An effective procedure to determine corrugation functions from atomic beam-diffraction intensities

169

Surface Science 108 (19811 169-180 North-Ho~and PubIish~g Company

AN EFFECTIVE PROCEDURE TO DETERMINE CORRUGATION FROM ATOMIC BEG-DIFFUSION

FUNCTIONS

INTENSIVES

K.H. RIEDER IBM Zurich Research Laboratory,

CH-8803 Riischlikon,

Switzerland

N. GARCIA * ~epartamen to de Fisica Funda~lentale, Institute de2 Estado Soiido, U~~~ers~dadAutonoma Madrid, Can to Blanco, E-Madrid 34, Spain

de

and V. CELL1 Physics ~ep~tment~

university of Virginia, Charlotte~il~e~ ~ir~~~a 22901,

USA

Received 15 October 1980; accepted for publication 1.5 January 1981

A computational method is described, which, starting from given difraction intensities, approaches effectively the best-fit corrugation function S(R). Because of the approximations involved, the procedure works well for smooth corrugations with amp~tudes not exceeding -10% of the lattice constant. The method rests on two crucial observations: (i) With the full knowledge of the scattering amplitudes AG = IAG I exp(ilpG) (absolute values plus phases), the corrugation function can be calculated to a high degree of accuracy from f(R) = (2&)-r Inl C AG exp(iG .R) I which is derived easily from the hard corrugated wall scattering (HCWS) equation by approximating kG by -&i (& and kG being the wavevectors of the incoming and diffracted beams, respectively). (ii) With only the IL4Gl’s (or intensities) known, approximate solutions of the HCWS equation can be obtained with a rough estimate of the relative phases of onZy a few intense diffraction beams; the estimate is readily performed by investigating systematically a coarse mesh of phases. In this way, approximate corrugations are found with which a full set of phases can be generated, which allows the calculation of an improved r(R); this step is repeated in a loop, until optimum agreement between calculated and given intensities is obtained. The effectiveness of the procedure is demonstrated for three one-dimensional model corrugations described by several Fourier coefficients. The method is finally applied to the case of Ha d~fra~tion from the quasi~nedimensional adsorbate corrugation Ni(ll0) + H(l X 2).

1. Introduction

The hard corrugated wall (HCW) model describes to a first and useful approximation the diffraction of light atoms from solid surfaces [l-3], for an incident * Present address: Department of Aeronautics and Astronautics, MIT, Cambridge, Massachusetts 02139, USA.

0 03~~028/81/0000-0000~$02.50

0 North-Holland

170

K.H. Rieder et al. /Corrugation functions

wavelength of the order of 1 A. In this approximation, all the information derivable from experiments is contained in the surface profile z = {(x, ,Y), which is a periodic function of the lateral variables x and y, with periodicity given by the surface reciprocal lattice vectors G. The kinematics of the Bragg diffraction process is sufficient to determine the set G, but often several models of reconstruction or adsorption configurations are compatible with the same periodicity. Knowledge of the detailed shape of {(x,y) is extremely valuable to discriminate among these alternative models, and to infer the relevant physical parameters, such as the location of adsorbate atoms and the shifts of the substrate atoms. Examples of the use of He diffraction for this type of surface crystallography are the determinations of several adsorbate phases for hydrogen on Ni(ll0) [4,5 1. The inversion of scattering data (diffracted intensities in this case) to obtain the shape of the scatterer is relatively straightforward in the classical limit. The classical approx~ation gives, roughly, the envelope of the pattern of diffracted intensities. If a single surface reflection occurs, this envelope is confined between Bi - 2cy,,, and 61 + ~cz,,,, where 61 is the angle of incidence (and of specular reflection), and LX,,, is determined by the maximum slope of the surface profile. The determination of {(x, y) is easy, classically, if double reflections can be neglected in some range Of@i, i.e., ifa,,,, < 40”. Under the same conditions, the Kirchhoff, or eikonal approbation is generally applicable [6]. The prevalent procedure has been to determine {(x, y) by trial and error [1,4,5], by comparing the experimental intensities with those calculated in the Kirchhoff approximation. A more systematic procedure, introduced by Cantini et al. [7], makes use of the Patterson transform method. Both these approaches become very cumbersome if the Fourier expansion of the surface profile,

