selected subject to cutoff criteria of maximum g length and their perturbation strength. These should be tested to check their effects on the calculated intensities. For a 100-kV electron traveling at a speed 1⁄3 that of light, the time it spends inside a TEM specimen is about 10⫺15 seconds. The typical phonon frequency is about 1012 Hz. Thus, the fast electron sees individual frozen configurations of a vibrating lattice. The experimentally observed diffraction pattern at a 1-second exposure time is an average of individual diffraction patterns from ⬃1012 instant configurations of the crystal. It can be shown that the average diffracted beam at a reciprocal lattice point is equal to that scattered by an average potential 7V(r)8 under both the kinematical and dynamical diffraction conditions. In general we have, 7V(r)8 ⫽ 0e0 4⑀o 兰 d r8 3 7Z(r8) ⫺ (r8)8 0 r ⫺ r8 0 , (11) Zq ⫽ ␦(q ⫺ g) 兺 Z exp (2iq · r ) i i i (15) ⫻ 7exp (2iq ·ui)8. Here u is the displacement from the equilibrium atomic position and g is a reciprocal lattice point. The quantity, 7exp (2iq · ui)8, is the temperature factor. For harmonic vibrations, it can be shown that (Willis and Pryor, 1975) T(g) ⫽ 7exp (2ig · u)8 ⫽ exp (⫺227(g · u)28). (16) For isotropic vibrations, this reduces to the familiar D-W factor T(g) ⫽ exp (⫺22g27u28) ⫽ exp (⫺Bg2/4), (17) with B ⫽ 82 7u28. For the anisotropic case, see Willis and Pryor (1975). Equation 12 is the basic relationship between potential and charge density as measured by electrons or x-rays for the study of crystal bonding. It is clear from equation 13 that the thermal effects on both potential and charge density can be described by the same temperature factors. EXPERIMENT The experimental setup using an in-column ⍀ energyfilter is illustrated in Figure 1. A probe (P) is formed on top of a thin crystal plate by de-magnifying the source REFINEMENT USING CBED Fig. 1. 223 Schematic diagram of experimental setup using a in-column imaging energy filter. using the condenser lens. A circular aperture is placed after the condenser lens, which forms a cone of illumination. A diffraction pattern consisting of disks is formed at the back focal plane of the objective lens. A point in the CBED disk and all conjugate points in other disks (related by reciprocal lattice vectors) constitute a diffraction pattern for one incident beam direction in the illuminating cone. In Figure 1, the filter accepts the image of the probe formed by the objective lens. In the LEO 912 electron microscope, there is an additional intermediate lens between the filter and objective lens. The filter forms a series of probe images at its exit plane, dispersed according to the beam energy. An energy-selection slit is placed around the zero-loss image to obtain the energy-filtered CBED pattern, which is recorded with a 2-D detector. The condensor aperture determines the CBED disk size. The disk should be comparable to the reciprocal-lattice spacing to avoid excessive overlap, which makes the interpretation of experimental data difficult. For large unit-cell crystals, the large angle CBED (LACBED) technique may be useful. LACBED selects a single diffracted beam by using a small selected-area aperture. LACBED also removes part of the large-angle inelastic scattering due to phonons, which is not possible with the current energy-filter (Morniroli et al., 1997). The diffracted beams pass through an energy filter, which is placed after the final intermediate lens and before the projector lens of the microscope. Alternatively, a post-column energy filter can be placed below the viewing screen (Krivanek et al., 1995). With the in-column filter, it is possible to have a combination of different detectors, and it also takes the full advantage of the projector lens of the microscope. Uhlemann and Rose (1994) made a theoretical comparison between different filters. Details about the ⍀-filter can be found in Lanio (1986), Rose and Krahl (1995), and Tsuno et al. (1997). The LEO 912 ⍀ filter uses the design of Lanio (1986). A description of the post-column Gatan Imaging Filter (GIF) is given by Krivanek et al. (1995). For the ⍀ filter, the filter takes an image (in diffraction mode) of the diffraction pattern at the entrance pupil plane and forms a diffraction pattern at the exit pupil plane and an image at the energy-slit aperture. Figure 2 shows a zero-loss energy-filtered Si CBED pattern and the corresponding spectrum (see inset in Fig. 2) using the 224 J.M. ZUO be 40 mm. At such a small CL, it is often difficult to adjust for lens distortions. Additionally, there are some residual distortions in the GIF, which are difficult to characterize. The