First principles calculations of Ag2Al stability and electronic properties

Acta metall, mater. Vol. 43, No. 5, pp. 2097 2102, 1995

Pergamon

0956-7151(94)00412-9

Copyright ¢~ 1995ElsevierScienceLtd Printed in Great Britain,All rights reserved 0956-7151/95$9.50+ 0.00

FIRST PRINCIPLES CALCULATIONS OF Ag2A1 STABILITY A N D ELECTRONIC PROPERTIES C. L. R O H R E R l, R. W. H Y L A N D Jr t, M. E. M c H E N R Y 2 and J. M. M a e L A R E N 3

~AluminumCompany of America, Alcoa Technical Center, Alloy Technology Division, Alcoa Center, PA [5069, 2Carnegie Mellon University, Materials Science and Engineering Department, Pittsburgh, PA 15213 and 3Tulane University, Physics Department, New Orleans, LA 70118, U.S.A. (Received 18 June 1994)

Abstract--The first principles layer Korringa-Kohn Rostoker computational method is used to distinguish between two proposed atomic configurations for the binary A1 Ag h.c.p, intermediate phase, y, of composition Ag2A1. A comparison between the calculated total energies of the two configurations is supported by a comparison of the resulting electron density of states to existingelectron spectroscopy data. The present calculations indicate that the lowest internal energy is obtained for the atomic configuration reported by Howe et al. from atomic resolution TEM studies of partially coherent plates of the 7' transition phase. The 7' phase is identical to the equilibrium ~, phase except for the absence of misfit dislocations at the broad faces of the plates. The h.c.p. 7' structure possesses basal planes alternating in composition from pure Ag to AgA12and exhibits both short range order within the basal planes as well as long range order over alternate basal planes.

INTRODUCTION

The influence of Ag on the properties of Al rich A 1 A g alloys is particularly intriguing because Ag does not affect the A1 lattice parameter [1]. Hence, simple thermodynamic models, such as the quasichemical regular solution approximation, should be useful for examining the energetics of the A1-Ag alloys since there is no volume change on mixing. However, the predictive capabilities of simple thermodynamic models to match A1-Ag alloy energetics with experiment have met with qualitative, but not quantitative success [2-4]. From such studies, it is clear that chemical effects must dominate A1 Ag alloy properties and that continuum level approximations must be interpreted cautiously. Recent developments in first principles quantum mechanical techniques make it possible to probe such chemical effects. The present work makes use of the layer Korringa Kohn-Rostoker (LKKR) method, developed and described by MacLaren et al. [5], to study the electronic properties of the h.c.p. Ag2AI phase in an effort to theoretically determine its most favorable atomic configuration for further calculations of the AI:AgzA1 interface energy. The calculations presented herein seek to distinguish between two possible, yet distinct crystal structures for h.c.p. Ag2AI that have been presented in the literature. The first structure, proposed by Neumann [6] based on X-ray diffraction results, possesses a strong tendency for compositional short range order in the basal planes and a slight modulation in composition between alternating basal planes, each layer having the approximate compo-

sition of 33% A1 and 67% Ag. A schematic diagram of the resulting structure is shown in Fig. 1(a). A later atomic resolution TEM study by Howe et al. [7] indicated a substantially different ordered structure. The structure proposed in [7] is shown in Fig. l(b). As in Neumann's proposed structure, the structure proposed by Howe et al. also possesses short range order within the basal planes, but the chemical composition in alternating basal planes was found to be vastly different, alternating from 100% Ag in one plane to 67% AI and 33% Ag in the next. In the present work, the internal energies of long range ordered versions of the two h.c.p, configurations have been calculated at OK and electron density of states (d.o.s.) have been compared with experimental spectroscopic information to lend further credence to the structural models and to elucidate details of the atomic bonding.

LKKR METHODOLOGY

While the quantum mechanical and mathematical formalisms for the LKKR technique have been thoroughly explained in [5], we briefly summarize certain aspects of the technique as they relate to the atomistic modeling of metals. Of most importance is the fact that, unlike the case for nearest neighbor pair potential models or volume dependent many body potential models, the only inputs to the LKKR code are the number of electrons possessed by each atom and the atomic configuration of the system of interest. No fitting to experimental data is required. Furthermore, quantum mechanical calculations such

2097

2098

ROHRER et al.: FIRST PRINCIPLES CALCULATIONS ON Ag2A1

b)

O Silver Fig. l. Idealized Ag2A1crystal structures from (a) Neumann [6] and (b) Howe [7]. Small (large) circles represent the A (B) plane in the h.c.p. ABAB stacking. as the LKKR method are the only computational means of probing atomic bonding. Crystal calculations are performed by first finding solutions to Schr6dinger's equation for the individual atomic species. Solving Schr6dinger's equation for a multi-electron atom has been made tractable by the implementation of density functional theory [8], a well tested theory stating that the one electron total energy can be written as a functional of the electron density as E [n (r)]~t(r) = Htk'i(r)= [T[n (r)] + Vn[n (r)] + V~[n (r)]] ~, (r).

