A re-examination of multilayer relaxation of Ag(110) by LEED structural analysis

269

Surface Science 218 (1989) 269-282 North-Holland, Amsterdam

A REEXAMINATION OF MULTILAYER BY LEED STRUCTURAL ANALYSIS M. LINDROOS,

C.J. BARNES,

RELAXATION

OF Ag(ll0)

M. VALDEN

Department of Physics, Tampere University of Technology, P.O. Box 527, SF-33101 Tampere, Finland

and D.A. KING Department of Physical Chemistry, Cambridge CB2 IEP, UK Received

8 November

Cambridge

1988; accepted

University, Lensfield Rod

for publication

6 March

1989

Although the existence of an oscillatory relaxation of the topmost two interlayer spacings of Ag(ll0) is well established by both low energy electron diffraction (LEED) and Rutherford backscattering spectroscopy (RBS) the magnitude of the first layer contraction and second layer expansion is still a matter of considerable dispute. In an attempt to clarify the situation we have performed an independent LEED study utilising a large normal incidence data set and 7 different reliability factors (R-factors) to judge the level of theory-experiment agreement. The distance dz,, between the first and second layers is smaller than the bulk interlayer distance by 7 f 2% and the distance dz,, between second and third layers is expanded by 1*2%. Furthermore, a contraction of 2 f 2% of the third interlayer spacing (dz,) has been extracted, with subsequent interlayer spacings equal to the bulk value within the precision of this analysis. These results are compared with theoretical embedded atom method and tight binding predictions of changes of up to the top four interlayer spacings. These results confii and extend the surface geometry determined in an earlier LEED study of Davis and Noonan. Finally, the multilayer relaxations obtained by LEED are compared with recent RRS studies and possible reasons for discrepancies discussed.

1. Introduction The determination of the deviation in spacing of the topmost atomic layers of single crystal surfaces from their bulk equilibrium spacing (relaxation) has and continues to be an area of considerable activity. Over the last decade a substantial experimental data base has been built up, with low energy electron diffraction (LEED) being by far the most utilised and successful technique. Another possible method to obtain information on surface relaxation which has recently emerged as a potential challenger to LEED is Rutherford back0039-6028/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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M. Lindroos et al. / Multilayer relaxation of Ag(ll0)

scattering spectrometry (RBS) using the so-called channeling and blocking configuration [l]. However, although the aforementioned techniques have together provided the bulk of information regarding surface relaxation, significant disagreement between independent studies still exists in a number of instances. The Ag(llO)-(1 X 1) surface is a case in point. The single existing LEED study taking the possibility of multilayer relaxation into account is that of Davis and Noonan [3] who obtain damped oscillatory relaxation of the first two interlayer spacings with a first layer contraction of 5.7% and second layer expansion of 2.2%. These values contrast strongly with those obtained in a recent ion scattering study of Holub-Krappe et al. [2], indicating considerably larger first layer contraction (9.5 + 2%) and second layer expansion (6.0 f 2.5%). The large first layer contraction found by Holub-Krappe et al. is supported by the earlier LEED study of Maglietta et al. [4] ( - 10%). However, the aforementioned LEED study [4] must be considered less comprehensive than that by Davis and Noonan [3] as only first layer relaxation was tested. A second study by Alff and Moritz adopting constant momentum transfer averaging rather than full dynamical treatment obtained a first layer contraction of 8% [5]. Finally, we mention the results of an independent RBS determination using single alignment geometry by Kuk and Feldman [6] who obtain a first layer contraction of 7.8 + 2.5% and a second layer expansion of 4.3 f 2.5%. Thus, it appears that the exact geometry of the Ag(llO)-(1 x 1) surface seems far from being settled, despite the relatively large number of studies existing. The range of determined geometries leaves considerable scope to debate which particular values should be adopted. We have recently completed our study of the surface geometries of Pd(llO)(1 x 1) and an alkali-induced Pd(llO)-(1 X 2) reconstruction [7]. As a next step we have expanded our studies to the Ag(ll0) surface. As a preliminary to studies of chemisorbed overlayer structures and surface reconstruction phenomena we have undertaken a re-determination of the geometry of Ag(llO)(1 x 1) using the video LEED technique. We have used a large high quality normal incidence data base in conjunction with a comprehensive multi-R-factor approach to determine the favoured surface geometry. In the next section we shall describe our measurement technique and theoretical methods. In the following section we will present our results. Finally we discuss our results and compare them with previously published studies of Ag(ll0) both experimental and theoretical.

