3.8 KiB
Playing with the scattering factor
In this tutorial we will play with an important parameter of any multiple scattering calculation: the scattering factor. When a electron wave is scattered by an atom, the electron trajectory is modified after this event. The particle will most likely continue its trajectory in the same direction (forward scattering), but, to a lesser extent and depending on the atom and on the electron energy, the direction of the scattered electron can change. The electron can even be backscattered. The angular distribution of the electron direction after the scattering event is the scattering factor.
In a paper published in 1998, T. Gerber et al. used the quite high backscattering factor of Rhodium atoms to probe the distance of Oxygen atoms adsorbed on a Rhodium surface. Some electrons coming from Oxygen atoms are ejected toward the Rhodium surface. They are then backscattered and interfere with the direct signal comming from Oxygen atoms (see the figure below). They demonstrated both experimentally and numerically with a sinle scattering computation that this lead to a very accurate probe of adsorbed species that can be sensitive to bond length changes of the order of ±0.02Å.
Interferences produced by the backscattering effect.
First, compute the scattering factor of both chemical species, Rh and O.
System Message: ERROR/3 (<stdin>, line 31)
Unknown directive type "literalinclude".
.. literalinclude:: sf.py
:linenos:
Running the above script should produce this polar plot. You can see that for Rhodium, the backscattering coefficient is still significant even at quite high kinetic energy (here 723 eV).
Polar representation of the scattering factor
Let an Oxygen atom being adsorbed at a distance z0 of an fcc site of the (111) Rh surface. and compute the θ − ϕ scan for different values of z0. You can see on the stereographic projection 3 bright circles representing fringes of constructive interference between the direct O(1s) photoelectron wave and that backscattered by the Rhodium atoms. The center of these annular shapes changes from bright to dark due to the variation of the Oxygen atom height above the surface which changes the path difference.
System Message: ERROR/3 (<stdin>, line 57)
Unknown directive type "only".
.. only:: html
.. figure:: RhO_fig2b.gif
:align: center
:width: 80%
Stereographic projections of O(1s) emission at :math:`E_k` = 723 eV for an
oxygen atom on top of a fcc site of 3 Rh atoms at various altitudes
:math:`z_0`
System Message: ERROR/3 (<stdin>, line 67)
Unknown directive type "only".
.. only:: latex
.. figure:: RhO_fig2b.png
:align: center
:width: 80%
Stereographic projections of O(1s) emission at :math:`E_k` = 723 eV for an
oxygen atom on top of a fcc site of 3 Rh atoms at various altitudes
:math:`z_0`
Here is the script for the computation. (:download:`download <RhO.py>`)
System Message: ERROR/3 (<stdin>, line 80); backlink
Unknown interpreted text role "download".System Message: ERROR/3 (<stdin>, line 82)
Unknown directive type "literalinclude".
.. literalinclude:: RhO.py
:linenos:
Note
After runing this script, you will get 20 images in your folder. You can merge them in one animated gif image like this:
System Message: WARNING/2 (<stdin>, line 89)
Cannot analyze code. Pygments package not found.
.. code-block:: bash
convert -delay 50 -loop 0 image*.png animation.gif
System Message: ERROR/3 (<stdin>, line 93)
Unknown directive type "seealso".
.. seealso::
X-Ray Photoelectron Diffraction in the Backscattering Geometry: A Key to Adsorption Sites and Bond Lengths at Surfaces
T. Gerber, J. Wider, E. Welti & J. Osterwalder, Phys. Rev. Lett. **81** (8) p1654 (1998)
`[doi] <https://doi.org/10.1103/PhysRevLett.81.1654>`__