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Determination of adsorbate orientation from symmetry rules in low-energy elastic electron scattering
Surface Science 217 (1989) L442-L450 North-Holland, Amsterdam
L442
SURFACE
SCIENCE
LETTERS
DE~~NA~ON OF ADSO~A~ O~ENTA~ON FROM SYMMETRY RULZS IN LOW-ENERGY ELASTIC ELEXTRON SCATIERING Carl WINSTEAD,
Vincent McKOY
A.A. Noyes Laboratory of Chemical Physics *, California Institute of Technology, Pasadena, CA 91125, USA
and Horia METIU
Received 20 .fanuary 1989; accepted for publication 12 April 1989
In this Letter we describe a particular elastic-scattering experiment which is simple to perform and which establishes a connection between the scattering pattern and the orientation of a molecule adsorbed on a solid surface. This approach is iIlustrated by examining the cases of CO and C,H,.
1. Introduction A recent series of theoretical studies [l-3] has implied that differential electron scattering cross sections can be used to determine the orientation of adsorbed molecules with respect to their substrates, concluding that such determinations are indeed possible. Experimental and theoretical results of Palmer et al. [4] for Oz physisorbed on graphite also support this conclusion, although the resonant scattering process studied there involves somewhat different considerations from the nonresonant scattering considered here and in the other studies [l-3]. In this paper we identify a particular type of elastic scattering experiment which, by exploiting symmetries of the scattering process, should be especially effective in deter~g adsorbate orientations. As we illustrate in the cases of CO and C2H4, this type of experiment may be sufficient by itself to fix the adsorbate orientation, and when it is not, it can limit the number of possibili* Contribution No. 7910.
0039-6028/S9/$03_50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division}
C. Winstead et al. / Adrorbate
orientation determined by elastic electron scattering
L443
ties that need to be considered further. A potential advantage of the proposed experiment is that it requires no relative motion of the source and detector.
2. Theory The most general scattering geometry is indicated in fig. 1. Electrons incident along the direction given by the polar coordinates (ei, pi) are scattered by the target and detected in the direction (or, Of). Because the adsorbed molecules are oriented, the cross section depends on all angles (Bi, ei, e,, I#+). Since in practice only a small set of combinations of these four angles can be explored, it is of some interest to find those combinations of angles which most readily yield the desired information about the target. Preceding studies [l-4] consider two types of experiment. In one, Bi and ei are held fixed, as is f?,, and the final azimuthal angle & is varied from O” to 360 O; in the second, ei and $i are again kept fixed, along with q+, while 8, is varied. Several such data sets are to be taken for various values of the angles held fixed. When the surface is made up of domains which lie at a variety of azimuthal angles, as was the case in the work of Palmer et al. [4], studies of the first type are less useful and the altitudinal-angle (0) dependence must be studied instead. Theoretical results obtained for a few small molecules indicate that these experiments can indeed determine the adsorbate orientation. However, there are obvious disadvantages to the procedure described. First of all, to obtain either type of data set, the electron detector must be able to move relative to the electron gun, while to obtain more than one set of either type, it must also be possible to move the gun relative to the target. Secondly, even if the apparatus allows such motion, it may be difficult to obtain reliable data in the near-backscattering and, for elastic scattering, near-specular directions, where readily identifiable maxima and minima frequently occur [l-3]. Finally, since the initial and final azimuthal angles relative to the adsorbate are not known
Fig. 1. Scattering geometry. Electrons emitted by the source S in the direction (ei, &) scatter from the surface-adsorbate target T in the direction (O,, er) toward the detector D. The z direction is taken to be normal to the surface.
L444
C. Winstead et al. / Adrorbate orientation determined by elastic electron scattering
to the experimenter, it is necessary to take multiple sets of data in order to locate those most easily compared to theory. A consideration of these drawbacks suggests the following modification. Let the altitudinal angles Bi and 8t remain fixed in such a way that 8, = 180 o - Bi. Referring to fig. 1, we see that this simply means that the source and detector point downward at the same angle. Further, let the final azimuthal angle I& be equal to oi + @, as +i is varied from 0 o to 360 o and @ is held constant. Note that this may be achieved simply by rotating the target, leaving the source and detector fixed. This type of experiment was mentioned in an earlier study [2] but was not considered in detail. As will be shown below, in many cases this can provide sufficient information to identify the adsorbate orientation. The symmetry under time reversal [5] implies that u(k;,
k,) = u( -k,,
4ri).