2. Experimental procedure and theoretical considerations The experiments were performed in an ion and titanium sublimation pumped UHV chamber fitted with a liquid nitrogen chilled cryopanel and capable of attaining base pressures of -C 3 X lo-” Torr. The system is

M. Lindrm et al. / M~tiIa~errelaxationojAg(l IO)

271

equipped with Varian 4-grid LEED optics which also served as a retarding field analyser for Auger electron spectroscopy (AES). The Ag sample was cut to within - lo of the (110) plane and polished using standard metallographic techniques. Final cleaning was performed in-situ by repeated cycles of argon ion bombardment and annealing at 900 K. The sample was deemed clean when no ~onta~n~ts could be observed by AES, at this stage the sample exhibited a bright and sharp p(1 x 1) LEED pattern. The sample temperature was measured with a chromel-alumel thermocouple attached to the edge of the sample. The sample holder was equipped with cooling facilities. Rapid cooling to < 100 K was possible by mounting the sample directly off a liquid nitrogen chilled reservoir. Intensities were measured with a commercial rapid data acquisition videoLEED analyser. I(E) spectra of 8 non-equivalent beams were collected at normal incidence using an energy increment of 3 eV. Normal incidence was attained by comparison of sets of beams symmetrically equivalent over an energy range of 200 eV. The experimental spectra of symmetrically equivalent beams were averaged whenever possible in order to minimise any residual effects of sample misalignment and as a means of increasing statistical accuracy. All spectra were normalized to constant primary beam current and background was subtracted when necessary to remove contributions from thermal diffuse scattering, which was increasingly important at high primary beam energies (200 ev). The backwood was subtracted by empirical fitting to intensity minima in spectra. Theoretical I(E) spectra were calculated using both a modified version of the CAVLEED program [8] and the package of Van Hove and Tong [9] for the clean Ag(llO)-(1 X 1) structure. Interlayer scattering was treated via the layer doubling technique. The ion-core-scattering potential was of muffin-tin form. We used the tabulated silver potential of Moruzzi et al. [lo]. Up to nine phase shifts were used. Phase shifts were temperature corrected using an effective Debye temperat~e of 215 K for bulk and 150 K for surface atomic layer. Up to 65 (h, k) beams were used in the plane wave expansion of the wave field between layers for non-symmetrized CAVLEED and 25 beams in symmetrized programs of Van Hove and Tong. These choices guaranteed that all propagating beams up to 300 eV were included in the calculations. The real part of the optical potential was initially set equal to 15.0 eV. For the purpose of R-factor analysis this parameter was treated as a variable and was rigidly shifted in 1 eV steps to obtain best theory-experiment agreement. This is acceptable as we are considering only normal incidence of incoming electrons. The imaginary part of the optical potential was fixed at 3.5 eV. Some attempts to vary this parameter to enhance theory-experiment agreement was made. However results indicated that no significant improvement in the level of agreement was obtained by changing this non-structural parameter.

212

hf. Lindroos et al. / Multilayer relaxation of Ag(ll0)

Theory-experiment agreement was tested by conventional R-factor analyses. InitiaIIy 7 different methods were used. The R-factors employed were those of Pendry (Rpe) [ll], Anderson et al. (R,,) [12], Zanazzi-Jona (R,) [13] and the four metric distances of Lindgren et al. (R,,), (R,,), (RHa) and (R,,) [14]. As some of the R-factors are highly sensitive to noise, it was necessary to smooth experimental and theoretical spectra before carrying out the R-factor analysis. This was achieved by convoluting both the experimental and theoretical data sets with a Lorentzian of width 1 eV.