(1)

In the above equation, q'~(r) is the electron wave function at position r, E is the total energy, and H is the Hamiltonian expressed as a sum of the electron kinetic energy, T, the nuclear~lectron interaction potential energy Vn, and the electron~electron interaction potential energy Ve. The electron density, n (r), is constructed by summing over the individual electron wave functions as shown n (r) = £l~/',(r)l 2.

(2)

Note that Ve also incorporates an exchange-correlation energy which corrects tbr two problems resulting from the assumption that each electron interacts with a uniform electron cloud. First, it accounts for the calculated interaction of the single electron i with itself, since electron i is inextricably embedded into n (r). Second, it accounts for the individual electron-

electron energy restrictions imposed by the Pauli exclusion principle including electron spin effects. The eigenvector solution to Schr6dinger's equation is the wave function, Yet it is clear that the Hamiltonian is dependent on the value of the wave function through its dependence on the electron density. Thus, Schr6dinger's equation is solved iteratively. An initial electron density is used to construct the initial potential included in the Hamiltonian. Solving Schr6dinger's equation, the resulting wave function solution is found, and a new electron density is formulated from this result. The cycle is continued until there is negligible difference between the input electron density and the resulting electron density; i.e. the energy eigenstates are determined self-consistently. The LKKR method emphasizes the wave-like scattering behavior of electrons and reformulates the atomic solution into a Green's function, more suitable for use in the scattering formalism. The crystal potential is described as a muffin tin, spherically symmetric potential wells centered at the nuclear cores. Space is partitioned into touching atom spheres and interstitial volume. To calculate the system total energy, the atomic eigenfunctions are solved exactly within the spherical muffin tins and are expanded into a specified number of plane waves in the interstitial regions. The plane wave solutions are constrained to match the radial solutions at the muffin tin boundaries. Increasing the spherical harmonics and plane wave basis sets increases the accuracy of the calculations. The topology of the muffin tin reflects the symmetry of the crystal. The LKKR technique takes advantage of the crystal translational symmetry by only obtaining solutions for the atoms in a single structural repeat unit. However, it only imposes this restriction in two dimensions. A two dimensional Fourier transform is used to repeat units in a single atomic layer periodically throughout the layer. The scattering properties of the symmetric repeat units are then summed appropriately using scattering theory to result in a unique layer scattering power. This is repeated for each unique layer, building the threedimensional crystal layer by layer by treating both intra- and inter-layer scattering. Interlayer scattering employs a unique layer doubling algorithm which does not require translational periodicity. In this way, two-dimensional disturbances, such as stacking faults or interfaces, can be embedded into bulk material. Note that the total energy of the entire crystal is still evaluated iteratively and self-consistently with the charge density adjusting to accommodate the new environment in a bulk or defective cryslal. Charge redistribution across interfaces and between atomic species is treated explicitly. Typical predictive capabilities of muffin-tin potentials and local density functional theory have been demonstrated by Janak, Moruzzi, and Williams [9] for a variety of crystalline species. Cohesive energies

2099

ROHRER et al.: FIRST PRINCIPLES CALCULATIONS ON Ag2A1

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Fig. 2. Total energy vs lattice parameter calculated for f.c.c. A1. are calculated to within 20% of experimental values, lattice parameters to within 2%, and bulk moduli to within 10%. The major disadvantages of such calculations are that they are limited to ground state properties and that atomic relaxation near structural or compositional defects must be hand selected since they are not energy minimization techniques in which atoms are allowed to adjust their positions to minimize their energy.