(la)
(Spin labels are omitted, for an unpolarized electron beam and no spin detection.) For elastic scattering the magnitudes 1ki ) and 1k, 1 are equal, so that in polar coordinates this symmetry becomes u(eiY
(Pi
*
ef9
+f
)
=
~(180~ - 13,, 180” + (it+
180“ - Bi, 180” + &).
(lb)
For the scattering geometry just proposed this implies o(~i++r=~i+@)=u(~i+@+1800++i+1800),
(lc)
where, for convenience, we do not specify the altitudinal angles Bi and 8,, which are related by 0, = 180 o - Bi. This equality shows that the time-reversed process involves scattering to the right of forward, and hence the final azimuth is decreased by Cprelative to the initial azimuth. Therefore the cross section obtained with the detector placed to the right of forward, plotted from will be identical to that obtained with the +i=@+180” to @+540°, detector placed to the left, plotted from & = 0” to 360”. Because of this relation, a single theoretical curve may be compared with either experiment. There is another, more important consequence of eq. (lc) when the adsorbate is so oriented as to possess a vertical plane of symmetry. As an illustration, consider the case of a diatomic molecule AB which is lying parallel to a flat, structureless surface; the vertical plane containing the molecular axis is then such a symmetry plane. If we call the direction from A to B 9 = 0 O, then the mirror symmetry of the elastic scattering cross section is expressed by “(Oi> +i + ‘r? (Pf) =a(ei,
-Gi + et, -&)a
It is easy to show from eqs. (lc)
(2)
and (2) that the particular experiment we propose yields a cross section which is symmetric about the angles (pi = 90” Q/2, 270” - @/2; consequently, these angles must be local maxima or
C. Winstead et al. / Adwrbate
orientation determined by elastic electron scattering
L445
Fig. 2. Overhead view of scattering from adsorbed AB molecule lying along the x axis. Two trajectories are shown: for the first (a) the initial azimuth exceeds 90 o - o/2 by an angle cx and for the second (b) the initial azimuth is less than 90 o - G/2 by a. Time reversal followed by reflection through x interchanges the two trajectories, provided that the altitudinal angles satisfy 0, =180° - Bi.
minima. These relationships should make it quite easy to locate the appropriate pair of angles from the data; all that is then needed is to decide which angle is 90” - Q/2 and which is 270 o - G/2, or in terms of our example, which end of the molecule is A and which B. To make such a decision one must use calculations of the kind reported in refs. [l-3]. Before proceeding it may be useful to clarify how this symmetry arises. Fig. 2 gives an overhead view of the situation; because we have chosen an arrangement for which neither the reflection nor time reversal affects the altitudinal angles Bi and f$, we need follow only the azimuths. The combination of reflection in the vertical plane containing AB (the xz-plane) with time reversal transforms the trajectory of fig. 2a into that shown in fig. 2b; thus, we have a(900
- @/2+cu-,90°+~/2+a)=a(900-@/2-a+900+@/2-a) (3a)
and the scattering pattern will be symmetric about 90“ - Q/2. Likewise we may show that ~(270~ - @/2+a+270°+@/2+a) = ~(270~ - Q/2 - CY+ 270 o + Q/2 - CI).
