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An improved procedure for inverting atom diffraction intensities for structural determination
305
Surface Science 133 (1983) 305-310 North-Holland Publishing Company
AN IMPROVED PROCEDURE FOR INVERTING ATOM DIFFRACTION INTENSITIES FOR STRUCTURAL DETERMINATION R. JAMES Department
*, D.S. KAUFMAN of Chemistry
and T. ENGEL
BG - IO, University of Washington, Seattle,
Received 5 January 1983; accepted for publication
Washington 98195, USA
7 June 1983
A method has been developed to obtain the surface corrugation function from atom diffraction intensities. Two procedures are outlined and it is shown that a stepwise inversion can be carried out which does not require assumptions about the corrugation amplitude or the phases of the diffracted beams. The rapid convergence of the procedure allows such inversions to be carried out on laboratory size computers. A few examples are discussed to show the utility of the method.
1. Introduction Atom diffraction from surfaces is a technique which has recently made contributions to our understanding of the atom-surface potential and surface structure [l]. In particular, the unique sensitivity of atom diffraction to the electron distribution at surfaces has the potential to give direct information on chemical bounds at surfaces. Experimental diffraction data yield the magnitude and direction of the surface unit cell vectors since this requires only a knowledge of the angular positions of the diffraction maxima. However, the structural information within the unit cell, which is contained in the corrugation function S(R), can only be obtained through an analysis of the diffraction intensities. A number of methods have been proposed to calculate diffraction intensities from a known corrugation function [l-4] in the hard wall approximation. More recent studies have used realistic soft potential to model the surface [5-71. The corrugated hard wall remains a good approximation to the scattering problem if the atomic structure of the surface rather than the details of the atom-surface potential are of primary interest. Initial analyses of surface structures [7-121 were carried out by expanding the (unknown) corrugation function in a Fourier series and systematically varying the magnitude of the expansion parameters until agreement was reached between the experimental and calculated intensities. However, this * Permanent address: Department California 94380, USA.
0039-6028/83/0000-0000/$03.00
of Mathematics,
California
State College, Stanislaus,
0 1983 North-Holland
Turlock,
R. James et at. / Inverting atom diffraction intensities
306
procedure is time consuming and it would be of advantage to develop a method which proceeds directly from the observed intensities to l(R). Cantini et al. 1131 have developed a method of this type using Patterson transform techniques and more recently. Rieder et al. [14] have developed an alternative inversion procedure which starts with the observed diffraction intensities. These authors assume phases of the most intense diffraction beams and the Kirchhoff approximation [ 151 is used to generate a starting function S’(R). An iterative procedure is then used to improve {O(R) until agreement between the experimental and observed ~ffraction intensities is achieved. In this study we report an improved inversion procedure which is also based on the hard wall model. It converges more rapidly than the procedure of ref. [14] and in our model studies has required considerably less computer time. Our method and that of ref. [14] are at present restricted to one dimensional corrugation functions.
2. Procedure We begin with the hard wall boundary equation in the Rayleigh ansatz CA,
exp(iG . R) exp[ik,,
Z(R)]
= - exp[ik,,
S(R)]
,
(1)
G
in which the attractive atom-surface potential is included only as a refractive index [I]. This limits the applicability of the method to cases in which no bound state resonances are encountered. This is not a severe restriction since these conditions can be met for He diffraction if sufficiently short wavelengths and angles of incidence di near normal are used. In eq. (l), G are the reciprocal lattice vectors of the surface, and the incoming and outgoing wavevectors ki and k, have been separated into their components parallel and normal to the surface respectively kj = (KS kizJt
(2)
k,=(K+G,k,,),
(3)
we expand 3(R) in a Fourier series
P(Ri) = c
g, cos(2snR,/a)
n
for a set of NG equidistant points Ri in the unit cell of length u. The number of points is chosen to be equal to the number of experimentally observed diffraction peaks. Choosing initial values for the coefficients {,, (they need not be very close to the true values for small corrugation amplitudes), we calculate the complex amplitudes AZ’ by solving the complex linear system
R. James et al. / Inverting atom diffraction intensities
CA,
exp(iG - Ri) exp[ik,,
l( R~)] = - exp[ik,
S( Ri)]
307
(5)
G
for i= l,..., NG. This is equivalent to the GR method discussed by Garcia [3]. The calculated diffraction intensities are given by
which has been
(6) The experimental results are taken into account diffraction amplitudes given by A’, = A:’ IA”c”P/A”c”‘l.
