Disordered surfaces: a smoothed He-target scattering potential for metal atoms adsorbed on metal surfaces

Chemical Physics ELSEVIER

Chemical Physics 216 (1997) 391-399

Disordered surfaces: a smoothed He-target scattering potential for metal atoms adsorbed on metal surfaces G. Petrella a'b, L. Cassidei a, F. Ciriaco a a Department of Chemistry, University ofBari, 4 Trav. Re David 200, 70126 Bari, Italy b Centro Studi Chimico Fisici sulla lnterazione Luce-Materia, 4 Trav. Re David 200, 70126 Bari, Italy

Received 24 July 1996

Abstract The theoretical interpretation of experimental results on He scattering from metal adatoms epitaxially grown on metal substrates makes use, in general, of an interaction potential that, as far as the He-adatom interaction is concerned, is a pairwise sum of Lennard-Jones (Vu) terms. This approach results in more corrugated metal surfaces than expected on the basis of both experimental data and qualitative considerations on the contribution of the electron densities of the metal adatoms to the repulsive part of the interaction potential. The present work suggests a very simple way to significantly reduce or eliminate the surface corrugation by averaging the He-adsorbate potential over a unit cell of the substrate lattice. The so obtained potential (VAv) was compared with Vu for the He-Ag/Pt(111) colliding system on the basis of turning point surfaces, specular and diffractive intensities vs. surface coverage and angular intensity distribution of scattered atoms, calculated under the sudden approximation. These data show that V^v, although too simple to correctly reproduce all the features of the He-adatom interaction, could represent a useful tool in the study of the growth of metal adlayers. (~) 1997 Elsevier Science B.V. All rights reserved.

1. Introduction The main problem concerning the theoretical research on disordered surface properties carried out by means o f the He scattering technique is that of defining the correct He-target interaction potentials. Their knowledge is absolutely necessary to make calculations whose results can be fruitfully compared with experimental data. The He-target potential is usually represented in the form [ 1 - 5 ] : V(r = (x,y, z)) = ~

Vn~-Ad(Ir -- r i l ) + V s ( z ) (1) i

i.e. a pairwise sum o f He-adatom interaction contributions and a He-surface interaction term, assuming

negligible the contribution from three-body terms. In Eq. (1) r and ri represent the coordinates of the He atom and of the i-th adatom, respectively. The pair contributions are usually Lennard-Jones (VLj) functions whose parameters are extracted either from gasphase interaction data or obtained by means o f empirical adjustments. When the number o f defects on the surface is higher than one, the interaction potential, calculated by means of Eq. (1), can result less precise and realistic than expected. This is evidenced by the fact that for metal atoms adsorbed on a metal surface at full substrate coverage, the specular intensity calculated using VLJ does not fully recover the value detected at O=0, i.e., nearly one, which is due to the fact that VLj generates not so smooth equipotential adlayer surfaces as those

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G. Petrella et al./Chemical Physics 216 (1997) 391-399

characterizing real close-packed metal surfaces [ 6,7]. In conclusion, suitable empirical adjustments, such as fitting the calculated cross section of an isolated defect to the experimental one, does not assure that the features of the surface are correctly reproduced by VLj in the whole range of coverage. Obviously unphysical corrugation could affect values of calculated intensities for specular, diffractive and diffusive processes as well as the angular distribution of scattered He atoms. Since interest in studies on homo- and heteroepitaxial growth involving metal adlayers is recently increasing more and more both through experimental [8,9] and theoretical approaches [5,10], the importance of the problem of detecting correct He-target potentials for these systems is increasing at the same rate. To this end this paper suggests a modification of the original Lennard-Jones potential to take account of the surface smoothness requirement. The modified potential (VAV) comes from averaging the He-adatom pairwise potential over a substrate unit cell (u.c.). In the paper equipotential surfaces for an Ag complete adlayer, an island of 19 Ag adatoms and an isolated Ag adatom on Pt( 111 ) calculated by using both VLj and VAV are shown. The two interactions are also compared on the basis of sudden scattering calculations on systems of randomly adsorbed Ag adatoms on P t ( l l l ) in the range of coverage from 0 to 1 ML. The structure of the paper is the following: Section 2 shows the method of calculation. The new formulation of the interaction potential is described in detail in Section 3. In Section 4 the results of calculations are discussed. Main conclusions are reported in the last section.