(3b)
When more than one plane of symmetry is present, there will be a corresponding number of pairs of directions about which the scattering cross section is symmetric, and some confusion is possible. Here calculation of the cross section is needed to establish which plane is which. When surface scattering cannot be neglected, the same scattering experiment may still give information about the adsorbate’s orientation if some
L446
C. Winstead et al. / Adsorbate orientation determined by elastic electron scattering
symmetry is preserved. For instance, a clean (110) surface has two inequivalent mirror planes. An AB molecule lying flat in one mirror plane breaks the symmetry with respect to the other one, while an AB molecule standing up preserves both. A third situation arises when surface scattering is strong and the orientation is already known from other evidence. It may then be possible to determine the binding site. If the adsorbate is known to be standing up on a (111) surface, for example, on-top binding preserves the six mirror planes present in the clean surface. Binding at the three-fold hollow preserves three mirror planes, while the bridge site preserves only two. It should be noted with regard to the preceding example that sites such as the three-fold hollow and the bridge occur in more than one orientation, so that, depending on the long-range ordering, if any, of the adsorbate, the total scattering pattern may be an average over site orientations, thus raising the pattern’s symmetry. (We are assuming that either there is no long-range order or that the coherence length of the electron beam is small compared to the adsorbate spacing, so that the total scattering pattern is an incoherent sum of the individual scattering events.) In the (111) surface example, the three binding sites would nonetheless remain distinguishable. Before proceeding to examples, it is worth mentioning that the same symmetry considerations we have discussed apply to elastic scattering of particles other than electrons. Also, inelastic scattering experiments of this type may yield cross sections with enough structure to determine the adsorbate orientation. Though the presence of mirror planes will not be directly indicated by symmetry in an inelastic cross section, if two such cross sections can be measured, one with the detector at & + @ and one with the detector at & - @, they will be mirror images of each other about the angles at which the source lies in the mirror plane.
3. Examples The symmetry rules governing our proposed experiment are exact if we can neglect scattering by the surface or if the adsorbate’s planes of symmetry coincide with symmetry planes of the surface. Thus we give examples not to illustrate that these rules hold but to show that, in cases of interest, the features of the scattering data are clear enough that identification of the relevant angles is easy. The theoretical method used to obtain the differential scattering cross section has been described previously [l]. Briefly, the Schwinger multichannel method [6-81 is used to generate an accurate scattering amplitude for the isolated, oriented molecule. The CO amplitude was obtained from earlier work [2]; details of the C,H, calculation will be published elsewhere 191. The surface is represented by a step potential, which acts as a partially-reflecting mirror, and scattering by the molecule and the surface are treated as
C. Winstead et al. / Adrorbate
0
60
orientation determined by elastic electron scattering
1x0
180
240
300
I.447
360
4. Fig. 3. Elastic scattering of 9.9 eV electrons by CO on a surface, plotted as a function of & with & = +i + @, Bi = 135 O, and 0, = 45 “. Results are shown for CO parallel to the surface (solid line) and for CO at 45” to the surface, with C closer (dashed line). The direction from C to 0 is defined as + = O“, and the distance from the surface to the C nucleus is 1.0 A in both cases. (a) @=90”;(b) @=120’.
separate events. As indicated earlier, the obvious limitations of this model restrict our attention to those overall, qualitative features of the scattering cross section which are unlikely to be artifacts. Fig. 3a shows the differential elastic scattering cross section obtained for CO at an electron energy of 9.9 eV. The scattering arrangement is that described above, with @ chosen to be 90 O, and Bi and f?, chosen to be 135 o
L448
C. Winstead et al. / Adsorbate orientation determined by elastic electron scattering