geometric distortion is almost absent in the ⍀ filter owing to its mid-plane symmetry (Rose and Krahl, 1995), in principle. For quantitative analysis, the energy-filtered CBED pattern is recorded digitally with an SSC or IP in an array of pixels. Both detectors have excellent linearity and dynamic range in each pixel. For a comparison between these two detectors, see Zuo (1996) and Zuo et al. (1996). Two important characteristics of a linear detector for quantitative analysis are the point-spread function (PSF) and detector quantum efficiency (DQE). The PSF determines the resolution of as-recorded images, while DQE measures the amount of noise introduced in the detection process. Methods for measuring these are given by Zuo (1996) and Weickenmeier et al. (1995). Figure 4 shows the measured modulation transfer function (MTF) and DQE for Gatan model 679 SSC. In general, the recorded image can be expressed as ⫽ Hf ⫹ n. Fig. 2. Energy-filtered Si  zone axis pattern using the ⍀-filter of Leo 912 electron microscope at 120 kV with 15 eV energy window. The corresponding spectrum is shown at lower right corner. The white lines indicate the position of the energy window. Arrows indicate the energy-filtering envelope. (The size of the envelope can be enlarged with free lens control). LEO 912 electron microscope with an energy slit 16 eV wide. The envelope seen in Figure 2 is due to the non-isochromaticity caused by the axial second-order aberrations at the energy-selection plane (Rose and Krahl, 1995). The effects of this aberration make a circle in the object plane into an ellipse; the center of which is displaced by a distance proportional to the square of the radius. This is illustrated in Figure 3. A disk forms a flattened cone-shaped aberration figure. Figure 3c shows the experimentally observed aberration figure for silicon. For a large object, the aberration figure overlaps on the energy-dispersion plane with those for different energy losses. With a finite energy slit, the primary beam energy of the filtered pattern differs at different radii from the optical axis (called non-isochromaticity). By using a narrower energy slit, it is possible to reduce the contributions from electrons of different energies. However, this also results in a smaller field of view. For energy filtering with the zero-loss peak, the problem is most severe for metals with large plasmon losses. In the CBED mode, the image is made of diffracted beams. They are sufficiently separated in the Leo 912 ⍀ filter for the filtering of a few reflections (see Fig. 2). We observed no significant difference between the zero-loss disk intensities with 8and 16-eV windows. The plasmon energy of silicon is about 18 eV. Krivanek et al. (1995) reported a better performance for isochromatic imaging using the GIF. The field of view of the GIF is limited by the smallest camera length (CL) available in the microscope and the size of the CCD used for detection. The GIF has a magnification of 15⫻, so to record a diffraction pattern of 600 mm CL with the GIF, the microscope CL needs to (18) Here and f are the recorded and original image, respectively, n is the noise, and H is the PSF operator. The effects of the PSF can be removed by deconvolution. However, the direct deconvolution of a recorded image using the inverse of the PSF leads to an excessive amplification of noise. To overcome this, we use the constrained image-restoration technique by employing a filter of type (Ren et al., 1997): Q(u, v) ⫽ H *(u, v) H *(u, v) H(u, v) ⫹ ␥P(u, v) (19) with P(u, v) as a smoothing function and ␥ adjusted to satisfy the constraint 00 g ⫺ Hf̂ 00 ⫽ 00 ⌬n 00 . (20) Here f̂ is the deconvoluted image and ⌬n is the estimated additional noise introduced in the detection process. There are a number of choices for P(u, v). The optimum in the least-square measure is the Wien filter, in which P(u, v) is the ratio of the power spectra of signal and noise. This requires a prior knowledge of the signal, which is often unknown. Alternatively, we may take P(u, v) ⫽ [⫺4(u2 ⫹ v2)]2, which makes the restored image as smooth as possible. The value of ␥, which measures the degree of removal of the PSF, with 0 as complete, depends on the amount of noise in the recorded image—the smaller the noise, the better the deconvolution. Other procedures for deconvolution are also available. The Richardson-Lucy method described by Snyder et al. (1993) is specifically targeted for Poisson processes, which can be applied to CCD images. Figure 5 shows the relative importance of energyfiltering and image deconvolution. The example used is the TiA1 (200) systematics. The top is the