- Email: [email protected]
Linear photoelectron diffraction: application of a rapid approximation for surface structural studies
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surface science letters ELSEVIER
Surface Science Letters 302 (1994) L336-L341
Surface Science Letters
Linear photoelectron diffraction: application of a rapid approximation for surface structural studies A.P. Kaduwela a, MA. Van Hove a**,C.S. Fadley a,b aMaterials Sciences Division, Lawrence Berkeley Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA b Department of Physics, University of California at Davis, Davis, CA 95616, USA
(Received 22 July 1993; accepted for publication 12 October 1993)
Abstract The linear superposition approximation proposed by Wander, Pendry and Van Hove for efficient low-energy electron diffraction calculations (“linear LEED”) is applied to photoelectron diffraction (“linear PD”). As with linear LEED, linear PD works very well for calculating the effect of displaced atoms. However, due to strong forward scattering at higher energies, linear PD requires that atoms do not move into or out of alignment. This limitation can be removed by suitable simple adjustments to the basic approximation, promising to make the method effective for structural searches of complex surfaces.
1. Introduction Photoelectron diffraction (PD) is a well-established tool for surface structure dete~ination [ 11. State-of-the-art simulations are done routinely at the fully-converged multiple-scattering limit [2-51. However, structure determinations and optimizations in PD are still performed in a trial-and-error manner due to the lack of more rapid search methods such as those that are available for low-energy electron diffraction (LEED) simulations 161. As the first step in developing such methods for PD, we investigate the applicability to multiple-scattering simulations of the linear superposition approximation of Wander et al. [7], as recently introduced successfully for multiple-scattering LEED. Fritzsche and Pendry [8]
* Co~es~nding author. Fax: i-1 510 486 4995; E-mail: [email protected].
have recently examined the same problem in a more restricted and more favorable case which avoids the commonly occurring complications of multiple forward scattering: we here pay special attention to this issue with 3-atom clusters that test all important effects.
2. The principle and benefits of linear dif%iaction To illustrate the principle of this linear superposition approximation, as applied to PD, consider a photoelectron emitter at the origin and two scatterers at positions rl and I”~, as shown in Fig. la. Let the total photoemission amplitude at the detector, including all multiple scattering, be A, =A(rl,r2) (the intensity would thus be proportional to 1A, I 2>. Now, consider the displacement of scatterer 1 alone by an arbitrary vector atl, as shown in Fig. lb, and let the new exact amplitude be A, = A(r, + 6r,r,). Similarly, dis-
0039-6028/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0039-6028(93)EOZl7-~
A.F. ~ad~~~~a ei al. / SurjacaceScience Letters302 f199#) L336-L34i
EMITTER
fb)
(d)
Fig. 1. Displacements of scatterers in a 3-atom Cu cluster.
placing the second scatterer alone by an independent Sr,, as in Fig. Ic, yields the amplitude A, =A(r,,r, + Sr,). The linear supe~~sition approximation [7] gives an estimate A,, of the exact scattering amplitude Air, + Srl,r2 + 6rJ for the combined displacements 6r, and Sr, shown in Fig. Id, as: Aiin =X4* -6 (A, --A,) + (.rl, -A,)*
(I) The correction A, leading from for the initial “reference” structure to Ali, for the doubly distorted structure is simpIy the linear superposition of the ampfitude changes due to the individual displacements, using the exact amplitudes Ai for the singly-distorted structures. This formalism remains unchanged for clusters of many atoms, and also generalizes easily to any number of displaced atoms, by addition of further terms (Aj -A,) in Eq. (1). Note that, if there were only one displaced atom, Eq. Cl) would reduce to a trivial statement of no practical use: it is useful only for co~~~~fft~~~ of atomic displacements. Eq. (I) is exact in the single-scattering (kinematic) limit. It is an approximation only in that
some multiple scattering paths are not treated exactly, namely those that scatter from at least two different displaced atoms. Examples of such paths in Fig. 1 travel from emitter to atom 1 to atom 2 to detector. Multiple ~u~~~~-~~ttering paths will play a major role in our further discussion, especially when forward focusing at higher energies [l] takes place. An important consideration is how the Iinear PD approximation behaves as a function of the magnitude of atomic displacements. For sufficiently small displacements, it clearly will converge to the correct result. The behavior for large displacements is affected by the above-mentioned multiple-scattering paths, However, such paths will only contribute strongly if: (1) the displaced atoms are very close to each other, especially at Iower energies where several large-angle scatterings are possible; or (2) the atoms are moved into or out of alignment with the emitter (especially at high kinetic energies where forward focusing dominates). The