and 45O, respectively. The solid curve in the figure results when the CO molecule lies parallel to the surface, along the x axis, with + x (+ = 0 o ) being the direction from C to 0. The predicted symmetry about & = 45 o and 225 o is quite evident. Moreover, the strong peaks in the cross section are noticeably shifted toward 225 O, away from 45 O, making it possible to determine which end of the molecule is 0 and which C, especially if the small maximum at 45 o is observable. The dashed curve in fig. 3a shows the scattering cross section for a CO molecule tilted at 45 o from the vertical in the +x direction, with C being the end closer to the surface. As expected, the scattering pattern shows the same symmetry as seen for horizontal CO, but there are marked differences in structure between the two patterns. Since a vertical CO molecule would give a completely flat scattering profile in this experiment, it should thus be possible to say whether the molecule is vertical or nearly vertical, whether it is tilted at about 45 O, or whether it is horizontal or nearly so. However, the two maxima in the dashed curve are nearly equal, making it difficult to say from experiment in which of the two possible directions the CO molecule tilts. This is an instance in which a more elaborate experiment proves necessary; for example, we could use the approach of ref. [3] and vary +r, while leaving & fixed. Results of the type shown in fig. 3a would still be valuable in identifying the approximate angle of tilt and narrowing to the two possibilities that must be distinguished in the further experiment. Fig. 3b contains CO scattering data analogous to that in fig. 3a, but for @ = 120 O, in order to illustrate the dependence of the symmetry on @. Note that the results are qualitatively the same as in fig. 3a, but that, as predicted by the symmetry rules, the cross sections are now symmetric about 30” and 210°. Fig. 4a shows the elastic scattering cross section obtained for the C,H, molecule at 10 eV, using the same Bi, 0,, and @ as in fig. 3a and assuming the molecule to lie parallel to the surface with the C-C bond along the x axis. Because there are now two vertical mirror planes, the scattering pattern is symmetric both about the pair (pi = 45 o and & = 225 O, and about the angles (pi = 135 o and 315 O. Again there is sufficient structure in the cross section that the symmetry is readily apparent. Moreover, it is clear that the maxima corresponding to & = 135 o and 315” are much stronger than those at 45 o and 225”. The scattering pattern thus clearly indicates that the C-C bond direction is $I = 0 o and not + = 90 O. While the electron scattering pattern may establish the adsorbate orientation, it may not be sufficiently sensitive to detect adsorption-induced changes in adsorbate geometry. An estimate of the effects of such distortion is provided by fig. 4b, which shows results for the same situation as in fig. 4a, but with the C,H, molecule distorted by bending the H atoms away from the surface: the C-C and C-H distances are left unchanged, but the HCH and
C. Winstead et al. / Adrorbate
orientation determined by elastic electron scattering
JA49
(b)
Fig. 4. (a) Elastic scattering of 10 eV electrons by adsorbed C,H,. The molecule plane is parallel to and 1 A above the surface, and the C-C bond is defined to be + = 0 O. Scattering angles are as in fig. 3a. (b) Scattering of 10 eV electrons by distorted C,H,. The molecular geometry is as described in the text; the scattering angles are as in fig. 3a.
HCC angles are both set at 109.5”. Comparison with fig. 4a shows that this distortion has little effect on the cross section and would therefore be undetectable by the present method. Though it is probably an advantage on the whole for the predictions of our theoretical model to be insensitive to small changes in bond lengths and angles, it is somewhat surprising that as large a distortion as is considered here has so little effect. In actual applications of
LA50
C. Winstead et al. / Adrorbate orientation determined by elastic electron scattering
electron scattering for determining adsorbate orientation, distortion must be borne in mind.
this insensitivity
to
4. Conclusions We have shown that adsorbate orientations can be determined from a particularly simple elastic electron scattering experiment. By using an arrangement that exploits basic symmetry relationships, we are able to establish the orientation of any mirror planes possessed by the scatterer from the symmetry of the cross section alone. Complete specification of the adsorbate orientation may require an additional experiment, but having first located the mirror planes should reduce the overall labor. While many adsorbate species of interest are expected to show at least one mirror plane, this approach should also work for less symmetric situations, provided the scattering pattern has recognizable strong features.
Acknowledgment This work was supported by the Army Research Office and the Air Force Office for Scientific Research under contract DAA G29-85-K-0117 through the University of California at Santa Barbara.
References [l] S. Nagano, Z.-P. Luo, H. Metiu, W.M. Huo, M.A.P. Lima and V. McKay, J. Chem. Phys. 85 (1986) 6153. [2] S. Nagano, Z.-P. Luo, H. Metiu, W.M. Huo, M.A.P. Lima and V. McKay, Surface Sci. 186 (1987) L548. [3] S. Nagano, Z.-P. Luo, H. Metiu, W.M. Huo and V. McKay, J. Chem. Phys. 88 (1988) 7970. [4] R.E. Palmer, P.J. Rous, J.L. Wilkes and R.F. Willis, Phys. Rev. Letters 60 (1988) 329. [5] R.G. Newton, Scattering Theory of Waves and Particles, 2nd ed. (Springer, New York, 1982) p. 452. [6] K. Takatsuka and V. McKay, Phys. Rev. A 30 (1984) 1734. [7] M.A.P. Lima, T.L. Gibson, K. Takatsuka and V. McKay, Phys. Rev. A 30 (1984) 1741. [8] R.R. Lucchese, K. Takatsuka and V. McKay, Phys. Rept. 131 (1986) 147. [9] C.L. Winstead, V. McKay and H. Metiu, to be published.
