- Email: [email protected]
Elastic constants of Si crystal determined by thermal diffuse electron scattering
ARTICLE IN PRESS
Ultramicroscopy 98 (2004) 159–163
Elastic constants of Si crystal determined by thermal diffuse electron scattering Renhui Wang*, Jianhua Yin, Jianian Gui, Jianbo Wang, Huamin Zou Department of Physics and Center for Electron Microscopy, Wuhan University, Wuhan 430072, China Received 19 April 2003; received in revised form 15 July 2003 Dedicated to Professor Fang-hua Li on the occasion of her 70th birthday
Abstract The method of determining elastic constants of crystals by measuring thermal diffuse X-ray scattering around some Bragg reflections, is extended to measuring thermal diffuse electron scattering for the first time, in a transmission electron microscope, equipped with a field-emission gun, an O-type energy filter and a multi-scan charge-coupled device. Quantitative diffuse electron scattering in the vicinity of the ð10 6% 0Þ Bragg reflection was measured on a Si crystal in order to obtain information about elastic constants. Values of the elastic constants ratios C12/C11=0.4246, C44/C11=0.4707 obtained by simplex fitting method are consistent with the values C12/C11=0.3856, C44/C11=0.4804 determined by other traditional methods. This method may be expected to open a new route to measuring elastic constants of polycrystalline, nanometer-scaled and composite materials. r 2003 Elsevier B.V. All rights reserved. PACS: 61.14. 62.20 Keywords: Diffuse electron scattering; Elastic constants determination; Si crystal
1. Introduction Determining elastic constants is a fundamental work for investigating physical and mechanical properties of materials. The classic methods of determining the elastic constants, such as the tensile test, resonant ultrasound spectroscopy, ultrasonic wave velocity measurement and so on, have significant limitations when the sample is *Corresponding author. Tel.: +86-27-87669170; fax: +8627-87654569. E-mail address: [email protected] (R. Wang).
very soft or when the experiments are made under high or low temperatures. Under such conditions, diffuse scatterings both of X-ray [1] and of neutron scattering [2,3] have been used as a powerful and unique approach to determine elastic constants. The fundamental theory of thermal diffuse scattering (TDS) from crystals was established in the 1940s. Wooster and his coworkers [1] developed the experimental technique for elastic constant determination by measuring diffuse X-ray scattering and Wu and Wang [4] determined the elastic constants of Zn by this method.
0304-3991/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2003.08.009
ARTICLE IN PRESS 160
R. Wang et al. / Ultramicroscopy 98 (2004) 159–163
Large-dimension single crystals are required by most of the above-mentioned methods and only general or average effect properties can be achieved when measured without microscopic information. With the recent great developments in hardware and software related to electron microscopy, e.g., the field-emission gun (FEG), charge-coupled device (CCD) camera, imaging plate system and energy filter system, together with the widespread and advanced workstations or personal computers having high-speed CPU and enough memory, it is now feasible to acquire and process the quantitative intensity distribution of electron diffraction. As a result, quantitative electron microscopy came to be (first) established by Vincent et al. [5] in 1984 to determine the positions of atoms in AuGeAs crystal. This was then expanded to be a powerful tool to investigate the local crystal structures, including both the atomic (unit cell parameters, atomic positions, etc.) and electronic structures (charge density distribution). It is now well established that the technique of quantitative convergent-beam electron diffraction (CBED) can be used for accurate measurements of low-order structure factors [6–8]. The measurements are based on quantitative comparison between experimentally measured elastic CBED pattern and computer simulations [9]. Then Zou et al. [10,11] extended this method to automated quasicrystal’s structure refinement. With the development of quantitative electron diffraction, it becomes feasible to investigate the elastic constants by using diffuse electron scattering on transmission electron microscopes (TEM). The advantages of the TEM study are: (1) the possibility of using samples containing defects, grain boundaries and second phases by selecting only a small perfect area, and (2) short time for recording a plenty large quantity of data. Lei et al. [12] are the first to determine the ratio of the two phason-type elastic constants of face-centered icosahedral Al–Pd–Mn quasicrystal by diffuse electron scattering. In the present paper, based on the general theory for TDS from crystals, quantitative diffuse electron scattering in the vicinity of some Bragg reflections was measured on a Si crystal. To reduce dynamic effects, the sample was tilted away from
the zone axis by small angles (1–4 ) so that almost no strong Bragg reflections were excited. In this case the experimental data may be processed within the framework of kinematical theory. The values of the ratios of elastic constants C12/C11=0.4246 and C44/C11=0.4707 were determined by quantitative comparison between experimentally measured intensity distribution of the quasi-elastically scattered electrons in selectedarea electron diffraction (SAED) patterns and the theoretically calculated intensity distribution of TDS. These ratios are consistent with the values C12/C11=0.3856 and C44/C11=0.4804 determined by other traditional methods.