first condition is not critical, as it is found to apply only for interatomic distances closer than typical bond lengths allow. In addition, Eq. (I) should be a very good appro~mation at high kinetic energies, as long as atoms donot move into or out of alignment with the emitter. The reason for this last point is that Eq. (1) does not turn on or off the multiple forward scattering that occurs when alignment sets in or is broken, respectively. We shah show that this is the major difference between, on the one hand, linear PD at high energies with aligned atoms and, on the other hand, linear PD and linear LEED at low energies or when no atoms are aligned (such that forward focusing is much Iess pronounced). At first sight, Eq. (1) is inefficient, since it replaces ooze full multiple-scattering calculation for the doubly distorted cluster by three full multiple-scattering calculations for A,, A,, and A,, and a trivia1 sum to combine them. However, when conducting a structural search in which many atomic displacements must be investigated, the linear diffraction scheme can reduce by orders of magnitude the number of required full multiple-scattering calculations. To see this, consider N positions each for M scatterers. The conventional method would demand NM full
A. P. Kad~mlela et at. /Surface Science Letters 302 (1994) L336-L34t fc)
(e)
0.2h0.3AJ
0.4A
3A)
T
0.5AC0.2A) 0.58(0.54)
EMITTER
EMITTER
EMITTER (d)
(f)
EMITTER
EMITTER
Fig. 2. Geometries tested. panels (a)-(d) show an aligned chain geometry: (a) forward-scattering for on-axis displacements, (b) back-scattering for on-axis displacements, (c) forward-scattering for off-axis displacements, and (d) back-scattering for off-axis displacements, Panels (e) and (f> show a non-aligned reference geometry. The arrows indicate direction of displacements and numbers next to them are the tolerance levels at 1000 and 100 eV (latter in parenthesis).
On-Axis
Movement
Ekin= I OQ
Ekin= 1000 eV Exact Linear
On-Axis
a
___ --..-
Calculation Approximation
1(&l
Movement
b
ev
Exact Linear
Calculation Approximation
56
R=0.0057 _4,.--....
WV
-_ _ c‘sl"G
40 Polar
60
100
120
Qr (Degree)
160
20
40
Polar
60
80
Angle
100
120
140
160
Cp (lkgree)
Fig. 3. Comparison of PD curves obtained with an exact calculation and the linear approximation at (a) 1000 eV and (b) 100 eV, for the geometry shown in Fig. 2a, as a function of the distance I, where 1= 2.56 A is the reference structure. The R-factor quantifies the overall disciepancy between each pair of exact and approximate curves.
A.P. Kaduwela et al. / Surface Science Letters 302 (1994) L336-L341
multiple-scattering calculations, whereas the linear method requires only MN full multiplescattering calculations, from which the desired NM linear combinations can be obtained in a comparatively negligible time [7f. 3. Computations We have explored the validity of the linear PD approximation for a small cluster like that of Fig. 1; this should exhibit all important features relevant to larger clusters. The scattering geometries tested are shown in Fig. 2. Each cluster has a Cu emitter and two Cu scatterers. Photoelectron emission is assumed to be from an s initial state to a p final state. Angular distributions of photoelectrons were simulated at both 100 and 1000 eV, using a fully converged multiple-scattering curved-wave formalism [4,5]. The polarization vector (Z> of photons was kept parallel to tke direction of the photoelectron wavevector Ck> (i.e., the detector). Neither damping due to inelastic scattering nor thermal effects were included in the calculations. We start with a linear chain, in which multiple forward scattering is important. Shown in Fig. 3a Movement
Off-Axis -
Exact
are polar angle distributions of photoelectrons at 1000 eV for the test geometry in Fig. 2a, representing a subsurface emitter. The undistorted reference structure yields A,, which*gives the full line at center {labeled I= 2.56 A). The other full-line distributions are calculated exactly, for distortions wherein the two scatterers are moved in opposite directions by the same amount, thus varying their separation 1. The dashed lines are the linear PD approximations of Eq. (1). Their agreement with the exact curves is excellent, as judged either visually or by an R-factor f91. The differences are primarily due to the doublescattering path from emitter to far scatterer to near scatterer to detector, which is seen to contribute more at small separations. For larger L, the approximation does not appreciably worsen; in fact, the agreement should slowly improve as the displaced atoms move farther apart. At 100 eV (see Fig. 3b) the difference between exact and appr~~mate curves is farger due to stronger backscattering in the multiple-scattering paths. In a back-scattering geometry, as shown in Fig. 2b, a similar level of agreement is observed (not illustrated here). Hence, the linear diffraction approxt imation is generally satisfactory if the scatterers
a
Calculation inear
20
40
Polar
60
80
Angle
100 120
!40 160
@ (Degree)
20
40
Polar
60
Approxim
80
Angle
LOO 120 140
160
Q (Degree)
Fig. 4. Same as Fig. 3 but for the geometry shown in Fig. Zc, as a function of the off-axis displacement d = I&, I = I&-, (: Ia) 1000 and (b) 100 eV.