by forming a new set of (7)
In the above equation the amplitudes A$ have been adjusted such that the calculated phases have been retained, but the experimental amplitudes have been substituted for the calculated quantities. Using these adjusted values of A, new values of S(Ri) are calculated from eq. (5) in an iteration procedure using Newton’s method. The previous set of {(Ri) are used as Starting values and convergence is rapid, usually requiring only 2 or 3 iterations. The imaginary part of {(Ri) is discarded and a least squares fitting procedure is used to calculate the coefficients S, in eq. (4). This procedure is repeated until s=C[Im5(Ri)12,
(8)
reach acceptably low values. In eq. (9), NG is the number of diffraction beams used in fitting the data. The increased speed over the method described in ref. [14] results from the elimination of the time consuming procedure which was used to arrive at a set of starting phases and from these [O(R), and the faster convergence of Newton’s method in calculating the A,. It brings the time for inversion calculations down to a value compatible with laboratory computers such as the LSI 1 l/23 which was used in these studies. 3. Application of the method We have tested the method described above for both simple and more complex corrugation functions. The values of A, were computed from the test corrugation function using eq. (5). These values were used to generate { PgxP> using eq. (6) and these data were inverted to retrieve l(R). For a corrugation function with a single coefficient, .X(x) = f{, cos(2ax/a). Table
(10)
1 lists the values of the starting parameter c, for which the inversion
308
R. James et al. / Inverting atom diffraction intensities
Table 1 Corrugation parameter 3,
Reciprocal lattice vectors used
Input parameters which converged to correct value
0.10
-1Oto6 -1Oto6 -1Oto6 -1Oto6
0 to 0.22 0. to 0.23 0.20 to 0.54 0.37 to 0.66
0.20 0.40 0.50
procedure converged to the correct value. In this example, a = 3.52 A, X = 0.57 A, Bi = 20’ and four G vectors lying outside the Ewald sphere have been included in the calculation: Since these calculations require less than one minute to achieve unitarity within 0.001 and R and S values below 1 x 10p4, a set of starting values for Ii can be rapidly tested. In all cases the final value of 3, was equal to the correct value to within 0.1% after at most 5 iterations. The Rayleigh assumption limits the convergence for the above parameter to S1 = 0.504 [ 151. We have also tested this procedure on the corrugation function given as model III in table 2 of ref. [14]. Using cf = 7.04 A, X = 0.57 A and 8 = 20’ and a test corrugation function with 1, = 0.23, 5; = 0.15 and 5, = 0.08 to generate the (PFP}, convergence to the correct values was obtained for nearly all values in the intervals 0.15 < l, -C0.26, 0.10 < l2 < 0.18 and 0 -Cl3 c 0.12. In these calculations which included four G vectors outside the Ewald sphere, unitarity was achieved to within 0.0001. Fewer than 20 iterations which require 8 min computational time were needed.