unfiltered CBED pattern, which shows a large amount of background in the line profile. The filtering removes most of the background. The remainder is mostly due to the REFINEMENT USING CBED 225 Fig. 3. The aperture aberration in uncorrected ⍀ energy filter. a: Aberration figure in energy selection plane. b: Corresponding circle in the object plane for ellipse in a. c: Experimentally observed spectrum for Silicon in image mode. limited resolution of the detector, which is removed by deconvolution. The deconvolution also makes edges sharper. For comparison, we show the fitted theoretical pattern at the bottom. The effects of filtering and deconvolution are large compared to the amount of noise expected from the detector. The quality of a CBED pattern depends on the sample and the microscope. Critical alignment of the microscope is required. Special attention is needed for the optimum objective lens setting, specimen z-position, correction for projector lens distortion, and the choice of probe size. For the use of an SSC, a highquality image is needed to correct for the gain variations. For a relatively flat sample with no defects, such as silicon, acquiring a good-quality CBED pattern in a non-zone-axis orientation for quantitative analysis is fairly easy. Acquiring a good zone-axis pattern is considerably more difficult; the strong dynamical channeling condition is very sensitive to defects and surface imperfections or contamination. Distortions of CBED patterns due to crystal faults such as stacking faults and dislocations are easy to spot. The deficiency line tends to split owing to the strain field associated with these faults. Thickness variation under the probe also distorts the CBED pattern; this can be reduced with a smaller probe. The ultimate check of the quality of CBED patterns is the refinement itself, which measures the degree of agreement between theory and 226 J.M. ZUO Fig. 4. Experimentally measured (a) MTF and (b) DQE of Gatan model 679 SSC. The dashed lines in b are projected DQE with zero noise in gain image. experiment. Additional assessment of experimental diffraction pattern quality can be made in a zone-axis orientation, where the symmetry-related points can be used to detect distortions. REFINEMENT Automated refinement from CBED intensities is achieved by finding the best fit between experiment and theory. The best fit is defined by using a goodness-of-fit (GOF) parameter: GOF ⫽ ⑀ 兺 h(I exp i , i, I it). tion of (I t ⫺ I exp)/. Box and Tiao (1992) defined a class of probability distributions (I exp 0 p, ) ⬀ 15 exp ⫺ ⌫[(1 ⫹ ␤)/2] Here the function h measures the agreement between the experimental intensity, Iexp, with estimated uncertainty and the theoretical intensity, It. The theoretical intensity is calculated with parameters, p. ⑀ is the normalization coefficient. In the maximum-likelihood approach, h is the logarithm of the probability distribu- 0 0 2 (22) 2/(1⫹␤) , with ⫺1 ⬍ ␤ ⱕ 1. For ␤ ⫽ 0, equation 22 reduces to the Gaussian distribution. On the basis of equation 22, we can define a series of GOF’s using different values of ␤: (21) i 6 ⌫[3(1 ⫹ ␤)/2] 1/(1⫹␤) I exp ⫺ I t( p) GOF ⫽ ⑀ 兺0 i I iexp ⫺ I it( p) i 0 2/(1⫹␤) . (23) For constant standard error, , the lowest GOF gives the maximum likelihood of the probability distribution of equation 22. The definition of equation 23 encompasses the two most commonly used measures in 227 REFINEMENT USING CBED Fig. 5. A comparison between the effects of energy-filtering and deconvolution. For reference, the bottom shows the fitted theoretical pattern. crystallography, the 2 (␤ ⫽ 0) and normalized R-factor (␤ ⫽ 1). For large ␤, the probability distribution falls off slower than the Gaussian, which gives a more robust measure against systematic errors in theory and experiment due to its long-tailed distribution. For cases where differences between theory and experiment are mostly normally distributed, except for a few outlier points, the 2 refinement can be distorted owing to the presence of these outlier points (for discussion on this, see Huber, 1981). Huber proposed an alternative measure considering a small percentage of outlier points: M⫽⑀ 兺 i hi 10 I iexp ⫺ I it( p) i 02 , (24) (for example 95%) is covered by the normal distribution. In next section we will show how the choice of GOF affects the refinement results using a test case of silicon. The experimentally recorded intensity in each detector pixel (i, j) is the sum of all intensities within the pixel. Intensity mixing between different beam directions is also expected from the finite spatial resolution of the detector and energy window of the filter; in particular, both