Helium-adsorbate cross section on highly corrugated substrates

G. Armand, L. Schwenger, and H.-J. ErnstCommisariat a` l’Energie Atomique Saclay, De´partement de Recherche sur l’Etat Condense´, les Atomes et les Mole´cules/Service de

Recherche sur les Surfaces et l’Irradiation de la Matie`re, 91191 Gif Sur Yvette, France~Received 6 October 1995!

The interaction of He atoms with adsorbates placed on strongly corrugated substrates is investigated boththeoretically and experimentally. An analysis of the scattering process within the transition matrix approachsuggests that the normalized differential cross section, associated with the attenuation of coherent diffractiondue to incoherent scattering in the presence of adsorbates, is expected to be largest for the most intensecoherent diffraction peaks. For the adsorption of K on Cu~115!, this trend is experimentally observed.

He scattering from weakly corrugated surfaces gives riseto specular scattering, which is often many orders of magni-tudes more intense than diffraction for higher-order Braggpeaks. Thus, weakly corrugated surfaces are almost perfectmirrors for He atoms. The deposition of small amounts ofadsorbates on the surface produces incoherent scattering,which leads, in turn, to a decrease of the coherently scattereddiffraction peak intensity. In the limit of high adsorbate di-lution, the attenuation of the relative specular peak height isa direct measure of the probability for incoherent scattering,and the relative specular peak intensity decreases linearlywith adsorbate coverage.1 Thus, one can define a total crosssection for incoherent scattering from the flat surface throughthe expression

Ssf52

1

n

d~ I /I 0!

du Uu50

, ~1!

wheren denotes the number of substrate atoms per unit area,and I /I 0 is the relative specular intensity at adsorbate cover-ageu. For practical purposes,I 0 is often taken as the inten-sity of the specularly reflected beam of the surface prior todeposition of the adsorbate.

On the other hand, He diffraction from strongly corru-gated surfaces gives rise to diffraction patterns which arecharacterized by a number of coherent diffraction peaks,whose intensities differ but are often of the same order ofmagnitude. Adsorbates on such a surface also produce inco-herent scattering, and consequently, coherent diffraction isattenuated. From an experimental point of view, then, thequestion arises whether one can use the relative attenuationof any diffraction peak for the determination of the crosssection. For this purpose, we introduce what we call in thefollowing ‘‘the normalized differential cross section,’’

SGc 52

1

n

d~ I G /I 0G!

du Uu50

. ~2!

Here,I 0G is the intensity of theGth diffraction peak prior toadsorption. With this definition, the normalized differentialcross section is identical to the total cross section@Eq. ~1!#, ifone considers diffraction from a mirrorlike surface. How-ever, in the case of diffraction from a corrugated surface thequestions then arise, first, whether the normalized differential

cross sections are the same for all diffraction peaks, andmoreover, whether the value of the cross section for thespecular peak is the same compared to the cross section in anotherwise identical but uncorrugated system.

Based on theoretical grounds, the answers are no. An ex-amination of the theoretical expressions shows~i! that on thecorrugated surface the normalized differential cross sectionwill be largest for the most intense peaks, and~ii ! that thetotal cross section of the adsorbate placed on a flat surface islarger than the specular peak cross section of the same ad-sorbate positioned on a corrugated surface of the same na-ture. We show here experimentally that the former expectedtrend is verified for the system K/Cu~115!.

THEORETICAL EXPECTATIONS

In order to justify the above statements, it is necessary tolook at the equations that give the reflection coefficients andthe cross sections.2,3 For simplicity, we discuss in the follow-ing at first the theoretical expectations in terms of the differ-ential cross section only. The ratio of an intensity measuredin a final statef , to the incident intensity, gives the reflectioncoefficient for transition from initial statei to the final statef . For the specular channel of a periodic surface, this quan-tity is given by

Rsi5U12iphsipi

U25112p

piIm~hsi!1

p2

pi2 uhsiu2, ~3!

wherehsi is theT-matrix element for the transition from theinitial to the final state, andpi the dimensionless normalcomponent of the incident wave vector.

With similar definition forhGi andpG , the reflection co-efficient for a higher-order diffraction channelG is given by

RGi5p2

pipGuhGiu2. ~4!

When an adsorbate is placed on a periodic surface, the re-flection coefficients are given, to first-order perturbation, by

Rsic1a5Rsi1

2p

piu Im~ Tsi!1

p2

pi2 u2 Re~hsi* Tsi!

1p2

pi2 u2uTsiu2, ~5!

PHYSICAL REVIEW B 15 FEBRUARY 1996-IIVOLUME 53, NUMBER 8

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for the specular beam on a corrugated surface and by

RGic1a5RGi1

p2

pipG@2u Re~hGi* TGi!1u2uTGiu2#, ~6!

for a higher-order diffraction peak. In these expressions,Tsiis the matrix element for scattering from one single adsorbateplaced on the corrugated surface and includes the contribu-tion of multiple scattering between the adsorbate and theuncovered corrugated surface to the reflection coefficient.