2. Method of calculation

2.1. Model system and interaction potentials The model system consists of Ag atoms adsorbed on a 30 x 30 Pt( 111 ) surface; the allowed sites have the same structure and lattice constant as the underlying substrate. Pt(111) was assumed to be flat, which is certainly reasonable for Pt( 111 ) [ 11 ] and, in general, for close-packed metals [ 6,7 ]. Targets were treated as static, nonvibrating systems: the limits and validity of this assumption can be found in the literature [ 12-15 ].

A potential widely present in the literature [ 12,1517] was used for the He-Pt( 111 ) interaction term: VS(Z ) = VHe/Pt(Z )

= Ds [e -2~(z-z') _ 2e-a(Z-Zm) ] ,

(2)

where Ds = 2.888 x 1 0 - 4 a . u . , o~ ---- 0.518 a.u. and Zm = 11.4648 a.u. [ 17]. As far as the He-Ag interaction is concerned, the calculations were made using two different potentials. The first one is a Lennard-Jones potential: V L I ( r H e - - r A d ) -- VLI(r)

=D

-2

,

(3)

where rile and rAd are the vectors identifying the positions of He and adatom, respectively. D and rm are 3.3241 x 10 -5 a.u. and 8.7086 a.u., respectively [ 18]. The modified potential VAv is described in the next section. The He scattering calculations were performed for an isolated Ag adatom, a hexagonal cluster of 19 Ag adatoms and random distributions of Ag adatoms on Pt( 111 ) in the whole range of surface coverage (0 < O ~< 1). In the last case for each surface coverage the averages of specular and diffractive intensity values were calculated over a suitable number of random distributions (from 30 to 50).

2.2. The sudden approximation The calculations were performed by using the sudden approximation. This method is widely used for gas-phase collisions, and has been proposed by Gerber and co-workers for collisions involving both ordered [ 19] and disordered solid surfaces [ 1,3,12,20]. Numerically exact wavepacket calculations on some colliding systems have been made by Yinnon and coworkers to test the reliability of the sudden approximation. They showed that, for realistic corrugations, the sudden approximation correctly reproduces shape and magnitude of the angular intensity distribution of scattered He atoms and that it holds better for normal incidence and when the momentum transfer normal to the surface (kz) is much higher than the momentum transfer in parallel to the surface plane [3]. kz >> I K - K ' I ,

(4)

393

G. Petrella et al./Chemical Physics 216 (1997) 391-399

where the K and K ~ are the initial and final components, parallel to the same plane, of the incident and refracted beam wavenumbers k, and k ~. The intensity of He atoms scattered from Pt( 111 ) surface is given in the sudden approximation by the equation: Ixx, = - ~1 / e i ( g - - K ' ) ' R e E i n ( R ) dR 2 ,

(5)

.I

where A is the Pt surface area, R = (x, y) represents the co-ordinates of the scattered atoms in the (x, y) plane parallel to the Pt surface, and r/(R) represents the phase shift calculated according to the WKB approximation [ 19] : oo

R) = i dz

[ k 2 - 2 m V ( x , y, z ) l h2 ] 112 _ kzzo,

ZO

(6) where kz is the z-component of k, z0 is the classical turning point of the integrand in Eq. (6) and m is the He mass. The calculations were performed for a normally incident He particle and collision wavenumber kz = 3 bohr - l . Boundary conditions were applied at the ends of the Pt ( 111 ) surface. The cross section for diffuse scattering from an isolated Ag adatom has been calculated for collision wavenumbers kz ranging from 2.29 to 4.26 bohr- l by the following equation: ~£dl=a[l--(/~)

] ,

s

(7)

where ( l l l o ) s represents the relative specular intensity, I0 and I being the intensities of He atoms reflected by a "clean" and an adsorbate covered surface, respectively.