f(R) =

5


contains more than a few coefficients {G, or more generally if a suitable representation of c(R) requires more than a couple of parameters to be determined. The purpose of this paper is to present a method of inversion which is quick and computationa~y more convenient. We concentrate on the problem of dete~in~g r(R), to the extent possible, from a single diffraction pattern, i.e., from the scattered intensities PG = lAG 1’ (kGz/ki,) for a given incident momentum parallel (Kr) The assumption is and perpendicular (ki,,) to the surface (with 01 = tan-‘(KJkt,)). made that, for small enough Bi, all the diffracted beams with an appreciable intensity he well above the horizon. This assumption is analogous to the single-hit assumption (or (Y<40”) in the semiclassical limit, but the treatment is more general, and the Kirchhoff approximation (in a simplified version) is used only for a first rough estimate. It is also assumed that bound-state resonances are negligible, otherwise of course, the simple HCW model fails [8,9]. These resonances are due to beams below the horizon, also known as evanescent beams or closed channels, and these are, in fact,

XX

Rieder et al. / Corrugation functions

171

weak if the rainbow angles, Be 4 2or,,, are well above the horizon. It is possible to include approximately the acceleration due to the attractive potential by letting ki, be the perpendicular momentum with respect to the bottom of the surface potential well. A difficulty in the theory, as in every inverse scattering problem, is that only the intensities IAG I2 are given by experiment, and nothing is known directly about the phase of AG. Previous experience has shown however that, within the class of “smooth” profdes, t(R) can be determined almost uniquely [lo].

2. Procedure Step I: Full knowledge of scattering amplitudes. We start from the hard wall scattering equation [ 1,6] g

A G exp(iG +R) exp [ik&(R)]

= -exp [tii, t(R)]

,

(1)

where r = (R, z) denotes the space coordinate, with z being the surface normal, f(R) the corrugation function, G the reciprocal lattice vectors, and ki and kc the wave vectors of the incoming and outgoing beams, respectively. The complex scattering amplitudes AG yield the intensities PG via pG = (kGz/kiz)

14 GIz

.

(2)

Eq. (1) has to be fulfilled for any point of the surface unit cell. As long as l(R) is a smooth function with a small amplitude, diffracted beams of appreciable intensity will occur near normal incidence only near the specular or, in other words, the rainbow angle will appear well above the horizon. In this case, we can replace kGzby -ki,without introducing a large error, and eq. (1) can be written as

expI-%, RR>1

exp(iG 1R) = -exp [iki, f(R)] ,

(2)

or GAG

exp(iG 1R) = -exp ]2iki,

Within this approximation, the unit cell from

L,(R) =

in/-

c(R)] .

the corrugation

(3) can be determined

for any point

R in

r;AGexp(iG*R)I 2iki,

,

(4)

provided full knowledge of the AG’S is available; withAG = 1.4~ 1exp(i@G), this means that both the absolute values of the At’s as well as their phases (PGhave to

K.H. Rieder et al. / Corrugationfunctions

172

be known. Further iteration

starting from {e(R) by using

In - C A G exp(iG * R) exp [i(kct - ki,) {n_1(R)] L(R)

=

I

G

(5)

2ikiz

may improve the quality of the c(R)determined in this manner. We illustrate the usefulness of this part of the procedure by analyzing lowing three one-dimensional model corrugation functions

C(x) =

Cl cos: C”

with different

vx

the fol-

(6) of the {, :

combinations

Model I:

{1=0.23&

<,=0.15&

ModelII:

c1=0.238,

c2 =O.O

Model IIl:

cl = 0.23 a,

cz = 0.15 A,

allother{,=O; A,

{3=0.15A,

allother<,=O;


all other <, = 0 .

The lattice constant was chosen to be a = 7.04 A. The AG’s were calculated for these three models by using the iterative procedure of Lopez, Yndurain and Garcia [ 111, which, for the small corrugation amplitudes of all three models, gives almost identical results for angles of incidence smaller than 40” as the eikonal approximation. The diffraction spectra corresponding to the three models are shown in figs. 1, 2 and 3 together with the corrugation functions. We use the tAG 1and (PGVaheS for 8i = 20”, for an incoming wavelength A = 2n/ki = 0.57 8, and maintain only beams for the calculation of eq. (4) which are larger than 1% of the total intensity (in the above cases beams -8 to t8). Using eqs. (4) and (5) {n(Xi) is determined in all three cases for 30 equidistant points xi within the unit cell of length a. The results of a Fourier analysis of this function for several iterations is given in table 1. The agreement between the r(x) of the initial model, and the c,(x) of the approximate iteration procedure is judged by using a reliability factor R, which is defined as

1 l/2

.

