Materials Science and Engineering A 412 (2005) 204–213
Transmission electron microscopy analysis of grain boundary precipitate-free-zones (PFZs) in an AlCuSiGe alloy A. Tolley a , D. Mitlin b , V. Radmilovic a,∗ , U. Dahmen a a
National Center for Electron Microscopy, Lawrence Berkeley National Laboratory, University of California, Berkeley, CA, USA b Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alta., Canada T6G 2G6 Received in revised form 29 June 2005; accepted 15 July 2005
Abstract We have characterized the elevated temperature (190 ◦ C) precipitation sequence near the grain boundaries of an AlCuSiGe alloy, comparing these results to the binary AlCu and the ternary AlSiGe. In the quaternary alloy, there is a graded microstructure that evolves with increasing distance from the boundaries, which is generally a superposition of the precipitate-free-zones (PFZs) in the binary AlCu and in the ternary AlSiGe. After aging for 3 h, this graded area consists of an approximately 140 nm wide region that is entirely precipitate free, followed by a 400 nm wide region that is denuded of Si–Ge and precipitates. Rather than containing the (Si–Ge)– pairs observed in the bulk, this 400 nm wide region contains only homogeneously nucleated . Only in the overaged condition (144 h) are the near grain boundary replaced by a coarse distribution of large plate-like . In the alloys, the solute depleted zones are much narrower than the total length of the PFZ. For example, in both AlCu and AlCuSiGe, the Cu depleted zone is only 30 nm wide. This underscores the need for vacancies during precipitation of not only and Si–Ge, but of as well. © 2005 Elsevier B.V. All rights reserved. Keywords: Aluminum alloy design; Transmission electron microscopy (TEM); Solid state precipitation; Microstructural evolution; Grain boundaries; Precipitate-free-zones (PFZs)
1. Introduction Al–Cu–Si–Ge based alloys have potential commercial application due to their combination of reasonably high strength, quick aging response and excellent high temperature stability. The alloys’ unique aging response may also allow it to be successfully welded. In previous work, we explained these favorable properties to be a result of a formation of dense distribution of thermally stable precipitates [1,2]. Bulk precipitation occurred in the following sequence: The room temperature microstructure (aged for at least two weeks) of AlCuSiGe consisted of GP zones and precipitates. Upon aging at 190 ◦ C, the Si–Ge precipitates quickly nucleated and grew (they were detected as early as 30 min during aging at 190 ◦ C). The majority of these precipitates adopted the morphology of needles ∗
Corresponding author. E-mail address:
[email protected] (V. Radmilovic).
0921-5093/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2005.07.017
lying along the 0 1 0Al with an orientation relationship ¯ SiGe and [0 1 0]Al ||[1 1 0]SiGe . (OR) of [1 0 0]Al ||[1¯ 1 1] Other particles grew as heavily twinned spheroids with varying orientation relationships including cube-on-cube. The Si–Ge particles then acted as nucleation sites for precipitates, resulting in a peak aged microstructure consisting of attached to Si–Ge. Compared to the normally found in binary Al–Cu alloys, the in AlCuSiGe had a much more equiaxed geometry and were finer in size. One issue not addressed in previous studies of the AlCuSiGe and AlCuSi microstructure [1–6], are the precipitate-free-zones (PFZs) adjacent to the grain boundaries. Because PFZs can cause catastrophic failure of otherwise strong materials, their control is a very important consideration in the development of any commercially viable alloy. For a sufficiently wide and soft PFZ, failure will occur by nucleation of cracks in the soft precipitate-free regions [7]. The width of a PFZ is determined by the concentration profile
A. Tolley et al. / Materials Science and Engineering A 412 (2005) 204–213
of vacancies as well as solute atoms near grain boundaries, and by their interactions [8–15]. In this work, we present our findings on PFZs in an Al–5.6 wt.% Cu–0.5Si–1.3Ge alloy aged at 190 ◦ C. This is the same composition and heat treatment for which we previously detailed the bulk precipitation sequence. Results from binary Al–Cu and ternary Al–Si–Ge alloys are also presented for comparison.