Ag2AI potentials used in the alloy self-consistency cycles, and (c) optimize the future A1 matrix potentials for AI:Ag2AI interface energy calculations. The converged AI total energy as a function of lattice parameter is shown in Fig. 2. Based on a fit of the data points to a quadratic equation using Mathematica, the calculated 0 K minimum energy lattice parameter for A1 is 3.98 A. The experimental room temperature value is 4.05/~. Therefore, the calculated lattice parameter implies a thermal expansion coefficient of approximately 58 x 10 6 K-~, a factor of 2.5 larger than the experimental room temperature value of 23 x 10 6 K ' [7]. The lattice parameter discrepancy is well within the 2% range found by Janak et al. [9] for comparable first principles calculations. The calculated bulk modulus is 0.815 x 10 H J / m 3, a reasonable match to the experimental value of 0.793 x 1011J/m 3 and well within Janak et al.'s 10% range. The corresponding Fermi energy, E r, was found to be 8.8140eV from the bottom of the valence band. CALCULATED RESULTS FOR THE AgzAI

STRUCTURES AND DISCUSSION DETAILS OF THE CALCULATIONS Two series of total energy calculations were carried out for each atomic configuration so as to find the minimum total energy as a function of lattice parameters a and c. In the first set of calculations, the near neighbor separation was varied assuming an ideal c/a ratio. In the second, the minimum energy neighbor separation from the first set of calculations was imposed in the basal planes while the c/a ratio was varied, changing the inter-plane near neighbor separation. Variation of the c/a ratio results in the structure no longer being ideally close packed. The use of muffin tins in the L K K R method requires an appropriate choice of muffin tin radii for a non-close packed structure. For the ideal close packed structures, the muffin tin radii of both A1 and Ag were taken to be equal, consistent with the fact that there should be no atomic size difference, and touching. For the non-ideal structures, all muffin tin radii were kept equal, but were made to touch in one direction; i.e. for c/a smaller than ideal, the muffin tins touched between neighboring basal planes, but not within basal planes and vice versa for c/a greater than ideal. The Hedin and Lundqvist [10] local density approximation was used for exchange-correlation effects. Calculations were performed using a spherical basis set expanded to include s, p and d angular m o m e n t u m contributions within the muffin tins and a 37 interstitial plane wave basis within the interplane regions. Total energies reported were calculated using the atomic sphere approximation. The total energy of pure f.c.c. AI was self-consistently converged at a series of lattice parameters to: (a) test the validity of the energy calculations with respect to experiment; (b) optimize the initial guess

The results of the two sets of 0 K total energy calculations for the two h.c.p. AgzAI atomic configurations are shown in Fig. 3. In this figure, the self-consistently converged energy is plotted as a function of Wigne~Seitz radius, rws, which is defined with respect to the atomic volume, ~, as follows 4

3

~ =Srq'wsThe atomic volume for the h.c.p, crystal is calculated as ~ @a2c.

It is clear that the same atomic volume, and therefore rw~, can be obtained with multiple choices of a and c. It is also clear from Fig. 3 that the total energy is multi-valued for a given rw~, also to be expected since variations in the c/a ratio effectively change the atomic coordination sphere. For the ideal h.c.p, crystal, each atom has 12 equidistant near

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Fig. 3. Total energy vs Wigner-Seitz radius, rws, for idealized versions of the two atomic arrangements proposed by Neumann and Howe for h.c,p. Ag2AI.

2100

ROHRER et al.: FIRST PRINCIPLES CALCULATIONS ON AgzAl T a b l e 1. C a l c u l a t e d m i n i m u m e n e r g y a n d a s s o c i a t e d lattice p a r a m e t e r s o f i d e a l i z e d N e u m a n n a n d H o w e A g 2 A I c r y s t a l s t r u c t u r e s . E n e r g i e s a r e in e V , L a t t i c e p a r a m e t e r s a r e in ~ . F o r c o m p a r i s o n , e x p e r i m e n t a l l y m e a s u r e d v a l u e s o f a a n d c at 77 K a r e 2 . 8 6 6 3 a n d 4 . 6 0 9 4 A [121 o r 2 . 8 8 5 a n d 4 . 5 8 2 ~ a t r o o m t e m p e r a t u r e [7] Idealized Neumann

