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He scattering from Rh(110)
Surface Science 282 (1993) 273-278 North-Holland
He scattering from Rh( 110) A.F. Bellman I, D. Cvetko 2, A. Morgante 3, M. Polli, F. Tommasini Laboratorio INFM-TAX,
4
Padriciano 99, 34012 Trieste, Itaiy
V.R. Dhanak, A. Lausi, K.C. Prince, R. Rosei 3 Sincrotrone Trieste SpcA, Padriciano 99, 34012 Trieste, Italy
P. Cortona 5 and M.G. Dondi 6 Unit; INFM and Centro FSBT/CNR
via Dodecaneso 33, 16146 Genova, Italy
Received 1 September 1992; accepted for publication 3 November 1992
High resohttion He diffraction and bound state resonance measurements carried out with energy selection of the scattered beam from a Rh(ll0) surface are reported and analyzed. As for Ag(llO), Cu(ll0) and Au(llO1, the lateral average of the electron density of Rh(ll0) is found to be in close agreement with that obtained by the su~~ition of atomic electron densities calculated within the degeneracy-dependent self-interaction corrected (D-SIC) local density approximation (LDA). The spread of the atomic electron densities in the surface plane exhibits behaviour similar to that observed for noble metals. The He-surface dispersion coefficient C is found experimentally to be 280 f 20 meV K.
1. Introduction The technique of He beam scattering provides quite accurate information about the surface electron density [l] and has recently been developed to a high level of accuracy by ~provements in both the e~e~rnent~ apparatus [Zl and the methods for the analysis of the experimental data [3,4]. For the surfaces of noble metals, He scatterAlso at: Dipartimento di Fisica dell’Universitl di Milano, Milano, Italy. Permanent address: Institute .I. Stefan, Ljubljana, Slovenia; present address: Sincrotrone Trieste SpcA, Padriciano 99, 34012 Trieste, Italy. Also at: Dipartimento di Fisica dell’Universitk di Trieste, Trieste, Italy. Also at: Dipartimento di Fisica deIl’Universitlt di Trento, Trento, Italy. Also at: Dipa~~ento di Fisica del~unive~it~ di Genova, Genova, Italy. Also at: Istituto di Fisica di fngegneria dell’Universit~ di Genova, Genova, Italy. 0039-6028/93/$06.00
ing yields electron densities in close agreement with the superposition of atomic electron densities calculated within the framework of the degeneracy-dependence self-interaction corrected (D-SIC) local density appro~mation (LDA) [S]. The densities seem to be spread to some extent in the surface plane, smoothing the surface corrugation and leaving the laterally averaged electron density unchanged [6,3,4]. Nevertheless, it is still not clear whether this reflects a real spread or arises from limited resolution of the He probe in sampling the surface electron density [7-91. The fact that the effect is also observed for the Au(llOX1 x 2) surface in the direction perpendicular to Jhe rows of close-packed atoms which are 8.15 A apart, supports the idea of a real spread, but further investigations may be necessary in order to reach a firm conclusion. In particular the study, within this scheme, of He scattering from d electron metal surfaces or from rare gas overlayers, which are expected to have
0 1993 - Elsevier Science Publishers B.V. All rights reserved
214
A.F. Bellman et al. / He scattering from Rh(llO)
more localized electron clouds than noble metals, should give further insight into the subject. In this paper, He diffraction patterns and bound state resonance measurements from the Rh(l10) surface are presented. With respect to previous measurements [lo], the present data show some new features and allow a quite accurate determination of the interaction potential and of the surface electron density. The experimental details and results are reported in section 2, while the analysis of the data is carried out in section 3 where the scattered intensities are compared to the scattering probabilities as calculated by the close coupled channel (CCC) method [ 111. The calculations are carried out for interaction potentials described as superpositions of pseudo-pairwise terms [3]. Two models are considered for the pseudo-pairwise potential, both based on the effective medium approximation [12,13] and on DSIC calculations of the electron density of Rh atoms 151.Model 1 was recently shown to be quite accurate for Ag and Cu surfaces [3] as well as for the missing row Au(llOX1 X 2) surface [4]. Model 2 is equivalent to a Tang and Toennies potential [14] and provides a more flexible account of the dispersion energy [15]. A discussion is given in section 4.
