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A method to determine long-range order parameters from electron diffraction intensities detected by a CCD camera
Ultramicroscopy 96 (2003) 105–116
A method to determine long-range order parameters from electron diffraction intensities detected by a CCD camera Takayoshi Kimotoa,*, Toshiyuki Takedab, Shigenari Shidac a
Materials Engineering Laboratory, National Institute for Materials Science, 1-2-1 Sengen, Tsukuba 305-0047, Japan b Institute of Applied Physics, The University of Tsukuba, Tennodai, Tsukuba 305-8577, Japan c Department of Materials Science, Iwaki Meisei University, Iwaki 970-8551, Japan Received 20 April 2002; received in revised form 1 October 2002
Abstract To determine long-range order parameters from electron diffraction intensities, the authors have developed a CCD camera system to detect precisely electron diffraction intensities, a method for quickly and precisely measuring specimen thickness, and a computer programming to calculate long-range order parameters from the ratio of superlattice and fundamental diffraction intensities. Thickness variation over a diffraction area is taken into consideration in the calculation of electron diffraction intensities on the basis of the multi-slice method, and long-range order parameters are calculated by the successive approximation method. The absorptive form factors are also calculated from experimental data of diffraction intensities by parameter fitting, and the effect of absorption on the calculation of long-range order parameters is examined. The values of Cu3 Au alloys aged at 523 and 653 K that were obtained by averaging long-range order parameters determined for several diffraction areas with the developed method are close to the reported data obtained by the X-ray diffraction method. The main causes for the deviation of longrange order parameters determined for several diffraction areas are also discussed. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Long-range order parameter; Electron diffraction intensity; Specimen thickness; Cu3 Au alloy
1. Introduction Long-range order (LRO) parameters have been quantitatively determined from the diffraction intensities in the X-ray diffraction method since the 1940s [1,2] and the neutron diffraction method since 1950s [3], because their diffraction intensities have been measurable. In the X-ray diffraction *Corresponding author. Tel.: +81-298-59-2733; fax: +81298-59-2701. E-mail address: [email protected] (T. Kimoto).
method, LRO parameters have been easily determined from the ratio of superlattice and fundamental reflection intensities by applying kinematical diffraction theory [1,2]. In the case of neutron diffraction, quantitative analysis for ordering has also been performed from the analysis of superlattice reflection intensities in magnetic alloys [3]. Generally, these methods are not suitable for obtaining LRO parameters in a selected microscopic area. The electron diffraction method, on the other hand, provides LRO parameters in a freely selected microscopic area,
0304-3991/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0304-3991(02)00403-5
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and therefore it could enable microscopic fluctuations of LRO parameters to be examined. The electron diffraction method to determine structure factors of alloys does not always allow us to determine their LRO parameters. For instance, superlattice reflections of ordering alloys usually have no critical voltage and therefore we cannot determine LRO parameters with the criticalvoltage method [4]. On the other hand, the intersecting Kikuchi-line method (IKL method) proposed by Gj^ness and H^ier [5] was used to determine LRO parameters for some oxides and semiconductors [6,7] in which clear Kikuchi lines can usually be observed. Because Kikuchi patterns from thin metal foils suffer from bending of the specimen, however, the IKL method had not been used to determine LRO parameters of ordering alloys until the work by Matsuhata et al. [8]. They determined LRO parameters of Cu3 Au by combining the IKL method with the convergent-beam electron diffraction (CBED) technique in order to reduce the influence of bending in a thin alloy [8]. The CBED method was also used by Tanaka et al. [9] to determine the relative LRO parameters in Sr2 Ta2 O7 by measuring intensities of the HOLZ line that occurs from ordering. However, the IKL and CBED methods require skillful experimental techniques in the case of ordering alloys because of the influence of bending in a thin TEM specimen. A method to determine LRO parameters from the ratio of superlattice and fundamental reflection intensities in the conventional electron diffraction pattern, which seems to be experimentally easier than the IKL and CBED methods, has not been successfully performed so far, because there has been no suitable method to detect precisely electron diffraction intensities for a long time. However, the recent development of a CCD camera and an imaging plate for a transmission electron microscope (TEM) enables us to measure precisely electron diffraction intensities because their dynamic range is much higher than that of conventional film. We therefore attempted to develop a method to determine the LRO parameters from the ratio of superlattice and fundamental reflection intensities detected by a CCD camera. The present paper describes the
development of this method and discusses the factors which affect its accuracy. The paper also describes the application of the method to determine LRO parameters in Cu3 Au alloys and compares the obtained result with the previously reported data by the X-ray diffraction method.
2. Development of experimental and calculation method 2.1. Experimental method Fig. 1 shows the cooled slow-scan type CCD camera which was specially designed and mounted on a JEM-2010 transmission electron microscope (TEM) to measure the electron diffraction intensities as precisely as possible. The first characteristic of the CCD camera is that it has a Photometrics 512 512 CCD device (TK512F Grade 1) of the highest dynamic range of 64,000.
Fig. 1. Specially designed CCD camera system for precise measurement of electron diffraction intensities.
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The size of active area of the CCD device is 13:8 13:8 mm; and its pixel size is 27 27 mm: Secondly, we employ a long ð13:5 cmÞ optical fiber plate as a photon transmission device in order to absorb X-rays coming from an accelerating tube of TEM which cause intensity noise in the CCD camera by irradiating the CCD device. The diameter of the outer clad of each optical fiber is 6 mm and that of the core is 5:7 mm: The thickness of the single-crystal YAG sintilator is 50 mm: The third characteristic is that it has a mechanical shutter that opens only while the diffraction intensities are being measured. This mechanical shutter protects the YAG scintillator from unnecessary irradiation damage by electrons that would significantly decrease the lifetime of the CCD camera. Other details about the CCD camera system are found elsewhere [10]. Measurement of specimen thickness in a diffraction area is very important in the present method. Because of inevitable specimen drift in TEM, therefore, the time lag between the measurement of specimen thickness and that of diffraction intensities should be as shorter as possible. In the present method it is also required that an area with a thickness range suitable for intensity recording can be localized during the experiment, as explained below. The method of specimen thickness measurement on the basis of CBED is known to be accurate, but it does not satisfy the latter requirement because the analysis takes time [11,12]. Therefore, a new method for specimen thickness measurement, which meets both requirements and is also as accurate as the CBED method, needed to be developed. It is well known that the transmitted electron beam current decreases exponentially with specimen thickness because of scattering absorption [13]. Fig. 2 shows this relationship that was experimentally obtained in Cu3 Au alloy. In Fig. 2, specimen thickness was obtained with the CBED method by analyzing excited ð3 1 1Þ reflection. The small electron beam of NBD (nano beam diffraction) mode is used as a transmitted electron beam and its size is less than 10 nm at the specimen position. After deriving the relationship as shown in Fig. 2, we can quickly measure specimen thickness in a small area only by
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Fig. 2. Thickness dependence of transmittance of electron beam in Cu3 Au alloy. Thickness was measured by the CBED method.