3. Results For the Ag(ll0) surface the optimum structure was obtained with the fifth interlayer spacing dq6, at its bulk value of 1.439 A. Pendry’s R-factor plots in fig. 1 display the response of the R-factor upon variation of dz,,, dz,,, dz,,, and dz,,. In each curve the other three distances are set at their optimum

1.3

1.4

dz - Distance

1.5

[i]

Fig. 1. Variation of Pendry reliability factor for the clean A&110) surface upon variations of dz,,, dr,, and dQ,, when others are kept at their optimum value. Dashed line indicates the bulk interplanar spacing.

d%,

M. Limbos et al. / Multilayer relaxation 01Ag(I 10)

5 0.

Energy

150.

[eV]

2 5 0.5 0. Energy

Fig. 2. Comparison between experimental and theoretical I(E) of Ag(llO)-(1 x 1).

150.

273

2 5 0.

[eVl curves for the optimum geometry

value. The minimum value of R,, 0.180, is reached at interlayer spacings of: dz,, = 1.338 A, dz,, = 1.453 A, and dz,, = 1.410 8. When these interlayer distances are compared with the bulk interlayer separation displayed by a dashed line, we obtained an oscillatory variation of -7% + 1% and -2% (negative values indicate contractions). We also tested the effect of the variation of the fourth interlayer spacing. Pendry’s R-factor indicated an expansion of 1%. However, other R-factor showed values of - 1% or the bulk value. This means that the change of the fourth interlayer spacing is small, our data indicating less than 1% from the bulk value. Fig. 2 depicts the experimental I(E) curves together with the optimum theoretical data. In the figure we also display with dashed lines some theoretical data outside our measured experimental range. The theoretical spectra are normalized so that the maximum peak height in the experimental energy range is the same in both experimental and theoretical spectra. Visual comparison reveals that our theoretical calculations reproduce ah experimental peaks. In addition to some minor differences in relative intensities, we have two noticeable disagreements, one in beam (0,l) and the other in the beam (1, 1). In the case of the (0,l) beam our experimental data set indicates a triplet structure centred at - 120 eV. However, theoretically we do not reproduce

214

M. Lindroos et al. / Multilayer relaxation of Ag(l10)

this structure very satisfactorily, rather a single peak with shoulders is generated. The same disagreement is present in previous theoretical results of Davis and Noonan [3] and Maglietta et al. [4]. In the case of the (1, 1) beam the converse is true. Experimentally, at - 150 eV we see a peak with two rather weak shoulders, while theoretically we generate a well resolved triplet. It may be thought that these effects could arise from sample misalignment or imperfect nulhng of magnetic fields leading to errors in the experimental data set. However, the agreement of our experimental data with that of Davis and Noonan in these particular energy ranges of the beams in question is excellent. Davis and Noonan also reproduce experimentally a well resolved triplet for the (0, 1) beam and a singlet structure in the (1, 1) beam. Davis and Noonan [3] have this part of the (1, 1) beam much more damped, probably due to higher temperatures used during measurements and correspondingly in calculations. It should be pointed out that even for these beams we see equal number of peaks in both experimental and theoretical spectra. To analyse possible origins of these disagreements we carried out two tests. As pointed out by Davis and Noonan [3] the effect of the errors caused by sample misalignment might be large for Ag(ll0). It is feasible that beam intensities may be extremely sensitive to the angle of incidence. To check whether this is the case we chose an angle of incidence 0.25 o off-normal for a range of different azimuthal angles. There were small changes in relative intensities of the three peaks of the (0, 1) beam, but changes were not large enough to explain the observed disagreement. It was noted in one azimuth that the Pendry R-factor was improved nearly 10% to 0.164. The theoretical off-normal angle producing the absolute minimum in R-factor was 0.4”. This change in angles, however did not alter the optimum geometry. The second possible explanation for the above disagreement might be the need, in the case of silver, to include relativistic effects. We repeated our calculations using a scalar relativistic potential [15] and averaging of phase shifts calculated using a relativistic potential [16]. Once again we saw minor changes in relative intensities, but no changes great enough to explain the noted discrepancies. To test the reliability of our analysis we display in fig. 3 Pendry’s R-factors for each individual beam, when dz,, and dz,, are varied. These results are comparable to previous studies if we notice that 1% corresponds to 0.014 A (see e.g. Van Hove and Koestner [17]). For dz,, we have a maximum deviation for the (2, 0) beam by 3% away from the averaged R-factor value, but all other beams coincide within 1% with the optimum topmost layer spacing. For dz,, only the (1, 2) beam deviates by more than 1% from the optimum spacing. Fig. 3 clearly demonstrates that our determination of the R-factor minimum is unique. To further expand our evaluation of the reliability of our results, in fig. 4 we show results of different R-factors for the same dz,, variations. This figure