R is zero for ideal agreement

(7)

of the initial intensities PG with the intensities P”Gobtained with the nth iteration. As can be seen from table 1, for all three models considered, the zero step [eq. (4)] gives Fourier coefficients quite close to the initial ones. Further iterations using eq. (5) improve their values only slightly, if at all. This illustrates that the procedure outlined works very well as long as the maximum amplitude of the corrugation function is small compared to the lattice constant (c,,/a < 0.10) which is the case for most corrugations investigated up to now. StepII: Knowledge of intensities PG (or lAGI) only. In the preceding section,

173

K.H. Rieder et al. / Corrugation functions [IX) MODEL

[al

I

(,

= 0.23a

c2

= 0.158

0.3 0.2 0.1 0

x : 0.57x 0.2

8,

= 20”

3 PG

0.1

4 5 6

:__:

0 -20

-10

0

7

10 2o

3o

8 4o

5o

6o

7o

Fig. 1. Upper part: Shape of the corrugation function for model I. The model parameters are given 111the figure. Lower part: Diffraction spectrum from the corrugation of model I. The best-fit spectrum obtained with the procedure outlined in the text is practically indistinguishable from this spectrum. 5(X,

MODEL 5, =

[aI

0.23a

II

c2 =0.08

c3 = 0.15a

0.4 0.3

0.3

PG

A = 0.578 8,=20”

0.2

5

0.1

-30

-20

-10

0

10

20

30

40 e

Fig. 2. Same as fig. 1 for model II.

50

60

174

K.H. Rieder et al. / Corrugationfunctions 5 (X)

MODEL Ill 5, = 0.23a

[al

c2

= 0.158

c3

= 0.08a

0.4

PG

0.2

-30

-20

-10

0

10

20

30

40

50

60

70

e

Fig. 3. Same as fig. 1 for model III.

it was shown that the corrugation function f(x) can be calculated directly with a rather high degree of accuracy, if both the absolute values of the scattering experiment amplitudes 1-4~ 1 as well as their phases cpc are known. In a real scattering experiment however, the information on the phases is completely lost, because only intensities PG can be measured, and they are related according to eq. (2) to the absolute values of the scattering amplitudes (AG I. The second important step of our procedure rests on the observation that eq. (1) can be fulfilled approximately with the phases of only a few intense beams roughly determined. The search for such a set of phases can be done rather quickly by investigating systematically a coarse mesh in phase space; usually four steps with Acpc = n/2 are sufficient. As for small angles of incidence Bi, the phases for beams +G and -G are about equal for structures with two-dimensional inversion symmetry, it is possible to vary pG and P_G in parallel. The phase of the specular beam can be set zero arbitrarily, because a translation of t(R) in the z direction does not influence the intensities. Therefore, with I intense beams +G and 1 intense beams -G, it suffices to investigate 4’ points in phase space. In all our examples, 1 never exceeded four, although the total number of beams with appreciable intensities was as high as 20. Two criteria, when an estimate of the phases fulfills eq. (1) satisfactorily well, can be used:

K.H. Rieder et al. /Corrugation

115

functions

Table 1 Fourier analysis of r,(x) obtained using eqs. (4) and (5) [step 1 (fuII knowledge of absolute values IAG I plus phases I~G)] with 30 equidistant points in the unit cell for the three different models cited; results are given for all values of n which give different Fourier coefficients fV; note that for model II, an oscillation occurs between even and odd iterations, the odd iterations giving much smaller R values

Model I 0 1 2

0.007 0.007 0.007

Model II 0 1 2 3 4

0.038 0.008 0.037 0.008 0.037

Model III 0

1 2 3

0.038 0.039 0.040 0.040

0.23

0.15

0.2198 0.2192 0.2191

0.1494 0.1497 0.1497

0.0072 0.0079 0.0080

0.23

0

0.15

0.2266 0.2295 0.2278 0.2297 0.2278

-0.0080 -0.0016 -0.008 -0.0016 -0.0088

0.1389 0.1521 0.1381 0.1521 0.1391

0.23

0.15

0.08

0.2111 0.2090 0.2087 0.2086

0.1406 0.1397 0.1395 0.1395

0.0795 0.0798 0.0798 0.0797

0.0049 0.0053 0.0054

0.0017 0.0016 0.0016

0.0021 0.0021 0.0021

0.0105 0.0030 0.0110 0.0031 0.0111

0.0010 -0.0013 0.0004 -0.0014 0.0004

0.0086 0.0031 0.0087 0.0031 0.0087

0.0076 0.0086 0.0087 0.0087

0.0077 0.0089 0.0090 0.0091

0.0060 0.0068 0.0069 0.0069

(i) The imaginary part of the corrugation function c,,(x) determined with the trial phases using eqs. (4) and (5) has to be small; we choose as a measure the quantity

S=

F

[WXxi)12.

