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Characterisation of CVD grown diamond and its residual stress state
Diamond and Related Materials 8 (1999) 1555–1559 www.elsevier.com/locate/diamond
Characterisation of CVD grown diamond and its residual stress state M. Hempel *, M. Ha¨rting Department of Physics, University of Cape Town, Rondebosch 7701, South Africa Accepted 30 November 1998
Abstract One of the most important quality factors in the judgement of thin diamond layers is the adhesion between the substrate and the layer which is limited by the residual stress state. The main reason for residual stress in coatings is the misfit in various properties of the layer and also the substrate, e.g. thermal expansion and crystal lattice type. The most common method of residual stress determination, based on X-ray diffraction, has to date found little application in the study of diamond films. In this paper, the method is applied to determine the residual stress in a CVD diamond film grown on a polycrystalline Al substrate. The results are interpreted with regard to the crystal structure and orientation of the layer, determined by X-ray diffraction and scanning electron microscopy. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Adhesion; CVD; Diamond; Residual stress
1. Introduction Residual stresses are those stresses which exist in a material without any kind of external loading. In general they are related to the presence of crystal defects and different phases in the material [1]. Additionally in the case of layered structures, misfit strains, arising from the different crystal structures and lattice parameters of the substrate and the overlayer, result in further stresses in both materials [2]. In their typical applications, whether as hard coatings for machine tools or as heat sinks and diffusion barriers in electronic devices, technological diamond films are subject to significant variable mechanical and thermal loads, e.g. see Ref. [3]. Large changes in temperature result in additional thermal stresses due to the different thermal expansion coefficients of the layer and the substrate. Mechanical stresses superimpose on the already present residual and thermal stresses in the materials [4]. Material failure occurs when the total, not the applied, stress exceeds the critical stress. Therefore for layered structures, residual stresses form one of the most important factors affecting the adhesion of the layer to the substrate. Primarily two methods have been used to study stress in diamond films: macroscopically from the radius of * Corresponding author. Fax: +27-216503342. E-mail address: [email protected] (M. Hempel )
curvature, and microscopically by the determination of Raman peak shifts [5]. Additionally, X-ray diffraction, which is the usual method in metal studies, has been applied [6,7]. Although both X-ray diffraction and Raman spectroscopy yield, in principle, the same information, the latter is more suited to studies of single crystals [7], whereas the former is vastly simplified in its application to polycrystals [4]. Both methods deliver an estimate of the macro-residual stress (homogenous over the measurement area), which is usually assumed to be biaxial in the plane of the sample. In this paper results are presented for the determination of residual stress, using the conventional sin2 Y X-ray diffraction method, in CVD grown diamond layer on a polycrystalline aluminium substrate. Results for other layers grown on single crystalline silicon substrates will be presented elsewhere. All samples were initially characterised using H–2H X-ray diffractometry and scanning electron microscopy (SEM ).
2. Characterisation The diamond layer reported here was produced by chemical vapour deposition on a polycrystalline aluminium substrate. Exact details have not been provided by the producers and consequently a detailed characterisation using optical and electronic microscopy and X-ray diffraction techniques was essential. The structure
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Fig. 1. Scanning electron micrograph of a CVD diamond film grown on aluminium.
Fig. 2. X-ray diffraction pattern, taken with Co Ka radiation, of a CVD diamond film grown on aluminium.