2. Experimental The sample is a highly pure Si single crystal. A TEM specimen with plane normal parallel to an [0 0 1] axis was prepared from the thin Si slice by mechanical grinding and polishing, dimpling and subsequent Ar ion milling. Examinations by TEM observations revealed a very high degree of structural perfection. The quantitative electron diffraction was carried out on a TEM of JEOL JEM-2010FEF TEM equipped with a FEG, an O energy filter system and a Gatan 1024 1024 pixels astronomy-grade multi-scan CCD camera, and operated at the accelerating voltage of 200 kV. This advanced TEM was recently installed in the Center for Electron Microscopy of Wuhan University [13]. Since the energy of phonons is usually less than 0.1 eV, a narrow energy slit with a width of 10–16 eV was used. Thus, not only elastically scattered electrons, but also quasielastically scattered electrons with an energy loss less than 5–8 eV (including the TDS electrons), are digitally recorded by the multi-scan CCD camera. The contribution to the SAED patterns comes from Bragg spots and distribution of diffuse scattering around them. We can use the advantage of the TEM to select perfect micro-regions of the studied crystals to avoid contributions from other defects. Moreover, in the experiment, the sample was tilted away from the zone axis by small angles (1–4 ) so that almost no strong Bragg reflections were
ARTICLE IN PRESS R. Wang et al. / Ultramicroscopy 98 (2004) 159–163
excited. The exposure time for recording the diffuse scattering patterns, being tens to hundreds of seconds, was so selected that (1) the diffuse scattering intensity may be recorded with enough statistics, and (2) the strongest intensity in the frame recorded by the CCD camera was close to and smaller than the dynamic linear range of the camera (214 D16000), after the transmitted spot was moved out of the frame. The experimental data were processed within the framework of kinematical theory since the dynamic effects are very weak under the present experimental conditions. A conventional Kikuchi pattern was taken along with each SAED pattern. Three coordinate systems, namely, the experimental imaging system, zone-axis system and standard crystallographic coordinate system, were involved in the present work. To know the relationship between the zone-axis and crystallographic coordinate systems, it is necessary to index the Kikuchi pattern or the corresponding SAED pattern. In the present paper both the zone-axis and the standard crystal coordinate systems are the same, with the three mutually orthogonal 4-fold axes of the diamond-structured Si crystal parallel to the coordinate axes. We use three Euler angles, namely, c; y and f; to define the orientation relationship between the experimental and zoneaxis coordinate systems. The approximate values of c; y and f can be evaluated from the corresponding Kikuchi pattern associated with the SAED pattern.
161
Fig. 1. Schematic diagram showing the relation between diffraction vector s, the reciprocal lattice vector Q, and lattice wave vector p: s=Q7p.