L442
SURFACE
SCIENCE
LETTERS
DE~~NA~ON OF ADSO~A~ O~ENTA~ON FROM SYMMETRY RULZS IN LOW-ENERGY ELASTIC ELEXTRON SCATIERING Carl WINSTEAD,
Vincent McKOY
A.A. Noyes Laboratory of Chemical Physics *, California Institute of Technology, Pasadena, CA 91125, USA
and Horia METIU
Received 20 .fanuary 1989; accepted for publication 12 April 1989
In this Letter we describe a particular elastic-scattering experiment which is simple to perform and which establishes a connection between the scattering pattern and the orientation of a molecule adsorbed on a solid surface. This approach is iIlustrated by examining the cases of CO and C,H,.
1. Introduction A recent series of theoretical studies [l-3] has implied that differential electron scattering cross sections can be used to determine the orientation of adsorbed molecules with respect to their substrates, concluding that such determinations are indeed possible. Experimental and theoretical results of Palmer et al. [4] for Oz physisorbed on graphite also support this conclusion, although the resonant scattering process studied there involves somewhat different considerations from the nonresonant scattering considered here and in the other studies [l-3]. In this paper we identify a particular type of elastic scattering experiment which, by exploiting symmetries of the scattering process, should be especially effective in deter~g adsorbate orientations. As we illustrate in the cases of CO and C2H4, this type of experiment may be sufficient by itself to fix the adsorbate orientation, and when it is not, it can limit the number of possibili* Contribution No. 7910.
0039-6028/S9/$03_50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division}
C. Winstead et al. / Adrorbate
orientation determined by elastic electron scattering
L443
ties that need to be considered further. A potential advantage of the proposed experiment is that it requires no relative motion of the source and detector.
2. Theory The most general scattering geometry is indicated in fig. 1. Electrons incident along the direction given by the polar coordinates (ei, pi) are scattered by the target and detected in the direction (or, Of). Because the adsorbed molecules are oriented, the cross section depends on all angles (Bi, ei, e,, I#+). Since in practice only a small set of combinations of these four angles can be explored, it is of some interest to find those combinations of angles which most readily yield the desired information about the target. Preceding studies [l-4] consider two types of experiment. In one, Bi and ei are held fixed, as is f?,, and the final azimuthal angle & is varied from O” to 360 O; in the second, ei and $i are again kept fixed, along with q+, while 8, is varied. Several such data sets are to be taken for various values of the angles held fixed. When the surface is made up of domains which lie at a variety of azimuthal angles, as was the case in the work of Palmer et al. [4], studies of the first type are less useful and the altitudinal-angle (0) dependence must be studied instead. Theoretical results obtained for a few small molecules indicate that these experiments can indeed determine the adsorbate orientation. However, there are obvious disadvantages to the procedure described. First of all, to obtain either type of data set, the electron detector must be able to move relative to the electron gun, while to obtain more than one set of either type, it must also be possible to move the gun relative to the target. Secondly, even if the apparatus allows such motion, it may be difficult to obtain reliable data in the near-backscattering and, for elastic scattering, near-specular directions, where readily identifiable maxima and minima frequently occur [l-3]. Finally, since the initial and final azimuthal angles relative to the adsorbate are not known
Fig. 1. Scattering geometry. Electrons emitted by the source S in the direction (ei, &) scatter from the surface-adsorbate target T in the direction (O,, er) toward the detector D. The z direction is taken to be normal to the surface.
L444
C. Winstead et al. / Adrorbate orientation determined by elastic electron scattering
to the experimenter, it is necessary to take multiple sets of data in order to locate those most easily compared to theory. A consideration of these drawbacks suggests the following modification. Let the altitudinal angles Bi and 8t remain fixed in such a way that 8, = 180 o - Bi. Referring to fig. 1, we see that this simply means that the source and detector point downward at the same angle. Further, let the final azimuthal angle I& be equal to oi + @, as +i is varied from 0 o to 360 o and @ is held constant. Note that this may be achieved simply by rotating the target, leaving the source and detector fixed. This type of experiment was mentioned in an earlier study [2] but was not considered in detail. As will be shown below, in many cases this can provide sufficient information to identify the adsorbate orientation. The symmetry under time reversal [5] implies that u(k;,
k,) = u( -k,,
4ri).