2. Experimental The He scattering apparatus, fully described in ref. 121,is designed for the study of surface vibrations by He energy loss spectroscopy. The angle of deflection at the surface is fixed at 110” and the time of flight (TOF) distributions of the scattered intensities are measured for different orientations of the crystal surface with respect to the beam direction. In the present study the TOF detection is used as an energy filter with resolution set at +0.3 and f2 meV for beam energies of 18.62 and 66.6 meV, respectively. The incidence and exit angles Oi and 0, are calculated as Oi = 55” -A and 0, = 55” + A where A is measured in the scattering plane with respect to the specular direction. The angular accuracy is close to O.Ol”, the angular divergence of the beam is 0.135” FWHM, while the FWHM velocity spread
4.0
4.5
5.0
5.5
6.0
6.5
7.0
kin (A-‘) Fig. 1. TOF spectrum measured at fiied angle of incidence Oi = 55” and at three different azimuthal angles y for a beam with large velocity spread. The x-axis is incident wavevector. Upper panel: y = 20”; middle panel: y = 48”; lower panel: y = 55”. The arrows indicate bound state resonance features involving the He-surface vibrational level E, and the (01) reciprocal lattice vector.
of the beam is 1.2% and 3% of the beam velocity with the source at liquid nitrogen and at room temperature, respectively. The crystal surface was prepared by sputtering, annealing and oxidation and hydrogen reduction cycles at 1000 K iterated until the specular and diffracted peaks showed angular widths at the limits of the experimental resolution. Bound state resonance features have been observed in the specular intensity as a function of the azimuthal angle, but, as first noted by Semerad et al. [16], more precise determinations of the bound state levels are obtained by the measurements shown in fig. 1. In this case the velocity spread of the beam is deliberately increased by reducing the pressure of the source and the TOF distribution of the specular intensity is measured with both the incidence angle and the azimuthal angle fixed. The azimuthal angle y is measured from the IYX direction. The times of arrival at which dips in the TOF spectra are detected correspond to the beam wavevectors entering the bound state resonances and yield the He-surface vibrational energy levels. The reciprocal lattice vector involved is the (01) in all cases. The analy-
A.F. Bellman et al. / He scattering fmm Rh@lO)
27.5
Table 1 Experimental and calculated bound state resonances (meV) for the He-RhtllO) system, and parameters for the potentials for the two models; units are: crN (meV), @ &-‘I, C (meV &I, Q CL> v
0 1 2 3 4
talc Ev
pP
Y
100
a)
b1
c)
d)
6.05 rir0.10 3.02&0.11 1.29f0.13 0.30 f0.15 -
3.20f0.02 1.47*0.02 0.55 iO.02 O.fS&O.O;!
6.125 3.212 1.448 0.551 0.176
6.102 3.223 1.451 0.550 0.171
ff~=8~
-
p = 2.57 C=284 u = 3.81
x@= =: 75 2.57 C = 274 D =c3.71
a) After ref. [lo]. b, Present work. “) Model 1. d, Model 2.
sisof several
patterns such as those shown in fig, 1, measured at di~~rent ~~uthal angles, gives the bound state energy levels reported in table 1. The present levels are deeper than those reported by Parschau et al. [IO] by about 0.2 meV, One more weakly bound level is detected but, due to the geometry of our scattering apparatus, the fundamental level detected by Parschau et al. cannot be observed. Diffraction patterns taken with beam energies 18.62 and 66.6 meV and surface temperature
A (deg.1 Fig. 2. Di$raction patterns measured with beam wavevector k = 5.972 A- ’ and surface temperature T, = 250°C along the [liO] azimuth (T-XI (upper panel) and along the [WI] azimuth (f-Y) (lower panel). The angle d is related to the incidence and exit angles as A = (ddi - @,1/Z. The ( f 1, 0) and (0, f 1) peaks are ma~i~ed relative to (0, 0) by factors 20 and 2 respectiveiy.
-20
x50
x2
-10
0 A (deg.)
10
20
Fig. 3. I3if$action patterns measured with beam waveveetor k = If.29 A-’ and surface temperature T, = 250°C along the [liOf azimuth (T-X_I (upper panel) and along the [Ml] a~imuth W-Y) (lower panel). The angle A is related to the incidence and exit angles as A = (ei - &j/2. The (+ 1, 01, the (0, f 11, the (0, f 2) and the (0, rl:3) peaks are magnified relative to (0, 0) by factors 20, 2, 50 and 100, respectively,
T, = 25O”C, are shown in fig. 2 and in fig. 3. The upper and the lower panels refer to the PX and PY directions respectively. The e~e~menta~ly measured intensities show some asymmetry, white the di~raction probabilities are s~et~c with respect to the specular position; this is consistent with a Monte Carlo simulation of the experiment carried out taking account of the dimensions of the collimators defining the beam and the detector, and of the velocity spread of the beam [4]. The simulation shows that the average of the integrated intensities of the symmetrically equivalent peaks is a measure of the diffraction probability in a given channel. The ratios of the diffraction probabilities, for a given (m,n) peak, to the probabil~~ of specular reflection derived from the data shown in figs. 2 and 3 and corrected for Debye-Waffer attenuation, are given in table 2 as Pz$ It is noted that the pe~end~cular momentum exchange is very similar for all peaks in the same pattern due to the scattering geometry with fixed total deflection, therefore the inaccuracies in the Debye-Wailer correction reflect in errors on the ratios between the peak intensities smaller than 2%.