particular Bragg peak Q can be written as kB T jF ðQÞj2 e2W ðQÞ ITDS ðQ þ pÞ ¼ Vc ðQ þ pÞT A1 ðpÞðQ þ pÞ;
ð1Þ
where kB is Boltzmann constant, T is temperature in K, Q is a reciprocal lattice vector, and p is a lattice wave vector as shown in Fig. 1. Q and p can also be understood as a 3 1 matrix and the uppercase T indicates the transpose of the matrix. Vc is the volume of an unit cell, F(Q) is the structure factor of the unit cell and e2W ðQÞ is the Debye–Waller factor. A(p) is the hydrodynamic matrix, which is related to p and elastic constants Cijkl of the crystal, and can be written as
3. Results
Aik ðpÞ ¼ pj Cijkl pl :
In our experiments, SAED patterns containing information of diffuse electron scattering around some Bragg reflections in a plane nearly perpendicular to [0 0 1] axis were recorded when the [0 0 1] axis of the Si crystal was tilted away from the incident beam by a small angle y (1.13 ). The Euler angles, which describe the relationship between the experimental system and the zone-axis coordinate system in this experiment, were f ¼ 13:6 , y ¼ 1:13 and c ¼ 3:85 . From the general formulae for TDS from a crystal [1,14], the TDS intensity per unit cell at a position p away from a
In the case of diamond-structured Si crystals, the three mutually orthogonal 4-fold axes are chosen as the bases of coordinate system and the elastic constant matrix [15] can be given as 1 0 c11 c12 c12 0 0 0 C B B c12 c11 c12 0 0 0 C C B Bc 0 0 C C B 12 c12 c11 0 ½C ¼ B ð3Þ C B 0 0 0 c44 0 0 C C B B 0 0 0 0 c44 0 C A @ 0 0 0 0 0 c44
ð2Þ
ARTICLE IN PRESS R. Wang et al. / Ultramicroscopy 98 (2004) 159–163
162
so the hydrodynamic matrix A(p) in (2) can be expressed as 0 ðC44 þ C12 Þp1 p2 C11 p21 þ C44 ðp22 þ p23 Þ B A ¼ @ ðC44 þ C12 Þp2 p1 C11 p22 þ C44 ðp23 þ p21 Þ ðC44 þ C12 Þp3 p1
ðC44 þ C12 Þp3 p2
w2 ¼
i;j
þ bÞ
Ii;jexp 2
Ii;jexp
þ
C44 ðp21
þ
p22 Þ
C A:
ð4Þ
ð5Þ
we applied a Standard Multidimensional Downhill Simplex method [16] to generate the fit of theoretical TDS intensity calculated with Eq. (1) to an experimental diffuse scattering intensity around the Bragg reflection spot ð10 6% 0Þ: In this case, only the ratios of the diffuse scattering ITDS ðQ þ pÞ to the integrated intensity of the Bragg scattering IBInt ðQÞ ¼
C11 p23
1
C44/C11=0.4804 determined by other traditional methods [17].
Based on the objective function X ½ðcIi;jtheo
ðC44 þ C12 Þp1 p3 ðC44 þ C12 Þp2 p3
ð2pÞ3 jF ðQÞj2 e2W ðQÞ ; Vc
namely, only the expressions kB T ðQ þ pÞT A1 ðpÞðQ þ pÞ ð2pÞ3 for different p are involved in calculation regardless of the absolute value of the factor jF ðQÞj2 e2W ðQÞ for each Bragg reflection. Fig. 2b shows the result of fitting to Fig. 2a. Values of the elastic constants ratios C12/C11=0.4246, C44/C11=0.4707 obtained by simplex fitting method are consistent with the values C12/C11=0.3856,
4. Discussion and conclusion In summary, the diffuse electron scattering in the vicinity of ð10 6% 0Þ Bragg reflection of a Si crystal was measured. The quantitative analysis of experimental data shows that the diffuse scattering intensity from Si crystal is related to ratios of its elastic constants C12/C11 and C44/C11. The fitted values of these ratios C12/C11=0.4246, C44/C11=0.4707 are consistent with the values C12/C11=0.3856, C44/C11=0.4804 determined by other traditional methods [17]. This is the first time the elastic constants of crystals are being determined by the diffuse electron scattering method. This method may be expected to open a new route to measuring elastic constants of polycrystalline, nanometer-scaled and composite materials. In the present work the calculation was performed under the first approximation, namely, we considered only the first order of TDS and neglected contributions of second and higher
Fig. 2. Diffuse electron scattering around reflection spot ð10 6% 0Þ in SAED pattern tilted away from the zone axis [0 0 1] by 1.13 . (a) Experimental iso-intensity contours and (b) Calculated iso-intensity contours with ratios C12 =C11 ¼ 0:4246 and C44 =C11 ¼ 0:4707: The unit of the numbers in the vertical and horizontal axes is pixel.
ARTICLE IN PRESS R. Wang et al. / Ultramicroscopy 98 (2004) 159–163
orders of TDS. Usually the second-order contribution to the TDS seldom amounts to more than 10% of the total effects and higher-order TDS is always negligible. Moreover, the calculation in the present work was carried out within the framework of kinematical theory, which is feasible under the present experimental conditions as described in Section 2. Unlike X-ray and neutron scattering, dynamic effect is much more important for electron scattering. More accurate results may be obtained by taking into account of the secondorder TDS and dynamic effects in detail.
