Diffraction intensities in helium scattering; Topographic curves

Surface Science 68 (1977) 399-407 0 North-Holland Publishing Company

DIFFRACTION CURVES

INTENSITIES

IN HELIIJhI SCATTERING;

TOPOGRAPHIC

N. GARCIA Section de Fisica Fun&mental, Institute de Fisica de1 Estado Solid0 (CSIC), Universidad Autonoma de Madrid, Gzntoblanco (Madrid), Spain and Departamento de hiatenuiticas, Universidad National de Educaion a Distancia Ciudad Universitar&z(Madrid), Spain

G. ARMAND and J. LAPUJOULADE Centre d’Etudes Nucleaires de &clay, Service de Physics Atomique, Section d’Etudes des Interactions Gaz-Solides, BP 2, 91 I90 Gif-sur- Yvette, France

Diffracted beam intensities have been calculated using the hard corrugated wall potential and the so-called GR method, for numerous values of the corrugation parameter and incident angle. Curves of equal intensity are drawn as a function of these two parameters (topographic curves). They give a clear description of diffraction phenomena. Comparing to experimental data, the potential corrugation is determined for alkali halide, for dense metallic surfaces and tentatively for tungsten (112) face. So the microscopic roughness seems to be the parameter which can explain the quite different behaviour of these surfaces regarding the scattering of helium. The potential corrugation of the repulsive part of the potential is discussed in terms of localization of delocalization of electrons in the vicinity of atom cores.

1. The model In the scattering of helium atoms by alkali halide surfaces, the diffraction phenomenon has been observed in different experimental situations [ l-4] In contrast, no diffraction peaks have been detected when helium atoms are scattered by clean metallic surfaces [5-91 if one excludes the (112) face of tungsten. In this case, in-plane first-order diffraction peaks have been observed when the incident plane contains the [liO] direction [IO,1 11. In these two experiments, out-ofplane measurements were not possible. In order to explain this strikingly different behavior, two kinds of arguments, which are not exclusive, have been prpoposed: (a) in the experimental condition, the Debye-Waller factor is always very much smaller for metallic surfaces than for alkali halide surfaces [ 121, and (b) the surface corrugation is lower on metallic 399

400

N. Garcia et al. /Diffraction

intensities in helium scattering

surfaces, that is to say the Fourier components of the potential are strongly weaker than those of alkali halide surface [ 131. The latter explanation is based on the results of diffraction peak intensities calculation, in which the interaction potential is represented by an infinitely hard corrugated surface (HCS model) [ 13-161. If the @! direction is normal to the surface plane and R (Xl’) denotes a position vector belonging to the surface, the potential is taken to be: VR, z,(W)

= O” =o

z
(1)

when for a square unit cell of side a z,(R)

= $Q&

[cos(%X/Q)

+ COS(~TY/Q)]

.

(2)

Assuming that only elastic events occur, the time-independent Schroedinger equation is solved on the condition that the total wave vanishes for z = z,(R), and one obtaines the following set of linear equations [ 13,141 * : ~~cexp{(kCI+k,)zl(R)tG.RI}=-l,

(3)

in which AG is the scattered amplitude in the channel given by the two-dimensional reciprocal lattice vector G, and k, and kcr are the normal wave vector components respectively for the incident beam and for the diffracted beam in channel G. The conservation of energy gives: k&

= k; -

(Ki + C)* .

where Ki is the incident particle wave vector component parallel to the surface. Any good solution of (3) must give results which satisfy the unitarity condition, i.e.: CP,.= F

1)

pF= (k&k,>

i&i*

,

F being a G vector for which k&Z = k&, > 0. The AG values have been first calculated in the framework of eikonal approximation [13]. The oscillating behavior of diffraction peak intensities, known as “rainbow pattern” and experimentally observed [4], is very well depicted. But in order to remove this approximation and to take multiple scattering into account, two methods have been used: (a) the Fourier transform methods [14,15], and (b) the so-called CR method, developed by Garcia et al. [14,16]. A set of eq. (3) is solved by taking a finite number N of non-equivalent points, regarding translational symmetry, in the unit cell (N R values) and N corresponding points in the reciprocal space (N G vectors). The number N is chosen large enough so that the uni* Eq. (3) corresponds to the Raleigh hypothesis.