consists of the aluminium substrate and a thick diamond layer capped with a thin gold film. From optical microscopy we have determined the diamond thickness to be 240 mm and the substrate thickness to be 120 mm. The thickness of the gold metallisation was determined by both electron microscopy and nuclear solid state techniques [8] to be approximately 50 nm. Furthermore, a cross-sectional view under the optical microscope indicates a fibrous or columnar growth of the diamond layer. A scanning electron micrograph is shown in Fig. 1. The diamond predominantly forms pyramidal crystal-
lites typically of size 10 to 30 mm, typical of thick diamond layers grown on polycrystalline substrates [9]. Besides reflections due to diamond, a conventional H–2H X-ray diffraction scan (Fig. 2) exhibits peaks from aluminium, aluminium carbide (Al C ) and possi4 3 bly graphite. The diffraction pattern was measured using ˚ ) in steps of 0.01°. The diffuse Co Ka radiation (1.8897 A background, which is particularly significant at lower diffraction angles, results from scattering by the polymethylmethacralite sample holder. In Fig. 2 several expected Al C reflections do not appear. This results 4 3
M. Hempel, M. Ha¨rting / Diamond and Related Materials 8 (1999) 1555–1559
from a strong texture discussed below, and their presence has been determined using the multiple axis stress diffractometer. For any orientation of the sample, we found no indication of oxygen related phases. In the H– 2H scan the Al peaks include a very small contribution from diffraction in the gold layer. At the higher diffraction angles used in the stress measurements these peaks can be separately resolved. The diffraction pattern ( Fig. 2) clearly shows a (110) texture in the diamond, with the (220) reflection being considerably more intense than the (111) reflection. In powder samples the (111) reflection is the most intense and has an intensity four times that of the (220) reflection. This suggests a columnar growth in the 110 direction with the pyramid surfaces being formed by (111) faces [9]. In contrast, the aluminium substrate shows no indication of a fibre texture, with all three major reflections equally intense. Nevertheless, considering the crystal structure of diamond in comparison with the fcc aluminium substrate, a fibre texture in the diamond could be expected even if the substrate has no texture. The lattice parameters of the two materials only match for the directions 111 in diamond and 100 in aluminium. In this case two lattice spacings of diamond fit along the edge of the aluminium unit cell. This can be seen by the proximity of the diamond (111) and aluminium (200) reflections. It could therefore be expected that the diamond seeds preferentially on (100) aluminium grains with the 110 direction perpendicular to the surface.
3. X-ray stress determination Residual stress analysis using X-ray diffraction relies on the measurement of the lattice strain Dd(W,Y )/d= e(W,Y ) as a function of the direction Y relative to the sample normal and the azimuth angle W (Fig. 3). The strain is related to the components of the stress tensor by the fundamental equation of X-ray residual stress analysis ( RSA) Dd(W,Y )/d=1/2 s [s cos2 W sin2 Y 2 11 +s sin2 W sin2 Y+s cos2 Y 22 33 +s sin2 W sin2 W+s cos W sin2 Y 12 13 +s sin W sin2 Y ]+s [s +s +s ] (1) 23 1 11 22 33 and to the measurement angle by Dd(W,Y )/d= −cot H [H(W,Y )−H )]. H(W,Y ) and H are the mea0 0 0 sured Bragg angle for the stress-free material respectively, and s and 1/2s are the X-ray elastic constants, 1 2 see for example Refs. [4,10,11]. To determine the two-dimensional stress state, Eq. (1) can be simplified by neglecting the tensor components s with i, j=3. In the usual sin2 Y method, a ij
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global average value for the stress can be determined from the gradient of a plot of strain against sin2 Y for constant W [4]. Every measurement is an average over the diffraction contribution of all lattice spacings of the reflecting grains in the irradiated volume V , where (hkl ) are the Miller (hkl) indices for the chosen reflection, around the measurement point (x, y, z) of the sample. In the presence of a three-dimensional stress state, in which the stress components normal to the sample surface are non-zero, the strain will not depend linearly on sin2 Y. Two different effects can be seen in the measured sin2 Y curves. Firstly, if the lattice strain varies over the penetration depth of the X-rays, the stress gradient will lead to a curvature in the measured curve. Secondly, if either of the shear components s or s are non-zero, strains measured 12 13 at negative Y tilts will differ from those measured at positive angles. This phenomenon is known as Y-splitting [10]. The stress determination was performed using the (331) reflection of Cu Ka radiation at a nominal Bragg angle of 70.61° [12] in conventional V-geometry. Different diffraction peaks up to a maximum tilt of 65°, in 5° intervals, were recorded with a position sensitive detector and multichannel analyser system.