3. Analysis As usual, the coordinate z is defined along the surface normal with the origin at the topmost
A.F. Bellman et al. / He scattering from Rh(l
276
Table 2 Experimental and calculated He diffraction probability from Rh(llO) relative to (m, n) peaks reported in figs. 2 and 3 with Pu(, = 1 (upper panel: energy of incident He beam E = 18.62 meV, lower panel: energy of incident He beam E = 66.5 meV; the best-fit choices for the parameters nr and r~r corresponding to both models 1 and 2 are reported in table 3) pW ,tl,n
pcalc m,n
a)
b)
c)
(0, 0) (LO) (0, 1) to, 2)
1.0 0.005 0.123 0.0011
1.0 0.0058 0.102 0.0032
1.0 0.0058 0.098 0.0030
(0, 0) (LO) (2,O) (0, 1) to, 2) to, 3)
1.0 0.0049 0.103 0.0037 0.0001
1.0 0.0042 0.~~8 0.126 0.0041 0.000078
1.0 0.0043 0.~ 0.123 0.0039 0.000074
ar Expe~mentai intensities of the peaks reported in figs. 2 and 3, normalized at the intensity of the specular peak. b, Normalized diffraction probabilities calculated with model 1 with the parameters A, /3, C and cr set as in table 1, column c. ‘) Normalized diffraction probabilities calculated with model 2 with the parameters x, p, C and CTset as in table 1, column d.
nuclear plane, while x is defined along the rows of close packed atoms, i.e. along TX. The lattice constants of Rh(ll0) are taken as 2.690 and 3.804 aiong IX and IY, respectively. The He-surface interaction potential is described as a superposition of pseudo-pairwise anisotropic terms u(r) = u(r) + w(r) where the short range contribution u(r) describes the entire pairwise potential in the region of internuclear separations including the repulsive wall and the well bottom, while w(r), negligibIe in this range, accounts for only the long range tails of the dispersive energy. In this way the summation of u(r) fully determines the periodic Fourier components of the atom-surface potential while the summation of w(r) contributes only to the laterally averaged potential [3]. Two choices for u(r) are considered, model 1 and model 2. Model 1, found to be quite satisfactory for Ag(ll0) and Cu(ll0) [3] as we11 as for Au(llOX1 x 2) 141,is ui( r) = ~~v~arN e--Pp( 1 - e-0.36p(p-uT)),
(1)
10)
where 17 and ny are anisotropy parameters, (Y= 30 eV 2 is the Esbjerg and Norskov constant [12] as determined by Manninen et al. [13] and p=
ZZ+(r/&2+(7)yY)2.
(2)
As for noble metals [3,4] the values of N and /3 are set by starting from D-SIC calculations of the electron densityOof the free met?1 atom. The values N = 2.67 AP3 and p = 2.57 A-‘, providing the fit shown in fig. 4 to the D-SIC electron density of Rh, are chosen. Note that @ is the only free parameter entering the lateral average of the superposition of terms ui(r), The tails of the dispersive energy neglected by u,(r) yield a contribution W,(z) to the laterally averaged potential, given by [61 W,(z) =c
fz-d/2)3+IT” i
xe~p[ _f( (z-~2))‘]i-‘, (3) where the atom-surface dispersive coefficient C, known for nobIe metals [17] is a further free parameter and d is the interlayer separation.
1
2
3
4
5
6
7
8
Fig. 4. Electron density of the Rh atom as a unction of the distance from the nucleus, given by D-SIC calculations (dots). The full line is given Fy the formula N$xp(- /3r) with N = 2.67 A-s and p = 2.57 A- t.
A.F. Belhan
et ai. / He scattering from Rh(ll0)
Model 2 is based on a power series expansion of the pairwise potential recently reported [U] and shown to be equivalent to a Tang and Toennies potential 1141.The short range term is given by
(4) where 0 is the atomic volume in the solid, p is given by eq. (2) and the coefficients a& are given by ai=
-1 f a2 = -x2
271
2, respectively. These values vary slightly with respect to those given by model 1. We conclude that the two models are substanti~ly equivalent if x and /? are fixed at the atomic values. A few different choices for x and p with x ranging between ;t5 and 75 and p ranging between 2.25 and 2.9 A-* have been considered in order to evaluate upper and lower bounds for the atomsurface dispersion coefficient C. The result is C = 280 + 20 meV A3. It is also noted that the anisotropy parameters are very similar in all cases considered.
