Tensor LEED

As described above, the intensity calculation for a certain structure should always be done from scratch. Even when the phase shifts are already known, the layer diffraction matrices must be computed and subsequently, the layers have to be stacked. Equivalently, large matrices have to be inverted at each energy of the spectrum as well as for each parameter set to be tested for the comparison with experiment This is even when the changes from one to the next trial structure are small, so that the spectra change only modestly. It is the basic concept of Tensor LEED (TL) to calculate these modest spectral changes by a perturbation 

Certainly, since spherical scattering symmetry applies only to the new atomic position, the matrix referring to the old position cannot be diagonal anymore. By using the spherical wave propagator Puli rj) (with I = (I, m)), which converts a set of spherical waves centered at rj to a set centered at rj + Sfj, we get 

We now want to describe the ampHtude change caused by a distortion of the surface, possibly including displacements of several atoms. Of course, we first need the wave field of the reference structure as this is to be perturbed. Imagine that we have a state e(koii) representing the plane wave with parallel momentum kon and energy e impinging on the surface. Assume further that we know the full operator G+, which by action on e(koii)) creates the wave field due to full multiple scattering at the reference stracture so that outside the crystal, the set of diffracted waves 

We now turn to the disturbed surface with G being the corresponding operator and g the (possibly new) surface parallel reciprocal lattice vectors. Formally, the new amplitudes of diffracted waves can be written as 

The necessary quantity G+ can be developed into a Born series using the atomic scattering modifications Stj, namely 

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Jentz D, Rizzi S, Barbieri A, Kelly D, Van Hove M A and Somorjai G A 1995 Surface structures of sulfur and carbon overlayers on Mo(IOO) a detailed analysis by automated tensor LEED Surf. Sc 329 14-31... 

Covers in great detail the praotioal applioation of low-energy eleotron diffraotion (LEED) for struotural studies, exoepting more reoent teohniques like tensor LEED and holography. 

Some more recent software uses the tensor LEED approximation of Rous and Pen-dry which can save a substantial amount of computer time [2.268-2.270]. In tensor LEED the amplitudes (0) of all escaping electron waves (spots) are first calculated conventionally as described above for a certain reference geometry. Then the derivatives of these amplitudes 5Ag/5ri with respect to small displacements of each atom i in this reference geometry are calculated. These derivatives are the constituents of the "tensor". The wave amplitude for a modified model geometry where atom i is displaced by the vector Aq is then approximately given by ... 

Subsequently, Mitchell s group in Vancouver, by means of a tensor-LEED study17 of the Cu (110)-(2 x 3)N surface structure, supported a reconstruction model in which the topmost layer is described as a pseudo-(100)-c(2 x 2)N overlayer with metal corrugation of about 0.52 A in the reconstructed layer. Each nitrogen adatom is almost coplanar with the local plane formed by the four neighbouring copper atoms. Of the four N atoms present in the unit mesh, three are also bonded to Cu atoms in the layer below and therefore are five coordinate. 

The interpretation of the HREELS spectrum and the structure belonging to the (2x2)-3CO LEED pattern has been the subject of some debate in the literature [57— 59], The CO stretch peak at the lower frequency had previously been assigned to a bridge-bonded CO [57], with obvious consequences for the way CO fills the (2x2) unit cell. A recent structural analysis from the same laboratory on the basis of tensor LEED has confirmed the structures of both the (V3xV3)R30° and the (2x2)-3CO as given in Fig. 8.14, i.e. with CO in linear and threefold positions in the (2x2)-3CO structure [58]. The assignments have also been supported by high-resolution XPS measurements [59],... 

A. Barbieri and M. A. Van Hove, Symmetrized Automated Tensor LEED Package, Version 4.1, Private communication (1999). 

Table 7. The coordinates of 18 atoms in the Si(100)2x2-Al stmcture as determined by tensor LEED [93 S8]. The atom numbers and axes are defined in Fig. 16.

Fig. 16. (a) Top and (b) side views of the parallel symmetric A1 dimer on the Si(lOO) surface. The coordinates of 18 atoms as determined by tensor LEED [93S8] are listed in Table 7. Experimentally determined and calculated values of Si dimer bond length (Lsi), A1 dimer bond length (Lj ), Al-Si back bond length (i Ai -sO and separation (AZ) between the planes containing A1 dimers and Si dimers are listed in Table 8. 

Rous PE (1992) The tensor LEED approximation and surface crystallography be low energy electron diffraction. Progress in Surface Science 39 3-63. 

Vogt J (2007) Tensor LEED study of the temperature dependent dynamics of the NaCl(lOO) single crystal surface. Phys Rev 875 125-423... 

J.D. Batteas, A. Barbieri, E.K. Starkey, M.A. Vanhove, G.A. Somorjai, The Rh(llO)-P2 Mg(2Xl)-20 surface-structure determined by automated tensor LEED—structure changes with oxygen coverage. Surf Sci. 339(1-2), 142-150 (1995)... 

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