Idealized Howe

I d e a l c/a M i n i m u m energ}, Lattice parameters

- 591261.9861 a = 2.8622, c = 4.6740

591261.9012 a = 2.867l, c = 4.6820

F i x e d a, v a r i e d c/a Minimum energy Lattice parameters

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neighbors. For c/'a less than ideal, the six inter-plane nearest neighbors are pulled closer than the six remaining intra-plane neighbors. Table 1 lists the minimum energies calculated for each curve and the corresponding lattice parameters. The data obtained with the ideal c/a ratio were fit with a quadratic equation. The data obtained by varying c/a were found to be better fit by a third degree polynomial. Note that the curvature at the energy minima of this second set of data gives the diagonal element in the c direction of the elastic tensor, C33. The calculated values of C,~ for the idealized Howe and Neumann structures were 1.8 and 1.3 x 10 n J/m 3, respectively. The experimental value is 1.81 × 1011J/m 3 [12], which compares favorably with that calculated for the Howe structure. The results indicate that for both atomic arrangements, the structure is more stable at c/a ratios less than ideal, in agreement with experimental information. The minimum energy c/a ratio for the idealized Howe structure was found to be 1.43, while that for the idealized Neumann structure was 1.51. These values can be compared to the value of 1.6081 measured by N e u m a n n and Chang at 7 7 K [12] and 1.588 measured by Howe at room temperature [7]. At this point, it is important to re-emphasize that both Neumann and Howe indicated that the Ag2AI structure exhibits short range order in the basal planes. In contrast, the work of Howe et al. [7] suggests the presence of long range order normal to the basal planes. Three-dimensional long range order was imposed on the configurations studied in the present work due to computational limitations on the number of atoms that can be treated independently. Therefore, the calculated results are for idealized structures and are representative of limiting cases. Furthermore, because it was not possible at present to carry out true energy minimization as a function of both lattice parameters, it is likely that the calculated minimum energy c/a ratios are not the absolute minimum energy values. Keeping these facts in mind, the calculations predict that the minimum energy idealized Howe structure is more stable than the minimum energy idealized Neumann structure by 0.21 eV per structural repeat unit (six atoms). To further probe the validity of the prediction that the Howe structure is more stable, the calculated d.o.s, for the two structures were compared with experimental spectroscopic data for materials having

the composition Ag2A1. The d.o.s, were generated at the established minimum energy c/a ratios for the respective structures. Figure 4 shows the calculated A1 and Ag total d.o.s.--summing s, p and d contributions for each a t o m - - f o r the idealized N e u m a n n structure and the mixed species layer of the idealized Howe structure. In both cases, the A1 total d.o.s. exhibits a local minimum coinciding with the top of the Ag 4d bands in the vicinity of 4 eV below E r. (The total d.o.s, for Ag in both structures is dominated by the 4d states. The s and p states make negligible contributions.) This depletion of AI states agrees with Watson et al.'s [13] predictions from a soft X-ray spectroscopy (SXS) study of the first transition metal series in AI. However, Watson et al. emphasized that their study did not allow them to differentiate between depletion of A1 states at the top of the transition metal d bands and enhancement of AI states at the bottom of the d bands. The L K K R study carried out in the present work indicates that both occur. This is clearly shown in Fig. 5 where the total At d.o.s, in the alloys is compared to the total A1 d.o.s. 5O

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FIRST PRINCIPLES CALCULATIONS ON Ag2AI

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Fig. 5. Comparison of calculated total d.o.s, for alloyed A1 ( ) and bulk AI ~ -) in (a) the idealized Neumann structure and (b) the idealized Howe structure. calculated for bulk f.c.c. A1. Depletion of alloyed A1 states below those of bulk AI near 4 eV below E r and enhancement of alloyed A1 states above those of bulk A1 in the vicinity of 7 9 eV below Ef were observed for both atomic configurations. The enhanced states at low energies are of further interest in that they represent a shift of the AI 3s electrons toward lower, more tightly bound energy levels. Note in Fig. 4 that the lowest energy A1 peaks for both atomic arrangements are found at energies below those occupied by Ag 4d electrons. Figure 6 shows the individual s,p and d contributions to the A1 total d.o.s. The lowest energy AI peaks in Fig. 4 can be attributed to the 3s electron states. This shift of A1 3s states to more tightly bound energies was also observed by Fuggle et al. [14] using SXS on Ag2A1. Fuggle et al.'s spectra for AgzA1 are reproduced in Fig. 7, peaks in the A1 L2,3 spectrum originating from A1 3s states and peaks in the AI K spectrum originating from A1 3p states. Note that their energy representation is reversed from that of the present work. They proposed that the A1 3s states are moved to lower energies rather than hybridized with the Ag 4d states. This supposition is supported by the present calculations for the idealized Howe structure. Mixing is observed in the Howe structure mainly between the AI 3p states and the Ag 4d states. A1 3p and Ag 4d mixing at the higher energies was also inferred by Marshall et al. [15] from SXS studies of AI : 20 at% Ag alloys, although they indicated that mixing did occur between A1 3s and Ag 4d states at