3. A smoothed potential for He--metal adatoms on a metal surface The repulsive part of the He-metal atom interaction potential is determined by the electrostatic repulsion between the charge densities of the two atoms. One can easily make a rough estimate of the contributions to the repulsion from single metal atom shells, since

data for electronic wave-functions are widely available in the literature [21,22]. Such an estimate for the He-Ag system shows that a contribution of only 3.7 Hartree comes from each 4d shell electron at a 6 bohr internuclear distance, a typical minimum approach distance for the H e - A g interaction. This is less than one hundredth of the total repulsion energy; even lower contributions come from inner shells. In the solid as in the gas phase the repulsion energy, in the range of interest to the He scattering technique, is therefore determined by the valence electrons [23 ]. Such qualitative considerations support the idea that the interaction between He and clusters of Ag atoms must be much smoother than accounted for by the Lennard-Jones model, since in the solid phase the valence electrons are smeared out on the overall surface. On the other hand the smoothness of the global interaction can be directly shown for full substrate coverage by the nearly unitary value of the scattering relative specular intensity [24]. For these reasons an improved potential should be built taking into account the following considerations and conditions: (i) As dispersion interaction potentials between He and adsorbed atoms or molecules are usually written in the Lennard-Jones potential form, it seems to be convenient to employ a Vu function as the starting point for the proposed VAV. (ii) VLj and VAv should give very similar results for isolated adatoms. (iii) Vu and VAV must have the same asymptotic behaviour. (iv) VAV is required to give negligible corrugation for substrates covered by full adatom layers. The last two requirements can be satisfied only assuming that the potential arising from an overlayer of metal adatoms (Vne-Ovl) is the average of VLj over all the points of the substrate surface:

x 10-6

VHe-Ovl(Z) = ~

V u ( x , y , z ) dx dy

A --q'/'O~- [1 (~)10-- (ff~)4] , (8) where S and A are the areas of the unit cell and the surface substrate, respectively. VHe-Ovlis therefore dependent only on the z-coordinate and the resulting

G. Petrella et al./Chemical Physics 216 (1997) 391-399

394

1

10.0( q.~/ou IO.lO

10.,30 z*.rn/al; lO.lO

9.90

0.~

9.70

0.70

9.50

0.$0

I 10.0

I 20.0

aO.O x/~

I 4,0.0

I $0.0

/ I

10.0

b

I

20.0

I

$0.0

x/~u

I

40.0

I

$0.0

b

Fig. 1. VLJ turning point data at kz = 3.0 b o h r - ] for 1 Ag adatom on P t ( l l l ) surface: (a) Ztum vs. x at y constant; ( b ) Zturn vs. x,y.

Fig. 2. VAv turning point data at kz = 3.0 bohr - l for 1 Ag adatom on Pt( 111 ) surface: (a) Ztum vs. x at y constant; (b) Ztum VS. X, y.

surface is therefore flat. One can partition the integral in Eq. (8) into contributions from the single unit cells centered at the adatom sites: VAV(rHe, r i ) ,

VHe-Ovl(ZHe) = ~

(9)

iEsites

where VAv(rae, ri) is the pairwise potential with the required properties: VAV( rne, rad ) = "~

VLJ(rile, X, y, Zad ) dx dy.

S(r~)

(10) It is worthwhile to notice that VAV is not flat except at very small distances from the metal adatom. When several adatoms are put together in wider clusters the

flatness range is enlarged, until at full coverage flatness is extended to the whole zne range. The choice of the u.c. should conform to the symmetry of the adsorbate lattice to preserve the symmetry features of the scattering event. In the present case the u.c. was a hexagonal cell of side 5.2381 bohr. It can be noticed that the shape and dimension of the u.c. can also be viewed as an adjustable parameter to obtain the desired amount of corrugation. The depth parameter D of VLj in Eq. (10) was corrected by a multiplicative factor 1.2 so that VLJ and VAV have the same single adatom cross section at the incident wave vector of 3 bohr -t.

G. Petrellaet al./Chemical Physics 216 (1997) 391-399

395

4. Results and discussion 10 ° The He scattering features o f VAV were compared with the corresponding ones o f VLJ for a few significant target models: an isolated A g adatom, random distributions o f A g adatoms and an isolated hexagonal cluster o f 19 A g adatoms.

(I/Io)s

\

"

10" X

•

X

(i) Case of an isolated Ag adatom.