4. Results and discussion Fig. 4 shows the dependence of the measured strain (d−d )/d =e on the square of the sine of the tilt 0 0 WY angle Y. The true value of d is not known, and the 0 ˚ [12] has been value of 0.816584 A assumed. As d only 0 acts as a scale factor the error introduced into the final stress s is less than 0.1%, which is smaller than the W experimental and statistical errors [4]. This is not the case for the intercept (s +s ) which is very sensitive 11 22 to errors in d and will not discussed here. Both 0 positive and negative Y angles are plotted in Fig. 4. The almost classic text book results shown in Fig. 4 are in fact, for this sample, very surprising. A linear sin2 Y curve only arises from a uniform two-dimensional stress state. For the diamond layer, this means that the material is uniformly stressed parallel to its surface through out the whole 240 mm of its thickness. There is no indication of any gradient with depth or any stress components perpendicular to the layer. Linear relationships are usually only expected for dense metals where the penetration depth of the X-rays is in the micron or submicron range. In this case, the penetration depth for the (331) reflection ranges from 53 mm at Y=65° to 292 mm at Y=0° [4], covering the whole depth of the diamond layer. Any gradient, even near the interface with the substrate, would be expected to be seen in a non-linear sin2 Y curve. Similarly, any normal stress components would lead to a splitting of the positive and
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M. Hempel, M. Ha¨rting / Diamond and Related Materials 8 (1999) 1555–1559
Fig. 3. Illustration of the measurement condition to determine residual stress states. (a) Definition of the measurement angles: (a1) coordinate system of the sample S , i=1,2,3, in which the strain tensor e is defined; (a2) azimuth angle W, sample rotation around the normal to the sample i ij surface; (a3) tilting by angle Y, together with W leads to the laboratory coordinate system with the measurement directions L in which the strain WY e is defined. (b) At a fixed Bragg angle H different measurement directions L bring different grains in measurement conditions, d is the WY (hlk) WY lattice spacing of a single grain [8].
Fig. 4. sin2 Y curve for the (331) reflection from a CVD diamond film grown on aluminium.
M. Hempel, M. Ha¨rting / Diamond and Related Materials 8 (1999) 1555–1559
negative Y curves. At present there is no satisfactory model to account for this stress state, but it may be possible that the presence of the other phases may play a role. The fibre texture in the diamond layer has an insignificant effect on the shape of the sin2 Y curve. Further measurements to study the stress states and textures of the carbide and graphite phases are being carried out. To determine the magnitude of the stress we have first estimated the X-ray elastic constants from the singe crystal elastic constants [13] according to the Reuss model [14]. The values obtained for the diamond (331) reflection were s =−4.9088×10−10 Pa−1 and 1 1/2s =0.9022×10−12 Pa−1. Using these values we find 2 a tensile stress in the layer of s =768 MPa and W s =738 MPa for the positive and negative Y measureW ments respectively. These values are within the estimated error of 30 MPa of each other, and are about 10% of the critical stress [3].
5. Conclusions Using X-ray diffraction methods we have studied the residual stress state in a textured polycrystalline diamond layer grown by chemical vapour deposition on an aluminium substrate. Although the magnitude of the stress is moderate, the form of the stress state is extraordinary. There is no indication of any stress gradient or of any
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component of the stress tensor perpendicular to the plane of the film. The layer is uniformly stressed parallel to the substrate interface throughout its whole thickness of 240 mm.
References [1] A. Sommerfeld, in: Vorlesung u¨ber theoretische Physik: Mechanik der deformierbaren Medien, 6th edn. Vol. II Akademische Verlagsgese, Geest & Portig, Leibzig, 1970. [2] A.P. Sutton, R.W. Balluffi, Interfaces in Crystalline Materials, Oxford University Press, New York, 1996. [3] J. Wilks, E. Wilks, Properties and Applications of Diamond, Butterworth-Heinemann, Oxford, 1991. [4] I.C. Noyan, J.B. Cohen, Residual Stress, Springer, New York, 1987. [5] D.S. Knight, W.B. White, J. Mater. Res. 4 (1988) 385. [6 ] N.S. Van Damme, D.C. Nagle, S.R. Winzer, Appl. Phys. Lett. 58 (1991) 2919. [7] J.W. Ager III, M.D. Drory, Phys. Rev. B 48 (1993) 2601. [8] D.T. Britton, M. Hempel, M. Ha¨rting, Appl. Surf. Sci. (1999) in press. [9] M.N.R. Ashfold, P.W. May, C.A. Rego, Chem. Soc. Rev. 23 (1994) 21. [10] V. Hauk, in: V. Hauk, H. Hougardi, E. Macherauch ( Eds.), Residual Stress Measurements, Calculation, Evaluation, DGM Informationsgesellschaft, Oberursel, 1991. [11] M. Ha¨rting, Acta Mater. 46 (1998) 1427. [12] W.Kraus, W. Nolze, Program ‘Powder Cell’, version 1.8d, 1997. [13] D.G. Gray (Ed.), American Institute of Physics Handbook, 3rd edn., McGraw-Hill, New York, 1972. [14] E. Eigenmann, E. Macherauch, Mat.-wiss u. Werkstofftech. 26 (1995) 199.