4. A procedure for the calculation of starting values for c(R) The model calculations described above show that for starting values of & suitably close to their true values, our method converges rapidly. However, it would be convenient to have a method which would avoid the necessity of choosing starting values for &. The results shown in table 1 indicate that for small li, convergence occurs for starting values quite different from the true value. We have utilized this behavior by slowly increasing the corrugation in our best fit functions S(R) in the following way. We begin with a set lAzP1 corresponding to the true corrugation S(R). We calculate a set of AZ’ from eq. (5) using {, = lo-* and S;,= 0, n > 1. These Ag’ are used to generate a new set of A& given by A& = VIA”,“’ IA”G”PI/JAg’l,
G * 0,
where initially 2-’ -Cm -C2-5. In generating the
(11) A&,
the calculated phases are
R. James ei al. / Inverting
atom diffraction intensities
309
retained and the intensities of all beams for which G * 0 are reduced by the factor m to mimic a corrugation of much smaller amplitude. A renormalized set of amplitudes A&, given by
G
are then used in eq. (5) to solve for S(Ri). The imaginary part is discarded and a least squares procedure is used to calculate new coefficients Si. There 3, are used to repeat the procedure outlined above until succesive iterations lead to no significant change in the &. At this point m is increased to a somewhat higher value. The entire procedure is repeated until m is unity at which point the & are the best fit parameters to the experimental data. It was found that the most rapid change in the coefficients Si is observed for 0.5 < m < 0.9 and in this range m is usually increased in small increments. To test this procedure we examined three special cases of 5( Ri) given by eq. (4) for the parameters a = 3.52 A, Bi = 30” and X = 0.57 A. For S, = C2 and [, = 0 for n 2 3, the input parameters were recovered for {, < 0.14 A corresponding to a peak-to-peak corrugation amplitude [,,, of 0.44 A. For S, = 2& and S;, = 0 for n 2 3, the input parameters were recovered for {, < 0.18 A corresponding to l,,, = 0.40 A. For l1 = S2 = l3 and {, = 0 for n z 4, the input parameters were recovered for S1 < 0.06 corresponding to lm,, = 0.26 A. Fewer than 165 iterations requiring approximately 50 min computational time were needed to satisfy unitarity to within 0.00001 and to reduce R and S to below 1 x lop3 in the worst cases and to below 1 x lo-’ in the best cases. From these test cases, it is clear that the range of corrugation functions for which the procedure converges to the correct value depends on the relative magnitudes of the parameters. The observed decrease in I,,,,, as the number of corrugation parameters increases is consistent with previous studies [ 151 which show that the convergence limit of the Rayleigh approximation decreases with the number of Fourier coefficients Si. However, the limits found for c,,,,, using this procedure indicates that it should be applicable to nearly all cases of clean metal surfaces as well as most cases of gas adsorption on metal surfaces. This inversion procedure has the important advantage that it requires no starting assumption to extract S(R) from the diffraction intensities.
5. Summary Two methods to invert atom diffraction intensity data to determine the one-dimensional corrugation function l(R) have been discussed. In the first of these, trial imput value of the corrugation function are required and an
310
R. James et al. / Inverting atom diffraction intensities
iterative method is used to converge to the correct value. A second method is discussed in which S(R) is determined from diffraction intensities without assumptions about the corrugation function or the phases of the diffracted beams. The range of applicability of the method is large enough that most cases of clean or adsorbate covered surfaces can be dealt with using either method.
Acknowledgements This research has been supported by the National Science Foundation under grant CHE-8109067. One of us (R.J.) has benefited from a NSF program which provides research opportunities for faculty members at small colleges.
References [I] T. Engel and K.H. Rieder, in: Structural Studies of Surfaces, Springer Tracts in Modern Physics, Vol. 91 (Springer, Berlin, 1982). [2] V. Garibaldi, A.C. Levi, R. Spadacini and G.E. Tommei, Surface Sci. 48 (1975) 649. [3] N. Garcia, J. Chem. Phys. 67 (1977) 897. [4] B. Salanon and G. Armand, Surface Sci. 112 (1981) 78. [5] G. Armand and J.R. Manson, Phys. Rev. Letters 43 (1979) 1839. [6] G. Armand, J. Physique 41 (1980) 1475. [7] A. Liebsch and J. Harris, Surface Sci. 111 (1981) L721; J. Harris and A. Liebsch, to be published. [S] N. Garcia, J. Barker and I. Batra, to be published. [9] N. Garcia, Phys. Rev. Letters 37 (1976) 912. [IO] P. Cantini, G.P. Felcher and R. Tatarek, in: Proc. 7th Intern. Vacuum Congr. and 3rd Intern. Conf. on Solid Surfaces, Vienna, 1977, Eds. R. Dobrozemsky et al., p. 1357. [ 111 J. Lapujoulade, Y. Le Cruer, M. Lefort, Y. Lejay and E. Maurel, in: Proc. 4th Intern. Conf. on Solid Surfaces and 3rd European Conf. on Surface Science, Cannes, 1980 [Suppl. Le Vide, Les Couches Minces 201 (1980)]. [12] T. Engel and K.H. Rieder, Surface Sci. 109 (1981) 140. [13] P. Cantini, R. Tatarek and G.P. Felcher, Phys. Rev. B19 (1979) 1161. [14] K.H. Rieder, N. Garcia and V. Celli, Surface Sci. 108 (1981) 169. [IS] P.M. Van den Berg and J.T. Fokkema, J. Opt. Sot. Am. 69 (1979) 27.