(la)
(Spin labels are omitted, for an unpolarized electron beam and no spin detection.) For elastic scattering the magnitudes 1ki ) and 1k, 1 are equal, so that in polar coordinates this symmetry becomes u(eiY
(Pi
*
ef9
+f
)
=
~(180~ - 13,, 180” + (it+
180“ - Bi, 180” + &).
(lb)
For the scattering geometry just proposed this implies o(~i++r=~i+@)=u(~i+@+1800++i+1800),
(lc)
where, for convenience, we do not specify the altitudinal angles Bi and 8,, which are related by 0, = 180 o - Bi. This equality shows that the time-reversed process involves scattering to the right of forward, and hence the final azimuth is decreased by Cprelative to the initial azimuth. Therefore the cross section obtained with the detector placed to the right of forward, plotted from will be identical to that obtained with the +i=@+180” to @+540°, detector placed to the left, plotted from & = 0” to 360”. Because of this relation, a single theoretical curve may be compared with either experiment. There is another, more important consequence of eq. (lc) when the adsorbate is so oriented as to possess a vertical plane of symmetry. As an illustration, consider the case of a diatomic molecule AB which is lying parallel to a flat, structureless surface; the vertical plane containing the molecular axis is then such a symmetry plane. If we call the direction from A to B 9 = 0 O, then the mirror symmetry of the elastic scattering cross section is expressed by “(Oi> +i + ‘r? (Pf) =a(ei,
-Gi + et, -&)a
It is easy to show from eqs. (lc)
(2)
and (2) that the particular experiment we propose yields a cross section which is symmetric about the angles (pi = 90” Q/2, 270” - @/2; consequently, these angles must be local maxima or
C. Winstead et al. / Adwrbate
orientation determined by elastic electron scattering
L445
Fig. 2. Overhead view of scattering from adsorbed AB molecule lying along the x axis. Two trajectories are shown: for the first (a) the initial azimuth exceeds 90 o - o/2 by an angle cx and for the second (b) the initial azimuth is less than 90 o - G/2 by a. Time reversal followed by reflection through x interchanges the two trajectories, provided that the altitudinal angles satisfy 0, =180° - Bi.
minima. These relationships should make it quite easy to locate the appropriate pair of angles from the data; all that is then needed is to decide which angle is 90” - Q/2 and which is 270 o - G/2, or in terms of our example, which end of the molecule is A and which B. To make such a decision one must use calculations of the kind reported in refs. [l-3]. Before proceeding it may be useful to clarify how this symmetry arises. Fig. 2 gives an overhead view of the situation; because we have chosen an arrangement for which neither the reflection nor time reversal affects the altitudinal angles Bi and f$, we need follow only the azimuths. The combination of reflection in the vertical plane containing AB (the xz-plane) with time reversal transforms the trajectory of fig. 2a into that shown in fig. 2b; thus, we have a(900
- @/2+cu-,90°+~/2+a)=a(900-@/2-a+900+@/2-a) (3a)
and the scattering pattern will be symmetric about 90“ - Q/2. Likewise we may show that ~(270~ - @/2+a+270°+@/2+a) = ~(270~ - Q/2 - CY+ 270 o + Q/2 - CI).