Characterisation of CVD grown diamond and its residual stress state M. Hempel *, M. Ha¨rting Department of Physics, University of Cape Town, Rondebosch 7701, South Africa Accepted 30 November 1998
Abstract One of the most important quality factors in the judgement of thin diamond layers is the adhesion between the substrate and the layer which is limited by the residual stress state. The main reason for residual stress in coatings is the misfit in various properties of the layer and also the substrate, e.g. thermal expansion and crystal lattice type. The most common method of residual stress determination, based on X-ray diffraction, has to date found little application in the study of diamond films. In this paper, the method is applied to determine the residual stress in a CVD diamond film grown on a polycrystalline Al substrate. The results are interpreted with regard to the crystal structure and orientation of the layer, determined by X-ray diffraction and scanning electron microscopy. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Adhesion; CVD; Diamond; Residual stress
1. Introduction Residual stresses are those stresses which exist in a material without any kind of external loading. In general they are related to the presence of crystal defects and different phases in the material [1]. Additionally in the case of layered structures, misfit strains, arising from the different crystal structures and lattice parameters of the substrate and the overlayer, result in further stresses in both materials [2]. In their typical applications, whether as hard coatings for machine tools or as heat sinks and diffusion barriers in electronic devices, technological diamond films are subject to significant variable mechanical and thermal loads, e.g. see Ref. [3]. Large changes in temperature result in additional thermal stresses due to the different thermal expansion coefficients of the layer and the substrate. Mechanical stresses superimpose on the already present residual and thermal stresses in the materials [4]. Material failure occurs when the total, not the applied, stress exceeds the critical stress. Therefore for layered structures, residual stresses form one of the most important factors affecting the adhesion of the layer to the substrate. Primarily two methods have been used to study stress in diamond films: macroscopically from the radius of * Corresponding author. Fax: +27-216503342. E-mail address: [email protected] (M. Hempel )
curvature, and microscopically by the determination of Raman peak shifts [5]. Additionally, X-ray diffraction, which is the usual method in metal studies, has been applied [6,7]. Although both X-ray diffraction and Raman spectroscopy yield, in principle, the same information, the latter is more suited to studies of single crystals [7], whereas the former is vastly simplified in its application to polycrystals [4]. Both methods deliver an estimate of the macro-residual stress (homogenous over the measurement area), which is usually assumed to be biaxial in the plane of the sample. In this paper results are presented for the determination of residual stress, using the conventional sin2 Y X-ray diffraction method, in CVD grown diamond layer on a polycrystalline aluminium substrate. Results for other layers grown on single crystalline silicon substrates will be presented elsewhere. All samples were initially characterised using H–2H X-ray diffractometry and scanning electron microscopy (SEM ).
2. Characterisation The diamond layer reported here was produced by chemical vapour deposition on a polycrystalline aluminium substrate. Exact details have not been provided by the producers and consequently a detailed characterisation using optical and electronic microscopy and X-ray diffraction techniques was essential. The structure
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Fig. 1. Scanning electron micrograph of a CVD diamond film grown on aluminium.
Fig. 2. X-ray diffraction pattern, taken with Co Ka radiation, of a CVD diamond film grown on aluminium.