2. Experimental procedure Bulk alloys of composition Al–5.6 wt.% Cu–0.5Si– 1.3 wt.% Ge, Al–5.31 wt.% Cu and Al–0.46 wt.% Si–1.29Ge were synthesized using conventional alloy methods previously described [1,2]. Analysis by vacuum emission spectroscopy (at FTI Anamet Laboratories) confirmed their compositions and detected no other elements present in quantities greater than 0.01%. Bulk AlCu, AlSiGe and AlCuSiGe samples were aged at 190 ◦ C. The temperature was monitored with an external thermocouple, and varied no more than 1 ◦ C. Transmission electron microscopy (TEM) samples were made from the aged specimens, and the details of the specimen preparation procedure are also described in references [1,2]. TEM characterization was carried out using JEOL 200CX and Philips CM200 FEG microscopes, operated at 200 kV. The former was used for diffraction contrast analysis, and the latter was used for energy dispersive X-ray spectroscopy (EDXS) and electron energy loss spectroscopy (EELS) analysis. The spectrum profiles were acquired using probe sizes in the range of 1.6–5 nm. Acquisition and processing was carried out with ESVision software (Copyright 2000, Emispec. Inc.). High resolution transmission electron microscopy images were obtained using a CM300 Philips microscope with an ultra twin lens. The width of the PFZ is defined as the average distance between the grain boundary and the region with extensive precipitation. This analysis was performed on micrographs with the grain boundary in edge-on orientation, and with the appropriate diffraction conditions to obtain a clear contrast from the precipitates. For each condition at least four grain boundaries were analyzed to obtain the average width. Diffraction patterns were simulated using Desktop MicroscopistTM (Copyright 2000, Lucana Laboratories Inc.), using the space groups and the lattice parameters of Al, Si, Ge, (metastable Al2 Cu) and (metastable Al3 Cu) precipitates [3,16].
3. Results 3.1. Diffraction pattern simulation We used selected area diffraction and centered dark field imaging to identify the precipitates in the three alloys. As
205
an example, fully indexed 0 0 1 Al zone axis simulations are shown in Fig. 1. Fig. 1A and B show the simulated patterns of in Al, without and with double diffraction. Fig. 1C and D show this for in Al, while Fig. 1E through H display simulated diffraction patterns for Si–Ge precipitates in Al. In Al–Cu, both and precipitates grow as plates on the {1 0 0} planes of Al, with the three variants of (1 0 0)Al ||(1 0 0) or , [0 0 1]Al ||[0 0 1] or [17] (Fig. 1A–D). The Si–Ge precipitates display various morphologies and orientation relationships, but are dominated by two major types [1,2,18]. One type consists of multiply twinned, spheroidal particles with at least one section having a cube–cube orientation relationship with the matrix (cube–cube OR is shown in Fig. 1E and F). The second type (the majority) are rods lying along 0 1 0Al directions that adopt one of the twelve vari¯ SiGe , ants of the orientation relationship (1 0 0)Al ||(1¯ 1 1) [0 1 0]Al ||[1 1 0]SiGe . Due to the limitation of the software package only three of these variants are shown in Fig. 1G and H. The Si–Ge rods also develop twins with prolonged aging [2]. 3.2. Summary of the bulk microstructural evolution We begin with a brief summary of the bulk microstructural evolution during aging at 190 ◦ C. These results are shown as a reference to contrast with the grain boundary findings. Figs. 2 and 3 show the changes in the symmetric 0 0 1 selected area diffraction (SAD) patterns of AlCu and AlCuSiGe alloys, respectively, as a function of aging time. The times chosen to illustrate the evolution of the microstructure were 30 min, 3 h and 144 h, which correspond to the underaged, peak aged and overaged conditions for the AlCuSiGe alloy. After aging AlCu for 30 min the bulk microstructure consisted of GP zones and precipitates (Fig. 2A). Aging for 3 h eliminated the GP zones, leaving only precipitates (Fig. 2B). Following prolonged aging (144 h) the precipitates were entirely replaced by (Fig. 2C). As expected, our findings for the binary alloy agree with the well established results [17]. After aging AlCuSiGe for 30 min the diffraction pattern consisted of streaks and 1 1 0 spots (Fig. 3A). Analytical and high resolution TEM analysis indicated that there were also some Si–Ge particles that had already precipitated at this aging condition. After 1 h (SAD not shown), extensive Si–Ge precipitation had occurred. After 3 h, the bulk microstructure consisted of and Si–Ge paired together (Fig. 3B). This microstructure persisted in the overaged condition (144 h) (Fig. 3C). The presence of both and Si–Ge precipitates is clearly apparent from the SAD patterns both Fig. 3B and C. After prolonged aging, the Si–Ge precipitates were so extensively twinned that the SADs developed spotty {1 1 1}Si–Ge and {2 2 0}Si–Ge ring patterns (Fig. 3C).