small- and large-angle thermal diffuse scattering are not removed by the energy filter. A general expression for the theoretical intensity considering all these factors is: I i,t j ⫽ with h(t) ⫽ 5 t2 t⬍a 2at ⫺ a2 tⱖa 兰兰 dx8dy8t(x8, y8) C(x ⫺ x8, y ⫺ y8) i j (26) ⫹ B(xi, yj). (25) This definition adopts the normal distribution for the center and exponential distribution for the tail. The parameter a is selected so that the majority of points Here, the diffracted intensity, t, convoluted with the response function, C, and the background intensity, B, are integrated over the pixel. Simple models of C are the delta and Gaussian functions, which can be implemented easily. The background function is often as- 228 J.M. ZUO Fig. 6. A simplified flow chart of EXTAL program. Two optimization cycles are employed for structural and geometric parameters respectively. The optimization for geometric parameters is repeated for each individual diffraction pattern. sumed constant within the CBED disk (Saunders and Bird, 1995), which seems sufficient for the refinement process. Depending on the experimental intensities collected, the parameters can be structure-factor amplitudes and/or phases, absorption coefficients, lattice constants, atomic position parameters, and D-W factors. Other parameters include crystal orientation, thickness, electron accelerating voltage, detector function, and residual diffuse background intensities. For structure-factor measurements, we can choose between systematic, three-beam, or zone-axis diffraction conditions. We prefer the systematic condition for its simplicity, short computational time, and good accuracy. Estimation of errors in refined parameters is quite involved in the case that the distribution is not normal. For discussions on this, see Box and Tiao (1992). In the normal case, the precision of the measured parameters is given by ⫺1 a2k ⫽ 2D kk , (27) and Dkl ⫽ 1 21 2 1 ⭸I it ⭸I it 兺 i 2 i ⭸pk ⭸pl . (28) The partial derivative of the intensity with respect to a parameter can be calculated numerically or by using perturbation theory. General formulas using the firstorder perturbation method for these derivatives are given by Zuo (1997b). There are a number of refinement algorithms (Zuo and Spence, 1991; Nuchter et al., 1995; Saunders and Bird, 1995; Birkeland 1997). The REFINE program by Zuo and Spence (1991) uses line scans of intensities. Modifications of this program have also been made to include non-parallel line scans (Nuchter et al., 1995; Birkeland 1997). The algorithm of Saunders and Bird (1995) uses a Fourier-transform method for the evaluation of derivatives and has been mostly applied to zone-axis patterns. We have developed an new algorithm EXTAL (Electron crys(X)TALlography) for quantitative analysis of 2-D CBED patterns (Zuo, 1997a; Zuo, 1998b). The simplified flowchart of the program is shown in Figure 6. This program is capable of dealing with multiple CBED patterns by separating the structural parameters of structure factors, atomic positions, and D-W factors from the geometrical parameters of crystal orientation, image position, camera length, and sample thickness. Since ED patterns are generally twodimensional owing to the short wavelength of the Fig. 7. a: Experimentally recorded energy-filtered Si (1 1 1) systematic CBED pattern. b: Intensities along lines indicated in a and the best theoretical fit. c: Difference between theory and experiment plotted in b normalized by the estimated error in experimental intensities. a b Fig. 7. 230 J.M. ZUO TABLE 1. Dependence of Si (1 1 1) and (2 2 2) Refinement Results on the Probability Distributions of Equation 22 Å2 U(1 1 1) 10⫺1 U8(1 1 1) 10⫺3 U(2 2 2) 10⫺3 ␤⫽0 ␤ ⫽ 0.25 ␤ ⫽ 0.5 ␤ ⫽ 0.75 ␤ ⫽ 1.0 Mean 0.4732(5) 0.832 0.893(54) 0.4737 0.872 0.912 0.4746 0.807 0.862 0.4727 0.829 0.990 0.4738 0.825 0.949 0.4736(7) 0.833(23) 0.921(50) incident electrons, using multiple diffraction patterns in different projections enables the measurement of three-dimensional parameters such as atomic positions and the parameters of the three-dimensional unit cell. The separation between structural and geometrical parameters is achieved using two optimization cycles. For each trial in the structural parameter optimization, an independent optimization is carried out for the geometrical parameters. This is repeated for each diffraction pattern included. Considerable saving of computer time is achieved by fixing the beam direction in the theoretical calculation and locating the corresponding directions in the 2-D pattern as suggested by Gribelyuk and Ruhle (1993). When additional image processing is applied to both experimental