The negative derivative of these expressions with respectto u gives the differential cross section. The sum of all thesequantities is equal to the total cross section. One gets, respec-tively, for the specular and nonspecular diffracted beams,

Ssc52

2p

piF Im~ Tsi!1

p

piRe~hsi* Tsi!G , ~7!

SGc 52

p2

pipG2 Re~hGi* TGi!. ~8!

A particular case is that of scattering from a flat surface.ThenRsi

f 51 ~elastic scattering considered only!; that is tosay,hsi50 and

Rsif1a511

2p

piu Im~Tsi!1

p2

pi2 u2uTsiu2, ~9!

Ssf52

2p

piIm~Tsi!, ~10!

whereTsi is the matrix element for scattering from an adsor-bate placed on the flat surface. It includes the contribution ofmultiple scattering occurring in this process.

In general one cannot obtain analytical expressions for theimaginary and real parts, or the phase, of anyT-matrix ele-ment. We can only get their values in a numerical calcula-tion. This is the reason why we cannot demonstrate our state-ment rigorously. One can only expect that they are true, bylooking at the mathematical properties of these equations.

We assume now that the reflection coefficient is less than1, which means that the intensity of the specular or a higher-order diffracted beam is less than the incident one. Then, onegets from Eq.~3!,

2pip

< Re~hsi!<pip, ~11!

pip

~212a!< Im~hsi!<pip

~211a!, ~12!

with

a5S 12p2

pi2 @Re~hsi!#

2D 1/2, ~13!

which shows that Im~hsi! is always negative. This is gener-ally true and this property is confirmed by numericalcalculations.2 Equation~9! confirms this statement: becausethe reflection coefficient should decrease with coverage,Im~Tsi! should always be negative. Therefore the cross sec-tion given by Eq.~10! is positive. The same argument ap-plied to Eq.~5! shows that

Im~ Tsi!1p

piRe~hsi* Tsi!,0. ~14!

While Im~Tsi! is certainly negative, one cannot say anythingabout the sign of the real part. Therefore the reflection coef-ficient for a given diffraction beam with adsorbate on thesurface@Eq. ~6!# may be greater or smaller than that of theclean surface. However, the corresponding differential crosssections will have the same sign.

We now compareS sf to S s

c, that is,

uIm~Tsi!u

and

UIm~ Tsi!1p

pi@~Rehsi!~ReTsi!1~ Imhsi!~ ImTsi!#U.

~15!

In the last expression the product of the two imaginary partsis positive as the imaginary part of each matrix element isnegative. This term yields a decrease of the absolute value ofthe cross section and, on the whole, that on the uncorrugatedsurface will be probably the largest. Thus, due to enhancedmultiple scattering, the decrease in coherently scattered in-tensity under the same kinematical conditions will be smalleron the corrugated surface than compared to the flat surface,and consequently, the differential cross section deduced forspecular diffraction on the corrugated surface will be smallerthan the total cross section for the flat surface.

The differential cross section for a diffraction channel onthe corrugated surface is given by Eq.~8!. In order to com-pare the differential cross section for two different diffractionpeaks, we have to look at the real parts,4

uhGiuuTGiucos~w2a!, ~16!

which is the product of the modulus of the matrix elementmultiplied by the cosine of their phase differences. We knowthat the modulus ofhGi is larger for the more intense diffrac-tion peak@Eq. ~4!#. One can expect that the same holds alsofor the modulus ofTGi . Under this assumption, and if thephase differences for the two beams are comparable, it fol-lows that the coherently diffracted beam with larger intensitywill show a stronger attenuation in intensity at a given ad-sorbate coverage; thus the differential cross section associ-ated with this diffraction channel will be larger than that forless intense ones.

In order to make contact with experiment, we considernow the normalized differential cross sections, which areequal to the differential cross sectionsS s

c ,S Gc divided byRsi

andRGi , respectively. Consequently, the kinematical factorsdisappear. Using the form~16! one gets

SGc 52

uTGiuuhGiu

cos~w2a!. ~17!

Under the above assumption, if the modulus ofTGi decreasesmore rapidly than the modulus ofhGi , by going from higherintensity peaks to the lower ones, the normalized differentialcross section will be the largest for the largest peak intensi-ties. Again this can be expected because theT-matrix ele-ments contain all the multiple scattering between corrugated

R4250 53G. ARMAND, L. SCHWENGER, AND H.-J. ERNST

surface and corrugated surface with adsorbate. Note that thetest on this last expectation, using the normalized differentialcross sections, is more severe than that just using the un-normalized ones.

EXPERIMENT

We now confront these theoretical expectations with thecross sections determined for each coherent diffraction peakof the Cu~115! surface under adsorption of small amounts ofK. This system has been chosen, since alkali-metal atomsinteract repulsively5 at low coverage, so that island formationis inhibited even at temperatures for which the adsorbate ismobile. Moreover, He diffraction from the clean Cu~115!surface perpendicular to the average step direction is charac-terized by the appearance of numerous diffraction peaks withsignificant variations in intensity~Fig. 1!, which indicatesthat the He surface potential is highly corrugated in this di-rection.