A.P. Knduwela ei al. /Surface
Science Letters 302 (I 994) L336-L341
are moved along the internuclear axis of an aligned chain. In our second class of tests, scatterers are moved perpendicularly to the chain axis by an amount d = 1t%, I = f 6r, I to break the alignment, as shown in Figs. 2c and 2d. Now the exact and approximate curves diverge rapidly as the atoms are moved away from the chain axis for a subsurface emitter, Fig. 2c, and for both energies, 1000 and 100 eV, as shown in Figs. 4a and 4b, respectively (the same qualitative behavior is found for a surface emitter, but is not illustrated here). The differences are pronounced for emission along the chain axis and they diminish for emission far away from that axis. This behavior is due to fo~ard-focusing effects that are incompletely canceled in Eq. (1). It is known from previous studies AS,101 that when all scatterers are aligned, multiple scattering terms along that axis can be significantly stronger than the singlescattering terms. However, when chains are bent, the single-scattering terms become dominant [5]. Hence, the multiple~s~ttering contribution in the reference term A, is much stronger than in the terms A, or A, when measured along the chain axis. Therefore, the terms A, and A, cannot remove the multiple forward terms of A,. When the detector is far away from the axial direction, multiple scattering effects are minimal even for large atomic displacements. The opposite behavior is observed when we take a reference structure which is not aligned at first and then move the atoms into alignment, see Figs. 2e-2f. As long as the atoms remain misaligned, the linear diffraction appro~mation gives excellent results. But when they become aligned, divergences appear. 4, IXscussion From the foregoing results, we can conclude that the linear PD approximation in general works very we11 as long as the displaced atoms don’t move too close together (a physically unlikely situation), and as long as they don’t move into or out of alignment (within about 209. Such behavior will remain valid in clusters of any size, including therefore real surfaces.
At displacements small compared to the wavelength, before interference oscillations in multipie-smattering paths set in, linear PD is a particutarly good appro~mation~ Multiple scattering tends to worsen the agreement, especially at low energies. We summarize in Fig. 2 the resulting tolerance limits for scatterer movements in this small-displacement regime, as based on our calculations. This regime is applicable to structural refinement, i.e. the accurate final optimization of atomic ~ord~nates. In this regime, linear PD competes with techniques like “tensor LEED” [6j (also capable of extension to I’D), which can be even more efficient. It is also possible to combine linear diffraction with tensor LEED to benefit from both [Ill. It should also be possible to improve the performance of linear PD when atomic alignment is made or broken by adding or subtracting contributions from multiple forward-scattering paths along more or less linear chains of atoms. For this purpose one could monitor the possible alignments in the cluster or surface while atoms are moved, and smoothly turn on or off a multiple fo~ard-scattering term that is calculated separately in each case. Such a term would no longer be a “linear” term in Eq. (11, since it would depend on &he new positions of both displaced atoms. One approach is to add a term S_4$’ to Eq. (1) which is calculated for the aligned chain and contains only multiple forward-scattering paths. This term will add otherwise improperly represented paths when atoms come into alignment, and will remove such paths when the atoms break the alignment. This approach will remain economical in a big cluster of atoms, if onIy the scattering along the isolated chain is calculated, ignoring other atoms in the duster, In a big cluster, multiple forward scattering in longer chains of atoms can be treated similarly. An approach that would work only when breaking an aligned geometry, is to keep track separately of single and multiple scattering, e.g. A” ==ns, +=A?. When alignment is broken, the term A! (which would only need to be computed for aligned geometries) could be turned off gradually by a factor A that decays with the scattering
A.P. Kbduwela et al. / Su@ace Science Letters 302 (1994) L336-L341
angle from emitter via first atom to second atom, e.g. proportionally to the angular dependence of the atomic scattering amplitude. Eq. (1) would then generalize to:
+(A,-hAi;).