and theoretical CBED patterns, the program has shown to be very effective in measuring lattice parameters in the presence of dynamical effects. The advantage of this program, compared with the REFINE program, is that multiple line or area scans can be taken, which allows the user flexibility in designating the experiment for the maximum sensitivity to the measured structural parameters. An example of using this program for structure-factor measurement is shown next. For an example of measuring lattice parameters, see Zuo et al. (1997a). REFINEMENT OF SI (111) AND (222) STRUCTURE FACTORS As an example, we describe the structure factor measurements of Si (111) and (222) using the methods described here. Silicon is a good test case because of existing X-ray pendellosung measurements. The experimental CBED pattern of silicon (1 11) systematics was recorded by tilting about 12° off the  zone at room temperature and with a beam energy of 196.7 kV. This is shown in Figure 7a. The pattern was taken on a Philips CM200 with a field-emission gun and equipped with a GIF (BEML, China). The characteristics of the CCD were measured following the procedures described by Zuo (1996). Figure 7a was deconvoluted using the method described in section 3 (Ren et al., 1997). The orientation was estimated by indexing the diffraction pattern using the IdealMicroscope program (Zuo et al., 1993). The amount of experimental data in figure 7a far exceeds the number of parameters to be determined, so we select only some of the pixels in the experimental pattern for fitting. The typical number of data points is about a few thousand. We chose these data points by line scans. These lines are drawn with consideration given to their sensitivity to certain parameters. For example, intensities along a line across a diffracted disk are sensitive to structure factors and specimen thickness, intensities along a line across the zero disk are sensitive to absorption parameters, and intensities along a line across ZOLZ and HOLZ lines are sensitive to orientation and magnification. The line scans used TABLE 2. Dependence of Si (1 1 1) and (2 2 2) Refinement Results on the M-Estimator of Equation 24 and Comparison With the Experimental X-Ray Results* Å2 A ⫽ 3.0 A ⫽ 2.0 A ⫽ 1.5 X ray U(1 1 1) 10⫺1 U8(1 1 1) 10⫺3 U(2 2 2) 10⫺3 0.4726 0.822 0.867 0.4731 0.823 0.914 0.4738 0.825 0.978 0.4736(4) 0.943(5) *For a list of X-ray structure factors of Si, see Zuo et al. (1997b). for silicon (1 11) systematics are shown in Figure 7a. The refinement program EXTAL was used with different GOF criteria. Theoretical intensity was calculated with 279 beams, selected using the beam selection criteria described in Zuo and Weickenmeier (1995). The refinement program was run repeatedly with different starting points to ensure that the real minimum was found. The refinement depends on the GOF criterion selected. To check the effects of this and to see which criterion may suit our purpose best, we refined Figure 7a with various GOF defined in Refinement. Table 1 shows the refinement results with different ␤ for the GOF of equation 23. The estimated error is quoted in parentheses for the normal distribution (␤ ⫽ 0). The refinement results depends on the selected GOF or ␤ value. The column ‘‘Mean’’ shows the average of five different refinements and its standard deviation. The difference between the (222) structure factors determined using the 2 and R-factor is somewhat large compared with the estimated error. Table 2 compares the results using the M estimator of Huber with different cutoff values of a. As a decreases, the results approach the ones obtained with the Rfactor. Compared with the X-ray value, the mean from Table 1 gives the best agreement. Of the individual GOFs, the R-factor seems the best. The results from the standard 2 are acceptable within the estimated error; although the (222) is on the low side compared with the X-ray value. To check the success of the selected GOF, we look at the distribution of the residual difference between experiment and theory. Figure 8 shows the histogram of the normalized difference obtained with 2, the Huber estimator of a ⫽ 2.0, and the normalized R-factor. For comparison, we also fitted the histogram with Gaussian (Fig. 8a and b) and exponential (Fig. 8c) distributions. The R-factor resulted in the largest number of points having deviations within the estimated error. In obtaining the mean from different distributions in Table 1, we assumed an equal probability for the distribution. A more rigorous averaging using Bayesian statistics would require the evaluation of the probability for the distribution with the given