a3 = eBrrj3(x -t ep0’3)o (51
As z&j extends to longer range then z+(t), the tails wz(r) are substantially smaller and yield a smaller contribution to the laterally averaged potential, given by [151
Ii * 11 -l
(6)
For both models the faterally averaged potential does not depend on the anisotropy parameters vx and q,. Therefore the other free parameters may be determined by fitting the measured bound state levels to the eigenvalues of the Schrodinger equation for the He atom in the laterally averaged potential. As shown in table 1, model 1 fits satisfactorily the bound state levels for (r = 3.81 A and C = 285 meV ;i”. For these values of the parameters, CCC calculations of the diffraction probabilities yield results in agreement with the experimental data as shown in table 2, for values of the anisotropy parameters nx = 0.84 and n,, = 0.75. For model 2, the bound state levels cannot determine the four parameters C, J3, x and (+. Therefore we make the choice x = 75 which is quite accurate for atomic pairs involving at least one open shell atom [15] and set p at the atomic value. In this case good fits of the energy levels and of the diffraction patterns are obtained for C = 280 meV A3, u = 3.74 A, n, = 0.83 and qY = 0.74, as shown in table 1 and table
4. Discussion and conclusion The fact that the superposition of D-SIC atomic electron densities is consistent with the He scattering data for Rh(ll0) as well as for Cu, Ag [3] and Au surfaces [41 may seem surprising when considering that the electrons at the Fermi surface of Rh have d rather than sp character. One would expect less spread of the atomic electron densities in the surface plane and hence values of the anisotropy parameters nx and nY closer to unity. From table 3, is appears that this is possibly the case along the rows of close packed atoms, in the fact 77, is substantially larger for Rh than for Ag in spite of the fa$ that the atoms are closer together in Rh (2.69 A apart) than in Ag (2.88 A). On the other hand, in the direction perpendicular to the rows, no substantial differences between Rh and noble metals are obTable 3 Anisotropy parameiers r),,Y and peak to peak corrug?tions i r,y of the 10m3 Ae3 electron density contour, in A, for He-noble metals (110) surfaces and for He-Rh (110) system
0.71 0.79
% 0.70 0.78 0.79
5.x
Cu a) Ag a) Au b,
0.014 0.03
G 0.06 0.17 1.15
Rh ‘) Rh d,
0.84 0.83
0.75 0.74
0.025 0.03
0.085 0.09
%
a) b, ‘) d,
Ref. Ref. This This
[3]. 141. work: model 1. work: model 2.
278
A.F. Bellman et al. / He scattering from Rh(ll0)
served. Note also that the fact that nY is substantially smaller than 7, strongly suggests that a spread really occurs. In conclusion the superposition of D-SIC atomic densities has been shown to yield a quite good approximation to the surface electron density of Rh(llO), provided that the atomic densities are properly spread in the surface plane. Moreover the dispersion coefficient C for the He-Rh(ll0) system has been obtained from the experiments as C = 280 f 20 meV A3. References [ll T. Engel and K.-H. Rieder, in: Structural Studies of Surfaces, Vol. 91 of Springer Tracts in Modern Physics (Springer, Berlin, 1982) p. 55. 121 D. Cvetko, A. Lausi, A. Morgante, F. Tommasini, K.C. Prince and M. Sastry, Meas. Sci. Technol. 3 (1992) 997. 131 P. Cortona, M.G. Dondi, A. Lausi and F. Tommasini, Surf. Sci. 276 (1992) 333.
[41 P. Cortona, M.G. Dondi, D. Cvetko, A. Lausi, A. Morgante, KC. Prince and F. Tommasini, Phys. Rev. B, in press. [51 P. Cortona, Phys. Rev. A 34 (1986) 769. I61P. Cortona, M.G. Dondi and F. Tommasini, Surf. Sci. Lett. 261 (1992) L35. 171 D.E. Eichenauer, U. Harten, J.P. Toennies and V. Celli, J. Chem. Phys. 86 (1987) 3693. if31Y. Takada and W. Kohn, Phys. Rev. B 37 (1988) 826. 191 J. Tersoff, Phys. Rev. Lett. 55 (1985) 140. 1101 G. Parschau, E. Kirster, A. Bischof and K.-H. Rieder, Phys. Rev. B 40 (1989) 6012. [ill G. Wolken, J. Chem. Phys. 58 (1973) 3047. 1121 N. Esbjerg and J.K. Norskov, Phys. Rev. Lett. 45 (1980) 807. [131 M. Manninen, J.K. Norskov, M.J. Puska and C. Umrigar, Phys. Rev. B 29 (1984) 2314. 1141 K.T. Tang and J.P. Toennies, J. Chem. Phys. 80 (1984) 3726. [151 D. Cvetko, A. Lausi, A. Morgante, F. Tommasini, P. Cortona and M.G. Dondi, J. Chem. Phys., submitted. 1161E. Semerad, P. Sequard-Base and E.M. Horl, Surf. Sci. 189/190 (1987) 975. [171 E. Zaremba and W. Kohn, Phys. Rev. B 13 (1976) 2270.