(8)

(ii) The R factor [eq. (7)] obtained from a Fourier analysis of the CJx) yielding {i(x) and a subsequent intensity calculation with C:(x) has to be small Once a set of p+c is found, which gives small values for both R and S, a full set of new phases cp&can be calculated from S,“, and with these new phases a further calculation can be performed. This procedure can be repeated until, for example, the R value reaches a minimum. If both S and R are large, the next point in the mesh of trial phases can be investigated. To illustrate the full calculational procedure in a transparent manner, a flow chart of the sequence of calculational steps is shown in fig. 4. To illustrate the usefulness of the method, we again use models I, II and II discussed under step 1. In each case, {n(Xi) was calculated for 30 equidistant points Xi within the unitcell length a_ = 7.04 A. Usually, n was either 1 or 2. The Fourier

176

K.H. Rieder et al. /Corrugation

Further

Fourier

Fig. 4. Flow chart of computational f(R) from given scattering intensities

derotlons

analysis

procedure

functions

eq (5)

of Re [(CX,)]

to approach

the best-fit

corrugation

function

PG.

analysis of tn(x) to yield c:(x) was extended to six Fourier coefficients, and the calculation of the intensities was performed using all six coefficients. The results are summarized for the three different models in table 2. For both models I and III, the phases cpl, cpz, q3 and (p4 were varied systematically in steps 7~/2 from 0 to 3n/2. For model II, only the two phases cp2and ~4 corresponding to the most intense beams were varied; all other phases were kept zero. In all cases considered, the loop iterations were done for the cF(x) corresponding to the two best R values obtained with the trial phases. Starting from both sets of phases, the iterations gave almost the same final values for the Fourier coefficients, which for all models considered were very close to the input Fourier coefficients. Those coefficients, which are initially chosen zero, always came out much smaller than 0.01 A! It is worthwhile mentioning that for model III, a further calculation was performed in which we maintained only the largest three Fourier coefficients in the loop calculations. The R factor obtained in this case was 0.0013 (!), and the Fourier coefficients were almost exactly the input parameters (C1 = 0.2297, C2 = 0.1502, f3 = 0.0799).