1 0 "z

Figs. 1 and 2 show turning point surfaces for one A g adatom on a Pt( 111 ) surface when the potential is calculated in the VLj and VAV forms, respectively. The potential felt by the particle in the region o f space accessible for kz = 3 bohr-1 is similar in the two cases. Therefore for isolated adatoms, as far as the features o f the diffractive event are concerned, both potentials give closely similar results in terms o f scattered intensity. Furthermore the corresponding diffusive cross section values for an isolated A g adatom at different colliding energies are similar for the two potentials and in good agreement with the experimental data [25] (Table 1).

x

1 0 .3

• ,

0.0

,

i

Useful information on the different features o f VLj and VAV potentials can be obtained by taking into account the specular and diffractive intensities as functions o f surface coverage. In Figs. 3a,b the specular, (lifo)s, and diffractive, (I/Io)D, intensity curves vs. ~9 are reported. As can be seen, VAV generates more reflecting surfaces than VLJ at all ~9 values. Accordingly, the VAV diffraction Table 1 k~/bohr- l

X1~u~/A 2

Xl~^v~/A 2

Xl~cxp~/~2

2.292 2.644 2.806 2.966 3.021 3.127 3.220 3.399 3.587 3.868 4.043 4.256

160.69 150.34 145.67 140.92 139.19 135.76 132.75 126.88 120.94 113.30 109.76 106.72

158.74 145.56 142.30 140.24 139.65 138.49 137.37 134.79 130.97 122.76 116.76 109.35

151.59 158.55 145.95 138.32 135.20 129.63 120.41 124.52 115.30 107.67 99.51 85.71

,

,

20.0

,

i

,

,

40.0

,

J

,

,

60.0

,

i

,

,

80.0

,

100.0

e% 10 o

b

(I/Io)D

/

10"

I 0 "z

X X

(ii) Case of averaged random distributions of Ag adatoms (0 < ~9 <<,1).

•

X

I0"

X

X

X

X

X

× ×

j<

I 0 "~ 0.0

20.0

40.0

60.0

80.0

100.0

0%

Fig. 3. (a) Specular intensity

(l/lo)s

vs. ~ at kz = 3.0 bohr -1

for He-Ag/Pt( 111 ) colliding system. (b) Diffractive intensity (I/Io)D vs. O at kz = 3.0 bohr -1 for He-Ag/Pt(lll) colliding system. (e) = ~t,J, ( x ) = VAV. intensities are lower than the VLj ones. The (I/lo)s plots for both potentials show a m i n i m u m at around ~9 = 0.4. This can be explained by considering that the main process up to ~9 = 0.4 is the increasing diffusion o f scattered atoms due to the increasing concentration o f A g adatoms on Pt( 111 ). For ~9 > 0.4 He diffusion should be considered as arising mostly from vacancies in an otherwise flat A g adlayer. The slope o f (I/lo)s vs. O curves depends on the interference between He reflected by the top o f A g adatoms/clusters and by P t ( l l l ) surface [ 2 6 ] . The different slopes at O = 0 and O = 1 indicate two different values o f A g and vacancy cross sections: XAg = 140.08 /~2 and Xvae = 76.1/~2.

396

G. Petrella et al./Chemical Physics 216 (1997) 391-399

10.30

10.30f

a

10.10

I0.i0

9.90

9.90

9.70

9.50

9.50

I0.0 I

20.0 I

I 30.0

40.0 I

x/aa

101.0

I

20.0

I

30.0

x/~a

4O1.0 b

Fig. 4. VLJ turning point data at kz = 3.0 bohr-I for Pt(lll) surface fully covered by Ag adatoms: (a) Ztumvs. x at y constant; (b) Zturn vs. x,y. At low coverages (O ~< 0.1) where single Ag adatoms are mostly present on the Pt(111) surface VAV and VLj yield similar specular intensity values, as expected on the basis o f the above mentioned considerations concerning an isolated Ag atom adsorbed on the surface. At intermediate and high surface coverages (0.1 < O ~< 1 ) the two potentials give quite different specular and diffractive intensity values. The lost specular intensity for VLj can be found in the diffractive channels, as shown in Fig. 3b and similar intensity data for the two potentials can be obtained summing the specular and diffractive contributions.