M. Lindraos et al. / ~~lti~ayer Aleutian of A&I IO)

275

A=(2,1) B=(l,O) C=(i,2) D=(2,0) E=(O,l) F=(l,l) G=(0,3) H=f0,2) ---=Ave.

0 0 0

1ja I

j

-10.0 dz12

#

t

f

v

,

8

I

I1

-5.0 -Displacement

0.0 [%]

dz 23 -displacement

5.0

[ %]

Fig. 3. Variation of Pendry reliability factor for individualbeams upon variations of (a) dz,, and (b) dz,,. Other layer distances are set at their optimum value. The total R-factor obtained by summationof beam R-factors weightedby energy range is displayedby a dashed line.

that the level of agreement between different R-factors, which all emphasize different spectral features, is excellent. Thus, through comparison of single beam R-factors and of the geometry selected by the different R-factor formulations we must conclude that the geometry obtained in this study is highly reliable by current LEED standards. The sensitivity of the LEED spectra to surface geometry variations on the topmost two interfayer spacings, dz,, and dz,, is presented in fig. 5. As this figure shows there are large variations in shape of peaks and in their energy positions with both dz,, and dz,, ilhtstrating the sensitivity of our analysis to the exact surface geometricd structure. We have also thoroughly analysed effects of many non-structural parameters. We obtain Debye temperatures with optimum values of 215 K for the bulk and 150 K for the surface layer. The enhancement factor for the surface layer is 1.1 compared with the bulk mean square vibration amplitude. This is indicates

hf. findroos ei al. / ~~1iIayer reiaxakw

276

of Agf I f 0)

o

=RACJ

I 'Rpe +=+&J x =%I H =R~2

-10.0

n

=RHa

-

=%evy

-510

dz12 -Displacement

[%I

Fig. 4. Behaviour of different reliability factors (see text) upon variations of dz,,. All other layer spacings are fiied at their optimum value.

considerably less than 1.65 obtained by Holub-Krappe et al. However, it has to be admitted that LEED is not an especially sensitive and thus very reliable method to determine surface ~bration~ properties. Finally, to test the effect of energy dependence of the real part of the optical potential we carried out a calculation, in which the kinetic energy of the diffracted electrons was replaced by the expression: #??,=I?~+ EL v,(E)

V;(&J,

is the original kinetic energy and for P’, we used = Vo\I(E- 20)/220

+ y,_

Energies are measured in electron-volts and both V, and V, were varied. The previously determined optimum surface geometry was used during this test. Behaviour of the Pendry R-factor as a function of V, is displayed in fig. 6. The parabolic oscillations are due to the R-factor program where we change the constant part of V,, i.e. Y,, by 1 eV. However, the main result of this test is a

a b C

e

f g_ 150. Energy

250. 50.

150.

[eV]

[eVl

Energy

50. Energy

150. [eV]

Fig. 5. Comparison between experimental (curve d) and theoretical Z(E) curves when topmost (curves a (+ 5%),b (0%) and c (- 5%)) or second (curves e(+ 5%),f(O%) and g(-5%)) interlayer spacing is varied with other spacings set at their optimum value.

rl

0

‘,,,,,,,

0.0

I,,,,,,

(I,,,,,

4.0

2.0 Vg

I,,,,,

6.0

Cevl

Fig. 6. Effect of adoption of an energy dependent real part of the optical potential on Pendry’s R-factor.