He scattering from Rh( 110) A.F. Bellman I, D. Cvetko 2, A. Morgante 3, M. Polli, F. Tommasini Laboratorio INFM-TAX,
4
Padriciano 99, 34012 Trieste, Itaiy
V.R. Dhanak, A. Lausi, K.C. Prince, R. Rosei 3 Sincrotrone Trieste SpcA, Padriciano 99, 34012 Trieste, Italy
P. Cortona 5 and M.G. Dondi 6 Unit; INFM and Centro FSBT/CNR
via Dodecaneso 33, 16146 Genova, Italy
Received 1 September 1992; accepted for publication 3 November 1992
High resohttion He diffraction and bound state resonance measurements carried out with energy selection of the scattered beam from a Rh(ll0) surface are reported and analyzed. As for Ag(llO), Cu(ll0) and Au(llO1, the lateral average of the electron density of Rh(ll0) is found to be in close agreement with that obtained by the su~~ition of atomic electron densities calculated within the degeneracy-dependent self-interaction corrected (D-SIC) local density approximation (LDA). The spread of the atomic electron densities in the surface plane exhibits behaviour similar to that observed for noble metals. The He-surface dispersion coefficient C is found experimentally to be 280 f 20 meV K.
1. Introduction The technique of He beam scattering provides quite accurate information about the surface electron density [l] and has recently been developed to a high level of accuracy by ~provements in both the e~e~rnent~ apparatus [Zl and the methods for the analysis of the experimental data [3,4]. For the surfaces of noble metals, He scatterAlso at: Dipartimento di Fisica dell’Universitl di Milano, Milano, Italy. Permanent address: Institute .I. Stefan, Ljubljana, Slovenia; present address: Sincrotrone Trieste SpcA, Padriciano 99, 34012 Trieste, Italy. Also at: Dipartimento di Fisica dell’Universitk di Trieste, Trieste, Italy. Also at: Dipartimento di Fisica deIl’Universitlt di Trento, Trento, Italy. Also at: Dipa~~ento di Fisica del~unive~it~ di Genova, Genova, Italy. Also at: Istituto di Fisica di fngegneria dell’Universit~ di Genova, Genova, Italy. 0039-6028/93/$06.00
ing yields electron densities in close agreement with the superposition of atomic electron densities calculated within the framework of the degeneracy-dependence self-interaction corrected (D-SIC) local density appro~mation (LDA) [S]. The densities seem to be spread to some extent in the surface plane, smoothing the surface corrugation and leaving the laterally averaged electron density unchanged [6,3,4]. Nevertheless, it is still not clear whether this reflects a real spread or arises from limited resolution of the He probe in sampling the surface electron density [7-91. The fact that the effect is also observed for the Au(llOX1 x 2) surface in the direction perpendicular to Jhe rows of close-packed atoms which are 8.15 A apart, supports the idea of a real spread, but further investigations may be necessary in order to reach a firm conclusion. In particular the study, within this scheme, of He scattering from d electron metal surfaces or from rare gas overlayers, which are expected to have
0 1993 - Elsevier Science Publishers B.V. All rights reserved
214
A.F. Bellman et al. / He scattering from Rh(llO)
more localized electron clouds than noble metals, should give further insight into the subject. In this paper, He diffraction patterns and bound state resonance measurements from the Rh(l10) surface are presented. With respect to previous measurements [lo], the present data show some new features and allow a quite accurate determination of the interaction potential and of the surface electron density. The experimental details and results are reported in section 2, while the analysis of the data is carried out in section 3 where the scattered intensities are compared to the scattering probabilities as calculated by the close coupled channel (CCC) method [ 111. The calculations are carried out for interaction potentials described as superpositions of pseudo-pairwise terms [3]. Two models are considered for the pseudo-pairwise potential, both based on the effective medium approximation [12,13] and on DSIC calculations of the electron density of Rh atoms 151.Model 1 was recently shown to be quite accurate for Ag and Cu surfaces [3] as well as for the missing row Au(llOX1 X 2) surface [4]. Model 2 is equivalent to a Tang and Toennies potential [14] and provides a more flexible account of the dispersion energy [15]. A discussion is given in section 4.