Fig. 5. VAVturning point data at kz = 3.0 bohr -t for Pt(lll) surface fully covered by Ag adatoms: (a) Zturnvs. x at y constant; (b) Zturn vs. x V. At full coverage o f the Pt( 111 ) substrate the specular intensity values for the two potentials are very different. The Vav potential allows a complete recovery of the (I/Io)s intensity at ~9 = 1, ( (I//10)s = 0.9998), in closely accord with the experimental data [ 24]. On the contrary the specular intensity calculated by means of VLJ is equal to 0.4105. Accordingly, the diffractive intensity values are very different at t9 = 1, for Vav ( I / l o ) o is very low (6.2 x 10-5), while for VLJ is 0.5505. In Figs. 4 and 5 the turning point surfaces for the two potentials at ~9 = 1 are reported. The corrugation is evidently high for the Lennard-Jones potential compared to the incident particle wavelength, while it is negligible in the case of Vav interaction.

G. Petrella et al./Chemical Physics 216 (1997) 391-399

,13°[

10.30

2turn/a~

lO.10

10.10

9.90

9.90

9.70

9.70

/I-

9.50

I 10.0

397

I 20.0

I 30.0

x/au

40.0

9.50

I 10.O

I 50.0

b

I 20.0

I 30.0

x/au

I 40.0

I 50.0

b

Fig. 6. VLj turning point data at kz = 3.0 bohr -1 for 19 Ag adatoms on P t ( l l l ) surface: (a) Ztum vs. x at y constant; (b)

Fig. 7. VAV turning point data at kz = 3.0 bohr -1 for 19 Ag adatoms on P t ( l l l ) surface: (a) ztum vs. x at y constant; (b)

Z t u m VS. X , y .

Zturn VS. X ,

(iii) Case o f an isolated cluster o f 1 9 Ag adatoms. Figs. 6 and 7 refer to equipotential surfaces relevant to a 19 A g adatom hexagonal cluster. Much o f the corrugation o f the cluster top is lost in the averaging o f the potential. However, some o f the structure is preserved. To have a deeper insight on the effects o f the residual structure on the scattering process, it is very useful to consider the angular intensity distribution o f scattered atoms. Values o f llcK, are shown against A K x in Fig. 8 for both potentials. The shapes o f the licic, curves show the same structures although the values corresponding to high AKx are depressed in the case o f VAV. This is easily understood when considering that high A K x values correspond to small x-displacements in the coordinate space, since VAv is

smoother than VIII on a small scale compared with the unit cell dimensions, except at the edge o f the cell. Whether or not the small cluster surfaces are well represented by VAV depends on the delocalization o f the valence A g electrons. Maps o f electron density contour would be determinant in such a question. Equally useful would be experimental data on scattering intensity angular distribution, since the weights o f the structures are different for the two potentials. In absence o f such data one can only say that the spherical distribution o f the 5s A g electrons, underlying the VIA potential, is rather poor when the atoms come to nearest neighbour distance, hence for islands. The model o f an electron in a box applied to the electronic levels in A g islands on Pd( 111 ) has been

y.

398

G. Petrella et al./Chemical Physics 216 (1997) 391-399

(Vl0)°'~ 0.025

0.02

0.015

0.005

~' J'

-2

-1

1

2

~

Fig. 8. Angular intensity distribution of scattered atoms vs. AKx for He scauedng from a 19 Ag atom cluster adsorbed on Pt( ! 11 ) at kz = 3.0. Full curve VAVdata, dashed curve VM data. successfully used [27,28], nonetheless the shape of the electron functions is a parameter much more sensible to dimensions and local disorder. In the sum, only experimental data can ascertain the right features o f the potential. Since it cannot be taken full account o f the influence o f the island dimensions on the electron distribution in model calculations on surfaces containing several adatoms, if the experimental data should support the authors' qualitative considerations, VAV would result a useful tool in the simulation o f processes o f recent interest such as epitaxial growth.

He scattering calculations in the sudden approximation were made on A g / P t ( l l l ) targets in the whole range o f surface coverage using both potentials. Equipotential curves for an isolated A g atom, a cluster of 19 A g atoms and an A g full monolayer adsorbed on Pt( 111 ) were also obtained for the two potentials. A comparison among the obtained results demonstrates that: ( i ) at low coverages both potentials show very similar scattering features; (ii) VAV is different from VLj at intermediate coverages. In lacking o f experimental data, qualitative considerations based on the electron densities o f the A g adatom shells at the He locations suggest that the potential proposed in the present paper is more realistic than the simple Lennard-Jones potential; (iii) at high coverages VAV and VLJ have different behaviour; the specular and diffractive intensity curves obtained using VAV are characterized by the correct trends. In view of this behaviour with respect to the scattering process, VAV, although certainly not reproducing all the features o f the He-target interaction should be preferred to VLj at least as soon as large compact structures characterize the adsorbate growth.