278

h4. Lindrooset al. / Multilayerrelaxationof Ag(ll0)

prediction of variation of the real part of the inner potential by 3.7 eV in the energy range between 20 and 240 eV. This value is in good agreement with the theoretical value of 4.1 eV derived from results of Alff and Moritz [5]. A decrease in the optimum Pendry R-factor to 0.153 was obtained. The surface geometry, however, remains unchanged by inclusion of the energy dependence of the real part of the inner potential.

4. Discussion In this study we have provided clear evidence for damped oscillatory oscillations of the first three interlayer spacings of Ag(llO)-(1 x 1). We obtain values for the magnitude of the relaxations of - 78, + 1% and - 2% for dz,,, dz,,, and dz,, respectively. We note that the values for dz,, and dz,, are in good agreement with the previous LEED study of Davis and Noonan, strongly confirming their original analysis and extending it in the form of extraction of third and fourth interlayer spacing changes. The present study has also attempted to increase the confidence in the precision of the analysis through the use of a range of different R-factors. We have conclusively shown that the final deduced geometry is reproducibly determined by all R-factors used, consistency of R-factor minima as function of dz12, dz,,, and dz,, is considered excellent. The coincidence of the individual beam R-factor minima as a function of dz,, and dz,, also strongly suggests that the correct geometry has been obtained [5]. We estimate a precision of &2% based on the above-mentioned extensive R-factor studies. The high level of theory-experiment agreement obtained, along with the coincidence of two independent multilayer LEED studies places this structural determination in the highly reliable category. Our experimental values of - 78, + 1% and -2% for changes in dz,, and dz 23, and dz,, with the fourth interlayer spacing close to the bulk value may be compared with previous experimental studies and theoretical predictions for the Ag(ll0) surface. Table 1 summarises LEED multilayer analyses and ion scattering studies along with theoretical predictions utilising both the embedded atom method [l&19] and tight binding calculations [20]. The two dynamic LEED analyses are in good agreement with each other while the theoretical predictions, although not ab initio, agree both with each other and dynamic LEED data. Our own multilayer LEED study, which would appear to be the most comprehensive analysis to date may be compared to the recent embedded atom calculations of Ning Ting et al. [19], who report the first four interlayer spacing changes. The theory-experiment agreement is seen to be remarkably good (to within f 1%) in this particular case. Separate comparison of both LEED multilayer analyses and theoretical predictions indicate a typical variation of 1 to 2% from study to study.

279

M. Lindroos et al. / Multilayer relaxation of Ag(il0)

Table 1 Summary of the multilayer relaxation parameters from LEED and ion scattering analyses along with theoretical predictions utilising the embedded atom and tight binding methods Ref.

Technique Dynamic LEED Ion scattering Ion scattering Dynamic LEED Embedded atom method (Foiles et al.) Embedded atom method (Ning Ting et al.) Tight binding