Usually, the cross section for incoherent scattering is de-termined by monitoring the intensity attenuation of a se-lected diffraction peak with coverage. For the present pur-pose, we are interested in the attenuation of each of thediffraction peaks in Fig. 1. Since the expected effect mightbe small, the major difficulty is to assure an identical inci-dent K flux for the experiments on different diffractionpeaks. Therefore, we proceed for the determination of thenormalized differential cross section~or rather the slope ofthe initial attenuation in intensity; we do not give here abso-lute values for the cross section, since the K coverage is notaccurately independently known! for each diffraction peak asfollows. Starting from the clean surface, we have continu-ously measured full angular distributions during depositionof K at low flux. This procedure has the advantage that theerror in the determination of the cross section due to possiblevariations of the incident K flux is minimized. Moreover, onecan easily check that the half widths of the coherent diffrac-tion peaks do not change, which is a prerequisite for the useof peak amplitudes only. We found in the coverage range ofthis experiment that the half widths stay constant for allpeaks within 6%.

Figure 2 shows the result of this experiment for selecteddiffraction peaks labeledA andB in Fig. 1. The slopes differby about 30%, and the absolute value of the slope of peakBis larger, which implies that the differential cross section forthe intense peak is indeed larger. In Fig. 3 we show a com-pilation of the slopes deduced from the initial decay in in-tensity for all diffraction peaks in Fig. 1. At least a tendencytowards the theoretical expectation is indeed reflected in ourdata. However, this experimental result has to be discussedwith respect to the following source of uncertainty. Gener-ally, the cross section could show an intrinsic angular depen-

FIG. 1. He diffraction pattern from Cu~115!. The scatteringplane is perpendicular to the average step direction, the incidentwave vector iski510.8 Å21, and the angle between source anddetector is 104°. Under these kinematical conditions, nine coherentdiffraction channels are open, with maximum intensity variation ofabout a factor 10. FIG. 2. Normalized~peak! intensity variation of the coherent

diffraction channels labeledA and B in Fig. 1 with time underidentical impinging K flux. The difference in slope and thus nor-malized differential cross section is about 30% for these two peaks,whose initial absolute intensities differ by about a factor 4.

FIG. 3. Compilation of the initial slopes deduced from the at-tenuation of each diffraction peak in Fig. 1 due to K adsorption vsinitial absolute peak intensity. The trend towards larger slopes andthus larger normalized differential cross sections for the most in-tense peaks is observed.

53 R4251HELIUM-ADSORBATE CROSS SECTION ON HIGHLY . . .

dence due to an anisotropic scattering potential.6 Indeed, onthe basis of the present data set, such an effect cannot becompletely ruled out, but the fact that diffraction peaks withlow intensity for both positive and negative parallel momen-tum transfer show a small cross section suggests that an in-trinsic angular dependence of the cross section does not pro-duce the observed phenomenon.

In conclusion, the normalized differential cross section foradsorbates positioned on strongly corrugated substrates hasbeen investigated both theoretically and experimentally.Theory suggests that due to multiple scattering effects, thenormalized differential cross section is largest for the mostintense coherent diffraction peaks. This trend is observed ex-perimentally for the He-K/Cu~115! system. Of course, fur-ther experimental work is necessary in order to establishwhether this is a general phenomenon. Moreover, it would be

useful to have an independent calibration of the K coveragein order to determine absolute values for the normalized dif-ferential cross section. This would allow for a direct com-parison of the values deduced here for the corrugated surfacewith the total cross section of a corresponding mirrorlikesurface, e.g., Cu~001!.7

ACKNOWLEDGMENTS

We would like to thank Dr. R. Biagi for useful discus-sions, and P. Lavie for technical assistance. We are indebtedto Professor J. R. Manson for several fruitful discussions, inparticular, concerning the definition of the normalized differ-ential and the total cross section. This work was in part sup-ported by the EC Human Capital and Mobility program un-der Grant No. CHRX-CT93-0110.

1B. Poelsema and G. Comsa,Scattering of Thermal Energy Atomsfrom Disordered Surfaces,Springer Tracts in Modern PhysicsVol. 115 ~Springer-Verlag, Berlin, 1989!.

2G. Armand and J. R. Manson, J. Phys.~Paris! 44, 473 ~1983!;Comput. Phys. Commun.80, 53 ~1994!.

3G. Armand and B. Salanon, Surf. Sci.217, 341 ~1989!.4The kinematical prefactorspi ,pG can be calculated and arenearly equal for two neighboring diffraction peaks.

5Physics and Chemistry of Alkali Metal Adsorption,edited by H.Bonzel, A. Bradshaw, and G. Ertl, Material Science MonographNo. 57 ~Elsevier, New York, 1989!.

6A. M. Lahee, J. R. Manson, J. P. Toennies, and Ch. Wo¨ll, Phys.Rev. Lett.57, 471 ~1986!.

7A. Reichmuth, A. P. Graham, H. G. Bullman, W. Allison, and G.E. Rhead, Surf. Sci.307-309, 34 ~1994!.

R4252 53G. ARMAND, L. SCHWENGER, AND H.-J. ERNST