data. This would require integration in the parameter space, which is difficult to do in this case. For a full discussion on the robust statistics in Bayesian approach, see Box and Tiao (1992). REFINEMENT USING CBED 231 Fig. 8. Histogram of residual difference, (I t ⫺ I exp)/, as obtained from (a) chi-square fit (b) M fit with a ⫽ 2.0, and (c) R-factor fit. For details, see text. The results here show that the refinement method is highly accurate for the structure-factor measurement of both strong and weak reflections. The accuracy for the Si (111) reflection is comparable to the X-ray pendellosung method. Unlike the X-ray method, the ED method discussed here is general. With the small probe of CBED, it can be applied to most crystals with small unit cells. There is no fit-for-all magic solution in choosing the best GOF criteria. Results here show that the refinement does depend on the GOF within the estimated error bar. The R-factor gives better results than 2 in this case. CHARGE DENSITY IN MgO One of the important applications of quantitative CBED is the study of crystal bonding by accurate measurement of low-order structure factors. An ex- ample of this is the study of the charge density in MgO (Zuo et al., 1997c; Zuo 1997b). For applications of charge-density measurements to important materials problems, see Holmestad et al. (1999). Magnesium oxide is a prototype of ionic bonding with its NaCl structure. There is a general lack of experimental knowledge on the charge distribution in ionic crystals. In studying MgO, we want to see whether the charge distribution of MgO can be described in the traditional picture of ionic bonding with a charge transfer of two electrons from Mg to O. We also want to see the charge distribution of oxygen. With oxides playing an increasingly important role in materials with new properties, such as high-Tc superconductivity and colossal magnetoresistence, understanding oxygen states and their dependence on crystal structure can provide an important clue to the mechanism for these exotic phenomena. 232 J.M. ZUO The MgO specimen was prepared by burning magnesium ribbons in air and collecting MgO smoke on a copper grid covered with a thin carbon film. A platelet of micrometer size and about 80 nm thick was selected from different shapes of MgO particles. The experiment was performed on a Philips 400T electron microscope. The intensities were measured by deflecting the CBED pattern over the entrance aperture of a Gatan 607 serial energy-loss spectrometer. This was done before the energy filters and parallel detectors were available. This setup still offers unsurpassed resolution and signalto-noise ratio when conditions in the microscope are stable during the scan. Details about the measurement have been published (Zuo, 1997b). Table 3 lists all experimentally measured structure factors together with theoretical models for comparison. The theoretical structure factors are calculated using different models. The Dirac-Fock (DF) spherical ions are calculated using the multi-configurational DF program of Rez et al., 1994 and a Watson ⫹2 well for oxygen, which is an atomic calculation. In this model, we found that the radius of 1.2 Å gives the best fit to experimental structure factors. Theoretical structure TABLE 3. Listing of Present Measured Low-Order Structure Factors of MgO and Comparison With Theory* hkl 111 200 220 311 222 400 331 420 422 R-factor Present DF-N1 DF-ION2 LDA3 GGA4 11.142(20) 52.89(3) 40.68(8) 12.41(12) 33.75(12) 29.01(8) 10.06(10) 25.1(2) 22.7(3) 12.389 52.030 41.073 12.309 34.005 28.993 9.6401 25.256 22.371 .011 11.090 53.040 41.062 12.633 33.800 28.790 9.7352 25.065 22.222 .0072 11.175 52.765 40.953 12.356 33.777 28.949 9.6025 25.205 22.348 .0063 11.082 52.918 41.072 12.401 33.865 29.012 9.6533 25.256 22.390 .0067 *Superimposed spherical 1atoms and 2ions calculated using Dirac-Fock method; Crystal structure factor calculated using LAPW and the 3LDA or the 4GGA. factors are calculated using the full-potential linearized augmented plane wave (LAPW) method as implemented in the WIEN95 package (Blaha et al., 1990) for crystals. The calculation uses approximated functionals for the exchange and correlation energies. Two approximations are used. One is the local density approximation (LDA) and the other is the generalized gradient approximation (GGA). The LDA gives a slightly better fit than the GGA, but the difference is small. The deformation map, formed by the difference between the crystal charge density and the charge density of superimposed atoms, is used to show the bonding charge distributions. Figure 9 shows the difference in charge density