(21 When A = 0, we almost recover the singfe-scattering limit, but the terms A, and A, still contain useful multiple-battering contributions, improving the approximation.
5, Conclusions
We have tested the applicab~~~ of the linear supe~osition appro~mation of Wander et al. 1’71 in PD simulations for realistic situations that are more demanding than assumed by Fritzsche and Pendry [8]. We find that the original linear approximation, Eq. (0, works quite well as long as the displaced atoms don’t move unphysically close together, and as long as they donot move &to or ow of ahgnment (within about 20”). Making and breaking alignment can be dealt with by using adjustments to the basic Eq. 0). The method works very web for small atomic displacements. For larger displacements the method remains very attractive for a rough structural search. This search could be followed up by a more refined structural optim~ation, which could be accomplished by a variety of methods, including linear PD itself.
This work was supported in part by the Director, Office of Energy Research, Office of Basic
Energy Sciences, Materials Sciences Division of the US Department of Energy under Contract No. DE-AC03-76SFOO098, by the San Diego Super~mputer Center, and by the National Energy Research Supercomputing Center. We thank R.Z. Ynzunza for his assistance in the preparation of this manu~ript.
References [t] C.S. Fadiey, Prog. Surf. Sci. 16 0984) 275; C.S. Fadiey, Phys. Ser. T 17 (1987) 39; W.F. Egelhoff, Solid State Mater. Sci. 16 (199012L3; S.A. Chambers, Adv. P&s. 40 fiQ91) 357; C.S. Fadley, in: Synchrotron Ra~ation Research: Advances in Surface Science, Ed. R.Z. Bachrach (Plenum, New York, 1992) pp. 421-517. f2] S.Y. Tong, H.C. Poon and D.R. Snider, Pbys. Rev. B 32 (1985) 2096. [3] J.J. Barton and D.A. Shirley, Phys. Rev. B 32 (1985) 1892; J.J. Barton, PhD Thesis, University of California at Berkeley (198% [4] J.J. Rehr and E.A. Albers, Phys. Rev. B 41 (1990) 8139. [5] A.P. KaduweIa, D.J. Friedman and C.S. Fadfey, J. Electron Spectrosc. Relat. Pbenom. 57 f1QQf) 223. 161 P.J. Rous, J.B. Pendry, D.K. SaIdin, K. Heinz& K. Muller and N. Bickel, Phys. Rev. Lett. 57 f1986f 2951; P.J. Rous, Prog. Surf. Sci. 39 (1992) 3; M.A. Van Hove, W. Moritz, H. Over, P-3. Rous, A. Wander, A. Barbieri, N, Materer, U. Starke and G.A. Somojai, Surf. Sci. Rep. 19 (1993) 191. [7] A. Wander, J.B. Pendry and M.A. Van Hove, Phys. Rev. B 46 (1992) 9897. [8] V. Fritzxhe and J.B. Pendry, to be published. f9] R.S. Saiki, A.P. Kaduwela, M. Sagurton, J. Osterwalder, D.J. Friedman, C.S. Fadfey and C.R. Brundle, Surf. Sci, 282 (1993) 33. [IO] M.-L Xu and M.A. Van Hove, Surf, Sci. 207 (1989) 215; M.-L. Xu, J.J. Barton and M.A. Van Hove, Phys. Rev. B 39 (198918275. [ll] S.P. Tear, private communication.
