Patterson2D

Surface structure determination

Detlef Smilgies

The 2D Patterson function

From a 2D data set I_hk with the usual corrections due to the Lorentz, polarization and area factor [Robinson, Smilgies], we can obtain a Patterson function by

P_2D(x,y) = S_hk I_hk exp{2pi (h x + k y)}

However, we have to take a bit of care with its interpretation. Within the Fourier language this Patterson function is derived from

P_2D(x,y) = FT { |S(q)|² d(q_z)}

The delta function in qz describes that we only considered a thin slice of q space close to q_z=0, the so-called in-plane reflections. Evaluation yields

P_2D(x,y) = { {r*r}(r) * (d(x) d(y) 1_z) } = ò {r*r}(r) dz

i.e. P_2D is the density autocorrelation projected along z. So all atoms within the surface layer show up regardless of their relative height above the surface. (Note that this is a general result that can also be used for the analysis of 3D data!)

Another complication arises from the fact that we have a data set with systematic absences due to overlap of the monolayer reflections with scattering from the substrate. Because of these regularly spaced missing reflections, the positivity of the Patterson function does no longer hold, the incomplete Patterson function will also contain negative peaks. It has been shown [Feidenhans'l] that only the positive peaks carry the information about the interatomic vectors, and only those should be used to construct a surface model.

Fourier difference method

If atoms making up a structure have very different Z values, often only information about the heavier atoms can be obtained from the Patterson function. If a first model of the surface structure containing only the heavy atoms looks promising, the Fourier difference method can be used to gain information about the lighter atoms. In this approach we assume that the phase of the reflections f_hk,model is essentially determined by the heavy atom locations, and that we can subtract the heavy atom contributions, to shed some light on the weak structures of the light atoms. This map is described by

F(x,y) = S_hk (f_hk,exp - f_hk,model) exp(2p i [ h x + k y] + f_hk,model)

where we write the calculateted structure factor of the model as

S_hk = f_hk,model exp( i f_hk,model )

The experimentally measured scattering factors f_hk are given by

f_hk = sqrt( I_hk )

F(x,y) is a density difference map, since we have regained the phase information from the model of the heavy atoms. The positive peaks in F(x,y) are candidate locations for the lighter atoms. If needed, this approach can be iterated.

Structure refinement

The final structure determination is done by refining the model obtained by the Patterson function and the Fourier difference method by least-square fitting of model intensities and experimentally determined intensities. At this point, also the substrate has to be included. Integer-order reflections that do not coincide with bulk Bragg reflections of the substrate provide information, on how the surface layer is located relative to the substrate, so that adsorptions sites can be identified. At this step it is very important to carefully analyze the symmetry requirements, as imposed by the known symmetry of the substrate and the measured reflections, in order to cut down the number of fitting parameters enough for a stable fit. A systematic description on how the (Ö5 ´ Ö5) structure of O on Mo(001) was solved, can be found in [Robinson].

The public domain ROD/ANA software package for data analysis and structure refinement by Elias Vlieg can be found at the ESRF Scientific Software web site (see references below). The site also contains various modifications of the original package suitable for thin films (Robach) and for molecular monolayers (Svensson/Smilgies). Manuals for the use of these packages can also be found on this site.

Direct methods

Nowadays also direct methods for surface structure analysis have been developed. The problem is a lot more difficult than in 3D, because of the systematic absences. Some of the most challenging surface structures have recently solved by direct methods. As an example the c(8x2) reconstruction of the InSb(001) surface [Kumpf].

References

- selected papers -

I.K. Robinson, D.-M. Smilgies, and P.J. Eng, "Cluster Formation in the Adsorbate-Induced Reconstruction of the O/Mo(001) Surface", J. Phys.: Condens. Matter 4, 5845 (1992).

D.-M. Smilgies, "Geometry-independent intensity correction factors for grazing-incidence diffraction", Rev. Sci. Instr.,
accepted (preprint).

C. Kumpf, L. D. Marks, D. Ellis, D. Smilgies, E. Landemark, M. Nielsen, R. Feidenhans'l, J. Zegenhagen, O. Bunk, J. H. Zeysing, Y. Su, and R.L. Johnson: "Subsurface dimerization in III-V semiconductor (001) surfaces", Physical Review Letters 86, 3586 (2001).

- reviews -

R. Feidenhans'l, Surf. Sci. Rep. 10, 105 (1989).
I.K. Robinson and D.J. Tweet, Rep. Prog. Phys. 55, 599 (1992).

- software and manuals -

ANA/ROD project at ESRF
E. Vlieg, A concise ROD manual
E. Vlieg, From beamtime to structure factors
O. Svensson and D. Smilgies, Manual for Svensson's extension of ROD