low energies. This possibility is supported by the present calculations of the idealized Neumann structure. Two A1 3s peaks were observed for this structure, one below the Ag 4d bands at 7.6 eV below E~. and a second peak at 7.0 eV below Er, still below the bulk of the Ag 4d bands, which is matched by an Ag 4d contribution. The fact that the Howe d.o.s, more closely corresponds with data from the Ag2A1 composition while the Neumann d.o.s, more closely corresponds with data from the AI : 20 at% Ag composition contributes to the support of the Howe structure as more representative of the actual atomic arrangement of Ag2AI. A further distinguishing feature between the results for the two atomic configurations is the sharp minimum in the AI total d.o.s, at the bottom of the Ag 4d bands which is exhibited for the idealized Neumann structure, but is not as pronounced for the idealized Howe structure. As can be seen for the Neumann structure in Fig. 6(a), the sharp dip observed in the AI total d.o.s, at - 7 . 3 eV is actually a gap between a strong peak in the 3s d.o.s, and a strong peak in the 3p d.o.s. The Neumann configuration apparently results in the elimination of AI sp hybridization. Figure 6(b) shows that in the Howe structure, the AI 3s and 3p states are mixed at the corresponding energy. The mixing of the 3s and 3p states is a more reasonable result and lends further support to the stability of the Howe structure over the Neumann structure.

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2102

ROHRER et al.: AIAg2

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Acknowledgements The Alcoa authors would like to thank K. B. Lippert for handling code portability issues and B. O. Hall for critically reviewing the manuscript. M. E. M. acknowledges NSF NYI award DMR-9258450 and the Alcoa Foundation for support of this work.

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REFERENCES I EF 2

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Fig. 7. XPS and SXS data reproduced from Fuggle et al. [14]. The XPS line [14] originates from the Ag 4d states. The SXS A1 L2,3 line [14] originates from A1 2p to 3s transitions. The SXS A1 K line [16] originates from A1 ls to 3p transitions. CONCLUSION The stabilities of two idealized atomic configurations for h.c.p. Ag2AI have been c o m p a r e d using a first principles c o m p u t a t i o n a l technique. Based o n m i n i m u m total energy arguments a n d a better m a t c h between calculated d.o.s, a n d experimental spectroscopic information, the atomic a r r a n g e m e n t proposed by Howe [7] has been judged to be more stable t h a n t h a t proposed by N e u m a n n [6]. Calculated results agreed with general experimental observations for Ag2A1 including: (1) (2) (3) (4)

the intra-basal plane near n e i g h b o r separation, the less t h a n ideal c/a ratio, the C33 c o m p o n e n t of the elastic tensor, the displacement of AI 3s states to more tightly b o u n d energies in the presence of Ag, (5) the mixing of A1 3p and Ag 4d states, a n d (6) the depletion of Al electron states in the vicinity of the top of the Ag 4d bands. The present work precedes future studies of the interracial properties between an AI matrix and a

I. H. J. Axon and W. Hume-Rothery, Proc. Roy. Soc. A193, 1 (1948). 2. M. Hillert, B. L. Averbach and M. Cohen, Acta metall. 4, 31 (1956). 3. R. V. Ramanujan, H. I. Aaronson, J. K. Lee and R. W. Hyland Jr, Acta metall, mater. 40, 3433 (1992). 4. R. V. Ramanujan, J. K. Lee and H. I. Aaronson, Acta metall, mater. 40, 342l (1992). 5. J. M. MacLaren, S. Crampin, D. D. Vvedensky and J. B. Pendry, Phys. Rev. B 40, 12164 (1989); J. M. MacLaren, S. Crampin, D. D. Vvendensky, R. C. Albers and J. B. Pendry, Comp. Phys. Comm. 60, 365 (1990). 6. J. P. Neumann, Acla metall. 14, 505 (1966). 7. J. Howe, U. Dahmen and R. Gronsky, Phil. Mag. 56A, 31 (1987). 8. P. Hohenberg and W. Kohn, Phys. Ret,. 136, B864 (1964). 9. J. F. Janak, V. L. Moruzzi and A. R. Williams, Phys. Re+;. B 12, 1257 (1975). I0. L. Hedin and B. I. Lundqvist, J. Phys. C 4, 2064 (1971). 11. Aluminum Properties and Physical Metallurgy (edited by J. E. Hatch), American Society for Metals, Ohio (1984). 12. J. P. Neumann and Y. A. Chang, Trans. Met. Soc. A I M E 242, 700 (1968). 13. L. M. Watson, Q. S. Kapoor and D. Hart in Proc. Int. Syrup. X-ray Spectra Electron. Struct. Matter H (edited by A. Faessler and G. Wiech) 135 (1973). 14. J. C. Fuggle, E. K/illne, L. M. Watson and D. J. Fabian, Phys. Ret:. B 16, 750 (1977). 15. C. A. W. Marshall, L. M. Watson, G. M. Lindsay, G. A. Rooke and D. J. Fabian, Phys. Lett. 28A, 579 (1969). 16. S. A. Nemnonov, V. G. Zyryanov, V. I. Minin, M. F. Sorokina, Physica status solidi (b) 43, 319 (1971).