2. Experimental The He scattering apparatus, fully described in ref. 121,is designed for the study of surface vibrations by He energy loss spectroscopy. The angle of deflection at the surface is fixed at 110” and the time of flight (TOF) distributions of the scattered intensities are measured for different orientations of the crystal surface with respect to the beam direction. In the present study the TOF detection is used as an energy filter with resolution set at +0.3 and f2 meV for beam energies of 18.62 and 66.6 meV, respectively. The incidence and exit angles Oi and 0, are calculated as Oi = 55” -A and 0, = 55” + A where A is measured in the scattering plane with respect to the specular direction. The angular accuracy is close to O.Ol”, the angular divergence of the beam is 0.135” FWHM, while the FWHM velocity spread
4.0
4.5
5.0
5.5
6.0
6.5
7.0
kin (A-‘) Fig. 1. TOF spectrum measured at fiied angle of incidence Oi = 55” and at three different azimuthal angles y for a beam with large velocity spread. The x-axis is incident wavevector. Upper panel: y = 20”; middle panel: y = 48”; lower panel: y = 55”. The arrows indicate bound state resonance features involving the He-surface vibrational level E, and the (01) reciprocal lattice vector.
of the beam is 1.2% and 3% of the beam velocity with the source at liquid nitrogen and at room temperature, respectively. The crystal surface was prepared by sputtering, annealing and oxidation and hydrogen reduction cycles at 1000 K iterated until the specular and diffracted peaks showed angular widths at the limits of the experimental resolution. Bound state resonance features have been observed in the specular intensity as a function of the azimuthal angle, but, as first noted by Semerad et al. [16], more precise determinations of the bound state levels are obtained by the measurements shown in fig. 1. In this case the velocity spread of the beam is deliberately increased by reducing the pressure of the source and the TOF distribution of the specular intensity is measured with both the incidence angle and the azimuthal angle fixed. The azimuthal angle y is measured from the IYX direction. The times of arrival at which dips in the TOF spectra are detected correspond to the beam wavevectors entering the bound state resonances and yield the He-surface vibrational energy levels. The reciprocal lattice vector involved is the (01) in all cases. The analy-
A.F. Bellman et al. / He scattering fmm Rh@lO)
27.5
Table 1 Experimental and calculated bound state resonances (meV) for the He-RhtllO) system, and parameters for the potentials for the two models; units are: crN (meV), @ &-‘I, C (meV &I, Q CL> v
0 1 2 3 4
talc Ev
pP
Y
100
a)
b1
c)
d)
6.05 rir0.10 3.02&0.11 1.29f0.13 0.30 f0.15 -
3.20f0.02 1.47*0.02 0.55 iO.02 O.fS&O.O;!
6.125 3.212 1.448 0.551 0.176
6.102 3.223 1.451 0.550 0.171
ff~=8~
-
p = 2.57 C=284 u = 3.81
x@= =: 75 2.57 C = 274 D =c3.71
a) After ref. [lo]. b, Present work. “) Model 1. d, Model 2.
sisof several
patterns such as those shown in fig, 1, measured at di~~rent ~~uthal angles, gives the bound state energy levels reported in table 1. The present levels are deeper than those reported by Parschau et al. [IO] by about 0.2 meV, One more weakly bound level is detected but, due to the geometry of our scattering apparatus, the fundamental level detected by Parschau et al. cannot be observed. Diffraction patterns taken with beam energies 18.62 and 66.6 meV and surface temperature
A (deg.1 Fig. 2. Di$raction patterns measured with beam wavevector k = 5.972 A- ’ and surface temperature T, = 250°C along the [liO] azimuth (T-XI (upper panel) and along the [WI] azimuth (f-Y) (lower panel). The angle d is related to the incidence and exit angles as A = (ddi - @,1/Z. The ( f 1, 0) and (0, f 1) peaks are ma~i~ed relative to (0, 0) by factors 20 and 2 respectiveiy.
-20
x50
x2
-10
0 A (deg.)
10
20
Fig. 3. I3if$action patterns measured with beam waveveetor k = If.29 A-’ and surface temperature T, = 250°C along the [liOf azimuth (T-X_I (upper panel) and along the [Ml] a~imuth W-Y) (lower panel). The angle A is related to the incidence and exit angles as A = (ei - &j/2. The (+ 1, 01, the (0, f 11, the (0, f 2) and the (0, rl:3) peaks are magnified relative to (0, 0) by factors 20, 2, 50 and 100, respectively,
T, = 25O”C, are shown in fig. 2 and in fig. 3. The upper and the lower panels refer to the PX and PY directions respectively. The e~e~menta~ly measured intensities show some asymmetry, white the di~raction probabilities are s~et~c with respect to the specular position; this is consistent with a Monte Carlo simulation of the experiment carried out taking account of the dimensions of the collimators defining the beam and the detector, and of the velocity spread of the beam [4]. The simulation shows that the average of the integrated intensities of the symmetrically equivalent peaks is a measure of the diffraction probability in a given channel. The ratios of the diffraction probabilities, for a given (m,n) peak, to the probabil~~ of specular reflection derived from the data shown in figs. 2 and 3 and corrected for Debye-Waffer attenuation, are given in table 2 as Pz$ It is noted that the pe~end~cular momentum exchange is very similar for all peaks in the same pattern due to the scattering geometry with fixed total deflection, therefore the inaccuracies in the Debye-Wailer correction reflect in errors on the ratios between the peak intensities smaller than 2%.