measuring the current of the fine transmitted electron beam with a Faraday cup attached to a TEM. 2.2. Calculation method In the developed computer program, LRO parameters are calculated from the ratio of superlattice and fundamental reflection intensities in the zone-axis diffraction pattern. Calculation of electron diffraction intensities to compute LRO parameters is based on the multi-slice method (Cowley–Moodie method) [14], because the method yields us the shortest calculation time in the case of the zone-axis diffraction pattern. Since specimen thickness usually varies with position over a diffraction area, we should calculate the diffraction intensity by taking account of the thickness variation over a diffraction area. However, there have been few attempts so far to calculate diffraction intensities with consideration of thickness variation over a diffraction area. Here we explain the derivation of our calculation method considering thickness variation in a
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diffraction area by taking the example of electropolished Cu3 Au alloy. It is experimentally found that the thickness increases steeply almost in proportion to the distance from the edge of a hole in an electropolished Cu3 Au alloy. Therefore it is assumed in the case of an electropolished Cu3 Au alloy that thickness increases proportionally with distance from specimen edge as shown in Fig. 3a). On this assumption the thickness at any point in a diffraction area is simply calculated from the values of distance between the point and specimen edge and thickness at the center of a diffraction area. We assume that a diffraction area is composed of a large number of slices, which are parallel to the specimen edge as shown in Fig. 3b). Thickness is assumed to be constant within each slice and the difference of thickness ti tiþ1 between adjacent slices i and i þ 1 is a lattice constant in the direction normal to the specimen surface. We then calculate diffraction intensity Ii from each slice with the multi-slice (Cowley–Moodie) method. The thickness of one slice parallel to the specimen surface in the multi-slice method is a lattice constant in the direction normal to the specimen surface and the number of diffracted waves for calculation is 17 17 ¼ 289: We finally calculate the diffraction intensity I for the total diffraction area by summing Ii Si as
follows: n X Ii Si ; I¼
ð1Þ
i¼1
where Si is the area of slice i: In this way, first we wrote a computer program to calculate the diffraction intensity, taking account of the thickness variation over a diffraction area. In the other cases besides an electropolished Cu3 Au alloy, thickness variation in a diffraction area is different and should be estimated on the basis of experimental data, but basically the diffraction intensities are calculated using the same assumption that a diffraction area is composed of a large number of pieces in which thickness is constant. Bragg and Williams defined LRO parameter S for binary compositions as follows [15]: ra X A S¼ ; ð2Þ 1 XA where ra is the average fraction of a-site occupied by the right atom A and XA is the concentration of atom A. The present computer program to calculate LRO parameters of a binary alloy was developed with C compiler on the basis of this definition. Since it was confirmed that the intensity ratio of the superlattice to fundamental reflection always increases monotonously with increasing S; the successive approximation method was adopted
Fig. 3. Explanation of the method to calculate electron diffraction intensity in consideration of thickness variation in a diffraction area: (a) Thickness distribution in a diffraction area in a wedge-shaped specimen; (b) Slicing of a diffraction area for calculation.
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in the computer program for calculating S: If the experimental intensity ratio is unexpectedly larger than the intensity calculated for S ¼ 1; LRO parameter S is assumed to be the value larger than 1.0 by the parabolic extension in the program.
3. Application of the developed method to Cu3 Au alloy 3.1. Experimental The developed method was used to determine the LRO parameters of Cu3 Au alloy, which have been measured so far by the X-ray diffraction method [16]. Cu3 Au alloy produced by repeated arc-melting of pure Cu (99.9999%) and Au (99.9985%) was homogenized at 1123 K for 30 h: Disk specimens for TEM observation were finally annealed at 523 K for 1369 h for ordering after rapid quenching from 1123 K: The other specimens were annealed at 653 K for 3 h after quenching from 1203 K: TEM specimens were prepared by electropolishing in the electrolyte of 25% HClO4 and 75% CH3 COOH at 25 V: Diffraction intensities in the ð0 0 1Þ pattern were measured at the camera length of 15–20 cm with the CCD camera system shown in Fig. 1. We tried to get ð0 0 1Þ diffraction patterns as symmetrical as possible by slightly tilting the electron beam after adjusting the tilt of the specimen. The diameter of a selected diffraction area was about 270 nm; and the diameter of the electron beam in NBD mode used to measure specimen thickness was less than 10 nm: It should be noted that the reflection intensity ratio is different between peak intensity and integrated intensity. All boundaries between CCD pixels, outer clads in fine optical fibers, and chicken wires in combined fiber plate are dead areas for counting photons which are produced in the YAG scintillator by irradiation of electrons. This results in the reduction of measurable integrated diffraction intensities by probably more than about 30%. Because their reduction rate depends on the width and height of intensity distribution curve, the ratio of measured integrated intensities between two diffraction spots is not correct. On the other hand, the reduction rate
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of the measured peak intensity for a diffraction spot which is obtained as a maximum intensity from one CCD pixel intensities does not depend on its intensity distribution curve, because the intensity change within the CCD pixel of maximum intensity is small for all diffraction spots. Therefore, the ratio of measured peak intensity between any two diffraction spots is thought to be almost the same as the true value. For these reasons, peak intensities which were counted from the diffraction patterns with commercial software (IPLab Spectrum) were used to calculate LRO parameters. Fig. 4 shows an example of the ð0 0 1Þ pattern of Cu3 Au alloy detected with the CCD camera system. In this figure detected peak reflection intensities, mean background intensities in the small region of four corners and main reflection index are shown. The background intensities are caused by both the electric noise from the CCD camera and electron inelastic scattering. As shown in Fig. 4, some asymmetry usually remains in reflection intensities in spite of all efforts to tilt the specimen and electron beam, probably because of slight bending of the TEM specimen within a diffraction area. In the case of Fig. 4, background intensity IB is derived as 259 which is the average of four background intensities at corners, and, for example, ð1 0 0Þ intensity I100 is derived by subtracting IB from the average of the four equivalent reflection intensities as follows: I100 ¼ ð4647 þ 3283 þ 3145 þ 4339Þ=4 259 A 3595: Deviation of this reflection DI100 was obtained as follows: DI100 ¼ f½ð4388 3595Þ2 þ ð3024 3595Þ2 þ ð2886 3595Þ2 þ ð4080 3595Þ2 =4g1=2 A 651: 3.2. Calculation of diffraction intensities considering thickness variation The thickness of the TEM specimen of electropolished Cu3 Au alloy relatively steeply increases with the increase in distance from the edge of a
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Fig. 4. Example of ð0 0 1Þ diffraction pattern in Cu3 Au alloy aged at 523 K for 1369 h: Peak reflection intensities, background intensities and main reflection indexes are shown.
hole made by electropolishing. Probably for this reason, the thickness at any point of the TEM specimen increases almost in proportion to the distance between the point and the specimen edge as shown in Fig. 5. This experimental fact enables us to easily estimate thickness variation in a diffraction area as shown in Fig. 2. Fig. 6 shows the calculated diffraction intensity ratio I100 =I200 of superlattice ð1 0 0Þ and fundamental ð2 0 0Þ reflection intensities in the ð0 0 1Þ diffraction pattern of Cu3 Au alloy, whose LRO parameter S is 1.0, as a function of thickness t0 at the center of a diffraction area. In this figure, we considered the thickness variation in a diffraction area, but did not consider the absorption effects. Calculations were made for the different thickness
variation Dt ¼ tmax tmin in a diffraction area from Dt ¼ 10 to 60 nm: It is recommended from Fig. 6 that a diffraction area, whose thickness t0 at the center is about 30, 60, 95, 120, 150 or 180 nm and Dt is as large as possible, should be experimentally chosen in the case of Cu3 Au to determine LRO parameter from the ratio I100 =I200 ; because the ratio changes slowly with t0 and the error in calculating LRO parameters which is caused by the error in measuring t0 is reduced. 3.3. Calculation of LRO parameters in Cu3 Au alloys In order to minimize the influence of experimental error in measuring thickness t0 at the center
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of a diffraction area upon the calculation of LRO parameters by using the ratio I100 =I200 ; diffraction patterns were taken from several different diffrac-
Fig. 5. Graph showing linear increase of thickness of TEM specimen with increasing distance from specimen edge in electropolished TEM specimen of Cu3 Au alloy. Inset drawing shows the positions of thickness measurement. Thickness was measured with the CBED method.