Q4 = n/2

= n/2,94

= 7d2

Trial phases ‘p1 = n, QZ = n, 93 = n, 94 = S/2 -+ further 39 loop steps

--t further 28 loop steps

91 = 8, Q2 = d&93

Trial phases

Model III: r~ - 0.23, J;z = OX,53

+ further 108 loop steps

Ql=O,92=n/2,93=0,94=0

Trial phases

-+ further 60 loop steps

Qt=0,~2=n/2,93=0,94=n/2

Trial phases

= 0.08

Model II: {I = 0.23, S2 = 0.0, KJ = 0.15

-+ further 12 steps

Qi=n,92=n/2,93=n/2,94=0

Trial phases

Trial phases 91 = 71992= n/2, Q3 = n/2, -+ further 100 loop steps

Model I: c = 0.23, r2 = 0.15

0.0530 0.0116

0.0470 0.0116

0.0400 0.0168

0.0210 0.0168

0.0480 0.0057

0.0350 0.0054

0.00655 0.00026

0.01350 0.00026

0.02290 0.00041

0.01273 0.00041

0.00472 0.00004

0.00365 0.00004

0.1609 0.2261

0.1775 0.2257

0.2405 0.2307

0.2267 0.2307

0.2129 0.2185

0.1684 0.2280

0.1405 0.1487

0.1234 0.1486

0.0315 0.0075

0.0179 0.0075

0.1305 0.1533

0.1476 0.1508

0.0736 0.0818

0.0809 0.0818

0.1421 0.1540

0.1489 0.1540

0.0117 0.0085

0.0200 0.0030

0.0200 0.0042

0.0255 0.0043

0.0180 0.0028

0.0282 0.0028

0.0331 0.005 2

0.0169 0.0022

0.0322 0.0043

0.0358 0.0043

-0.0283 -0.0015

-0.0123 -0.0015

0.0108 0.0036

0.0242 0.0023

0.0145 0.0055

0.0127 0.0056

0.0234 0.0090

0.001 0.0090

0.0025 0.0056

0.0100 0.0047

-

Table 2 Results of the calculationat procedure outlmed in section 2, step 2 (knowledge of intensities PG only) (fig. 4) for three different model corrugations; the best trial phases were determined by a systematic variation in steps of n/2 (full loop in fig. 4); the phases ‘p1, 1pz,‘p3and Q4 were varied for models I and III; for model II, only Qpzand 94 were varied; the quality of the result is judged from the values of the quantities R and S (see text); the further loop steps correspond to the dashed loop in fig. 4 -.

K.H. Rieder et al. / Corrugation functions

178

Aa

= 704a.

Fig. 5. Graphical representation of the three best-fit functions for diffraction data of Hz from Ni(l10) + H(1 X 2) (see fig. 6). The numbering corresponds to that of table 3.

3. A case from real life: Hz scattering

from the adsorbate system H(l X 2) + Ni(ll0)

In fig. 6, we show the diffraction spectrum of a Hz beam with h = 0.87 A and Bi = 25 So from the quasi-one-dimensional adsorbate structure H(l X 2) on Ni( 110) [ 121. Using the procedure outlined above by systematically varying the phases pl, cp2and q3 and a subsequent loop iteration, we found three sets of possible best-fit Fourier coefficients. Starting from different sets cpl, cp2 and p3, slightly different parameters were obtained within one set of solutions. The results are comprised in table 3. A further refinement of the Fourier coefficients was performed in the usual way to vary systematically the parameters in steps of 0.01 a near the values found

’

l.O2 5

O-40

’

+ Uf1x71 ,,\.

’

’

’

’

’

’

’

40

50

60

I

-.

AH2 = 0.83&

0.8 -

0, D A 0.6 5 .G E cn 0.4 5 b = x 02*

’

hlllllnl ,..\,,“,

8, = 22.5”

I -

Experiment Theory (Corruqo t1on 3)

I

I

-30

-20

I -10

0

10

20

30

70

8

Fig. 6. Experimental and best-fit diffraction spectra for Hz scattering from the quasi-onedimensional adsorbate structure Ni(ll0) + H(1 X 2).

K.H. Rieder et al. / Corrugationfunctions

179

K.H. Rieder et al. / Corrugation functions

180

with the iterative procedure. The best-fit R values could be slightly decreased in this way (table 3). It is remarkable that three sets of Fourier coefficients give the same R values. [The three functions c(x) are plotted in fig. 5.1 Clearly, two of them (1 and 2) are closely related and the difference in the shape of these spectra indicates the loss in accuracy in determining c(x) with intensities which due to experimental error are only unprecisely known. The third function (3) differs appreciably from the other two. With the intensity data shown in fig. 6 alone, it would not be possible therefore to decide which of these two classes of corrugations is the correct one. However, analysis of data at other angles of incidence and other wavelengths yielded only the third function and therefore this has to be regarded as the true one. The best-fit intensities for function 3 are shown in fig. 6 together with the experimental data. The above result proves that our procedure is able to approach the different possible solutions c(R) for a given set of intensities.

References [l] N. Garcia, J. Chem. Phys. 67 (1977) 887. [2] N. Garcia and N. Cabrera, Phys. Rev. B18 (1978) 576. [3] R.I. Masel, R.P. Merrill and W.H. Miller, J. Chem. Phys. 65 (1976) [4] [S] [6] [7] [8] [9] [lo] [ 111 [ 121

2690.

K.H. Rieder and T. Engel, Phys. Rev. Letters 43 (1978) 373. K.H. Rieder and T. Engel, Phys. Rev. Letters 4.5 (1980) 824. U. Garibaldi, A.C. Levi, R. Spadacini and G.E. Tommei, Surface Sci. 48 (1975) P. Cantini, R. Tatarek and C.P. Felcher, Phys. Rev. B19 (1978) 1161. H. Chow and ED. Thompson, Surface Sci. 58 (1976) 225. V. Celli, N. Garcia and J. Hutchison, Surface Sci. 87 (1979) 112. K.H. Rieder, A. Baratoff and U.T. Hochli, Surface Sci. 100 (1980) L475. C. Lopez, F.J. Yndurain and N. Garcia, Phys. Rev. B18 (1978) 970. T. Engel and K.H. Rieder, Surface Sci., to be published.

659.