N. Garcia et al. / Diffaction intensities in helium scattering

401

tarity condition is satisfied. The calculated results [ 161 are in good agreement with the experimental ones of Boato et al. 141. Thus, provided that resonances with bound states do not influence the diffracted beam intensities (as experimentally occurs for He thermal beams [4]), the potential model and the calculation method seem to give a good description of the experimental results. It is important to notice that although the calculations this paper are done with the Rayleigh hypothesis [17] (eq. (3)) the results presented (& < 0.12; &,/2 rv 0.06 < 0.072) are the exact ones, as has been shown recently with the exact solution by Cabrera and Garcia 1181. That is to say, for the corrugation regime presented in this paper, the Rayleigh hypothesis leads to the exact solutions of the scattering of a wave by an HCS model. Therefore, we have calculated these quantities for different values of the corrugation parameter /3e and incident angle @t. The rest&s describe the overall evolution of the phenomena in function of these parameters. Thus, with the available experimental data, we can determine the fi, value for different materials. Conversely, if for a given material one can estimate the /3e value, the expected diffraction peak intensities can be deduced and the most favorable incident angle chosen. With these possibilities in mind, results are presented and discussed in the following section.

2. Conditions of calculation; results We consider an helium incident beam scattered by a surface, the unit cell of which is a square of side a. The particle in¢ wave vector and angle are respectively denoted by ki and 8i (the direction oi = 0 coincides with the normal to the surface). The incident plane contains the [ 1 lo] direction. ki is chosen SO that &a = 3 1.2, giving respectively in the cases of LiF(OOl), NaCl(OO1) and Cu(OO1) surfaces, a ki value equal to 11,7.8 and 12.2 A-‘. These conditions correspond to the experiments of refs. (31, [4] and [9] for LiF and copper. In the case of NaCl [3], the incident wave vector is slightly different being approximately equal to 11 A-‘. FOJ all cases, the number N is equal to 196. This is exactly the number of channels included in the calculation. With kta = 3 1.2, the number of open channels is of the order of 70 for all the values of Bi. The diffraction peak intensities have been calculated under these conditions for different values of the parameters PO and Bi. For /?o less than 0.12 and for all Bt values the unitarity condition is fulfilled as the sum (4) is lying between 0.9999 and 1. Thus, the set of equal intensity curves in function of /Ia and Bt can be drawn for different diffracted beams. The resulting diagram consists of these curves which are called topographic. Such topographic curves are presented for in-plane beams 00,

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N. Garcia et al. /Diffraction intensitiesin helium scattering

@degree

gdegree

Fig. 1. (a)-(c)

Curves of equal intensity of topographic curves in function of corrugation parameter po and incident angle Bi, for COO),(10) and fl0) beams. (d) Topographic curves for the ratio Iio/Ioo. ‘ho values 0, 0, A, * and m are obtained respectively from refs. [4], [ 31, [ 31, [IO] and [ 111. The shaded region correponds to dense or compact metallic surfaces.

10, io res~e~tive1~ in figs la, b and c, and for out-of-piane respectively in figs. 2a, b, c and d. Xn these two figures, the 1000. Complementarily, fig. Id depicts the topographic curves I-i&e,, which will be used later to determine the corrugation

-beams II, f 1, 22, 22 incident beam is scaled for the intensity ratio of different materials.

3. Discussion 3.1. General behavior The specular 00 beam is very much greater than the others for very small & values and for aif PO when approaching grazing incidence. A large “valley” appears for the three in-plane beams when complete extinction of these beams occurs. These intensities vanish in approximately the same PO, 0i region. As it could be expected from the unitarity condition, the out-of*plane beams then reach their maxima. It is now interesting, in order to get a &ear insight into diffractioa phenomena,

N. Garcia et al. jL?iffraction intensities in helium scattering

$1 -?k--SO

e*sree Fig. 2. (a)-(d) beams.