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tion areas whose thickness t0 was about 95 nm and thickness variation Dt was larger than 50 nm: Table 1 shows the experimental conditions and results, and calculation results for Cu3 Au alloy aged at 523 K for 1369 h after quenching from 1123 K: As shown in this table, diffraction intensities of the ð0 0 1Þ pattern were detected at seven different diffraction areas. The diameter of a diffraction area was about 270 nm; and the thickness range in a diffraction area was estimated by using the developed method explained above and the geometry of Fig. 3a). Table 2 shows similar results of five different diffraction areas in Cu3 Au alloy aged at 653 K for 3 h after rapid quenching from 1203 K: In these tables, LRO parameters were calculated without considering absorption effects from the ratio of I100 =I200 by using atomic scattering factors calculated by the Thomas–Fermi–Dirac (TFD) method which considers relativistic effects [17]. The reason why we chose the TFD method to calculate atomic scattering factors and not Hartree–Fock (HF) method [18] is that the former is believed to be more suitable for heavy atoms than the latter. The average value S and standard deviation DS of LRO parameters for seven data points in the Cu3 Au alloy aged at 523 K for 1396 h are
Fig. 6. Calculated intensity ratio I100 =I200 of ð1 0 0Þ to ð2 0 0Þ reflection as a function of thickness t0 at the center of a diffraction area in the case of S ¼ 1 in Cu3 Au alloy. Calculation was made in consideration of thickness variation in a diffraction area for several cases of thickness difference Dt ¼ tmax tmin :
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Table 1 Summary of experimental and calculation results in Cu3 Au alloy which was aged at 523 K for 1369 h after quenching from 1123 K Sampling area number
P-1 P-2 P-3 P-4 P-5 P-6 P-7
Thickness range in a diffraction area (nm) tmin tmax ðt0 71=2DÞ
65–147 ð106741Þ 55–135 ð95740Þ 59–130 ð95735Þ 69–129 ð99730Þ 53–142 ð97744Þ 72–123 ð97726Þ 57–146 ð102744Þ
Measured diffraction intensities Average7Deviation
Calculated LRO parameter Ratio
ð1 0 0Þ I100 7DI100 ðDI100 =I100 ð%ÞÞ
ð2 0 0Þ I200 7DI200 ðDI200 =I200 ð%ÞÞ
ð1 0 0Þ to ð2 0 0Þ I100 =I200
35007215 ð6:1%Þ 32057337 ð10:5%Þ 35957651 ð18:1%Þ 3946795 ð2:4%Þ 30737524 ð17:1%Þ 44087270 ð6:1%Þ 33667290 ð8:6%Þ
1438274375 ð30:4%Þ 1333174668 ð35:0%Þ 1774474158 ð23:4%Þ 1784575514 ð30:9%Þ 1246375092 ð40:9%Þ 1619477500 ð46:3%Þ 1578975040 ð31:9%Þ
0.2434
0.982
0.2404
0.971
0.2211
0.918
0.2466
1.000
0.2466
0.978
0.2722
1.165
0.2132
0.926
Data were taken at 300 K for P-1–P4, 110 K for P-5, 122 K for P-6 and 134 K for P-7.
Table 2 Summary of experimental and calculation results in Cu3 Au alloy which was aged at 653 K for 3 h after rapid quenching from 1203 K Sampling area number
P-1 P-2 P-3 P-4 P-5
Thickness range in a diffraction area (nm) tmin tmax ðt0 71=2DÞ
46–138 ð92746Þ 39–168 ð104765Þ 50–138 ð94744Þ 75–105 ð90715Þ 67–103 ð857176Þ
Measured diffraction intensities
Calculated LRO parameter
Average7 Deviation
Ratio
ð1 0 0Þ I100 7DI100 ðDI100 =I100 ð%ÞÞ
ð2 0 0Þ I200 7DI200 ðDI200 =I200 ð%ÞÞ
ð1 0 0Þ to ð2 0 0Þ I100 =I200
13807273 (19.8%) 11387217 (19.1%) 30367583 (19.2%) 24547208 (8.5%) 24137168 (7.0%)
906276087 (67.2%) 1178177718 (65.5%) 1795277231 (40.3%) 1110378786 (79.1%) 1111676012 (54.1%)
0.1523
0.802
0.0966
0.590
0.1691
0.842
0.2210
0.674
0.2171
0.782
Data were taken at 300 K:
S ¼ 0:991 and DS ¼ 0:076; respectively, as derived from Table 1. Those for five data points in the Cu3 Au alloy aged at 652 K for 3 h are S ¼ 0:738 and DS ¼ 0:093; respectively, as derived from Table 2.
3.4. Absorption effects on the calculation of LRO parameters Absorption effects or inelastic scattering effects of fast electrons by crystals were not considered in
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the calculation of LRO parameters shown in Tables 1 and 2. If the effect of absorption on the diffraction intensity of ð1 0 0Þ reflection and that of ð2 0 0Þ reflection is different, the intensity ratio I100 =I200 is affected by the absorption effects because thickness in the diffraction area is not small. Therefore, the effects of absorption on the calculation of LRO parameters were evaluated. It is generally accepted that the absorption effects can be treated mathematically by introducing imaginary part Vi besides real part Vr into the crystal potential V as follows [19]: V ¼ Vr þ iVi :
ð3Þ
The absorption effect increases with increasing ratio Vi =Vr (absorption coefficient). Many theoretical calculations of Vi =Vr have been carried out so far in the case of Bragg reflections, but few calculations have been performed in the present case of zone-axis orientation. It is expected that the latter is different from the former except for g ¼ ð0 0 0Þ: Therefore, we tried to obtain Vi =Vr for zone-axis orientation by parameter fitting from the experimental data. Because a calculated absorption coefficient Vi =Vr increases almost parabolically with jgj in many cases of Bragg reflection, it has been reported that Vi =Vr may be written into a parametric fit of the form for small value of jgj [20,21]: Vi =Vr ¼ ajgj2 bjgj þ c;
ð4Þ
where a; b and c are fitting parameters. We supposed that Eq. (4) stands for ga0 in the case of zone-axis orientation, and determined these parameters by minimizing the sum of the deviations of calculated from experimental diffraction intensities Sg jIðgÞcal IðgÞexp j=jIðgÞexp j for g ¼ ð0 0 0Þ; ð1 0 0Þ; ð1 1 0Þ; ð2 0 0Þ; ð2 1 0Þ and ð2 2 0Þ from one of seven experimental data shown in Table 1. In this calculation we used Vi =Vr ¼ 0:173 for Cu and 0.239 for Au as those for g ¼ ð0 0 0Þ which were calculated by Radi [22] in the case of Bragg reflection. It was found that the minimization of sum Sg jIðgÞcal IðgÞexp j=jIðgÞexp j by parameter fitting was most successful when we used the data of P4. The following were derived from the parameter fitting using the data point of P4 in
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Table 3 Summary of calculated LRO parameter in Cu3 Au alloy which was aged at 523 K for 1369 h Intensity ratio used for calculation
Calculated average LRO parameter Without absorption
ð1 0 0Þ ð1 1 0Þ ð1 0 0Þ ð1 1 0Þ
to to to to
ð2 0 0Þ ð2 0 0Þ ð2 2 0Þ ð2 2 0Þ
I100 =I200 I110 =I200 I100 =I220 I110 =I220
S S S S
¼ 0:991 ¼ 1:156 ¼ 0:675 ¼ 0:569
With absorption S S S S
¼ 0:953 ¼ 0:928 ¼ 0:973 ¼ 0:980
The LRO parameter shown were obtained by averaging seven calculated values from seven data point with and without consideration of absorption effects.
Table 1: Vi =Vr ¼ 0:044jgj2 0:23jgj þ 0:31:
ð5Þ
We calculated the LRO parameters for seven experimental data shown in Table 1 with and without consideration of absorption effects from the different intensity ratio of I100 =I200 ; I110 =I200 ; I100 =I220 and I110 =I200 ; respectively. The calculations of the LRO parameters considering the absorption effects were performed by using Vi =Vr given by Eq. (5) for ga0 and the values calculated by Radi [22] for g ¼ ð0 0 0Þ: After the calculations we averaged seven calculated LRO parameters to compare these eight calculation conditions. Table 3 compares these averaged LRO parameters calculated for different calculation conditions.