Surface Science 133 (1983) 305-310 North-Holland Publishing Company
AN IMPROVED PROCEDURE FOR INVERTING ATOM DIFFRACTION INTENSITIES FOR STRUCTURAL DETERMINATION R. JAMES Department
*, D.S. KAUFMAN of Chemistry
and T. ENGEL
BG - IO, University of Washington, Seattle,
Received 5 January 1983; accepted for publication
Washington 98195, USA
7 June 1983
A method has been developed to obtain the surface corrugation function from atom diffraction intensities. Two procedures are outlined and it is shown that a stepwise inversion can be carried out which does not require assumptions about the corrugation amplitude or the phases of the diffracted beams. The rapid convergence of the procedure allows such inversions to be carried out on laboratory size computers. A few examples are discussed to show the utility of the method.
1. Introduction Atom diffraction from surfaces is a technique which has recently made contributions to our understanding of the atom-surface potential and surface structure [l]. In particular, the unique sensitivity of atom diffraction to the electron distribution at surfaces has the potential to give direct information on chemical bounds at surfaces. Experimental diffraction data yield the magnitude and direction of the surface unit cell vectors since this requires only a knowledge of the angular positions of the diffraction maxima. However, the structural information within the unit cell, which is contained in the corrugation function S(R), can only be obtained through an analysis of the diffraction intensities. A number of methods have been proposed to calculate diffraction intensities from a known corrugation function [l-4] in the hard wall approximation. More recent studies have used realistic soft potential to model the surface [5-71. The corrugated hard wall remains a good approximation to the scattering problem if the atomic structure of the surface rather than the details of the atom-surface potential are of primary interest. Initial analyses of surface structures [7-121 were carried out by expanding the (unknown) corrugation function in a Fourier series and systematically varying the magnitude of the expansion parameters until agreement was reached between the experimental and calculated intensities. However, this * Permanent address: Department California 94380, USA.
0039-6028/83/0000-0000/$03.00
of Mathematics,
California
State College, Stanislaus,
0 1983 North-Holland
Turlock,
R. James et at. / Inverting atom diffraction intensities
306
procedure is time consuming and it would be of advantage to develop a method which proceeds directly from the observed intensities to l(R). Cantini et al. 1131 have developed a method of this type using Patterson transform techniques and more recently. Rieder et al. [14] have developed an alternative inversion procedure which starts with the observed diffraction intensities. These authors assume phases of the most intense diffraction beams and the Kirchhoff approximation [ 151 is used to generate a starting function S’(R). An iterative procedure is then used to improve {O(R) until agreement between the experimental and observed ~ffraction intensities is achieved. In this study we report an improved inversion procedure which is also based on the hard wall model. It converges more rapidly than the procedure of ref. [14] and in our model studies has required considerably less computer time. Our method and that of ref. [14] are at present restricted to one dimensional corrugation functions.
2. Procedure We begin with the hard wall boundary equation in the Rayleigh ansatz CA,
exp(iG . R) exp[ik,,
Z(R)]
= - exp[ik,,
S(R)]
,
(1)
G
in which the attractive atom-surface potential is included only as a refractive index [I]. This limits the applicability of the method to cases in which no bound state resonances are encountered. This is not a severe restriction since these conditions can be met for He diffraction if sufficiently short wavelengths and angles of incidence di near normal are used. In eq. (l), G are the reciprocal lattice vectors of the surface, and the incoming and outgoing wavevectors ki and k, have been separated into their components parallel and normal to the surface respectively kj = (KS kizJt
(2)
k,=(K+G,k,,),
(3)
we expand 3(R) in a Fourier series
P(Ri) = c
g, cos(2snR,/a)
n
for a set of NG equidistant points Ri in the unit cell of length u. The number of points is chosen to be equal to the number of experimentally observed diffraction peaks. Choosing initial values for the coefficients {,, (they need not be very close to the true values for small corrugation amplitudes), we calculate the complex amplitudes AZ’ by solving the complex linear system
R. James et al. / Inverting atom diffraction intensities
CA,
exp(iG - Ri) exp[ik,,
l( R~)] = - exp[ik,
S( Ri)]
307
(5)
G
for i= l,..., NG. This is equivalent to the GR method discussed by Garcia [3]. The calculated diffraction intensities are given by
which has been
(6) The experimental results are taken into account diffraction amplitudes given by A’, = A:’ IA”c”P/A”c”‘l.