Topographic

curves for the (II),

(ii),

403

,25 50 9 degree

9

(22) and (22) out-of-plane diffracted

to examine the evolution of intensities as @i varies given a value of the corrugation parameter Be. Doing so, one can distinguish three PO regions as far as the evolution of the in-plane peaks are concerned: (a) For low Be value, approximately for &-,less than 0.03, the specular beam increases with Bit whereas the 10 and i0 decrease. (b) For &, lying between 0.03 and 0.09, a region where a maximum appears for 10 and i0 beams (scaled 120), these beams first increase with Bit reach a maxima and decrease for large incident angle. The specular peak intensity increases monotonically, but its intensity at normal incidence is drastically lowered in such a way that it can completely vanish. (c) For high POvalue, but obviously less than 0.12, the 00 peak intensity decreases first, goes to extinction and increases again. The 10 beam has a very low intensity in this region as well as the f0 for Bi less than 45-50”. Beyond this value it increases slightly, going after that to very low value for grazing incidence. The out-of-plane beams have a different behavior. The 11 and i-i decrease monotonically as Bi increases for PO approximately less than 0.045. For Be greater than this value, they have a maxima for Bi value around 40-50” (scaled 130). On the other hand, the 22 and 22 decrease monotonicaIly. All these out-of-plane

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N. Garcia et al. /Diffraction

intensities in helium scattering

beams have very low intensities for low fl,, value as it could be expected: the particles are scattered into in-plane beams mainly in the specular. These results outline the necessary in an experiment to measure the different intensities for different incident angles, not only for the in-plane beams but also the out-of-plane, especially if the former are not very intense. For illustration of this point, suppose a material with a flu value about 0.08-0.09. The 10 and i0 beams remain practically undetectable for all 0i values, as we have to multiply the intensities given in the figures by a Debye-Waller factor. On the contrary, the 1 I and iI peaks reach a maxima which is as large as the specular for Bi equal to 40-60”. On the other hand, the 22 and 72 peaks are also greater than the specular in the neighborhood of normal incidence. From only in-plane measurements in this case, one could conclude that the surface potential is very smooth.

In order to obtain the corrugation potential of a material, we have to compare the experimental data with our calculated results. For lithium fluoride (001) face, we consider first the extended results of Boato et al. [4], the Debye-Walter factor being unknown. Nevertheless, it can be shown that this factor has practically the same value for the specular and i0 beams if the incident angle is less than 45”, because the incident wave vector modulus ki is large compared to the modulus of the smallest reciprocal vector G (G/k, N 0.2) and the crystal temperature is low (Ts = 80 K). With this in mind, the points corresponding to the measured ratio lie/Ioo for 01= O”, 15”, 27”, 30” and 40” are put in fig. Id (0). They fall on practically the same line, giving a mean /3e value equal to 0.10. The point at 8i = 65” lies on a higher flu value. However, in this case the Debye-Waller factor ratio for these two beams must certainly be less than one and consequently, the ratio Ire/Z,, for T, = 0, that is to say the calculated one, should be greater than the experimental ratio. This correction would lead to a translation of the representative point to lower &, values. The same correction should be applied to the point (0) deduced from the experimental data of Bledsoe and Fischer [3). It could be more important in this case as the crystal temperature is equal to 300 K. Concluding this analysis, one can say that the corrugation parameter for (001) face of LiF is equal to 0.10 + 0.01, leading to a full corrugation potential (@) of 0.28-0.31 A. A more precise calculation introducing a new Fourier component in the potential has been made [ 161 giving a better fitting of the data [4]. As far as the NaClfOOl) face is concerned, we have just one point (A) 131. It is difficult in this case to estimate the Debye-Waller correction. Compared to the LiF case (o), the surface temperature and incident angle are the same, the G/ki ratio (0.145) and the bulk Debye temperatue are lower to reduce and increase respectively the difference between the Debye-Waller factor of i0 and 00 beams. Thus, one can tantatively say that the fin value for this surface has probably around 0.085