between the experimental data and superimposed neutral atoms. The continuous lines show excess electrons and the broken lines show the depletion of electrons. Figure 9 clearly shows the charge transfer from Mg to O. The charge distribution around the oxygen is slightly non-spherical. A multiple analysis of the MgO charge density shows a small hexadecapole component. It was found that the charge distribution in MgO can be in the full ionic form. However, the distribution of transferred electrons is significantly broader than predicted by theory. This suggests some effects due to the polarization of charge in the ionic lattice due to thermal vibrations. For details about this, see Zuo et al. (1997c). CONCLUSIONS Recent developments in quantitative ED have shown that it is possible to extract quantitative structure information from CBED patterns. The success relies on quantitative analysis of experimental and theoretical intensities. The experimental setup for the measurement of intensities was found sufficient. The theoretical intensity is calculated taking the full account of dynamical effects. Experimentally measured structure factors provide an accurate description of charge distributions Fig. 9. MgO charge density difference map between crystal and superimposed neutral atoms in a-b plane for (a) experiment and (b) theory using LAPW and LDA. REFINEMENT USING CBED in real crystals. For MgO, the experimental charge density can be approximated by a superposition of spherical Mg2⫹ and O2⫺ ions. The additional difference between theory and experiment is due to the details of the charge distribution around oxygen. ACKNOWLEDGMENTS Many thanks to Dr. R. Holmestad for providing Silicon diffraction patterns and Prof. John Spence for many discussions. REFERENCES Anstis GR. 1996. Corrections to atomic scattering factors for high energy electrons arising from atomic vibrations. Acta Cryst A52:450– 455. Bethe H. 1928. Ann Phys (Leipzig) 87:55–69. Birkeland CR, Holmestad R, Marthinsen K, Høier R. 1996. Efficient beam selection criteria in quantitative convergent beam electron diffraction. Ultramicroscopy 66:89–99. Birkeland CR. 1997. Quantitative methods in electron diffraction and microscopy. PhD thesis, Norwegian University for Science and Technology. Blackman M. 1939. Proc R Soc London Ser A173:68–82. Blaha P, Schwartz K, Sorantin P, Trickey SB. 1990. Comput Phys Commun 59:399–411. Blaha P, Schwarz K, Dufek P, Augustyn R. 1995. WIEN 95, Technical University of Vienna. (Improved and updated unix version of the original copy-righted WIEN-code, which was published by. Box GEP, and Tiao GC. 1992. Bayesian Inference in Statistical Analysis. Addison-Wesley. Gribelyuk MA, Ruhle M. 1993. Fast calibration of CBED patterns for quantitative analysis. Proc 51st MSA. San Francisco: San Francisco Press. p 674–675. Holmestad R, Zuo JM, Spence JCH, Høier R, Horita Z. 1995. Effect of Mn doping on charge density in ␥-TiA1 by quantitative convergent beam electron diffraction. Phil Mag A 72:579–601. Holmestad R, Morniroli JP, Zuo JM, Spence JCH, Avilov A. 1997. Quantitative CBED study of SiC 4H. Inst Phys Conf Series (EMAG). Holmestad R, Birkeland CR, Marthinsen K, Hoier R, Zuo JM. 1999. Use of quantitative CBED in materials science. Microsc Res Tech, this issue. Huber PJ. 1981. Robust statistics. New York: John Wiley & Sons. Krivanek OL, Friedman SL, Gubbens AJ, Kraus B. 1995. An imaging filter for biological applications. Ultramicroscopy 59:267–282. Lanio S. 1986. High-resolution imaging magnetic energy filters with simple structure. Optik 73:99–107. MacGillavry CH. 1940. Physica (Utrecht) 7:329–337. Metherell AJF. 1975. Electron microscopy and materials science. In: Valdre U, Reudl R, editors. Commission of European Communities, Director General, Scientific and Technical Information, Luxembourg. Möllenstedt G. 1989. My early work on convergent beam electron diffraction. Phys Status Solidi A116:13–22. Morniroli JP, Cordier P, Lappellen EV, Zuo JM, Spence JCH. 1997. Application of the Convergent Beam Imaging (CBIM) technique to the analysis of crystal defects. Micros Microanal Microstruct 8:187– 202. Nüchter W, Weickenmeier AL, Mayer J. 1995. High precision measurement of Debye-Waller factors for NiA1 by convergent beam electron diffraction. In: Inst Phys Conf 147. p 129–132. Ogura N, Yoshida K, Kojima Y, Saito H. 1994. Development of the 25 micron pixel imaging plate system for TEM. Proc 13th ICEM, Paris. p 219–221. Peng LM. 1997. Anisotropic thermal vibrations and dynamical electron diffraction by crystals, Acta Cryst A5 3:663–672. Ren G, Zuo JM, Peng LM. 1997. Accurate measurements of crystal structure factors using a FEG electron microscope. Micron 28:459– 467. Rez P. 1978a. Theory of inelastic scattering in electron microscopy of thin crystals. D Phil, Univ of Oxford, England. Rez P. 1978b. Virtual inelastic scattering in high-energy electron diffraction. Acta Cryst A34:48–51. Rose H, Krahl D. 1995. Electron optics of imaging energy filters. In: Reimer L, editor. Energy filtering transmission electron microscopy. Berlin: Springer. p 43–149. 