Acknowledgement

The present paper was supported by the italian "Ministero per l'Universith e la Ricerca Scientifica e Tecnologica" by means o f "60%" and "40%" contributions.

References 5. C o n c l u s i o n s

Pairwise Lennard-Jones interaction potentials available in the literature for H e - m e t a l adatoms adsorbed on a metal substrate do not properly take into account the smoothing o f the surface when adsorbate atoms come in contact. A s a matter o f fact specular intensity for a full coverage surface, calculated through VLJ, is lower than 1, contrary to experimental data and theoretical considerations. In the present paper a formulation o f the H e - m e t a l adatom interaction is presented which yields negligible corrugation in the above mentioned conditions.

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G. Petrella et al./Chemical Physics 216 (1997) 391-399 [8] G. Rosenfeld, N.N. Lipkin, W. Wulfhekel, J. Kliever, K. Morgenstem, B. Poelsema and G. Comsa, Appl. Phys. A 61 (1995) 455. [9] H. Brune, H. Roder, C. Boragno and K. Kern, Phys. Rev. Lett. 73 (1994) 1955. [101 P. Blandin, C. Massobrio and P. Ballone, Phys. Rev. B 49 (1994) 16637. [ 11 ] B. Poelsema and G. Comsa, Faraday Discuss. Chem. Soc. 80 (1985) 16. [ 12] A.T. Yinnon, R. Kosloff, R.B. Gerber, B. Poelsema and G. Comsa, J. Chem. Phys. 88 (1988) 3722. [13] A.T. Yinnon, R. Kosloff and R.B. Gerber, Chem. Phys. 87 (1984) 441. [ 14] A.T. Yinnon, R. Kosloff, R.B. Gerber, D.K. Dacoi and H. Rabitz, J. Chem. Phys. 84 (1986) 5955. [15] A.T. Yinnon, R. Kosloff and R.B. Gerber, Chem. Phys. 88 (1988) 7209. [ 16] G. Petrella, A.T. Yinnon and R.B. Gerber, Chem. Phys. Lett. 158 (1989) 250. [ 17] M.A. Krzyzowski, P. Zeppenfeld and G. Comsa, J. Chem. Phys. 103 (1995) 8705. [181 A.T. Yinnon, D.A. Lidar (Hamburger), I. Farbmann, R.B. Gerber, E Zeppenfeld, M.A. Krzyzowski and G. Comsa, in preparation; The values of the He-Ag/Pt( 111 ) potential parameters were kindly provided by R.B. Gerber, D.A. Lidar (Hamburger) and T.A. Yinnon.

399

[ 19] R.B. Gerber, A.T. Yinnon and J.N. Murrel, Chem. Phys. 31 (1978) I. [20] J.I. Gersten, R.B. Gerber, D.K. Dacol and H. Rabitz, J. Chem. Phys. 78 (1983) 4277. [21 ] A.D. McLean and R.S. McLean, At. Data Nucl. Data Tables 26 (1981) 197. [22] E. Clementi and C. Roetti, At. Data Nucl. Data Tables 14 (1974) 177. [23] P. Bedrossian, B. Poelsema, G. Rosenfeld, L.C. Jorritsma, N.N. Lipkin and G. Comsa, Surf. Sci. 334 (1995) I. [24] A.E Becker, G. Rosenfeld, B. Poelsema and G. Comsa, Phys. Rev. Lett. 70 (1993) 477. [251 A.T. Yinnon, D.A. Hamburger and R.B. Gerber, Chem. Phys. Lett. 253 (1966) 223; The numerical cross sections values were kindly provided by T.A. Yinnon by private communication. [26] B. Poelsema and G. Comsa, in: Springer Tracts in Modem Physics, Vol. 115, Scattering of thermal energy atoms from disordered surfaces (Springer, Berlin, 1989). [27] R. Fischer, S. Schuppler, N. Fischer, Th. Fauster and W. Steinrnann, Phys. Rev. Lett. 70 (1993) 654. [28] R. Fischer, Th. Fanster and W. Steinmann, Phys. Rev. B 48 (1993) 15496.