Acknowledgements This work was supported by National Natural Science Foundation of China. Grant No: 59901013 and 50171048. References [1] W.A. Wooster, Diffuse X-ray Reflections from Crystals, Clarendon, Oxford, 1962. [2] M. de Boissieu, M. Boudard, B. Hennion, R. Bellissent, S. Kycia, A. Goldman, C. Janot, M. Audier, Phys. Rev. Lett. 75 (1995) 89.
163
[3] M. Boudard, M. de Boissieu, A. Letoublon, B. Hennion, R. Bellissent, C. Janot, Europhys. Lett. 33 (1996) 199. [4] Wu Dechang, Wang Renhui, Acta Phys. Sin. 22 (1966) 533 (in Chinese). [5] R. Vincent, D.M. Bird, J.W. Steeds, Philos. Mag. A 50 (1984) 745. [6] J.M. Zuo, J.C.H. Spence, M. O’Keeffe, Phys. Rev. Lett. 81 (1988) 353. [7] M. Sauders, D.M. Bird, N.J. Zaluzec, W.G. Burges, A.P. Preston, C.J. Humphreys, Ultromicroscopy 60 (1995) 311. [8] R. Hoier, L.N. Bakken, K. Marthinsen, R. Holmestad, Ultramicroscopy 49 (1993) 159. [9] J.C.H. Spence, J.M. Zuo, Electron Microdiffraction, Plenum, New York, 1992. [10] Zhao Chun, Zou Huamin, et al., Wuhan Univ. J. Nat. Sci. 5 (2000) 051. [11] Aihua Fang, Huamin Zou, et al., J. Phys.: Condensed Matter 15 (2003) 4947. [12] Jianlin Lei, Renhui Wang, Jianhua Yin, Xiaofeng Duan, J. Alloys Compounds 342 (2002) 326. [13] http://cem.whu.edu.cn [14] Wang Renhui, Hu Chengzheng, Gui Jianian, Quasicrystal Physics, China Science Press, Beijing, 2003 (in Chinese). [15] Wang Renhui, Guo Kexin (K.H.Kuo), Symmetry Groups in Crystallography, China Science Press, Beijing, 1990 (in Chinese). [16] W.H. Press, B.P. Flannery, S.A. TeuKolsky, W.T. Vettering, Numerical Recipes, Cambridge University Press, Cambridge, 1986. [17] J.P. Hirth, J. Lothe, Theory of Dislocations, McGrawHill, New York, 1968.
Ultramicroscopy 98 (2004) 159–163
Elastic constants of Si crystal determined by thermal diffuse electron scattering Renhui Wang*, Jianhua Yin, Jianian Gui, Jianbo Wang, Huamin Zou Department of Physics and Center for Electron Microscopy, Wuhan University, Wuhan 430072, China Received 19 April 2003; received in revised form 15 July 2003 Dedicated to Professor Fang-hua Li on the occasion of her 70th birthday
Abstract The method of determining elastic constants of crystals by measuring thermal diffuse X-ray scattering around some Bragg reflections, is extended to measuring thermal diffuse electron scattering for the first time, in a transmission electron microscope, equipped with a field-emission gun, an O-type energy filter and a multi-scan charge-coupled device. Quantitative diffuse electron scattering in the vicinity of the ð10 6% 0Þ Bragg reflection was measured on a Si crystal in order to obtain information about elastic constants. Values of the elastic constants ratios C12/C11=0.4246, C44/C11=0.4707 obtained by simplex fitting method are consistent with the values C12/C11=0.3856, C44/C11=0.4804 determined by other traditional methods. This method may be expected to open a new route to measuring elastic constants of polycrystalline, nanometer-scaled and composite materials. r 2003 Elsevier B.V. All rights reserved. PACS: 61.14. 62.20 Keywords: Diffuse electron scattering; Elastic constants determination; Si crystal
1. Introduction Determining elastic constants is a fundamental work for investigating physical and mechanical properties of materials. The classic methods of determining the elastic constants, such as the tensile test, resonant ultrasound spectroscopy, ultrasonic wave velocity measurement and so on, have significant limitations when the sample is *Corresponding author. Tel.: +86-27-87669170; fax: +8627-87654569. E-mail address: [email protected] (R. Wang).
very soft or when the experiments are made under high or low temperatures. Under such conditions, diffuse scatterings both of X-ray [1] and of neutron scattering [2,3] have been used as a powerful and unique approach to determine elastic constants. The fundamental theory of thermal diffuse scattering (TDS) from crystals was established in the 1940s. Wooster and his coworkers [1] developed the experimental technique for elastic constant determination by measuring diffuse X-ray scattering and Wu and Wang [4] determined the elastic constants of Zn by this method.