206
A. Tolley et al. / Materials Science and Engineering A 412 (2005) 204–213
Fig. 1. Simulated diffraction patterns of different precipitates in Al seen along an 0 0 1 zone axis. Patterns in the right column include double diffraction (filled spots: Al reflections; open spots: precipitate reflections; crosses: double diffraction). (A and B) (metastable Al3 Cu) precipitates in Al showing the three unique orientation variants; (C and D) (metastable Al2 Cu) precipitates in Al showing the three unique orientation variants; (E and F) Si–Ge precipitates in Al, ¯ SiGe , [0 1 0]Al ||[1 1 0]SiGe ; (G and H) Si–Ge precipitates in Al showing the cube–cube orientation relationship. with the orientation relationship (1 0 0)Al ||(1¯ 1 1)
A. Tolley et al. / Materials Science and Engineering A 412 (2005) 204–213
207
Fig. 2. Representative selected area diffraction patterns of bulk AlCu, taken in the [0 0 1]Al zone axis: (A) 30 min; (B) 3 h; (C) 144 h.
3.3. Microstructure near the grain boundaries Fig. 4A–C show the AlCu microstructure adjacent to a grain boundary for aging times of 30 min, 3 h and 144 h. At all three aging conditions the PFZ is approximately 100 nm wide (depending on the grain boundary examined the width was between 75 and 150 nm). The only observable difference between the precipitates near the grain boundary and those in the bulk is that during the early stage of aging (30 min) is coarser near the PFZ. After aging for 3 and 144 h the precipitates near the PFZ are similar in structure, size and distribution to the precipitates in the bulk. After 3 h all the precipitates are , while after 144 h all the precipitates are . Fig. 5 shows a dark field image of a grain boundary in ternary AlSiGe, after aging for 3 h at 190 ◦ C. The dark field image was obtained using g = 2 2 0SiGe reflection, near the [0 0 1]Al zone axis. At this aging condition there exists a 400 nm wide PFZ, which is marked in the figure. Figs. 6–8 detail the microstructure of AlCuSiGe aged for 30 min. Fig. 6 shows a g = 0 0 2 dark field image of the AlCuSiGe microstructure after aging for 30 min. The
microstructure near the PFZ consists of a distribution of precipitates. The width of the PFZ varied more than in the binary AlCu, with an approximate value of 200 nm. Relative to the AlCu, here the precipitates in the proximity of the grain boundary were finer, yet more coarsely distributed. It is unlikely that this spacing variation is an artifact due to a thickness difference between the two foils, since the intensities of the Kikuchi lines obtained from the samples were similar. Several edge-on and face-on Si–Ge rods (arrowed) were also detected in the proximity of the PFZ. This is consistent with previous work on a binary Al–Ge alloy, where authors demonstrated that preferential growth of rod-shaped Ge precipitates (versus other morphologies such as triangular plates) occurred in regions with low vacancy concentrations [19]. Interestingly, this is not consistent with our current observation of large Si–Ge plates near the grain boundaries in the AlSiGe alloy. In the quaternary alloy aged for 30 min, a more dense distribution of Si–Ge precipitates begins at a distance farther than 1 m from the grain boundary (not shown). Fig. 7 shows an annular dark field image of grain boundary precipitates, where EDXS was used to identify the
208
A. Tolley et al. / Materials Science and Engineering A 412 (2005) 204–213
Fig. 3. Representative selected area diffraction patterns of bulk AlCuSiGe, taken in the [0 0 1]Al zone axis: (A) 30 min; (B) 3 h; (C) 144 h.