4. Discussions The LRO parameters at seven positions in the Cu3 Au alloy aged at 523 and 652 K were determined with the developed electron diffraction method. The average value S and standard deviation DS of LRO parameters in the Cu3 Au alloy aged at 523 K; which were calculated for seven data from the intensity ratio I100 =I200 of ð1 0 0Þ superlattice to ð2 0 0Þ fundamental reflection intensity without consideration of absorption effect, are S ¼ 0:991 and DS ¼ 0:076; respectively
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(Table 1). Those for five data of the Cu3 Au alloy aged at 652 K are S ¼ 0:738 and DS ¼ 0:093; respectively (Table 2). Keating and Warren determined the LRO parameters of a single crystal of Cu3 Au alloy with the X-ray diffraction method [17]. According to their paper, the measured LRO parameter of Cu3 Au aged for ordering at 523 K for 52 h is S ¼ 0:972; and that of Cu3 Au aged at 653 K for 1:5 h is S ¼ 0:775: Therefore, the average LRO parameters determined from the ratio I100 =I200 are close to the values of LRO parameters determined by the X-ray diffraction method. On the other hand, the average LRO parameters which were calculated from the intensity ratios I110 =I200 ; I110 =I200 and I110 =I200 without consideration of absorption effects are S ¼ 1:156; 0:675 and 0.569, respectively (Table 3). They are all far from the values by the X-ray diffraction method. We also calculated LRO parameters for seven data in the Cu3 Au alloy aged at 523 K considering absorption effects from the different intensity ratio I100 =I200 ; I110 =I200 ; I100 =I220 and I110 =I220 ; respectively. In this calculation we used absorption coefficients for gað0; 0; 0Þ which were determined by calculation from the experimental diffraction intensities. As shown in Table 3, the average LRO parameter is 0.953 for the calculation from I100 =I200 ; 0.928 from I110 =I200 ; 0.973 from I100 =I220 ; and 0.980 from I110 =I220 : Therefore, they are close to each other, and all of them are close to the value 0.972 by the X-ray diffraction method. This implies that the present calculation of LRO parameters considering absorption effects is successful and that the absorption coefficients determined from diffraction intensities are reasonable. The large difference between LRO parameters calculated with consideration of absorption effects and those without consideration of them in the case of calculation from the ratio I110 =I200 ; I100 =I220 or I110 =I220 implies that absorption effects are usually essential to calculate diffraction intensities in thick diffraction areas like P1 to P7 in Table 1. The small difference between them in the case of calculation from the ratio I100 =I200 must be very special, and this implies that the reduction rate of diffraction intensities by the effects of absorption happen to be almost the same for the ð1 0 0Þ and
ð2 0 0Þ reflections. This allows us to discuss about causes of the deviation of calculated LRO parameters by taking examples shown in Tables 1 and 2. Although the average LRO parameter S determined by the present method is reasonable, the deviation DS of the LRO parameter is not necessarily small as shown in Tables 1 and 2. The deviation DS ¼ 0:076 is about 7.67% of S ¼ 0:991 in Cu3 Au aged at 523 K (Table 1) and the deviation DS ¼ 0:093 is about 12.6% of S ¼ 0:738 in Cu3 Au aged at 652 K (Table 2). The error in measuring specimen thickness t0 at the center of a diffraction area is thought to be the main cause for the deviation DS of calculated LRO parameters, because they are extremely sensitive to specimen thickness t0 : From calculations it was found that the change of 1% in thickness t0 causes on average the change of about 0.8% in the calculated LRO parameters in the case of Cu3 Au aged at 523 K and the change of about 0.5% in the case of Cu3 Au aged at 653 K: Therefore, the experimental error of about 9.6% in thickness t0 corresponds to the calculation error of 7.67% of S (or DS ¼ 0:076) in the case of Cu3 Au aged at 523 K (Table 1), and that of about 25.5% corresponds to the error of 12.6% of S (or DS ¼ 0:093) in the case of Cu3 Au aged at 653 K (Table 2). Judging from Figs. 2 or 5, experimental error in measuring specimen thickness t0 at the center of a diffraction area is around 10%. Therefore, it is reasonable to suppose that the experimental error in measuring thickness t0 is the main cause of the dispersion DS in the case of Cu3 Au aged at 523 K (Table 1). In the case of Cu3 Au aged at 653 K (Table 2), on the other hand, other experimental error besides thickness measurement error could be another main cause of the large dispersion of 12.6% of S (or DS ¼ 0:093), because 25.5% is too large as an error in measuring specimen thickness. As derived from Table 1, the average ratio of the deviation to the average value of superlattice ð1 0 0Þ reflection intensity is DI100 =I100 ¼ 9:8%; and that of fundamental ð2 0 0Þ intensity is DI200 =I200 ¼ 29:6%: From Table 2 it was derived that those are DI100 =I100 ¼ 14:2% and DI200 =I200 ¼ 50:5%: Therefore, the degree of symmetry of diffraction intensity in Cu3 Au aged at 653 K
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(Table 2) is much lower than that in Cu3 Au aged at 523 K (Table 1). This suggests that the large deviation from symmetry in the diffraction pattern is another main cause for the large dispersion of S in the case of Cu3 Au aged at 653 K (Table 2). It should be noted that the ratio DI100 =I100 is much smaller than DI200 =I200 in all cases, and it seems that there is no relationship between DI100 =I100 and DI200 =I200 for each sampling area as shown in Tables 1 and 2. It was found that to get a perfect four-fold symmetric diffraction pattern is almost impossible. It is thought that slight bending of TEM specimen in a diffraction area is one of the main causes of this inevitable imperfection of symmetry in a diffraction pattern. Wedge shape TEM specimens as shown in Fig. 5 could also destroy a perfect symmetry between IðgÞ and IðgÞ due to the differences in orientation between top and bottom faces of wedge foils. It is confirmed from the discussion above that the error in measuring specimen thickness t0 and the imperfection of symmetry in a diffraction pattern are main causes of deviation DS of calculated LRO parameters. It is well expected that the deviation DS and the errors of averaged value S caused by them are decreased by increasing the number of data points to take from one specimen. Therefore, the accuracy of LRO parameters determined by the present method is thought to be improved by increasing number of data points. Since it is possible to get diffraction patterns repeatedly from one specimen position in the present method, we can also obtain the more accurate LRO parameter from one position by averaging LRO parameters calculated from the more number of diffraction patterns. It should be noted that averaging of LRO parameters which are calculated from several ratios of I100 =I200 ; I110 =I200 ; I100 =I220 ; I110 =I220 and so on with consideration of absorption effects also reduces the errors of calculated LRO parameters. As far as computer programming, including of incident beam direction as parameter may improve the accuracy of the calculation of LRO parameters. Of course, improvements of experimental methods in measuring specimen thickness and adjustment of symmetry of four-folded diffraction patterns also improve the accuracy.
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5. Conclusions A system to determine long-range order (LRO) parameters from electron diffraction intensities was developed in the present research. A CCD camera system to detect precisely electron diffraction intensities, a new method for quickly measuring specimen thickness using scattering absorption of a fine electron beam in NBD mode, and a computer program to calculate LRO parameters from the ratio of superlattice and fundamental diffraction intensities were developed. The computer program to calculate LRO parameters was devised to be applicable to the case of a symmetric diffraction pattern. In the developed computer program, diffraction intensities were calculated using the multi-slice method and thickness variation in a diffraction area was taken into consideration for the calculation of diffraction intensities. The computer program to calculate absorption form factors from the diffraction intensities was also written in order to examine the effects of absorption on the calculation of LRO parameters. The successive approximation method was applied in the computer program to calculate LRO parameters from diffraction intensities. The developed system was used to determine LRO parameters of Cu3 Au alloy and the obtained results were compared with those by X-ray diffraction in order to verify the system. The diffraction area was chosen such that its thickness conditions minimized the influence of experimental error in measuring specimen thickness in the calculation of LRO parameters. Specimen thickness increased steeply and proportionally with increasing the distance between the position and the specimen edge in the electropolished Cu3 Au alloy. Seven diffraction areas were examined in the Cu3 Au alloy aged at 523 K for 1369 h; and the average LRO parameter S and standard deviation DS; which were calculated from the ratio I100 =I200 of superlattice ð1 0 0Þ to fundamental ð2 0 0Þ reflection intensity without consideration of absorption effects, were S ¼ 0:991 and DS ¼ 0:076 in the Cu3 Au alloy aged at 523 K: Five diffraction areas were examined in the Cu3 Au alloy aged at 653 K for 3 h; and the similarly calculated average LRO
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parameter was S ¼ 0:785 and the standard deviation was DS ¼ 0:093: Absorption coefficients were calculated from the experimental data of diffraction intensities. The influence of absorption effects on the calculation of LRO parameters is very small for the calculation of LRO parameters from the ratio I100 =I200 of ð1 0 0Þ to ð2 0 0Þ reflection intensity. On the other hand, the calculated LRO parameters were significantly influenced by the absorption effects for the calculation from the ratios I110 =I200 ; I110 =I220 and I110 =I220 : Since it is confirmed that the average LRO parameters in the Cu3 Au alloy aged at 523 and 653 K are in good agreement with the reported data by the X-ray diffraction method, it is reasonable to suppose that the present system is adequate for determining the LRO parameters of alloys. It is thought from the calculation of LRO parameters for different thickness conditions that the experimental errors in measuring specimen thickness are the main cause of the deviation of the calculated LRO parameter DS ¼ 0:076 in the Cu3 Au alloy aged at 523 K: It is also thought from the calculation that both the thickness measurement errors and the comparatively large deviation from the perfect symmetric diffraction intensity are the main causes of the large dispersion of the calculated LRO parameter DS ¼ 0:093 in the Cu3 Au alloy aged at 653 K:
Acknowledgements The authors would like to thank Dr. Wei Sun, Prof. Kenichi Ohshima and Prof. Denjiro
Watanabe for their stimulating discussions and encouragement.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
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A method to determine long-range order parameters from electron diffraction intensities detected by a CCD camera Takayoshi Kimotoa,*, Toshiyuki Takedab, Shigenari Shidac a
Materials Engineering Laboratory, National Institute for Materials Science, 1-2-1 Sengen, Tsukuba 305-0047, Japan b Institute of Applied Physics, The University of Tsukuba, Tennodai, Tsukuba 305-8577, Japan c Department of Materials Science, Iwaki Meisei University, Iwaki 970-8551, Japan Received 20 April 2002; received in revised form 1 October 2002
Abstract To determine long-range order parameters from electron diffraction intensities, the authors have developed a CCD camera system to detect precisely electron diffraction intensities, a method for quickly and precisely measuring specimen thickness, and a computer programming to calculate long-range order parameters from the ratio of superlattice and fundamental diffraction intensities. Thickness variation over a diffraction area is taken into consideration in the calculation of electron diffraction intensities on the basis of the multi-slice method, and long-range order parameters are calculated by the successive approximation method. The absorptive form factors are also calculated from experimental data of diffraction intensities by parameter fitting, and the effect of absorption on the calculation of long-range order parameters is examined. The values of Cu3 Au alloys aged at 523 and 653 K that were obtained by averaging long-range order parameters determined for several diffraction areas with the developed method are close to the reported data obtained by the X-ray diffraction method. The main causes for the deviation of longrange order parameters determined for several diffraction areas are also discussed. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Long-range order parameter; Electron diffraction intensity; Specimen thickness; Cu3 Au alloy
1. Introduction Long-range order (LRO) parameters have been quantitatively determined from the diffraction intensities in the X-ray diffraction method since the 1940s [1,2] and the neutron diffraction method since 1950s [3], because their diffraction intensities have been measurable. In the X-ray diffraction *Corresponding author. Tel.: +81-298-59-2733; fax: +81298-59-2701. E-mail address: [email protected] (T. Kimoto).