233 Saunders M, Bird DM. 1995. Measurement of low-order structure factors for silicon from zone-axis CBED patterns. Ultramicroscopy 60:311–320. Snyder DL, Hammoud AM, White RL. 1993. Image recovery from data acquired with a charge-coupled-device camera. J Opt Soc Am 10:1014–1023. Spence JCH. 1993. On the measurement of structure factor amplitudes and phases by electron diffraction. Acta Cryst A49:231–260. Spence JCH, Zuo JM. 1988. A large dynamical range, parallel detection system for electron diffraction and imaging with large dynamical range. Rev Sci Instr 59:2102–2108. Spence JCH, Zuo JM. 1992. Electron microdiffraction. New York: Plenum Press. Tomokiyo Y, Matsumura S, Okuyama T, Yasunaga T, Kuwano N, Oki K. 1993. Dynamical diffraction effects on HOLZ pattern geometry in Si-Ge alloys and determination of local lattice parameter. Ultramicroscopy 54:276–285. Tomokiyo Y, Kimura S, Zuo JM, Spence JCH. 1996. Structural refinements for ␣-A12O3 by energy filtered convergent beam electron diffraction. Proc 6th Asia-Pacific Conf Electr Microsc Barber D, et al., editors. Hong Kong: Chinetek Promotion. p 115–116. Tsuda K, Tanaka M. 1995. Refinement of crystal structure parameters using convergent-beam electron diffraction: the low-temperature phase of SrTiO3. Acta Cryst A51:7–15. Uhlemann S, Rose H. 1994. Comparison of the performance of existing and proposed imaging energy filters. ICEM-13 Paris 163–165. Weickenmeier AL, Kohl H. 1991. Computation of absorptive formfactors for high-energy electron-diffraction. Acta Cryst A47:590– 597. Weickenmeier AL, Kohl H. 1998. The influence of anisotropic thermal vibrations on absorptive form factors for high-energy electron diffraction. Acta Cryst A54:283–289. Weickenmeier AL, Nüchter W, Mayer J. 1995. Quantitative characterization for point spread function and detection quantum efficiency for a YAG scintillator slow scan CCD camera. Optik 99:147–154. Willis BTM, Pryor AW. 1975. Thermal vibrations in crystallography. Cambridge: Cambridge University Press. Yoshioka H. 1957. J Phys Soc Jpn 12:618–628. Zuo JM. 1992. Automated lattice parameter measurements from HOLZ lines and their use for measurements of oxygen content in YBa2CuO7-delta from nanometer-sized regions. Ultramicroscopy 41: 211–224. Zuo JM. 1993. Automated structure-factor refinement from convergentbeam electron-diffraction patterns. Acta Cryst A49:429–435. Zuo JM. 1996. Electron detection characteristics of slow-scan CCD camera. Ultramicroscopy 66:21–33. Zuo JM. 1997a. Extracting structure information from CBED patterns. Microscopy Microanal 3 (Suppl 2):1151–1152. Zuo JM. 1997b. Quantitative electron diffraction and charge density. Proc Asian Sci Semin New Direction TEM Nano-Characterization Materials. Fukuoka, Japan: Kyushu University Press. p 103–118. Zuo JM. 1998a. Quantitative convergent beam electron diffraction. Trans Jpn Inst Metals 39:938–946. Zuo JM, Spence JCH. 1991. Automated structure factor refinement from convergent beam electron diffraction patterns. Ultramicroscopy 35:185–196. Zuo JM, Weickenmeier AL. 1995. On the beam selection and convergence in the Bloch wave method. Ultramicroscopy 57:375–383. Zuo JM, Spence JCH, O’Keeffe M. 1988. Bonding in GaAs. Phys Rev Lett 61:353–356. Zuo JM, Spence JCH, Hoier R. 1989. Accurate Structure factor phase determination by electron diffraction in non-centrosymmetric crystals. Phys Rev Lett 62:547–550. Zuo JM, Zhu HR, Spence A. 1993. Simulating electron microscope diffraction mode with a Macintosh-based program. Proc. 51st MSA, 1210–1211. (Information about the program can be obtained from EMLab, 16203 S. 26th Place, Phoenix, AZ 85048.) Zuo JM, McCartney MR, Spence JCH. 1996. Performance of imaging plates for electron recording. Ultramicroscopy 66:35–47. Zuo JM, Kim M, Holmestad R. 1997a. A new approach to lattice parameter measurements using dynamical electron diffraction and pattern matching. J Electr Microsc 42:121–127. Zuo JM, Blaha P, Schwarz K. 1997b. Charge density of silicon: experimental test of exchange and correlation potentials. J Phys Condensed Matter 9:7541–7561. Zuo JM, O’Keeffe M, Rez P, Spence JCH. 1997c. Charge density of MgO: Implications of precise new measurements for theory. Phys Rev Lett 78:4777–4780.