- 5.7 -7.8rk2.5 -9.5&Z -7 *2 -5

2.2 + 4.3 f 2.5 +&O&2.5 +1.0*2 +0.3

- 6.87

+ 2.19

-1.04

-6.2

+ 0.7

< 0.7

-2*2

Of2

+0.54

[31 WI [21 This work WI P91 (201

Thus, it appears that there is a disturbing discrepancy between the LEED multilayer studies and the recent RBS study of Holub-Krappe et al. [2], who obtain a significantly larger first layer contraction and particularly second layer expansion. The LEED multilayer studies are consistent, and also agree rather well with the embedded atom results. On the other hand, the large first layer contraction favoured by Holub-Krappe et al. is supported by the earlier LEED study of Maglietta et al. [4]. The inclusion of first layer relaxation only in the aforementioned LEED study leads naturally to the speculation that it is quite possible, upon allowing deeper layers to relax away from bulk equilibrium positions, that the top layer ~ntra~tion may be reduced. The only sure way to test this assumption would be a full multilayer reanalysis of the experimental data of Maglietta et al. However, we note that in the present study we obtain a first layer contraction of -7% (exactly the same as upon inclusion of multilayer relaxation) if we fix second and deeper layers at bulk equilibrium sites. This is not supportive of the assumption that the deduced first layer contraction will change considerably upon multilayer analysis. To further complicate the situation, a second RBS He+ scattering study of Kuk and Feldman [6] obtain values intermediate between the multilayer LEED studies and the channeling and blocking study of Holub-Krappe et al. However, as pointed out by Holub-gape et al. the single alignment study of Kuk and Feldman is sensitive mainly to dz,, rather than dz,, and dz,, components individually. Furthermore, the values obtained by Knk and Feldman are critically dependent on the choice of the surface vibrational amplitude, perhaps introducing further uncertainty. It thus seems that the most reliable and precise structural determinations of the geometry of clean Ag(ll0) namely the RBS double alignment study and the two independent LEED multilayer analyses are in significant disagreement in comparison to recent LEED/ion scattering studies of other fee (110)

280

M. Linahos et al. / Multilayer relaxation ojAg(l IO)

surfaces. In particular we note that the most recent LEED and ion scattering studies (channeling and blocking) are in good quantitative agreement for both the Cu and Ni(ll0) surfaces [21]. One possible reason for this discrepancy may be traced to differences in sampling regions of the two techniques. LEED puts strong emphasis on the geometry of well ordered regions of the crystal surface while RBS effectively samples the geometric structure of the entire area upon which the ion beam is incident. The Ag(ll0) and Pd(ll0) surfaces seem to be particularly unstable towards surface reconstruction. It has recently been demonstrated that low coverages of alkali metal drive (1 x 2) surface reconstruction of both surfaces [7]. Recent calculations also predict that even when clean, the energy balance between the (1 x 1) and (1 x 2) phases of Ag(ll0) and Pd(ll0) is extremely delicate [22]. It is thus perhaps not unrealistic to expect that surface disorder may exist at the aforementioned surfaces even when clean. Observations of disorder have in fact been suggested from diffraction experiments on those surfaces in particular [4,23-251. In addition, recent calculations of Guillope and Legrand [20] also indicate that the Ag(ll0) may undergo an order-disorder transition just below room temperature, hence suggesting the possibility that disorder exists in the Ag(ll0) surface at 300 K. Our LEED measurements were performed well below the predicted disorder transition temperature and those of Davis and Noonan above. The geometry obtained via these two independent analyses agree rather well. Thus, it appears that if such a transition exists below room temperature LEED I(E) spectra are not particularly sensitive to the phenomenon. However, it remains a possibility that RBS may be more sensitive particularly if lateral shiftings of the top layer Ag atoms are involved. Preliminary results of Guillope and Legrand indicate that Pd(ll0) behaves rather similarly to Ag(ll0). Thus, it would seem to be of interest to perform an ion scattering study of the clean Pd(llO)-(1 x 1) surface for which two independent multilayer LEED analyses which are in good agreement already exist [7,26].

5. Conclusions An oscillatory relaxation of the clean Ag(llO)-(1 x 1) surface has been confirmed by an independent multilayer LEED analysis. This study obtains an inwardly relaxed first interlayer separation of Adz,, = 7 f 2% and an outwardly relaxed second interlayer spacing of Adz,, = 1 f 2%. These values are in excellent agreement with an earlier multilayer LEED study of Davis and Noonan. A third layer contraction of AdzM = -2 f 2% has been also determined. The variation of geometric parameters favoured by adoption of a range of 7 R-factors and non-structural parameters has been critically tested

M. Lindroos et al. / Multilayer relaxation of Ag(l IO)

with satisfying results regarding the consistency predicted.