0304-3991/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2003.08.009
ARTICLE IN PRESS 160
R. Wang et al. / Ultramicroscopy 98 (2004) 159–163
Large-dimension single crystals are required by most of the above-mentioned methods and only general or average effect properties can be achieved when measured without microscopic information. With the recent great developments in hardware and software related to electron microscopy, e.g., the field-emission gun (FEG), charge-coupled device (CCD) camera, imaging plate system and energy filter system, together with the widespread and advanced workstations or personal computers having high-speed CPU and enough memory, it is now feasible to acquire and process the quantitative intensity distribution of electron diffraction. As a result, quantitative electron microscopy came to be (first) established by Vincent et al. [5] in 1984 to determine the positions of atoms in AuGeAs crystal. This was then expanded to be a powerful tool to investigate the local crystal structures, including both the atomic (unit cell parameters, atomic positions, etc.) and electronic structures (charge density distribution). It is now well established that the technique of quantitative convergent-beam electron diffraction (CBED) can be used for accurate measurements of low-order structure factors [6–8]. The measurements are based on quantitative comparison between experimentally measured elastic CBED pattern and computer simulations [9]. Then Zou et al. [10,11] extended this method to automated quasicrystal’s structure refinement. With the development of quantitative electron diffraction, it becomes feasible to investigate the elastic constants by using diffuse electron scattering on transmission electron microscopes (TEM). The advantages of the TEM study are: (1) the possibility of using samples containing defects, grain boundaries and second phases by selecting only a small perfect area, and (2) short time for recording a plenty large quantity of data. Lei et al. [12] are the first to determine the ratio of the two phason-type elastic constants of face-centered icosahedral Al–Pd–Mn quasicrystal by diffuse electron scattering. In the present paper, based on the general theory for TDS from crystals, quantitative diffuse electron scattering in the vicinity of some Bragg reflections was measured on a Si crystal. To reduce dynamic effects, the sample was tilted away from
the zone axis by small angles (1–4 ) so that almost no strong Bragg reflections were excited. In this case the experimental data may be processed within the framework of kinematical theory. The values of the ratios of elastic constants C12/C11=0.4246 and C44/C11=0.4707 were determined by quantitative comparison between experimentally measured intensity distribution of the quasi-elastically scattered electrons in selectedarea electron diffraction (SAED) patterns and the theoretically calculated intensity distribution of TDS. These ratios are consistent with the values C12/C11=0.3856 and C44/C11=0.4804 determined by other traditional methods.
2. Experimental The sample is a highly pure Si single crystal. A TEM specimen with plane normal parallel to an [0 0 1] axis was prepared from the thin Si slice by mechanical grinding and polishing, dimpling and subsequent Ar ion milling. Examinations by TEM observations revealed a very high degree of structural perfection. The quantitative electron diffraction was carried out on a TEM of JEOL JEM-2010FEF TEM equipped with a FEG, an O energy filter system and a Gatan 1024 1024 pixels astronomy-grade multi-scan CCD camera, and operated at the accelerating voltage of 200 kV. This advanced TEM was recently installed in the Center for Electron Microscopy of Wuhan University [13]. Since the energy of phonons is usually less than 0.1 eV, a narrow energy slit with a width of 10–16 eV was used. Thus, not only elastically scattered electrons, but also quasielastically scattered electrons with an energy loss less than 5–8 eV (including the TDS electrons), are digitally recorded by the multi-scan CCD camera. The contribution to the SAED patterns comes from Bragg spots and distribution of diffuse scattering around them. We can use the advantage of the TEM to select perfect micro-regions of the studied crystals to avoid contributions from other defects. Moreover, in the experiment, the sample was tilted away from the zone axis by small angles (1–4 ) so that almost no strong Bragg reflections were
ARTICLE IN PRESS R. Wang et al. / Ultramicroscopy 98 (2004) 159–163
excited. The exposure time for recording the diffuse scattering patterns, being tens to hundreds of seconds, was so selected that (1) the diffuse scattering intensity may be recorded with enough statistics, and (2) the strongest intensity in the frame recorded by the CCD camera was close to and smaller than the dynamic linear range of the camera (214 D16000), after the transmitted spot was moved out of the frame. The experimental data were processed within the framework of kinematical theory since the dynamic effects are very weak under the present experimental conditions. A conventional Kikuchi pattern was taken along with each SAED pattern. Three coordinate systems, namely, the experimental imaging system, zone-axis system and standard crystallographic coordinate system, were involved in the present work. To know the relationship between the zone-axis and crystallographic coordinate systems, it is necessary to index the Kikuchi pattern or the corresponding SAED pattern. In the present paper both the zone-axis and the standard crystal coordinate systems are the same, with the three mutually orthogonal 4-fold axes of the diamond-structured Si crystal parallel to the coordinate axes. We use three Euler angles, namely, c; y and f; to define the orientation relationship between the experimental and zoneaxis coordinate systems. The approximate values of c; y and f can be evaluated from the corresponding Kikuchi pattern associated with the SAED pattern.