“.“:‘: ~~:@p
: : :.v:::.:.:.:. ..... .......................~. :(
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.
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. . .
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surface science letters ELSEVIER
Surface Science Letters 302 (1994) L336-L341
Surface Science Letters
Linear photoelectron diffraction: application of a rapid approximation for surface structural studies A.P. Kaduwela a, MA. Van Hove a**,C.S. Fadley a,b aMaterials Sciences Division, Lawrence Berkeley Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA b Department of Physics, University of California at Davis, Davis, CA 95616, USA
(Received 22 July 1993; accepted for publication 12 October 1993)
Abstract The linear superposition approximation proposed by Wander, Pendry and Van Hove for efficient low-energy electron diffraction calculations (“linear LEED”) is applied to photoelectron diffraction (“linear PD”). As with linear LEED, linear PD works very well for calculating the effect of displaced atoms. However, due to strong forward scattering at higher energies, linear PD requires that atoms do not move into or out of alignment. This limitation can be removed by suitable simple adjustments to the basic approximation, promising to make the method effective for structural searches of complex surfaces.
1. Introduction Photoelectron diffraction (PD) is a well-established tool for surface structure dete~ination [ 11. State-of-the-art simulations are done routinely at the fully-converged multiple-scattering limit [2-51. However, structure determinations and optimizations in PD are still performed in a trial-and-error manner due to the lack of more rapid search methods such as those that are available for low-energy electron diffraction (LEED) simulations 161. As the first step in developing such methods for PD, we investigate the applicability to multiple-scattering simulations of the linear superposition approximation of Wander et al. [7], as recently introduced successfully for multiple-scattering LEED. Fritzsche and Pendry [8]
* Co~es~nding author. Fax: i-1 510 486 4995; E-mail: [email protected].
have recently examined the same problem in a more restricted and more favorable case which avoids the commonly occurring complications of multiple forward scattering: we here pay special attention to this issue with 3-atom clusters that test all important effects.
2. The principle and benefits of linear dif%iaction To illustrate the principle of this linear superposition approximation, as applied to PD, consider a photoelectron emitter at the origin and two scatterers at positions rl and I”~, as shown in Fig. la. Let the total photoemission amplitude at the detector, including all multiple scattering, be A, =A(rl,r2) (the intensity would thus be proportional to 1A, I 2>. Now, consider the displacement of scatterer 1 alone by an arbitrary vector atl, as shown in Fig. lb, and let the new exact amplitude be A, = A(r, + 6r,r,). Similarly, dis-
0039-6028/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0039-6028(93)EOZl7-~
A.F. ~ad~~~~a ei al. / SurjacaceScience Letters302 f199#) L336-L34i
EMITTER
fb)
(d)
Fig. 1. Displacements of scatterers in a 3-atom Cu cluster.
placing the second scatterer alone by an independent Sr,, as in Fig. Ic, yields the amplitude A, =A(r,,r, + Sr,). The linear supe~~sition approximation [7] gives an estimate A,, of the exact scattering amplitude Air, + Srl,r2 + 6rJ for the combined displacements 6r, and Sr, shown in Fig. Id, as: Aiin =X4* -6 (A, --A,) + (.rl, -A,)*
(I) The correction A, leading from for the initial “reference” structure to Ali, for the doubly distorted structure is simpIy the linear superposition of the ampfitude changes due to the individual displacements, using the exact amplitudes Ai for the singly-distorted structures. This formalism remains unchanged for clusters of many atoms, and also generalizes easily to any number of displaced atoms, by addition of further terms (Aj -A,) in Eq. (1). Note that, if there were only one displaced atom, Eq. Cl) would reduce to a trivial statement of no practical use: it is useful only for co~~~~fft~~~ of atomic displacements. Eq. (I) is exact in the single-scattering (kinematic) limit. It is an approximation only in that
some multiple scattering paths are not treated exactly, namely those that scatter from at least two different displaced atoms. Examples of such paths in Fig. 1 travel from emitter to atom 1 to atom 2 to detector. Multiple ~u~~~~-~~ttering paths will play a major role in our further discussion, especially when forward focusing at higher energies [l] takes place. An important consideration is how the Iinear PD approximation behaves as a function of the magnitude of atomic displacements. For sufficiently small displacements, it clearly will converge to the correct result. The behavior for large displacements is affected by the above-mentioned multiple-scattering paths, However, such paths will only contribute strongly if: (1) the displaced atoms are very close to each other, especially at Iower energies where several large-angle scatterings are possible; or (2) the atoms are moved into or out of alignment with the emitter (especially at high kinetic energies where forward