method, LRO parameters have been easily determined from the ratio of superlattice and fundamental reflection intensities by applying kinematical diffraction theory [1,2]. In the case of neutron diffraction, quantitative analysis for ordering has also been performed from the analysis of superlattice reflection intensities in magnetic alloys [3]. Generally, these methods are not suitable for obtaining LRO parameters in a selected microscopic area. The electron diffraction method, on the other hand, provides LRO parameters in a freely selected microscopic area,
0304-3991/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0304-3991(02)00403-5
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and therefore it could enable microscopic fluctuations of LRO parameters to be examined. The electron diffraction method to determine structure factors of alloys does not always allow us to determine their LRO parameters. For instance, superlattice reflections of ordering alloys usually have no critical voltage and therefore we cannot determine LRO parameters with the criticalvoltage method [4]. On the other hand, the intersecting Kikuchi-line method (IKL method) proposed by Gj^ness and H^ier [5] was used to determine LRO parameters for some oxides and semiconductors [6,7] in which clear Kikuchi lines can usually be observed. Because Kikuchi patterns from thin metal foils suffer from bending of the specimen, however, the IKL method had not been used to determine LRO parameters of ordering alloys until the work by Matsuhata et al. [8]. They determined LRO parameters of Cu3 Au by combining the IKL method with the convergent-beam electron diffraction (CBED) technique in order to reduce the influence of bending in a thin alloy [8]. The CBED method was also used by Tanaka et al. [9] to determine the relative LRO parameters in Sr2 Ta2 O7 by measuring intensities of the HOLZ line that occurs from ordering. However, the IKL and CBED methods require skillful experimental techniques in the case of ordering alloys because of the influence of bending in a thin TEM specimen. A method to determine LRO parameters from the ratio of superlattice and fundamental reflection intensities in the conventional electron diffraction pattern, which seems to be experimentally easier than the IKL and CBED methods, has not been successfully performed so far, because there has been no suitable method to detect precisely electron diffraction intensities for a long time. However, the recent development of a CCD camera and an imaging plate for a transmission electron microscope (TEM) enables us to measure precisely electron diffraction intensities because their dynamic range is much higher than that of conventional film. We therefore attempted to develop a method to determine the LRO parameters from the ratio of superlattice and fundamental reflection intensities detected by a CCD camera. The present paper describes the
development of this method and discusses the factors which affect its accuracy. The paper also describes the application of the method to determine LRO parameters in Cu3 Au alloys and compares the obtained result with the previously reported data by the X-ray diffraction method.
2. Development of experimental and calculation method 2.1. Experimental method Fig. 1 shows the cooled slow-scan type CCD camera which was specially designed and mounted on a JEM-2010 transmission electron microscope (TEM) to measure the electron diffraction intensities as precisely as possible. The first characteristic of the CCD camera is that it has a Photometrics 512 512 CCD device (TK512F Grade 1) of the highest dynamic range of 64,000.
Fig. 1. Specially designed CCD camera system for precise measurement of electron diffraction intensities.
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The size of active area of the CCD device is 13:8 13:8 mm; and its pixel size is 27 27 mm: Secondly, we employ a long ð13:5 cmÞ optical fiber plate as a photon transmission device in order to absorb X-rays coming from an accelerating tube of TEM which cause intensity noise in the CCD camera by irradiating the CCD device. The diameter of the outer clad of each optical fiber is 6 mm and that of the core is 5:7 mm: The thickness of the single-crystal YAG sintilator is 50 mm: The third characteristic is that it has a mechanical shutter that opens only while the diffraction intensities are being measured. This mechanical shutter protects the YAG scintillator from unnecessary irradiation damage by electrons that would significantly decrease the lifetime of the CCD camera. Other details about the CCD camera system are found elsewhere [10]. Measurement of specimen thickness in a diffraction area is very important in the present method. Because of inevitable specimen drift in TEM, therefore, the time lag between the measurement of specimen thickness and that of diffraction intensities should be as shorter as possible. In the present method it is also required that an area with a thickness range suitable for intensity recording can be localized during the experiment, as explained below. The method of specimen thickness measurement on the basis of CBED is known to be accurate, but it does not satisfy the latter requirement because the analysis takes time [11,12]. Therefore, a new method for specimen thickness measurement, which meets both requirements and is also as accurate as the CBED method, needed to be developed. It is well known that the transmitted electron beam current decreases exponentially with specimen thickness because of scattering absorption [13]. Fig. 2 shows this relationship that was experimentally obtained in Cu3 Au alloy. In Fig. 2, specimen thickness was obtained with the CBED method by analyzing excited ð3 1 1Þ reflection. The small electron beam of NBD (nano beam diffraction) mode is used as a transmitted electron beam and its size is less than 10 nm at the specimen position. After deriving the relationship as shown in Fig. 2, we can quickly measure specimen thickness in a small area only by
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Fig. 2. Thickness dependence of transmittance of electron beam in Cu3 Au alloy. Thickness was measured by the CBED method.
measuring the current of the fine transmitted electron beam with a Faraday cup attached to a TEM. 2.2. Calculation method In the developed computer program, LRO parameters are calculated from the ratio of superlattice and fundamental reflection intensities in the zone-axis diffraction pattern. Calculation of electron diffraction intensities to compute LRO parameters is based on the multi-slice method (Cowley–Moodie method) [14], because the method yields us the shortest calculation time in the case of the zone-axis diffraction pattern. Since specimen thickness usually varies with position over a diffraction area, we should calculate the diffraction intensity by taking account of the thickness variation over a diffraction area. However, there have been few attempts so far to calculate diffraction intensities with consideration of thickness variation over a diffraction area. Here we explain the derivation of our calculation method considering thickness variation in a
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diffraction area by taking the example of electropolished Cu3 Au alloy. It is experimentally found that the thickness increases steeply almost in proportion to the distance from the edge of a hole in an electropolished Cu3 Au alloy. Therefore it is assumed in the case of an electropolished Cu3 Au alloy that thickness increases proportionally with distance from specimen edge as shown in Fig. 3a). On this assumption the thickness at any point in a diffraction area is simply calculated from the values of distance between the point and specimen edge and thickness at the center of a diffraction area. We assume that a diffraction area is composed of a large number of slices, which are parallel to the specimen edge as shown in Fig. 3b). Thickness is assumed to be constant within each slice and the difference of thickness ti tiþ1 between adjacent slices i and i þ 1 is a lattice constant in the direction normal to the specimen surface. We then calculate diffraction intensity Ii from each slice with the multi-slice (Cowley–Moodie) method. The thickness of one slice parallel to the specimen surface in the multi-slice method is a lattice constant in the direction normal to the specimen surface and the number of diffracted waves for calculation is 17 17 ¼ 289: We finally calculate the diffraction intensity I for the total diffraction area by summing Ii Si as
follows: n X Ii Si ; I¼
ð1Þ
i¼1
where Si is the area of slice i: In this way, first we wrote a computer program to calculate the diffraction intensity, taking account of the thickness variation over a diffraction area. In the other cases besides an electropolished Cu3 Au alloy, thickness variation in a diffraction area is different and should be estimated on the basis of experimental data, but basically the diffraction intensities are calculated using the same assumption that a diffraction area is composed of a large number of pieces in which thickness is constant. Bragg and Williams defined LRO parameter S for binary compositions as follows [15]: ra X A S¼ ; ð2Þ 1 XA where ra is the average fraction of a-site occupied by the right atom A and XA is the concentration of atom A. The present computer program to calculate LRO parameters of a binary alloy was developed with C compiler on the basis of this definition. Since it was confirmed that the intensity ratio of the superlattice to fundamental reflection always increases monotonously with increasing S; the successive approximation method was adopted
Fig. 3. Explanation of the method to calculate electron diffraction intensity in consideration of thickness variation in a diffraction area: (a) Thickness distribution in a diffraction area in a wedge-shaped specimen; (b) Slicing of a diffraction area for calculation.