161
Fig. 1. Schematic diagram showing the relation between diffraction vector s, the reciprocal lattice vector Q, and lattice wave vector p: s=Q7p.
particular Bragg peak Q can be written as kB T jF ðQÞj2 e2W ðQÞ ITDS ðQ þ pÞ ¼ Vc ðQ þ pÞT A1 ðpÞðQ þ pÞ;
ð1Þ
where kB is Boltzmann constant, T is temperature in K, Q is a reciprocal lattice vector, and p is a lattice wave vector as shown in Fig. 1. Q and p can also be understood as a 3 1 matrix and the uppercase T indicates the transpose of the matrix. Vc is the volume of an unit cell, F(Q) is the structure factor of the unit cell and e2W ðQÞ is the Debye–Waller factor. A(p) is the hydrodynamic matrix, which is related to p and elastic constants Cijkl of the crystal, and can be written as
3. Results
Aik ðpÞ ¼ pj Cijkl pl :
In our experiments, SAED patterns containing information of diffuse electron scattering around some Bragg reflections in a plane nearly perpendicular to [0 0 1] axis were recorded when the [0 0 1] axis of the Si crystal was tilted away from the incident beam by a small angle y (1.13 ). The Euler angles, which describe the relationship between the experimental system and the zone-axis coordinate system in this experiment, were f ¼ 13:6 , y ¼ 1:13 and c ¼ 3:85 . From the general formulae for TDS from a crystal [1,14], the TDS intensity per unit cell at a position p away from a
In the case of diamond-structured Si crystals, the three mutually orthogonal 4-fold axes are chosen as the bases of coordinate system and the elastic constant matrix [15] can be given as 1 0 c11 c12 c12 0 0 0 C B B c12 c11 c12 0 0 0 C C B Bc 0 0 C C B 12 c12 c11 0 ½C ¼ B ð3Þ C B 0 0 0 c44 0 0 C C B B 0 0 0 0 c44 0 C A @ 0 0 0 0 0 c44
ð2Þ
ARTICLE IN PRESS R. Wang et al. / Ultramicroscopy 98 (2004) 159–163
162
so the hydrodynamic matrix A(p) in (2) can be expressed as 0 ðC44 þ C12 Þp1 p2 C11 p21 þ C44 ðp22 þ p23 Þ B A ¼ @ ðC44 þ C12 Þp2 p1 C11 p22 þ C44 ðp23 þ p21 Þ ðC44 þ C12 Þp3 p1
ðC44 þ C12 Þp3 p2
w2 ¼
i;j
þ bÞ
Ii;jexp 2
Ii;jexp
þ
C44 ðp21
þ
p22 Þ
C A:
ð4Þ
ð5Þ
we applied a Standard Multidimensional Downhill Simplex method [16] to generate the fit of theoretical TDS intensity calculated with Eq. (1) to an experimental diffuse scattering intensity around the Bragg reflection spot ð10 6% 0Þ: In this case, only the ratios of the diffuse scattering ITDS ðQ þ pÞ to the integrated intensity of the Bragg scattering IBInt ðQÞ ¼
C11 p23
1
C44/C11=0.4804 determined by other traditional methods [17].