A? Garcia et al. /Diffraction intensities in helium scattering

405

giving a corrugation potential of 0.34 A. New experimental results are needed in order to obtain a more precise value. The case of metals is quite different as no diffraction peak has been observed except on the (112) face of tungsten. In a recent experiment on the (100) face of copper [9], a very narrow specular peak has been detected (0.5”) which is composed of elastically scattered particles, as shown by velocity analysis. Its intensity has been measured for different incident angles and surface temperature. Taking into account the particle acceleration in the stationary attractive part of the potential (Beeby [ 121) the usual Debye-Waller plot gives the value of the “apparent” surface Debye temperature interpreted in terms of simultaneous interaction of the incident helium aton with the four copper atoms belonging to a surface unit cell. This plot gives also the intensity of the 00 beam, supposint the crystal at rest, equal to 1000 + 100. Fig. la leads to a POvalue less than 0.01 for this surface. Therefore, the Iie/Zue ratio would be lower than 0.02 for a crystal at rest, and in the above experimental condition, the i0 beam intensity must be less than 0.004 of the incident beam intensity. As the experimental sensitivity is O.OOS,the diffracted beam, if it exists, could not be observed. A preliminary calculation with a rectangular unit cell has shown that this difference introduces a not too significant variation of the calculated intensities. There-, fore we have included in fig. Id the points relative to the (112) face of tungsten, the incident plane containing the [liO] direction following the data of Stickney [lo] (.) and Merrill [ 1 l] (m). The points fall in the same PO range. The correction for the Debye-Waller factor could be important here as the crystal temperature is high (1200 K). Following the preceding discussion, this correction tends to increase the ~c/Zoe ratio and the points would be translated a little to higher /3e value. However, one can tentatively adscribe to this face for the [ liO] direction a corrugation parameter of the order of 0.06. The corresponding full corrugation potential would be of the order of 0.27 A. When the incident plane contains the [ 1 li] direction, no diffraction peaks have been detected. As the rugosity perpendicaulr to this direction is great, and according to these calculated results, one may think that out-of-plane beams must be detected. Unfortunately, the experimental device did not allow such as measurement.

4. Conclusions The topographic curves calculated with hard corrugated wall potential and presented above, give a clear description of diffraction phenomena. They allow one to predict the evolution of in-plane or out-of-plane diffracted beam intensities if one can a priori determine the corrugation potential of the studied surface. Unfortunately, such a determination is quantitatively impossible as we are not able, at the moment, to calculate the potential from first principles. Thus, only com-

406

N. Garcia et al, f Diffraction intensities in helium scattering

parison with experimental results could give the potential corrugation value &a or the first Fourier components of the potential. In this way we have found for: LiF(OO1):

/3,a=0.30A,

/3,=0.11

NaCl(OO1):

&)a ‘v 0.34 A )

p(-)= 0.085 ;

cu(001):

&,a

< 0.025-0.03

a ,

;

fle = 0.01 :

(or dense faces of metals) W(112) [ liO] :

&a

= 0.27 a ,

PO10.06 .