focusing dominates). The first condition is not critical, as it is found to apply only for interatomic distances closer than typical bond lengths allow. In addition, Eq. (I) should be a very good appro~mation at high kinetic energies, as long as atoms donot move into or out of alignment with the emitter. The reason for this last point is that Eq. (1) does not turn on or off the multiple forward scattering that occurs when alignment sets in or is broken, respectively. We shah show that this is the major difference between, on the one hand, linear PD at high energies with aligned atoms and, on the other hand, linear PD and linear LEED at low energies or when no atoms are aligned (such that forward focusing is much Iess pronounced). At first sight, Eq. (1) is inefficient, since it replaces ooze full multiple-scattering calculation for the doubly distorted cluster by three full multiple-scattering calculations for A,, A,, and A,, and a trivia1 sum to combine them. However, when conducting a structural search in which many atomic displacements must be investigated, the linear diffraction scheme can reduce by orders of magnitude the number of required full multiple-scattering calculations. To see this, consider N positions each for M scatterers. The conventional method would demand NM full
A. P. Kad~mlela et at. /Surface Science Letters 302 (1994) L336-L34t fc)
(e)
0.2h0.3AJ
0.4A
3A)
T
0.5AC0.2A) 0.58(0.54)
EMITTER
EMITTER
EMITTER (d)
(f)
EMITTER
EMITTER
Fig. 2. Geometries tested. panels (a)-(d) show an aligned chain geometry: (a) forward-scattering for on-axis displacements, (b) back-scattering for on-axis displacements, (c) forward-scattering for off-axis displacements, and (d) back-scattering for off-axis displacements, Panels (e) and (f> show a non-aligned reference geometry. The arrows indicate direction of displacements and numbers next to them are the tolerance levels at 1000 and 100 eV (latter in parenthesis).
On-Axis
Movement
Ekin= I OQ
Ekin= 1000 eV Exact Linear
On-Axis
a
___ --..-
Calculation Approximation
1(&l
Movement
b
ev
Exact Linear
Calculation Approximation
56
R=0.0057 _4,.--....
WV
-_ _ c‘sl"G
40 Polar
60
100
120
Qr (Degree)
160
20
40
Polar
60
80
Angle
100
120
140
160
Cp (lkgree)
Fig. 3. Comparison of PD curves obtained with an exact calculation and the linear approximation at (a) 1000 eV and (b) 100 eV, for the geometry shown in Fig. 2a, as a function of the distance I, where 1= 2.56 A is the reference structure. The R-factor quantifies the overall disciepancy between each pair of exact and approximate curves.
A.P. Kaduwela et al. / Surface Science Letters 302 (1994) L336-L341
multiple-scattering calculations, whereas the linear method requires only MN full multiplescattering calculations, from which the desired NM linear combinations can be obtained in a comparatively negligible time [7f. 3. Computations We have explored the validity of the linear PD approximation for a small cluster like that of Fig. 1; this should exhibit all important features relevant to larger clusters. The scattering geometries tested are shown in Fig. 2. Each cluster has a Cu emitter and two Cu scatterers. Photoelectron emission is assumed to be from an s initial state to a p final state. Angular distributions of photoelectrons were simulated at both 100 and 1000 eV, using a fully converged multiple-scattering curved-wave formalism [4,5]. The polarization vector (Z> of photons was kept parallel to tke direction of the photoelectron wavevector Ck> (i.e., the detector). Neither damping due to inelastic scattering nor thermal effects were included in the calculations. We start with a linear chain, in which multiple forward scattering is important. Shown in Fig. 3a Movement
Off-Axis -
Exact
are polar angle distributions of photoelectrons at 1000 eV for the test geometry in Fig. 2a, representing a subsurface emitter. The undistorted reference structure yields A,, which*gives the full line at center {labeled I= 2.56 A). The other full-line distributions are calculated exactly, for distortions wherein the two scatterers are moved in opposite directions by the same amount, thus varying their separation 1. The dashed lines are the linear PD approximations of Eq. (1). Their agreement with the exact curves is excellent, as judged either visually or by an R-factor f91. The differences are primarily due to the doublescattering path from emitter to far scatterer to near scatterer to detector, which is seen to contribute more at small separations. For larger L, the approximation does not appreciably worsen; in fact, the agreement should slowly improve as the displaced atoms move farther apart. At 100 eV (see Fig. 3b) the difference between exact and appr~~mate curves is farger due to stronger backscattering in the multiple-scattering paths. In a back-scattering geometry, as shown in Fig. 2b, a similar level of agreement is observed (not illustrated here). Hence, the linear diffraction approxt imation is generally satisfactory if the scatterers
a
Calculation inear
20
40
Polar
60
80
Angle
100 120
!40 160
@ (Degree)
20
40
Polar
60
Approxim
80
Angle
LOO 120 140
160
Q (Degree)
Fig. 4. Same as Fig. 3 but for the geometry shown in Fig. Zc, as a function of the off-axis displacement d = I&, I = I&-, (: Ia) 1000 and (b) 100 eV.