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in the computer program for calculating S: If the experimental intensity ratio is unexpectedly larger than the intensity calculated for S ¼ 1; LRO parameter S is assumed to be the value larger than 1.0 by the parabolic extension in the program.
3. Application of the developed method to Cu3 Au alloy 3.1. Experimental The developed method was used to determine the LRO parameters of Cu3 Au alloy, which have been measured so far by the X-ray diffraction method [16]. Cu3 Au alloy produced by repeated arc-melting of pure Cu (99.9999%) and Au (99.9985%) was homogenized at 1123 K for 30 h: Disk specimens for TEM observation were finally annealed at 523 K for 1369 h for ordering after rapid quenching from 1123 K: The other specimens were annealed at 653 K for 3 h after quenching from 1203 K: TEM specimens were prepared by electropolishing in the electrolyte of 25% HClO4 and 75% CH3 COOH at 25 V: Diffraction intensities in the ð0 0 1Þ pattern were measured at the camera length of 15–20 cm with the CCD camera system shown in Fig. 1. We tried to get ð0 0 1Þ diffraction patterns as symmetrical as possible by slightly tilting the electron beam after adjusting the tilt of the specimen. The diameter of a selected diffraction area was about 270 nm; and the diameter of the electron beam in NBD mode used to measure specimen thickness was less than 10 nm: It should be noted that the reflection intensity ratio is different between peak intensity and integrated intensity. All boundaries between CCD pixels, outer clads in fine optical fibers, and chicken wires in combined fiber plate are dead areas for counting photons which are produced in the YAG scintillator by irradiation of electrons. This results in the reduction of measurable integrated diffraction intensities by probably more than about 30%. Because their reduction rate depends on the width and height of intensity distribution curve, the ratio of measured integrated intensities between two diffraction spots is not correct. On the other hand, the reduction rate
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of the measured peak intensity for a diffraction spot which is obtained as a maximum intensity from one CCD pixel intensities does not depend on its intensity distribution curve, because the intensity change within the CCD pixel of maximum intensity is small for all diffraction spots. Therefore, the ratio of measured peak intensity between any two diffraction spots is thought to be almost the same as the true value. For these reasons, peak intensities which were counted from the diffraction patterns with commercial software (IPLab Spectrum) were used to calculate LRO parameters. Fig. 4 shows an example of the ð0 0 1Þ pattern of Cu3 Au alloy detected with the CCD camera system. In this figure detected peak reflection intensities, mean background intensities in the small region of four corners and main reflection index are shown. The background intensities are caused by both the electric noise from the CCD camera and electron inelastic scattering. As shown in Fig. 4, some asymmetry usually remains in reflection intensities in spite of all efforts to tilt the specimen and electron beam, probably because of slight bending of the TEM specimen within a diffraction area. In the case of Fig. 4, background intensity IB is derived as 259 which is the average of four background intensities at corners, and, for example, ð1 0 0Þ intensity I100 is derived by subtracting IB from the average of the four equivalent reflection intensities as follows: I100 ¼ ð4647 þ 3283 þ 3145 þ 4339Þ=4 259 A 3595: Deviation of this reflection DI100 was obtained as follows: DI100 ¼ f½ð4388 3595Þ2 þ ð3024 3595Þ2 þ ð2886 3595Þ2 þ ð4080 3595Þ2 =4g1=2 A 651: 3.2. Calculation of diffraction intensities considering thickness variation The thickness of the TEM specimen of electropolished Cu3 Au alloy relatively steeply increases with the increase in distance from the edge of a
110
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Fig. 4. Example of ð0 0 1Þ diffraction pattern in Cu3 Au alloy aged at 523 K for 1369 h: Peak reflection intensities, background intensities and main reflection indexes are shown.
hole made by electropolishing. Probably for this reason, the thickness at any point of the TEM specimen increases almost in proportion to the distance between the point and the specimen edge as shown in Fig. 5. This experimental fact enables us to easily estimate thickness variation in a diffraction area as shown in Fig. 2. Fig. 6 shows the calculated diffraction intensity ratio I100 =I200 of superlattice ð1 0 0Þ and fundamental ð2 0 0Þ reflection intensities in the ð0 0 1Þ diffraction pattern of Cu3 Au alloy, whose LRO parameter S is 1.0, as a function of thickness t0 at the center of a diffraction area. In this figure, we considered the thickness variation in a diffraction area, but did not consider the absorption effects. Calculations were made for the different thickness
variation Dt ¼ tmax tmin in a diffraction area from Dt ¼ 10 to 60 nm: It is recommended from Fig. 6 that a diffraction area, whose thickness t0 at the center is about 30, 60, 95, 120, 150 or 180 nm and Dt is as large as possible, should be experimentally chosen in the case of Cu3 Au to determine LRO parameter from the ratio I100 =I200 ; because the ratio changes slowly with t0 and the error in calculating LRO parameters which is caused by the error in measuring t0 is reduced. 3.3. Calculation of LRO parameters in Cu3 Au alloys In order to minimize the influence of experimental error in measuring thickness t0 at the center
T. Kimoto et al. / Ultramicroscopy 96 (2003) 105–116
of a diffraction area upon the calculation of LRO parameters by using the ratio I100 =I200 ; diffraction patterns were taken from several different diffrac-
Fig. 5. Graph showing linear increase of thickness of TEM specimen with increasing distance from specimen edge in electropolished TEM specimen of Cu3 Au alloy. Inset drawing shows the positions of thickness measurement. Thickness was measured with the CBED method.