Based on the objective function X ½ðcIi;jtheo
ðC44 þ C12 Þp1 p3 ðC44 þ C12 Þp2 p3
ð2pÞ3 jF ðQÞj2 e2W ðQÞ ; Vc
namely, only the expressions kB T ðQ þ pÞT A1 ðpÞðQ þ pÞ ð2pÞ3 for different p are involved in calculation regardless of the absolute value of the factor jF ðQÞj2 e2W ðQÞ for each Bragg reflection. Fig. 2b shows the result of fitting to Fig. 2a. Values of the elastic constants ratios C12/C11=0.4246, C44/C11=0.4707 obtained by simplex fitting method are consistent with the values C12/C11=0.3856,
4. Discussion and conclusion In summary, the diffuse electron scattering in the vicinity of ð10 6% 0Þ Bragg reflection of a Si crystal was measured. The quantitative analysis of experimental data shows that the diffuse scattering intensity from Si crystal is related to ratios of its elastic constants C12/C11 and C44/C11. The fitted values of these ratios C12/C11=0.4246, C44/C11=0.4707 are consistent with the values C12/C11=0.3856, C44/C11=0.4804 determined by other traditional methods [17]. This is the first time the elastic constants of crystals are being determined by the diffuse electron scattering method. This method may be expected to open a new route to measuring elastic constants of polycrystalline, nanometer-scaled and composite materials. In the present work the calculation was performed under the first approximation, namely, we considered only the first order of TDS and neglected contributions of second and higher
Fig. 2. Diffuse electron scattering around reflection spot ð10 6% 0Þ in SAED pattern tilted away from the zone axis [0 0 1] by 1.13 . (a) Experimental iso-intensity contours and (b) Calculated iso-intensity contours with ratios C12 =C11 ¼ 0:4246 and C44 =C11 ¼ 0:4707: The unit of the numbers in the vertical and horizontal axes is pixel.
ARTICLE IN PRESS R. Wang et al. / Ultramicroscopy 98 (2004) 159–163
orders of TDS. Usually the second-order contribution to the TDS seldom amounts to more than 10% of the total effects and higher-order TDS is always negligible. Moreover, the calculation in the present work was carried out within the framework of kinematical theory, which is feasible under the present experimental conditions as described in Section 2. Unlike X-ray and neutron scattering, dynamic effect is much more important for electron scattering. More accurate results may be obtained by taking into account of the secondorder TDS and dynamic effects in detail.
Acknowledgements This work was supported by National Natural Science Foundation of China. Grant No: 59901013 and 50171048. References [1] W.A. Wooster, Diffuse X-ray Reflections from Crystals, Clarendon, Oxford, 1962. [2] M. de Boissieu, M. Boudard, B. Hennion, R. Bellissent, S. Kycia, A. Goldman, C. Janot, M. Audier, Phys. Rev. Lett. 75 (1995) 89.
163
[3] M. Boudard, M. de Boissieu, A. Letoublon, B. Hennion, R. Bellissent, C. Janot, Europhys. Lett. 33 (1996) 199. [4] Wu Dechang, Wang Renhui, Acta Phys. Sin. 22 (1966) 533 (in Chinese). [5] R. Vincent, D.M. Bird, J.W. Steeds, Philos. Mag. A 50 (1984) 745. [6] J.M. Zuo, J.C.H. Spence, M. O’Keeffe, Phys. Rev. Lett. 81 (1988) 353. [7] M. Sauders, D.M. Bird, N.J. Zaluzec, W.G. Burges, A.P. Preston, C.J. Humphreys, Ultromicroscopy 60 (1995) 311. [8] R. Hoier, L.N. Bakken, K. Marthinsen, R. Holmestad, Ultramicroscopy 49 (1993) 159. [9] J.C.H. Spence, J.M. Zuo, Electron Microdiffraction, Plenum, New York, 1992. [10] Zhao Chun, Zou Huamin, et al., Wuhan Univ. J. Nat. Sci. 5 (2000) 051. [11] Aihua Fang, Huamin Zou, et al., J. Phys.: Condensed Matter 15 (2003) 4947. [12] Jianlin Lei, Renhui Wang, Jianhua Yin, Xiaofeng Duan, J. Alloys Compounds 342 (2002) 326. [13] http://cem.whu.edu.cn [14] Wang Renhui, Hu Chengzheng, Gui Jianian, Quasicrystal Physics, China Science Press, Beijing, 2003 (in Chinese). [15] Wang Renhui, Guo Kexin (K.H.Kuo), Symmetry Groups in Crystallography, China Science Press, Beijing, 1990 (in Chinese). [16] W.H. Press, B.P. Flannery, S.A. TeuKolsky, W.T. Vettering, Numerical Recipes, Cambridge University Press, Cambridge, 1986. [17] J.P. Hirth, J. Lothe, Theory of Dislocations, McGrawHill, New York, 1968.