consists of the aluminium substrate and a thick diamond layer capped with a thin gold film. From optical microscopy we have determined the diamond thickness to be 240 mm and the substrate thickness to be 120 mm. The thickness of the gold metallisation was determined by both electron microscopy and nuclear solid state techniques [8] to be approximately 50 nm. Furthermore, a cross-sectional view under the optical microscope indicates a fibrous or columnar growth of the diamond layer. A scanning electron micrograph is shown in Fig. 1. The diamond predominantly forms pyramidal crystal-
lites typically of size 10 to 30 mm, typical of thick diamond layers grown on polycrystalline substrates [9]. Besides reflections due to diamond, a conventional H–2H X-ray diffraction scan (Fig. 2) exhibits peaks from aluminium, aluminium carbide (Al C ) and possi4 3 bly graphite. The diffraction pattern was measured using ˚ ) in steps of 0.01°. The diffuse Co Ka radiation (1.8897 A background, which is particularly significant at lower diffraction angles, results from scattering by the polymethylmethacralite sample holder. In Fig. 2 several expected Al C reflections do not appear. This results 4 3
M. Hempel, M. Ha¨rting / Diamond and Related Materials 8 (1999) 1555–1559
from a strong texture discussed below, and their presence has been determined using the multiple axis stress diffractometer. For any orientation of the sample, we found no indication of oxygen related phases. In the H– 2H scan the Al peaks include a very small contribution from diffraction in the gold layer. At the higher diffraction angles used in the stress measurements these peaks can be separately resolved. The diffraction pattern ( Fig. 2) clearly shows a (110) texture in the diamond, with the (220) reflection being considerably more intense than the (111) reflection. In powder samples the (111) reflection is the most intense and has an intensity four times that of the (220) reflection. This suggests a columnar growth in the 110 direction with the pyramid surfaces being formed by (111) faces [9]. In contrast, the aluminium substrate shows no indication of a fibre texture, with all three major reflections equally intense. Nevertheless, considering the crystal structure of diamond in comparison with the fcc aluminium substrate, a fibre texture in the diamond could be expected even if the substrate has no texture. The lattice parameters of the two materials only match for the directions 111 in diamond and 100 in aluminium. In this case two lattice spacings of diamond fit along the edge of the aluminium unit cell. This can be seen by the proximity of the diamond (111) and aluminium (200) reflections. It could therefore be expected that the diamond seeds preferentially on (100) aluminium grains with the 110 direction perpendicular to the surface.
3. X-ray stress determination Residual stress analysis using X-ray diffraction relies on the measurement of the lattice strain Dd(W,Y )/d= e(W,Y ) as a function of the direction Y relative to the sample normal and the azimuth angle W (Fig. 3). The strain is related to the components of the stress tensor by the fundamental equation of X-ray residual stress analysis ( RSA) Dd(W,Y )/d=1/2 s [s cos2 W sin2 Y 2 11 +s sin2 W sin2 Y+s cos2 Y 22 33 +s sin2 W sin2 W+s cos W sin2 Y 12 13 +s sin W sin2 Y ]+s [s +s +s ] (1) 23 1 11 22 33 and to the measurement angle by Dd(W,Y )/d= −cot H [H(W,Y )−H )]. H(W,Y ) and H are the mea0 0 0 sured Bragg angle for the stress-free material respectively, and s and 1/2s are the X-ray elastic constants, 1 2 see for example Refs. [4,10,11]. To determine the two-dimensional stress state, Eq. (1) can be simplified by neglecting the tensor components s with i, j=3. In the usual sin2 Y method, a ij
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global average value for the stress can be determined from the gradient of a plot of strain against sin2 Y for constant W [4]. Every measurement is an average over the diffraction contribution of all lattice spacings of the reflecting grains in the irradiated volume V , where (hkl ) are the Miller (hkl) indices for the chosen reflection, around the measurement point (x, y, z) of the sample. In the presence of a three-dimensional stress state, in which the stress components normal to the sample surface are non-zero, the strain will not depend linearly on sin2 Y. Two different effects can be seen in the measured sin2 Y curves. Firstly, if the lattice strain varies over the penetration depth of the X-rays, the stress gradient will lead to a curvature in the measured curve. Secondly, if either of the shear components s or s are non-zero, strains measured 12 13 at negative Y tilts will differ from those measured at positive angles. This phenomenon is known as Y-splitting [10]. The stress determination was performed using the (331) reflection of Cu Ka radiation at a nominal Bragg angle of 70.61° [12] in conventional V-geometry. Different diffraction peaks up to a maximum tilt of 65°, in 5° intervals, were recorded with a position sensitive detector and multichannel analyser system.