by forming a new set of (7)
In the above equation the amplitudes A$ have been adjusted such that the calculated phases have been retained, but the experimental amplitudes have been substituted for the calculated quantities. Using these adjusted values of A, new values of S(Ri) are calculated from eq. (5) in an iteration procedure using Newton’s method. The previous set of {(Ri) are used as Starting values and convergence is rapid, usually requiring only 2 or 3 iterations. The imaginary part of {(Ri) is discarded and a least squares fitting procedure is used to calculate the coefficients S, in eq. (4). This procedure is repeated until s=C[Im5(Ri)12,
(8)
reach acceptably low values. In eq. (9), NG is the number of diffraction beams used in fitting the data. The increased speed over the method described in ref. [14] results from the elimination of the time consuming procedure which was used to arrive at a set of starting phases and from these [O(R), and the faster convergence of Newton’s method in calculating the A,. It brings the time for inversion calculations down to a value compatible with laboratory computers such as the LSI 1 l/23 which was used in these studies. 3. Application of the method We have tested the method described above for both simple and more complex corrugation functions. The values of A, were computed from the test corrugation function using eq. (5). These values were used to generate { PgxP> using eq. (6) and these data were inverted to retrieve l(R). For a corrugation function with a single coefficient, .X(x) = f{, cos(2ax/a). Table
(10)
1 lists the values of the starting parameter c, for which the inversion
308
R. James et al. / Inverting atom diffraction intensities
Table 1 Corrugation parameter 3,
Reciprocal lattice vectors used
Input parameters which converged to correct value
0.10
-1Oto6 -1Oto6 -1Oto6 -1Oto6
0 to 0.22 0. to 0.23 0.20 to 0.54 0.37 to 0.66
0.20 0.40 0.50
procedure converged to the correct value. In this example, a = 3.52 A, X = 0.57 A, Bi = 20’ and four G vectors lying outside the Ewald sphere have been included in the calculation: Since these calculations require less than one minute to achieve unitarity within 0.001 and R and S values below 1 x 10p4, a set of starting values for Ii can be rapidly tested. In all cases the final value of 3, was equal to the correct value to within 0.1% after at most 5 iterations. The Rayleigh assumption limits the convergence for the above parameter to S1 = 0.504 [ 151. We have also tested this procedure on the corrugation function given as model III in table 2 of ref. [14]. Using cf = 7.04 A, X = 0.57 A and 8 = 20’ and a test corrugation function with 1, = 0.23, 5; = 0.15 and 5, = 0.08 to generate the (PFP}, convergence to the correct values was obtained for nearly all values in the intervals 0.15 < l, -C0.26, 0.10 < l2 < 0.18 and 0 -Cl3 c 0.12. In these calculations which included four G vectors outside the Ewald sphere, unitarity was achieved to within 0.0001. Fewer than 20 iterations which require 8 min computational time were needed.