composition on the individual particles. The grain boundaries contained both (equilibrium form of Al2 Cu) and Si–Ge precipitates. The number of counts in the Si K␣, and Ge K␣ peaks was normalized to the number of Al K counts. This was done to eliminate any interpretation error due to the changes in specimen thickness from preferential thinning of grain boundaries during sample preparation. The average widths of the solute denuded region for Si and Ge are 115 and 135, respectively, and are much narrower than the region that is Si–Ge precipitate-free. Fig. 8 shows the Si and Ge composition profiles across a grain boundary in AlCuSiGe aged for 30 min, obtained by EDXS. The number of counts in the Si K␣, and Ge K␣ peaks was normalized to the number of Al K counts. This was done to eliminate the contribution of changes in specimen thickness due to preferential polishing at the grain boundary. The widths of the solute denuded region for Si and Ge, obtained by Gaussian fit of experimental data, are 115 ± 14 and 135 ± 10 nm, respectively, and are much narrower than the region that is Si–Ge precipitate-free. Figs. 9 and 10 detail the Al–Cu–Si–Ge microstructure after aging for 3 h, while Fig. 11 shows the microstructure
after 144 h. Fig. 9 shows the different kinds of precipitates present in the proximity of a grain boundary. Fig. 9A is a bright field image of the microstructure adjacent to a grain boundary. Fig. 9B is a dark field image obtained using g = 0 0 2¯ . Because this reflection overlaps with the streaked 0 0 2 reflection, both types of precipitates appear bright in the image. Using a 0 1 1¯ reflection (Fig. 9C) it is possible to image only the precipitates while excluding . Similarly, the Si–Ge rods can be imaged in dark field using a 1 1¯ 1SiGe reflection (Fig. 9D), which displays SiGe rods oriented along two of the 0 0 1 matrix directions. Rods that are seen end-on appear as small bright dots. It is clear that the absence of near the grain boundary correlates with the absence of Si–Ge precipitates in the same region. Near this grain boundary, the width of the Si–Ge precipitate-free region is approximately 550 nm, which is slightly wider than for the AlSiGe alloy aged for 3 h. Fig. 10 presents a higher magnification dark field image showing and precipitates adjacent to a grain boundary. The grain boundary is on the left, outside the field of view. Again, because this reflection overlaps with the streaked reflection, both types of precipitates appear bright in the
A. Tolley et al. / Materials Science and Engineering A 412 (2005) 204–213
209
Fig. 4. (A) Dark field image of AlCu microstructure adjacent to a grain boundary after aging for 30 min. (B) Dark field image of AlCu microstructure adjacent to a grain boundary after aging for 3 h. (C) Dark field image of AlCu microstructure adjacent to a grain boundary after aging for 144 h. One orientation variant of is imaged near the [0 1 1] zone axis using g = 1¯ 1 2θ reflection.
Fig. 5. Dark field image (g = 2 2 0SiGe ) of a grain boundary in AlSiGe, after aging for 3 h. The width of the precipitate-free-zone is indicated.
image. As the distance from the grain boundary increases the precipitates are replaced by . Fig. 11 illustrates the role of Si–Ge precipitates in promoting a dense distribution of fine precipitates. Fig. 11a shows a dark field image of precipitates, imaged near their [1 1 0] zone axis. The sample shown is in the overaged condition (144 h) when all the precipitates have been replaced by . Fig. 11B shows a dark field image of Si–Ge precipitates in the same region. It is evident from the two images that the fine and densely distributed precipitates extend as far towards the grain boundary as the Si–Ge precipitates. In the areas where there are no SiGe precipitates the is large and coarsely distributed. The variation in the Cu content with the distance from a grain boundary was investigated using composition profiles established using jump ratios of EELS spectra. The spectra were averaged over a 50–100 nm wide trace across the grain boundary. Fig. 12a shows the variation of Cu content across a grain boundary in AlCu, aged for 30 min and for 3 h. Fig. 12b shows data for AlCuSiGe, under the same aging conditions. The quantitative feature of the data is the width of the Cu denuded region, while the actual position of the curves is arbitrary. In both alloys, the Cu-depleted zone was approximately 30 nm wide. The EELS data indicates that there is
210
A. Tolley et al. / Materials Science and Engineering A 412 (2005) 204–213
Fig. 8. EDS composition analysis of Si and Ge depletion at a grain boundary in AlCuSiGe aged for 30 min. The profiles indicate approximately the same width of the depleted zone for both elements.
Fig. 6. Dark field image (g = 0 0 2 ) of AlCuSiGe microstructure adjacent to a grain boundary after aging for 30 min. Due to the streaking and the double diffraction of Si–Ge patterns, some Si–Ge precipitates also appear bright in the image (arrowed).
no effect of aging on the depletion profiles (within the range tested). The Cu-depleted region is significantly narrower than the 140 nm and PFZ. This indicates that the remaining and denuded region (Figs. 9 and 11) corresponds to a vacancy-depleted zone.