3. Analysis As usual, the coordinate z is defined along the surface normal with the origin at the topmost
A.F. Bellman et al. / He scattering from Rh(l
276
Table 2 Experimental and calculated He diffraction probability from Rh(llO) relative to (m, n) peaks reported in figs. 2 and 3 with Pu(, = 1 (upper panel: energy of incident He beam E = 18.62 meV, lower panel: energy of incident He beam E = 66.5 meV; the best-fit choices for the parameters nr and r~r corresponding to both models 1 and 2 are reported in table 3) pW ,tl,n
pcalc m,n
a)
b)
c)
(0, 0) (LO) (0, 1) to, 2)
1.0 0.005 0.123 0.0011
1.0 0.0058 0.102 0.0032
1.0 0.0058 0.098 0.0030
(0, 0) (LO) (2,O) (0, 1) to, 2) to, 3)
1.0 0.0049 0.103 0.0037 0.0001
1.0 0.0042 0.~~8 0.126 0.0041 0.000078
1.0 0.0043 0.~ 0.123 0.0039 0.000074
ar Expe~mentai intensities of the peaks reported in figs. 2 and 3, normalized at the intensity of the specular peak. b, Normalized diffraction probabilities calculated with model 1 with the parameters A, /3, C and cr set as in table 1, column c. ‘) Normalized diffraction probabilities calculated with model 2 with the parameters x, p, C and CTset as in table 1, column d.
nuclear plane, while x is defined along the rows of close packed atoms, i.e. along TX. The lattice constants of Rh(ll0) are taken as 2.690 and 3.804 aiong IX and IY, respectively. The He-surface interaction potential is described as a superposition of pseudo-pairwise anisotropic terms u(r) = u(r) + w(r) where the short range contribution u(r) describes the entire pairwise potential in the region of internuclear separations including the repulsive wall and the well bottom, while w(r), negligibIe in this range, accounts for only the long range tails of the dispersive energy. In this way the summation of u(r) fully determines the periodic Fourier components of the atom-surface potential while the summation of w(r) contributes only to the laterally averaged potential [3]. Two choices for u(r) are considered, model 1 and model 2. Model 1, found to be quite satisfactory for Ag(ll0) and Cu(ll0) [3] as we11 as for Au(llOX1 x 2) 141,is ui( r) = ~~v~arN e--Pp( 1 - e-0.36p(p-uT)),
(1)
10)
where 17 and ny are anisotropy parameters, (Y= 30 eV 2 is the Esbjerg and Norskov constant [12] as determined by Manninen et al. [13] and p=
ZZ+(r/&2+(7)yY)2.
(2)
As for noble metals [3,4] the values of N and /3 are set by starting from D-SIC calculations of the electron densityOof the free met?1 atom. The values N = 2.67 AP3 and p = 2.57 A-‘, providing the fit shown in fig. 4 to the D-SIC electron density of Rh, are chosen. Note that @ is the only free parameter entering the lateral average of the superposition of terms ui(r), The tails of the dispersive energy neglected by u,(r) yield a contribution W,(z) to the laterally averaged potential, given by [61 W,(z) =c
fz-d/2)3+IT” i
xe~p[ _f( (z-~2))‘]i-‘, (3) where the atom-surface dispersive coefficient C, known for nobIe metals [17] is a further free parameter and d is the interlayer separation.
1
2
3
4
5
6
7
8
Fig. 4. Electron density of the Rh atom as a unction of the distance from the nucleus, given by D-SIC calculations (dots). The full line is given Fy the formula N$xp(- /3r) with N = 2.67 A-s and p = 2.57 A- t.
A.F. Belhan
et ai. / He scattering from Rh(ll0)
Model 2 is based on a power series expansion of the pairwise potential recently reported [U] and shown to be equivalent to a Tang and Toennies potential 1141.The short range term is given by
(4) where 0 is the atomic volume in the solid, p is given by eq. (2) and the coefficients a& are given by ai=
-1 f a2 = -x2
271
2, respectively. These values vary slightly with respect to those given by model 1. We conclude that the two models are substanti~ly equivalent if x and /? are fixed at the atomic values. A few different choices for x and p with x ranging between ;t5 and 75 and p ranging between 2.25 and 2.9 A-* have been considered in order to evaluate upper and lower bounds for the atomsurface dispersion coefficient C. The result is C = 280 + 20 meV A3. It is also noted that the anisotropy parameters are very similar in all cases considered.