A.P. Knduwela ei al. /Surface
Science Letters 302 (I 994) L336-L341
are moved along the internuclear axis of an aligned chain. In our second class of tests, scatterers are moved perpendicularly to the chain axis by an amount d = 1t%, I = f 6r, I to break the alignment, as shown in Figs. 2c and 2d. Now the exact and approximate curves diverge rapidly as the atoms are moved away from the chain axis for a subsurface emitter, Fig. 2c, and for both energies, 1000 and 100 eV, as shown in Figs. 4a and 4b, respectively (the same qualitative behavior is found for a surface emitter, but is not illustrated here). The differences are pronounced for emission along the chain axis and they diminish for emission far away from that axis. This behavior is due to fo~ard-focusing effects that are incompletely canceled in Eq. (1). It is known from previous studies AS,101 that when all scatterers are aligned, multiple scattering terms along that axis can be significantly stronger than the singlescattering terms. However, when chains are bent, the single-scattering terms become dominant [5]. Hence, the multiple~s~ttering contribution in the reference term A, is much stronger than in the terms A, or A, when measured along the chain axis. Therefore, the terms A, and A, cannot remove the multiple forward terms of A,. When the detector is far away from the axial direction, multiple scattering effects are minimal even for large atomic displacements. The opposite behavior is observed when we take a reference structure which is not aligned at first and then move the atoms into alignment, see Figs. 2e-2f. As long as the atoms remain misaligned, the linear diffraction appro~mation gives excellent results. But when they become aligned, divergences appear. 4, IXscussion From the foregoing results, we can conclude that the linear PD approximation in general works very we11 as long as the displaced atoms don’t move too close together (a physically unlikely situation), and as long as they don’t move into or out of alignment (within about 209. Such behavior will remain valid in clusters of any size, including therefore real surfaces.
At displacements small compared to the wavelength, before interference oscillations in multipie-smattering paths set in, linear PD is a particutarly good appro~mation~ Multiple scattering tends to worsen the agreement, especially at low energies. We summarize in Fig. 2 the resulting tolerance limits for scatterer movements in this small-displacement regime, as based on our calculations. This regime is applicable to structural refinement, i.e. the accurate final optimization of atomic ~ord~nates. In this regime, linear PD competes with techniques like “tensor LEED” [6j (also capable of extension to I’D), which can be even more efficient. It is also possible to combine linear diffraction with tensor LEED to benefit from both [Ill. It should also be possible to improve the performance of linear PD when atomic alignment is made or broken by adding or subtracting contributions from multiple forward-scattering paths along more or less linear chains of atoms. For this purpose one could monitor the possible alignments in the cluster or surface while atoms are moved, and smoothly turn on or off a multiple fo~ard-scattering term that is calculated separately in each case. Such a term would no longer be a “linear” term in Eq. (11, since it would depend on &he new positions of both displaced atoms. One approach is to add a term S_4$’ to Eq. (1) which is calculated for the aligned chain and contains only multiple forward-scattering paths. This term will add otherwise improperly represented paths when atoms come into alignment, and will remove such paths when the atoms break the alignment. This approach will remain economical in a big cluster of atoms, if onIy the scattering along the isolated chain is calculated, ignoring other atoms in the duster, In a big cluster, multiple forward scattering in longer chains of atoms can be treated similarly. An approach that would work only when breaking an aligned geometry, is to keep track separately of single and multiple scattering, e.g. A” ==ns, +=A?. When alignment is broken, the term A! (which would only need to be computed for aligned geometries) could be turned off gradually by a factor A that decays with the scattering
A.P. Kbduwela et al. / Su@ace Science Letters 302 (1994) L336-L341
angle from emitter via first atom to second atom, e.g. proportionally to the angular dependence of the atomic scattering amplitude. Eq. (1) would then generalize to:
+(A,-hAi;).