4. Discussion
Fig. 7. Annular dark field image of grain boundary precipitates in AlCuSiGe aged 30 min. The precipitates were identified using EDXS.
The majority of the 550 nm wide Si–Ge PFZ corresponds to a region that is vacancy depleted (compare Figs. 8 and 9). In many systems, vacancies play a role in lowering the nucleation barrier for precipitation [20]. In AlCuSiGe, vacancies are essential for the nucleation of Si–Ge precipitates. Precipitation does not occur in the vacancy-depleted regions even though there is adequate Si and Ge solute available. There can be many scenarios for the role of vacancies in the formation of diamond cubic particles in an fcc matrix. Because the Si–Ge structure is much more open than aluminum (relative ˚ 3 versus 1 atom/16.6 A ˚ 3 ), vacancies density of 1 atom/22.6 A may become incorporated into the precipitate [21,22]. Alternatively, vacancies may reduce the stress in the matrix during the nucleation and/or growth of Si–Ge. A compressive stress is developed in the matrix when a number of Al atoms are replaced by the same number of atoms constituting the diamond cubic Si–Ge precipitate. This stress may then be relieved by a vacancy flux into the region. We previously reported that the GP zones and the do promote the formation of Si–Ge precipitates in the bulk alloy [2]. This was explained by the well established prediction that that the zones will strain the Al matrix normal to their habit plane. The effect of GP zones on the Al matrix has been described in two models [23,24]. According to Gerold’s model [23] the relative displacement (an /a) of the Al planes around a zone varies from −10% at the adjacent 1 0 0Al plane
A. Tolley et al. / Materials Science and Engineering A 412 (2005) 204–213
211
Fig. 9. AlCuSiGe microstructure adjacent to a grain boundary after aging for 3 h: (A) Bright field image; (B–E) dark field images taken using the following reflections—(B) g = 0 0 2¯ θ , (C) g = 0 0 1¯ θ , (D) g = 1 1¯ 1SiGe , (0 0 1 Si–Ge rod precipitates), and (E) g = 2¯ 2 0SiGe (cube–cube Si–Ge precipitates).
Fig. 10. High magnification dark field image of AlCuSiGe microstructure adjacent to a grain boundary after aging for 3 h. The grain boundary is on the left, outside the field of view.
to −2% at the tenth 1 0 0Al plane. According to Toman’s model [24] the displacement varies from −3.9% at the adjacent plane to −0.3% at the third plane. The presence of GP zones would thus offset the compressive stress due to the Si–Ge precipitation. However, this work experimentally demonstrates that the mere pre-existence of GP zones and of is not sufficient to nucleate Si–Ge particles and that vacancies are essential. A comparison of Figs. 8, 9, 11 and 12 indicates that in AlCuSiGe, the and PFZ’s correspond to the regions of both solute and vacancy depletion. In general, vacancies diffuse faster than solute, resulting in a wider vacancy-depleted region compared to the solute-depleted region. The rate of zone ( may be considered as GPII zone) formation in Al–Cu alloys critically depends on the excess vacancy concentration [25,26]. In AlCuSiGe, there is a region where the presence of Cu solute is not a sufficient requisite for or formation within the time frame examined. Judging from the widths of the two precipitate-free regions, it appears that the minimum vacancy concentration needed to form
212
A. Tolley et al. / Materials Science and Engineering A 412 (2005) 204–213
Fig. 11. Dark field images of AlCuSiGe microstructure adjacent to a grain boundary after aging for 144 h. (A) One orientation variant of is imaged near the [1 1 0] zone axis using a g = 0 0 2 reflection. (B) Nearly equiaxed Si–Ge precipitates in a cube–cube orientation relationship with the matrix are imaged using a g = 220SiGe reflection.
Fig. 12. (a) EELS composition profile analysis of Cu depletion at a grain boundary in AlCu, after aging 30 min and 3 h. (b) EELS composition profile analysis of Cu depletion at a grain boundary in AlCuSiGe, after aging 30 min and 3 h.
GP and is less than required for the precipitation of Si–Ge.