During an experiment with dense faces of metals, the crystal temperature is often equal or greater than room temperature in order to avoid surface contamination . The corrugation is low. Therefore, only the in-plane 10 and i0 beams can be detected, but their intensities seem to be below the experimental sensitivity. So, to date, no diffracted peaks have been observed in a scattering experiment from these clean faces, On the contrary, the hard wall corrugation of W(l12) face in the [ liO] direction seems to be PON 0.06 and for LiF(OO1) PON 0.12. Consequently, diffracted beams are observed with this metallic surface even at high temperature. These important differences between materials could be understood qualitatively if one considers that the surface potential is due to the interaction of incident atom electrons with the surface electronic structure. Particularly, the short range repulsive part is, roughly speaking, dependent on the distance between the helium electron shell and the surface electronic charges. The electrons are probably strongly localized in the vicinity of ion cores on alkali halide surfaces and partly delocalized on metalhc surfaces. So, for surfaces of the same density such as (001) faces of lithium fluoride and Copper, the potential corrugation is certainly more important in the former than in the latter. On a tungsten surface, the electron delocalization can also occur. In the [liO] direction of a (112) face, however, this delocalization is certainly lower than on the copper (100) face, as the distance between nearest neighbor atoms is larger (W 4.5 A - Cu 2.55 a). This greater distance produces a larger potential corrugation. Along the [I lr] direction, the nearest neigbor distance (2.7 A) is equivalent to that of copper atom on a (100) face. In this direction, the potential corrugation is certainly low, reducing the possibility of multiple scattering effects which are, on the contrary, not hindered on the LiF surface. Therefore, from the point of view of diffraction, the behavior of W(112) face must be intermediate between the behavior of dense metallic and alkali halide surfaces. On the latter, particularly with LiF, it is well known that numerous in-plane and out-of-plane beams have been observed. Assuming a better experimental sensitivity, in-plane 10 and i0 beams can be detected on dense metallic surfaces. On W(112) face, we expect that not only these in-plane beams, but also out-of-plane 11 and ii beams can be observed especially when the incident plane contains the [ 1 Ii] direction.

401

Acknowledgements We are indebted to Professors N. Cabrera and J. Solana for many discussions. We would also like to thank Drs. D.A. Degras and C. Manus for discussions and critical reading of the manuscript.

References [l] D.R. O’Keefe, J.N. Smith Jr., R.L. Palmer and H. Saltsburg, J. Chem. Phys. 52 (1970) 4447; Surface Sci. 20 (1970) 27. [2] B.R. Williams, J. Chem. Phys. 55 (1971) 3220. [3] J.R. Bledsoe and S.S. Fisher, Surface Sci. 46 (1974) 129. [4] G. Boato, P. Cantini and L. Mattera, Surface Sci. 55 (1976) 141. [5] R.B. Subbarao and D.R. Miller, J. Chem. Phys. 10 (1969) 4679. [6] J.N. Smith, Jr., H. Saltsburg and R.L. Palmer, in: 6th Rarefield Gas Dynamics (Academic Press, New York, (1969) p. 1141. [7] S. Yamamoto and R.E. Stickney, J. Chem. Phys. 53 (1970) 1594. [S] A.G. Stall, D.L. Smith and R.P. Merrill, J. Chem. Phys. 54 (1971) 163. [9] G. Armand, J. Lapujoulade and Y. Lejay, J. Phys. Lettres 37 (1976) L187; Surface Sci. 63 (1977) 143. [lo] D.V. Tendulkar and R.E. Stickney, Surface Sci. 27 (1971) 516. [ll] A.G. Stall and R.P. Merril, Surface Sci. 40 (1973) 405. 1121 J.L. Beeby, J. Phys. C4 (1971) L359. [13] V. Garibaldi, A.C. Levi, R. Spadacini and G.E. Tommei, Surface Sci. 48 (1975) 649. [14] N. Garcia, J. Ib&ez, J. Solana and N. Cabrera, Solid State Commun. 20 (1976) 1159; Surface Sci. 60 (1976) 385. [ 151H. Chow and E.D. Thompson, Surface Sci. 54 (1976) 269; 59 (1976) 225. [ 161 N. Garcia, Phys. Rev. Letters 37 (1976) 912; J. Chem. Phys. (15 July 1977). [ 171 J.W. Strutt (Lord Rayleigh), Proc. Roy. Sot. (London) A79 (1907) 399. [ 181 N. Cabrera and N. Carcii, Topical Invited Paper to the 7th Intern. Vacuum Congr. and 3rd Intern. Conf. on Solid Surfaces, Vienna, 1977, to be published.