4. A procedure for the calculation of starting values for c(R) The model calculations described above show that for starting values of & suitably close to their true values, our method converges rapidly. However, it would be convenient to have a method which would avoid the necessity of choosing starting values for &. The results shown in table 1 indicate that for small li, convergence occurs for starting values quite different from the true value. We have utilized this behavior by slowly increasing the corrugation in our best fit functions S(R) in the following way. We begin with a set lAzP1 corresponding to the true corrugation S(R). We calculate a set of AZ’ from eq. (5) using {, = lo-* and S;,= 0, n > 1. These Ag’ are used to generate a new set of A& given by A& = VIA”,“’ IA”G”PI/JAg’l,
G * 0,
where initially 2-’ -Cm -C2-5. In generating the
(11) A&,
the calculated phases are
R. James ei al. / Inverting
atom diffraction intensities
309
retained and the intensities of all beams for which G * 0 are reduced by the factor m to mimic a corrugation of much smaller amplitude. A renormalized set of amplitudes A&, given by
G
are then used in eq. (5) to solve for S(Ri). The imaginary part is discarded and a least squares procedure is used to calculate new coefficients Si. There 3, are used to repeat the procedure outlined above until succesive iterations lead to no significant change in the &. At this point m is increased to a somewhat higher value. The entire procedure is repeated until m is unity at which point the & are the best fit parameters to the experimental data. It was found that the most rapid change in the coefficients Si is observed for 0.5 < m < 0.9 and in this range m is usually increased in small increments. To test this procedure we examined three special cases of 5( Ri) given by eq. (4) for the parameters a = 3.52 A, Bi = 30” and X = 0.57 A. For S, = C2 and [, = 0 for n 2 3, the input parameters were recovered for {, < 0.14 A corresponding to a peak-to-peak corrugation amplitude [,,, of 0.44 A. For S, = 2& and S;, = 0 for n 2 3, the input parameters were recovered for {, < 0.18 A corresponding to l,,, = 0.40 A. For l1 = S2 = l3 and {, = 0 for n z 4, the input parameters were recovered for S1 < 0.06 corresponding to lm,, = 0.26 A. Fewer than 165 iterations requiring approximately 50 min computational time were needed to satisfy unitarity to within 0.00001 and to reduce R and S to below 1 x lop3 in the worst cases and to below 1 x lo-’ in the best cases. From these test cases, it is clear that the range of corrugation functions for which the procedure converges to the correct value depends on the relative magnitudes of the parameters. The observed decrease in I,,,,, as the number of corrugation parameters increases is consistent with previous studies [ 151 which show that the convergence limit of the Rayleigh approximation decreases with the number of Fourier coefficients Si. However, the limits found for c,,,,, using this procedure indicates that it should be applicable to nearly all cases of clean metal surfaces as well as most cases of gas adsorption on metal surfaces. This inversion procedure has the important advantage that it requires no starting assumption to extract S(R) from the diffraction intensities.
5. Summary Two methods to invert atom diffraction intensity data to determine the one-dimensional corrugation function l(R) have been discussed. In the first of these, trial imput value of the corrugation function are required and an
310
R. James et al. / Inverting atom diffraction intensities
iterative method is used to converge to the correct value. A second method is discussed in which S(R) is determined from diffraction intensities without assumptions about the corrugation function or the phases of the diffracted beams. The range of applicability of the method is large enough that most cases of clean or adsorbate covered surfaces can be dealt with using either method.
Acknowledgements This research has been supported by the National Science Foundation under grant CHE-8109067. One of us (R.J.) has benefited from a NSF program which provides research opportunities for faculty members at small colleges.
References [I] T. Engel and K.H. Rieder, in: Structural Studies of Surfaces, Springer Tracts in Modern Physics, Vol. 91 (Springer, Berlin, 1982). [2] V. Garibaldi, A.C. Levi, R. Spadacini and G.E. Tommei, Surface Sci. 48 (1975) 649. [3] N. Garcia, J. Chem. Phys. 67 (1977) 897. [4] B. Salanon and G. Armand, Surface Sci. 112 (1981) 78. [5] G. Armand and J.R. Manson, Phys. Rev. Letters 43 (1979) 1839. [6] G. Armand, J. Physique 41 (1980) 1475. [7] A. Liebsch and J. Harris, Surface Sci. 111 (1981) L721; J. Harris and A. Liebsch, to be published. [S] N. Garcia, J. Barker and I. Batra, to be published. [9] N. Garcia, Phys. Rev. Letters 37 (1976) 912. [IO] P. Cantini, G.P. Felcher and R. Tatarek, in: Proc. 7th Intern. Vacuum Congr. and 3rd Intern. Conf. on Solid Surfaces, Vienna, 1977, Eds. R. Dobrozemsky et al., p. 1357. [ 111 J. Lapujoulade, Y. Le Cruer, M. Lefort, Y. Lejay and E. Maurel, in: Proc. 4th Intern. Conf. on Solid Surfaces and 3rd European Conf. on Surface Science, Cannes, 1980 [Suppl. Le Vide, Les Couches Minces 201 (1980)]. [12] T. Engel and K.H. Rieder, Surface Sci. 109 (1981) 140. [13] P. Cantini, R. Tatarek and G.P. Felcher, Phys. Rev. B19 (1979) 1161. [14] K.H. Rieder, N. Garcia and V. Celli, Surface Sci. 108 (1981) 169. [IS] P.M. Van den Berg and J.T. Fokkema, J. Opt. Sot. Am. 69 (1979) 27.