(3b)
When more than one plane of symmetry is present, there will be a corresponding number of pairs of directions about which the scattering cross section is symmetric, and some confusion is possible. Here calculation of the cross section is needed to establish which plane is which. When surface scattering cannot be neglected, the same scattering experiment may still give information about the adsorbate’s orientation if some
L446
C. Winstead et al. / Adsorbate orientation determined by elastic electron scattering
symmetry is preserved. For instance, a clean (110) surface has two inequivalent mirror planes. An AB molecule lying flat in one mirror plane breaks the symmetry with respect to the other one, while an AB molecule standing up preserves both. A third situation arises when surface scattering is strong and the orientation is already known from other evidence. It may then be possible to determine the binding site. If the adsorbate is known to be standing up on a (111) surface, for example, on-top binding preserves the six mirror planes present in the clean surface. Binding at the three-fold hollow preserves three mirror planes, while the bridge site preserves only two. It should be noted with regard to the preceding example that sites such as the three-fold hollow and the bridge occur in more than one orientation, so that, depending on the long-range ordering, if any, of the adsorbate, the total scattering pattern may be an average over site orientations, thus raising the pattern’s symmetry. (We are assuming that either there is no long-range order or that the coherence length of the electron beam is small compared to the adsorbate spacing, so that the total scattering pattern is an incoherent sum of the individual scattering events.) In the (111) surface example, the three binding sites would nonetheless remain distinguishable. Before proceeding to examples, it is worth mentioning that the same symmetry considerations we have discussed apply to elastic scattering of particles other than electrons. Also, inelastic scattering experiments of this type may yield cross sections with enough structure to determine the adsorbate orientation. Though the presence of mirror planes will not be directly indicated by symmetry in an inelastic cross section, if two such cross sections can be measured, one with the detector at & + @ and one with the detector at & - @, they will be mirror images of each other about the angles at which the source lies in the mirror plane.
3. Examples The symmetry rules governing our proposed experiment are exact if we can neglect scattering by the surface or if the adsorbate’s planes of symmetry coincide with symmetry planes of the surface. Thus we give examples not to illustrate that these rules hold but to show that, in cases of interest, the features of the scattering data are clear enough that identification of the relevant angles is easy. The theoretical method used to obtain the differential scattering cross section has been described previously [l]. Briefly, the Schwinger multichannel method [6-81 is used to generate an accurate scattering amplitude for the isolated, oriented molecule. The CO amplitude was obtained from earlier work [2]; details of the C,H, calculation will be published elsewhere 191. The surface is represented by a step potential, which acts as a partially-reflecting mirror, and scattering by the molecule and the surface are treated as
C. Winstead et al. / Adrorbate
0
60
orientation determined by elastic electron scattering
1x0
180
240
300
I.447
360
4. Fig. 3. Elastic scattering of 9.9 eV electrons by CO on a surface, plotted as a function of & with & = +i + @, Bi = 135 O, and 0, = 45 “. Results are shown for CO parallel to the surface (solid line) and for CO at 45” to the surface, with C closer (dashed line). The direction from C to 0 is defined as + = O“, and the distance from the surface to the C nucleus is 1.0 A in both cases. (a) @=90”;(b) @=120’.
separate events. As indicated earlier, the obvious limitations of this model restrict our attention to those overall, qualitative features of the scattering cross section which are unlikely to be artifacts. Fig. 3a shows the differential elastic scattering cross section obtained for CO at an electron energy of 9.9 eV. The scattering arrangement is that described above, with @ chosen to be 90 O, and Bi and f?, chosen to be 135 o
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C. Winstead et al. / Adsorbate orientation determined by elastic electron scattering