281

of the geometric structure

Acknowledgements The authors would like to thank Dr. M. Bowker for the loan of the Ag(ll0) sample used in these experiments and Mr. D.J. Holmes for his part in the preparation and cleaning of the sample. M.L. and C.J.B. gratefully acknowledge Tampere University of Technology and Neste foundation for financial support and the department of Chemistry (Liverpool University) for additional funding and hospitality offered to one of us. (C.J.B.) during a summer visit in 1987 during which time the measurements were made. The SERC is also gratefully acknowledged for an equipment grant.

References [l] J.F. van der Veen, Surface Sci. Rept. 5 (1985) 199. [2] E. Holub-Krappe, K. Horn, J.W.M. Fret&en, R.L. Krans and J.F. van der Veen, Surface Sci. 188 (1987) 335. [3] H.L. Davis and J.R. Noonan, in: Determination of Surface Structure by LEED, Eds. P.M. Marcus and F. Jona (Plenum, New York, 1984) p. 215; H.L. Davis and J.R. Noonan, Surface Sci. 126 (1983) 245. [4] M. Maglietta, E. Zanazzi, F. Jona, D.W. Jepsen and P.M. Marcus, J. Phys. C (Solid State Phys.) 10 (1977) 3287. [5] M. Alff and W. Mot-its, Surface Sci. 80 (1979) 24. [6] Y. Kuk and L.C. Feldman, Phys. Rev. B 30 (1984) 5811. [7] C.J. Barnes, M. Lindroos and D.A. King, Surface Sci. 210 (1988) 108; C.J. Barnes, M.Q. Ding, M. Lindroos, R.D. Diehl and D.A. King, Surface Sci. 162 (1985) 59; RD. Diehl, M. Lindroos, A. Kearsley, C.J. Barnes and D.A. King, J. Phys. C 18 (1985) 4069. [8] D.T. Titterington and G.C. Kinniburgh, Computer Phys. Commun. 20 (1980) 237. [9] M.A. Van Hove and S.Y. Tong, Surface Crystallography by LEED (Springer, Berlin, 1979). [lo] V. Moruzzi, J. Janak and A. Williams, Calculated Electronic Properties of Metals (Pergamon, New York, 1978). [ll] J.B. Pendry, J. Phys. C (Solid State Phys.) 13 (1980) 937. [12] J.N. Anderson, H.B. Nielsen, L. Petersen and D.L. Adams, J. Phys. C (Solid State Phys.) 17 (1984) 173. [13] E. Zanazzi and F. Jona, Surface Sci. 62 (1977) 61. [14] S.A. Lindgren, L. Wallden, J. Rundgren and P. Westrin, Phys. Rev. B 29 (1984) 576. [15] R. Pankaluoto, private communication. [16] R. Feder and W. Moritz, Surface Sci. 77 (1978) 505. [17] M.A. Van Hove and RJ. Koestner, in: Determination of Surface Structure by LEED, Eds. P.M. Marcus and F. Jona, (Plenum, New York, 1984) p. 357. [18] S.M. Foiles, M.I. Baskes and M.J. Daw, Phys. Rev. B 33 (1986) 7983. [19] Ning Ting, Yu Qingliang and Ye Yiying, Surface Sci. 206 (1988) L857. [20] M. Guillope and B. Legrand, Surface Sci. 215 (1989) 577.

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et al. / Multilayer relaxation of Ag(l IO)

[21] T. Gustafsson, M. Cope1 and P. Felter, The Structure of Surfaces II, Eds. J.F. van der Veen and M.A. Van Hove (Springer, Berlin, 1988). [22] K.W. Jacobson and J.K. Nsrskov, Phys. Rev. Letters 60 (1988) 2496. [23] S.M. Francis and N. Richardson, Phys. Rev. B 33 (1986) 662. [24] M. Wolf, A. Goschnick, J. Loboda-Cackovic, M. Grunze, W.N. Unertl and J.H. Block, Surface Sci. 182 (1987) 489. [25] G.A. Held, J.L. Jordan-Sweet, P.M. Horn, A. Mak and R.J. Birgeneau, Phys. Rev. Letters 59 (1987) 2075. [26] M. Skottke, R.J. Behm, G. ErtI, V. Penka and W. Moritz, J. Phys. Chem. 87 (1987) 6191.