4. Results and discussion Fig. 4 shows the dependence of the measured strain (d−d )/d =e on the square of the sine of the tilt 0 0 WY angle Y. The true value of d is not known, and the 0 ˚ [12] has been value of 0.816584 A assumed. As d only 0 acts as a scale factor the error introduced into the final stress s is less than 0.1%, which is smaller than the W experimental and statistical errors [4]. This is not the case for the intercept (s +s ) which is very sensitive 11 22 to errors in d and will not discussed here. Both 0 positive and negative Y angles are plotted in Fig. 4. The almost classic text book results shown in Fig. 4 are in fact, for this sample, very surprising. A linear sin2 Y curve only arises from a uniform two-dimensional stress state. For the diamond layer, this means that the material is uniformly stressed parallel to its surface through out the whole 240 mm of its thickness. There is no indication of any gradient with depth or any stress components perpendicular to the layer. Linear relationships are usually only expected for dense metals where the penetration depth of the X-rays is in the micron or submicron range. In this case, the penetration depth for the (331) reflection ranges from 53 mm at Y=65° to 292 mm at Y=0° [4], covering the whole depth of the diamond layer. Any gradient, even near the interface with the substrate, would be expected to be seen in a non-linear sin2 Y curve. Similarly, any normal stress components would lead to a splitting of the positive and
1558
M. Hempel, M. Ha¨rting / Diamond and Related Materials 8 (1999) 1555–1559
Fig. 3. Illustration of the measurement condition to determine residual stress states. (a) Definition of the measurement angles: (a1) coordinate system of the sample S , i=1,2,3, in which the strain tensor e is defined; (a2) azimuth angle W, sample rotation around the normal to the sample i ij surface; (a3) tilting by angle Y, together with W leads to the laboratory coordinate system with the measurement directions L in which the strain WY e is defined. (b) At a fixed Bragg angle H different measurement directions L bring different grains in measurement conditions, d is the WY (hlk) WY lattice spacing of a single grain [8].
Fig. 4. sin2 Y curve for the (331) reflection from a CVD diamond film grown on aluminium.
M. Hempel, M. Ha¨rting / Diamond and Related Materials 8 (1999) 1555–1559
negative Y curves. At present there is no satisfactory model to account for this stress state, but it may be possible that the presence of the other phases may play a role. The fibre texture in the diamond layer has an insignificant effect on the shape of the sin2 Y curve. Further measurements to study the stress states and textures of the carbide and graphite phases are being carried out. To determine the magnitude of the stress we have first estimated the X-ray elastic constants from the singe crystal elastic constants [13] according to the Reuss model [14]. The values obtained for the diamond (331) reflection were s =−4.9088×10−10 Pa−1 and 1 1/2s =0.9022×10−12 Pa−1. Using these values we find 2 a tensile stress in the layer of s =768 MPa and W s =738 MPa for the positive and negative Y measureW ments respectively. These values are within the estimated error of 30 MPa of each other, and are about 10% of the critical stress [3].
5. Conclusions Using X-ray diffraction methods we have studied the residual stress state in a textured polycrystalline diamond layer grown by chemical vapour deposition on an aluminium substrate. Although the magnitude of the stress is moderate, the form of the stress state is extraordinary. There is no indication of any stress gradient or of any
1559
component of the stress tensor perpendicular to the plane of the film. The layer is uniformly stressed parallel to the substrate interface throughout its whole thickness of 240 mm.
References [1] A. Sommerfeld, in: Vorlesung u¨ber theoretische Physik: Mechanik der deformierbaren Medien, 6th edn. Vol. II Akademische Verlagsgese, Geest & Portig, Leibzig, 1970. [2] A.P. Sutton, R.W. Balluffi, Interfaces in Crystalline Materials, Oxford University Press, New York, 1996. [3] J. Wilks, E. Wilks, Properties and Applications of Diamond, Butterworth-Heinemann, Oxford, 1991. [4] I.C. Noyan, J.B. Cohen, Residual Stress, Springer, New York, 1987. [5] D.S. Knight, W.B. White, J. Mater. Res. 4 (1988) 385. [6 ] N.S. Van Damme, D.C. Nagle, S.R. Winzer, Appl. Phys. Lett. 58 (1991) 2919. [7] J.W. Ager III, M.D. Drory, Phys. Rev. B 48 (1993) 2601. [8] D.T. Britton, M. Hempel, M. Ha¨rting, Appl. Surf. Sci. (1999) in press. [9] M.N.R. Ashfold, P.W. May, C.A. Rego, Chem. Soc. Rev. 23 (1994) 21. [10] V. Hauk, in: V. Hauk, H. Hougardi, E. Macherauch ( Eds.), Residual Stress Measurements, Calculation, Evaluation, DGM Informationsgesellschaft, Oberursel, 1991. [11] M. Ha¨rting, Acta Mater. 46 (1998) 1427. [12] W.Kraus, W. Nolze, Program ‘Powder Cell’, version 1.8d, 1997. [13] D.G. Gray (Ed.), American Institute of Physics Handbook, 3rd edn., McGraw-Hill, New York, 1972. [14] E. Eigenmann, E. Macherauch, Mat.-wiss u. Werkstofftech. 26 (1995) 199.