and 45O, respectively. The solid curve in the figure results when the CO molecule lies parallel to the surface, along the x axis, with + x (+ = 0 o ) being the direction from C to 0. The predicted symmetry about & = 45 o and 225 o is quite evident. Moreover, the strong peaks in the cross section are noticeably shifted toward 225 O, away from 45 O, making it possible to determine which end of the molecule is 0 and which C, especially if the small maximum at 45 o is observable. The dashed curve in fig. 3a shows the scattering cross section for a CO molecule tilted at 45 o from the vertical in the +x direction, with C being the end closer to the surface. As expected, the scattering pattern shows the same symmetry as seen for horizontal CO, but there are marked differences in structure between the two patterns. Since a vertical CO molecule would give a completely flat scattering profile in this experiment, it should thus be possible to say whether the molecule is vertical or nearly vertical, whether it is tilted at about 45 O, or whether it is horizontal or nearly so. However, the two maxima in the dashed curve are nearly equal, making it difficult to say from experiment in which of the two possible directions the CO molecule tilts. This is an instance in which a more elaborate experiment proves necessary; for example, we could use the approach of ref. [3] and vary +r, while leaving & fixed. Results of the type shown in fig. 3a would still be valuable in identifying the approximate angle of tilt and narrowing to the two possibilities that must be distinguished in the further experiment. Fig. 3b contains CO scattering data analogous to that in fig. 3a, but for @ = 120 O, in order to illustrate the dependence of the symmetry on @. Note that the results are qualitatively the same as in fig. 3a, but that, as predicted by the symmetry rules, the cross sections are now symmetric about 30” and 210°. Fig. 4a shows the elastic scattering cross section obtained for the C,H, molecule at 10 eV, using the same Bi, 0,, and @ as in fig. 3a and assuming the molecule to lie parallel to the surface with the C-C bond along the x axis. Because there are now two vertical mirror planes, the scattering pattern is symmetric both about the pair (pi = 45 o and & = 225 O, and about the angles (pi = 135 o and 315 O. Again there is sufficient structure in the cross section that the symmetry is readily apparent. Moreover, it is clear that the maxima corresponding to & = 135 o and 315” are much stronger than those at 45 o and 225”. The scattering pattern thus clearly indicates that the C-C bond direction is $I = 0 o and not + = 90 O. While the electron scattering pattern may establish the adsorbate orientation, it may not be sufficiently sensitive to detect adsorption-induced changes in adsorbate geometry. An estimate of the effects of such distortion is provided by fig. 4b, which shows results for the same situation as in fig. 4a, but with the C,H, molecule distorted by bending the H atoms away from the surface: the C-C and C-H distances are left unchanged, but the HCH and
C. Winstead et al. / Adrorbate
orientation determined by elastic electron scattering
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(b)
Fig. 4. (a) Elastic scattering of 10 eV electrons by adsorbed C,H,. The molecule plane is parallel to and 1 A above the surface, and the C-C bond is defined to be + = 0 O. Scattering angles are as in fig. 3a. (b) Scattering of 10 eV electrons by distorted C,H,. The molecular geometry is as described in the text; the scattering angles are as in fig. 3a.
HCC angles are both set at 109.5”. Comparison with fig. 4a shows that this distortion has little effect on the cross section and would therefore be undetectable by the present method. Though it is probably an advantage on the whole for the predictions of our theoretical model to be insensitive to small changes in bond lengths and angles, it is somewhat surprising that as large a distortion as is considered here has so little effect. In actual applications of
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electron scattering for determining adsorbate orientation, distortion must be borne in mind.
this insensitivity
to
4. Conclusions We have shown that adsorbate orientations can be determined from a particularly simple elastic electron scattering experiment. By using an arrangement that exploits basic symmetry relationships, we are able to establish the orientation of any mirror planes possessed by the scatterer from the symmetry of the cross section alone. Complete specification of the adsorbate orientation may require an additional experiment, but having first located the mirror planes should reduce the overall labor. While many adsorbate species of interest are expected to show at least one mirror plane, this approach should also work for less symmetric situations, provided the scattering pattern has recognizable strong features.
Acknowledgment This work was supported by the Army Research Office and the Air Force Office for Scientific Research under contract DAA G29-85-K-0117 through the University of California at Santa Barbara.
References [l] S. Nagano, Z.-P. Luo, H. Metiu, W.M. Huo, M.A.P. Lima and V. McKay, J. Chem. Phys. 85 (1986) 6153. [2] S. Nagano, Z.-P. Luo, H. Metiu, W.M. Huo, M.A.P. Lima and V. McKay, Surface Sci. 186 (1987) L548. [3] S. Nagano, Z.-P. Luo, H. Metiu, W.M. Huo and V. McKay, J. Chem. Phys. 88 (1988) 7970. [4] R.E. Palmer, P.J. Rous, J.L. Wilkes and R.F. Willis, Phys. Rev. Letters 60 (1988) 329. [5] R.G. Newton, Scattering Theory of Waves and Particles, 2nd ed. (Springer, New York, 1982) p. 452. [6] K. Takatsuka and V. McKay, Phys. Rev. A 30 (1984) 1734. [7] M.A.P. Lima, T.L. Gibson, K. Takatsuka and V. McKay, Phys. Rev. A 30 (1984) 1741. [8] R.R. Lucchese, K. Takatsuka and V. McKay, Phys. Rept. 131 (1986) 147. [9] C.L. Winstead, V. McKay and H. Metiu, to be published.