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tion areas whose thickness t0 was about 95 nm and thickness variation Dt was larger than 50 nm: Table 1 shows the experimental conditions and results, and calculation results for Cu3 Au alloy aged at 523 K for 1369 h after quenching from 1123 K: As shown in this table, diffraction intensities of the ð0 0 1Þ pattern were detected at seven different diffraction areas. The diameter of a diffraction area was about 270 nm; and the thickness range in a diffraction area was estimated by using the developed method explained above and the geometry of Fig. 3a). Table 2 shows similar results of five different diffraction areas in Cu3 Au alloy aged at 653 K for 3 h after rapid quenching from 1203 K: In these tables, LRO parameters were calculated without considering absorption effects from the ratio of I100 =I200 by using atomic scattering factors calculated by the Thomas–Fermi–Dirac (TFD) method which considers relativistic effects [17]. The reason why we chose the TFD method to calculate atomic scattering factors and not Hartree–Fock (HF) method [18] is that the former is believed to be more suitable for heavy atoms than the latter. The average value S and standard deviation DS of LRO parameters for seven data points in the Cu3 Au alloy aged at 523 K for 1396 h are
Fig. 6. Calculated intensity ratio I100 =I200 of ð1 0 0Þ to ð2 0 0Þ reflection as a function of thickness t0 at the center of a diffraction area in the case of S ¼ 1 in Cu3 Au alloy. Calculation was made in consideration of thickness variation in a diffraction area for several cases of thickness difference Dt ¼ tmax tmin :
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Table 1 Summary of experimental and calculation results in Cu3 Au alloy which was aged at 523 K for 1369 h after quenching from 1123 K Sampling area number
P-1 P-2 P-3 P-4 P-5 P-6 P-7
Thickness range in a diffraction area (nm) tmin tmax ðt0 71=2DÞ
65–147 ð106741Þ 55–135 ð95740Þ 59–130 ð95735Þ 69–129 ð99730Þ 53–142 ð97744Þ 72–123 ð97726Þ 57–146 ð102744Þ
Measured diffraction intensities Average7Deviation
Calculated LRO parameter Ratio
ð1 0 0Þ I100 7DI100 ðDI100 =I100 ð%ÞÞ
ð2 0 0Þ I200 7DI200 ðDI200 =I200 ð%ÞÞ
ð1 0 0Þ to ð2 0 0Þ I100 =I200
35007215 ð6:1%Þ 32057337 ð10:5%Þ 35957651 ð18:1%Þ 3946795 ð2:4%Þ 30737524 ð17:1%Þ 44087270 ð6:1%Þ 33667290 ð8:6%Þ
1438274375 ð30:4%Þ 1333174668 ð35:0%Þ 1774474158 ð23:4%Þ 1784575514 ð30:9%Þ 1246375092 ð40:9%Þ 1619477500 ð46:3%Þ 1578975040 ð31:9%Þ
0.2434
0.982
0.2404
0.971
0.2211
0.918
0.2466
1.000
0.2466
0.978
0.2722
1.165
0.2132
0.926
Data were taken at 300 K for P-1–P4, 110 K for P-5, 122 K for P-6 and 134 K for P-7.
Table 2 Summary of experimental and calculation results in Cu3 Au alloy which was aged at 653 K for 3 h after rapid quenching from 1203 K Sampling area number
P-1 P-2 P-3 P-4 P-5
Thickness range in a diffraction area (nm) tmin tmax ðt0 71=2DÞ
46–138 ð92746Þ 39–168 ð104765Þ 50–138 ð94744Þ 75–105 ð90715Þ 67–103 ð857176Þ
Measured diffraction intensities
Calculated LRO parameter
Average7 Deviation
Ratio
ð1 0 0Þ I100 7DI100 ðDI100 =I100 ð%ÞÞ
ð2 0 0Þ I200 7DI200 ðDI200 =I200 ð%ÞÞ
ð1 0 0Þ to ð2 0 0Þ I100 =I200
13807273 (19.8%) 11387217 (19.1%) 30367583 (19.2%) 24547208 (8.5%) 24137168 (7.0%)
906276087 (67.2%) 1178177718 (65.5%) 1795277231 (40.3%) 1110378786 (79.1%) 1111676012 (54.1%)
0.1523
0.802
0.0966
0.590
0.1691
0.842
0.2210
0.674
0.2171
0.782
Data were taken at 300 K:
S ¼ 0:991 and DS ¼ 0:076; respectively, as derived from Table 1. Those for five data points in the Cu3 Au alloy aged at 652 K for 3 h are S ¼ 0:738 and DS ¼ 0:093; respectively, as derived from Table 2.
3.4. Absorption effects on the calculation of LRO parameters Absorption effects or inelastic scattering effects of fast electrons by crystals were not considered in
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the calculation of LRO parameters shown in Tables 1 and 2. If the effect of absorption on the diffraction intensity of ð1 0 0Þ reflection and that of ð2 0 0Þ reflection is different, the intensity ratio I100 =I200 is affected by the absorption effects because thickness in the diffraction area is not small. Therefore, the effects of absorption on the calculation of LRO parameters were evaluated. It is generally accepted that the absorption effects can be treated mathematically by introducing imaginary part Vi besides real part Vr into the crystal potential V as follows [19]: V ¼ Vr þ iVi :
ð3Þ
The absorption effect increases with increasing ratio Vi =Vr (absorption coefficient). Many theoretical calculations of Vi =Vr have been carried out so far in the case of Bragg reflections, but few calculations have been performed in the present case of zone-axis orientation. It is expected that the latter is different from the former except for g ¼ ð0 0 0Þ: Therefore, we tried to obtain Vi =Vr for zone-axis orientation by parameter fitting from the experimental data. Because a calculated absorption coefficient Vi =Vr increases almost parabolically with jgj in many cases of Bragg reflection, it has been reported that Vi =Vr may be written into a parametric fit of the form for small value of jgj [20,21]: Vi =Vr ¼ ajgj2 bjgj þ c;
ð4Þ
where a; b and c are fitting parameters. We supposed that Eq. (4) stands for ga0 in the case of zone-axis orientation, and determined these parameters by minimizing the sum of the deviations of calculated from experimental diffraction intensities Sg jIðgÞcal IðgÞexp j=jIðgÞexp j for g ¼ ð0 0 0Þ; ð1 0 0Þ; ð1 1 0Þ; ð2 0 0Þ; ð2 1 0Þ and ð2 2 0Þ from one of seven experimental data shown in Table 1. In this calculation we used Vi =Vr ¼ 0:173 for Cu and 0.239 for Au as those for g ¼ ð0 0 0Þ which were calculated by Radi [22] in the case of Bragg reflection. It was found that the minimization of sum Sg jIðgÞcal IðgÞexp j=jIðgÞexp j by parameter fitting was most successful when we used the data of P4. The following were derived from the parameter fitting using the data point of P4 in
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Table 3 Summary of calculated LRO parameter in Cu3 Au alloy which was aged at 523 K for 1369 h Intensity ratio used for calculation
Calculated average LRO parameter Without absorption
ð1 0 0Þ ð1 1 0Þ ð1 0 0Þ ð1 1 0Þ
to to to to
ð2 0 0Þ ð2 0 0Þ ð2 2 0Þ ð2 2 0Þ
I100 =I200 I110 =I200 I100 =I220 I110 =I220
S S S S
¼ 0:991 ¼ 1:156 ¼ 0:675 ¼ 0:569
With absorption S S S S
¼ 0:953 ¼ 0:928 ¼ 0:973 ¼ 0:980
The LRO parameter shown were obtained by averaging seven calculated values from seven data point with and without consideration of absorption effects.
Table 1: Vi =Vr ¼ 0:044jgj2 0:23jgj þ 0:31:
ð5Þ
We calculated the LRO parameters for seven experimental data shown in Table 1 with and without consideration of absorption effects from the different intensity ratio of I100 =I200 ; I110 =I200 ; I100 =I220 and I110 =I200 ; respectively. The calculations of the LRO parameters considering the absorption effects were performed by using Vi =Vr given by Eq. (5) for ga0 and the values calculated by Radi [22] for g ¼ ð0 0 0Þ: After the calculations we averaged seven calculated LRO parameters to compare these eight calculation conditions. Table 3 compares these averaged LRO parameters calculated for different calculation conditions.