5. Conclusions We have detailed the grain boundary microstructure of AlCuSiGe during aging at 190 ◦ C, using AlCu and AlSiGe as a baseline. TEM results indicate that the solute depleted zones are significantly narrower than the total length of the PFZ. In the AlCuSiGe alloy, the Si and Ge solute depleted zones are
approximately 140 nm wide, while the Si–Ge PFZ is approximately one half of a micron wide. The Cu depleted zone is only 30 nm wide, whereas the PFZ is 140 nm wide, while the PFZ is approximately one half micron wide. These results indicate the need for vacancies during precipitation of all three phases. In the bulk, the Si–Ge precipitates catalyze early formation of by acting as heterogeneous nucleation sites. Where they are absent, AlCuSiGe ages analogously to the binary AlCu. There, the microstructure consists of only , whereas the bulk evolves to be a dense distribution of fine –SiGe pairs.
A. Tolley et al. / Materials Science and Engineering A 412 (2005) 204–213
Only after prolonged aging are , which are the near grain boundaries, become replaced by a coarse distribution of large plate-like . This study demonstrates that compared to the bulk, it is more difficult to use alloy additions to favorably alter the precipitation sequence near vacancy and solute sinks. Acknowledgements The National Center for Electron Microscopy is supported by the Director, Office of Science, U.S. Department of Energy under Contract No. DE-AC02-05CH11231. References [1] D. Mitlin, V. Radmilovic, U. Dahmen, J.W. Morris Jr., Metall. Mater Trans. A 32A (2001) 197. [2] D. Mitlin, V. Radmilovic, J.W. Morris Jr., U. Dahmen, Metall. Mater Trans. A 34A (2003) 735. [3] D. Mitlin, V. Radmilovic, J.W. Morris Jr., Metall. Mater Trans. A 31A (2000) 2697. [4] A.K. Mukhopadhyay, Metall. Mater. Trans. A. 32A (2001) 1949. [5] A.W. Zhu, E.A. Starke, Mater. Sci. Forum 396 (4) (2002) 735.
213
[6] V. Radmilovic, U. Dahmen, B. Dracup, M.K. Miller, D. Mitlin, J.W. Morris Jr., Mater. Sci. Forum 396 (4) (2002) 905. [7] I.J. Polmear, Light Alloys: Metallurgy of the Light Metals, Chapman and Hall, New York, 1989, p. 30. [8] W.M. Lomer, A.H. Cottrell, Phil. Mag. 46 (1955) 711. [9] R.L. Peck, K.H. Westmacott, Met. Sci. J. 5 (1971) 155. [10] M.A. Chapman, R.G. Faulkner, Acta Metall. 31 (1983) 677. [11] G. Thomas, J. Nutting, J. Inst. Metals 88 (1959–1960) 81. [12] H.S. Rosenbaum, D. Turnbull, Acta Metall. 7 (1959) 664. [13] G. Thomas, M.J. Whelan, Phil. Mag. 4 (1959) 511. [14] P.B. Hirsch, J. Silcox, R.E. Smallman, K.H. Westmacott, Phil. Mag. 3 (1958) 897. [15] G. Thomas, Phil. Mag. 4 (1959) 1213. [16] B.D. Cullity, Elements of X-Ray Diffraction, second ed., AddisonWesley Publishing Company Inc., 1978, p. 506. [17] J.M. Silcock, T.J. Heal, H.K. Hardy, J. Inst. Met. 82 (1953-54) 1519. [18] D. Mitlin, V. Radmilovic, J.W. Morris Jr., Mater. Sci. Eng. A 301 (2001) 231. [19] S. Hinderberger, S.Q. Xiao, K.H. Westmacott, U. Dahmen, Z. Metallk. 87 (1996) 161. [20] K.C. Russell, Scr. Metall. 3 (1969) 313. [21] J. Douin, U. Dahmen, K.H. Westmacott, Phil. Mag. B 63 (1991) 867. [22] S.Q. Xiao, S. Hinderberger, K.H. Westmacott, U. Dahmen, Phil. Mag. B 73 (1996) 1261. [23] V. Gerold, Z. Metallk. 45 (1954), pp. 593, 599. [24] K. Toman, Acta Cryst. 10 (1957) 187. [25] J.M. Silcock, Phil. Mag. 4 (1959) 1187. [26] L.A. Girifalco, H. Herman, Acta Metall. 13 (1965) 583.