a3 = eBrrj3(x -t ep0’3)o (51
As z&j extends to longer range then z+(t), the tails wz(r) are substantially smaller and yield a smaller contribution to the laterally averaged potential, given by [151
Ii * 11 -l
(6)
For both models the faterally averaged potential does not depend on the anisotropy parameters vx and q,. Therefore the other free parameters may be determined by fitting the measured bound state levels to the eigenvalues of the Schrodinger equation for the He atom in the laterally averaged potential. As shown in table 1, model 1 fits satisfactorily the bound state levels for (r = 3.81 A and C = 285 meV ;i”. For these values of the parameters, CCC calculations of the diffraction probabilities yield results in agreement with the experimental data as shown in table 2, for values of the anisotropy parameters nx = 0.84 and n,, = 0.75. For model 2, the bound state levels cannot determine the four parameters C, J3, x and (+. Therefore we make the choice x = 75 which is quite accurate for atomic pairs involving at least one open shell atom [15] and set p at the atomic value. In this case good fits of the energy levels and of the diffraction patterns are obtained for C = 280 meV A3, u = 3.74 A, n, = 0.83 and qY = 0.74, as shown in table 1 and table
4. Discussion and conclusion The fact that the superposition of D-SIC atomic electron densities is consistent with the He scattering data for Rh(ll0) as well as for Cu, Ag [3] and Au surfaces [41 may seem surprising when considering that the electrons at the Fermi surface of Rh have d rather than sp character. One would expect less spread of the atomic electron densities in the surface plane and hence values of the anisotropy parameters nx and nY closer to unity. From table 3, is appears that this is possibly the case along the rows of close packed atoms, in the fact 77, is substantially larger for Rh than for Ag in spite of the fa$ that the atoms are closer together in Rh (2.69 A apart) than in Ag (2.88 A). On the other hand, in the direction perpendicular to the rows, no substantial differences between Rh and noble metals are obTable 3 Anisotropy parameiers r),,Y and peak to peak corrug?tions i r,y of the 10m3 Ae3 electron density contour, in A, for He-noble metals (110) surfaces and for He-Rh (110) system
0.71 0.79
% 0.70 0.78 0.79
5.x
Cu a) Ag a) Au b,
0.014 0.03
G 0.06 0.17 1.15
Rh ‘) Rh d,
0.84 0.83
0.75 0.74
0.025 0.03
0.085 0.09
%
a) b, ‘) d,
Ref. Ref. This This
[3]. 141. work: model 1. work: model 2.
278
A.F. Bellman et al. / He scattering from Rh(ll0)
served. Note also that the fact that nY is substantially smaller than 7, strongly suggests that a spread really occurs. In conclusion the superposition of D-SIC atomic densities has been shown to yield a quite good approximation to the surface electron density of Rh(llO), provided that the atomic densities are properly spread in the surface plane. Moreover the dispersion coefficient C for the He-Rh(ll0) system has been obtained from the experiments as C = 280 f 20 meV A3. References [ll T. Engel and K.-H. Rieder, in: Structural Studies of Surfaces, Vol. 91 of Springer Tracts in Modern Physics (Springer, Berlin, 1982) p. 55. 121 D. Cvetko, A. Lausi, A. Morgante, F. Tommasini, K.C. Prince and M. Sastry, Meas. Sci. Technol. 3 (1992) 997. 131 P. Cortona, M.G. Dondi, A. Lausi and F. Tommasini, Surf. Sci. 276 (1992) 333.
[41 P. Cortona, M.G. Dondi, D. Cvetko, A. Lausi, A. Morgante, KC. Prince and F. Tommasini, Phys. Rev. B, in press. [51 P. Cortona, Phys. Rev. A 34 (1986) 769. I61P. Cortona, M.G. Dondi and F. Tommasini, Surf. Sci. Lett. 261 (1992) L35. 171 D.E. Eichenauer, U. Harten, J.P. Toennies and V. Celli, J. Chem. Phys. 86 (1987) 3693. if31Y. Takada and W. Kohn, Phys. Rev. B 37 (1988) 826. 191 J. Tersoff, Phys. Rev. Lett. 55 (1985) 140. 1101 G. Parschau, E. Kirster, A. Bischof and K.-H. Rieder, Phys. Rev. B 40 (1989) 6012. [ill G. Wolken, J. Chem. Phys. 58 (1973) 3047. 1121 N. Esbjerg and J.K. Norskov, Phys. Rev. Lett. 45 (1980) 807. [131 M. Manninen, J.K. Norskov, M.J. Puska and C. Umrigar, Phys. Rev. B 29 (1984) 2314. 1141 K.T. Tang and J.P. Toennies, J. Chem. Phys. 80 (1984) 3726. [151 D. Cvetko, A. Lausi, A. Morgante, F. Tommasini, P. Cortona and M.G. Dondi, J. Chem. Phys., submitted. 1161E. Semerad, P. Sequard-Base and E.M. Horl, Surf. Sci. 189/190 (1987) 975. [171 E. Zaremba and W. Kohn, Phys. Rev. B 13 (1976) 2270.