4. Discussions The LRO parameters at seven positions in the Cu3 Au alloy aged at 523 and 652 K were determined with the developed electron diffraction method. The average value S and standard deviation DS of LRO parameters in the Cu3 Au alloy aged at 523 K; which were calculated for seven data from the intensity ratio I100 =I200 of ð1 0 0Þ superlattice to ð2 0 0Þ fundamental reflection intensity without consideration of absorption effect, are S ¼ 0:991 and DS ¼ 0:076; respectively
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(Table 1). Those for five data of the Cu3 Au alloy aged at 652 K are S ¼ 0:738 and DS ¼ 0:093; respectively (Table 2). Keating and Warren determined the LRO parameters of a single crystal of Cu3 Au alloy with the X-ray diffraction method [17]. According to their paper, the measured LRO parameter of Cu3 Au aged for ordering at 523 K for 52 h is S ¼ 0:972; and that of Cu3 Au aged at 653 K for 1:5 h is S ¼ 0:775: Therefore, the average LRO parameters determined from the ratio I100 =I200 are close to the values of LRO parameters determined by the X-ray diffraction method. On the other hand, the average LRO parameters which were calculated from the intensity ratios I110 =I200 ; I110 =I200 and I110 =I200 without consideration of absorption effects are S ¼ 1:156; 0:675 and 0.569, respectively (Table 3). They are all far from the values by the X-ray diffraction method. We also calculated LRO parameters for seven data in the Cu3 Au alloy aged at 523 K considering absorption effects from the different intensity ratio I100 =I200 ; I110 =I200 ; I100 =I220 and I110 =I220 ; respectively. In this calculation we used absorption coefficients for gað0; 0; 0Þ which were determined by calculation from the experimental diffraction intensities. As shown in Table 3, the average LRO parameter is 0.953 for the calculation from I100 =I200 ; 0.928 from I110 =I200 ; 0.973 from I100 =I220 ; and 0.980 from I110 =I220 : Therefore, they are close to each other, and all of them are close to the value 0.972 by the X-ray diffraction method. This implies that the present calculation of LRO parameters considering absorption effects is successful and that the absorption coefficients determined from diffraction intensities are reasonable. The large difference between LRO parameters calculated with consideration of absorption effects and those without consideration of them in the case of calculation from the ratio I110 =I200 ; I100 =I220 or I110 =I220 implies that absorption effects are usually essential to calculate diffraction intensities in thick diffraction areas like P1 to P7 in Table 1. The small difference between them in the case of calculation from the ratio I100 =I200 must be very special, and this implies that the reduction rate of diffraction intensities by the effects of absorption happen to be almost the same for the ð1 0 0Þ and
ð2 0 0Þ reflections. This allows us to discuss about causes of the deviation of calculated LRO parameters by taking examples shown in Tables 1 and 2. Although the average LRO parameter S determined by the present method is reasonable, the deviation DS of the LRO parameter is not necessarily small as shown in Tables 1 and 2. The deviation DS ¼ 0:076 is about 7.67% of S ¼ 0:991 in Cu3 Au aged at 523 K (Table 1) and the deviation DS ¼ 0:093 is about 12.6% of S ¼ 0:738 in Cu3 Au aged at 652 K (Table 2). The error in measuring specimen thickness t0 at the center of a diffraction area is thought to be the main cause for the deviation DS of calculated LRO parameters, because they are extremely sensitive to specimen thickness t0 : From calculations it was found that the change of 1% in thickness t0 causes on average the change of about 0.8% in the calculated LRO parameters in the case of Cu3 Au aged at 523 K and the change of about 0.5% in the case of Cu3 Au aged at 653 K: Therefore, the experimental error of about 9.6% in thickness t0 corresponds to the calculation error of 7.67% of S (or DS ¼ 0:076) in the case of Cu3 Au aged at 523 K (Table 1), and that of about 25.5% corresponds to the error of 12.6% of S (or DS ¼ 0:093) in the case of Cu3 Au aged at 653 K (Table 2). Judging from Figs. 2 or 5, experimental error in measuring specimen thickness t0 at the center of a diffraction area is around 10%. Therefore, it is reasonable to suppose that the experimental error in measuring thickness t0 is the main cause of the dispersion DS in the case of Cu3 Au aged at 523 K (Table 1). In the case of Cu3 Au aged at 653 K (Table 2), on the other hand, other experimental error besides thickness measurement error could be another main cause of the large dispersion of 12.6% of S (or DS ¼ 0:093), because 25.5% is too large as an error in measuring specimen thickness. As derived from Table 1, the average ratio of the deviation to the average value of superlattice ð1 0 0Þ reflection intensity is DI100 =I100 ¼ 9:8%; and that of fundamental ð2 0 0Þ intensity is DI200 =I200 ¼ 29:6%: From Table 2 it was derived that those are DI100 =I100 ¼ 14:2% and DI200 =I200 ¼ 50:5%: Therefore, the degree of symmetry of diffraction intensity in Cu3 Au aged at 653 K
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(Table 2) is much lower than that in Cu3 Au aged at 523 K (Table 1). This suggests that the large deviation from symmetry in the diffraction pattern is another main cause for the large dispersion of S in the case of Cu3 Au aged at 653 K (Table 2). It should be noted that the ratio DI100 =I100 is much smaller than DI200 =I200 in all cases, and it seems that there is no relationship between DI100 =I100 and DI200 =I200 for each sampling area as shown in Tables 1 and 2. It was found that to get a perfect four-fold symmetric diffraction pattern is almost impossible. It is thought that slight bending of TEM specimen in a diffraction area is one of the main causes of this inevitable imperfection of symmetry in a diffraction pattern. Wedge shape TEM specimens as shown in Fig. 5 could also destroy a perfect symmetry between IðgÞ and IðgÞ due to the differences in orientation between top and bottom faces of wedge foils. It is confirmed from the discussion above that the error in measuring specimen thickness t0 and the imperfection of symmetry in a diffraction pattern are main causes of deviation DS of calculated LRO parameters. It is well expected that the deviation DS and the errors of averaged value S caused by them are decreased by increasing the number of data points to take from one specimen. Therefore, the accuracy of LRO parameters determined by the present method is thought to be improved by increasing number of data points. Since it is possible to get diffraction patterns repeatedly from one specimen position in the present method, we can also obtain the more accurate LRO parameter from one position by averaging LRO parameters calculated from the more number of diffraction patterns. It should be noted that averaging of LRO parameters which are calculated from several ratios of I100 =I200 ; I110 =I200 ; I100 =I220 ; I110 =I220 and so on with consideration of absorption effects also reduces the errors of calculated LRO parameters. As far as computer programming, including of incident beam direction as parameter may improve the accuracy of the calculation of LRO parameters. Of course, improvements of experimental methods in measuring specimen thickness and adjustment of symmetry of four-folded diffraction patterns also improve the accuracy.
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5. Conclusions A system to determine long-range order (LRO) parameters from electron diffraction intensities was developed in the present research. A CCD camera system to detect precisely electron diffraction intensities, a new method for quickly measuring specimen thickness using scattering absorption of a fine electron beam in NBD mode, and a computer program to calculate LRO parameters from the ratio of superlattice and fundamental diffraction intensities were developed. The computer program to calculate LRO parameters was devised to be applicable to the case of a symmetric diffraction pattern. In the developed computer program, diffraction intensities were calculated using the multi-slice method and thickness variation in a diffraction area was taken into consideration for the calculation of diffraction intensities. The computer program to calculate absorption form factors from the diffraction intensities was also written in order to examine the effects of absorption on the calculation of LRO parameters. The successive approximation method was applied in the computer program to calculate LRO parameters from diffraction intensities. The developed system was used to determine LRO parameters of Cu3 Au alloy and the obtained results were compared with those by X-ray diffraction in order to verify the system. The diffraction area was chosen such that its thickness conditions minimized the influence of experimental error in measuring specimen thickness in the calculation of LRO parameters. Specimen thickness increased steeply and proportionally with increasing the distance between the position and the specimen edge in the electropolished Cu3 Au alloy. Seven diffraction areas were examined in the Cu3 Au alloy aged at 523 K for 1369 h; and the average LRO parameter S and standard deviation DS; which were calculated from the ratio I100 =I200 of superlattice ð1 0 0Þ to fundamental ð2 0 0Þ reflection intensity without consideration of absorption effects, were S ¼ 0:991 and DS ¼ 0:076 in the Cu3 Au alloy aged at 523 K: Five diffraction areas were examined in the Cu3 Au alloy aged at 653 K for 3 h; and the similarly calculated average LRO
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parameter was S ¼ 0:785 and the standard deviation was DS ¼ 0:093: Absorption coefficients were calculated from the experimental data of diffraction intensities. The influence of absorption effects on the calculation of LRO parameters is very small for the calculation of LRO parameters from the ratio I100 =I200 of ð1 0 0Þ to ð2 0 0Þ reflection intensity. On the other hand, the calculated LRO parameters were significantly influenced by the absorption effects for the calculation from the ratios I110 =I200 ; I110 =I220 and I110 =I220 : Since it is confirmed that the average LRO parameters in the Cu3 Au alloy aged at 523 and 653 K are in good agreement with the reported data by the X-ray diffraction method, it is reasonable to suppose that the present system is adequate for determining the LRO parameters of alloys. It is thought from the calculation of LRO parameters for different thickness conditions that the experimental errors in measuring specimen thickness are the main cause of the deviation of the calculated LRO parameter DS ¼ 0:076 in the Cu3 Au alloy aged at 523 K: It is also thought from the calculation that both the thickness measurement errors and the comparatively large deviation from the perfect symmetric diffraction intensity are the main causes of the large dispersion of the calculated LRO parameter DS ¼ 0:093 in the Cu3 Au alloy aged at 653 K:
Acknowledgements The authors would like to thank Dr. Wei Sun, Prof. Kenichi Ohshima and Prof. Denjiro
Watanabe for their stimulating discussions and encouragement.
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