(21 When A = 0, we almost recover the singfe-scattering limit, but the terms A, and A, still contain useful multiple-battering contributions, improving the approximation.
5, Conclusions
We have tested the applicab~~~ of the linear supe~osition appro~mation of Wander et al. 1’71 in PD simulations for realistic situations that are more demanding than assumed by Fritzsche and Pendry [8]. We find that the original linear approximation, Eq. (0, works quite well as long as the displaced atoms don’t move unphysically close together, and as long as they donot move &to or ow of ahgnment (within about 20”). Making and breaking alignment can be dealt with by using adjustments to the basic Eq. 0). The method works very web for small atomic displacements. For larger displacements the method remains very attractive for a rough structural search. This search could be followed up by a more refined structural optim~ation, which could be accomplished by a variety of methods, including linear PD itself.
This work was supported in part by the Director, Office of Energy Research, Office of Basic
Energy Sciences, Materials Sciences Division of the US Department of Energy under Contract No. DE-AC03-76SFOO098, by the San Diego Super~mputer Center, and by the National Energy Research Supercomputing Center. We thank R.Z. Ynzunza for his assistance in the preparation of this manu~ript.
References [t] C.S. Fadiey, Prog. Surf. Sci. 16 0984) 275; C.S. Fadiey, Phys. Ser. T 17 (1987) 39; W.F. Egelhoff, Solid State Mater. Sci. 16 (199012L3; S.A. Chambers, Adv. P&s. 40 fiQ91) 357; C.S. Fadley, in: Synchrotron Ra~ation Research: Advances in Surface Science, Ed. R.Z. Bachrach (Plenum, New York, 1992) pp. 421-517. f2] S.Y. Tong, H.C. Poon and D.R. Snider, Pbys. Rev. B 32 (1985) 2096. [3] J.J. Barton and D.A. Shirley, Phys. Rev. B 32 (1985) 1892; J.J. Barton, PhD Thesis, University of California at Berkeley (198% [4] J.J. Rehr and E.A. Albers, Phys. Rev. B 41 (1990) 8139. [5] A.P. KaduweIa, D.J. Friedman and C.S. Fadfey, J. Electron Spectrosc. Relat. Pbenom. 57 f1QQf) 223. 161 P.J. Rous, J.B. Pendry, D.K. SaIdin, K. Heinz& K. Muller and N. Bickel, Phys. Rev. Lett. 57 f1986f 2951; P.J. Rous, Prog. Surf. Sci. 39 (1992) 3; M.A. Van Hove, W. Moritz, H. Over, P-3. Rous, A. Wander, A. Barbieri, N, Materer, U. Starke and G.A. Somojai, Surf. Sci. Rep. 19 (1993) 191. [7] A. Wander, J.B. Pendry and M.A. Van Hove, Phys. Rev. B 46 (1992) 9897. [8] V. Fritzxhe and J.B. Pendry, to be published. f9] R.S. Saiki, A.P. Kaduwela, M. Sagurton, J. Osterwalder, D.J. Friedman, C.S. Fadfey and C.R. Brundle, Surf. Sci, 282 (1993) 33. [IO] M.-L Xu and M.A. Van Hove, Surf, Sci. 207 (1989) 215; M.-L. Xu, J.J. Barton and M.A. Van Hove, Phys. Rev. B 39 (198918275. [ll] S.P. Tear, private communication.