14.1 VLEED: Multibeam Resonant Diffraction
267
For electrons close to the binding energy, the path is about 150 atomic unit long.
However, if the energy of electron is small or large compared with the barrier height,
then the path is shortened to around 50 atomic unit. The other pair of reflection
and transmission coefficients was found by reversing the direction of integration.
The Runga-Kutta technique was used to integrate through the barrier. This approach
provides both ψ and ψ
. The initial wave function was taken to be a plane wave.
The partial-wave phase shifts were determined by numerically integrating the radial
Schrödinger’s equation for the ion cores and matching the solutions at the muffin-tin
radius.
14.1.4 Multi-Atom Calculation Code
The code used previously to do VLEED calculation was developed for a single atom
per unit cell and so it was unsuitable for systems where the surface had reconstructed
or where there was more than one atom per unit cell. To extend the analysis to such
complex systems, a multi-atom code is necessary with the following modifications
to the previous one-atom code [3]:
• Based on the code of Malmstrom and Rundgren [8], Lindroos had developed a code
to calculate the effect of the SPB on the I-E curves by counting in the reflection
and transmission coefficients of the SPB. A modification of the code of Lindroos
et al. [9] by dealing with the layer-scattering matrices to make it suitable for such
low energies. The Kambe’s method was used to sum the scattering from atoms in
the layers.
• The Runga-Kutta method was used to integrate the Schrödinger’s equation from
the surface layer out to distance from the substrate. The package by Malmstrom
and Rundgren calculated the reflection and transmission coefficients of a SPB by
integrating the image-like part of the potential to large distances from the substrate.
All these attempts to find the effect of the SPB assumed that the SPB could be
represented by a one-dimensional function of distance from the substrate.
• A subroutine sums the effect of the SPB interference to the substrate lattice scattering. Once the reflection and transmission coefficients are calculated, reflection
and transmission matrices for the SPB layer are produced. The scattering of a single layer from all incident beams into all exciting beams is represented by a (n ×
n) matrix, where n is the number of active beams. In this convention, the barrier
matrices must be diagonal, as the barrier is one-dimensional so a beam incident in
one direction cannot be scattered into a beam in other directions. To save space,
Thurgate represented the reflection and transmission matrices as vectors and wrote
a subroutine to add the SPB effect to the substrate scattering. This was done by
summing to infinity all multiple-scattering effects between the substrate and the
SPB.
267
For electrons close to the binding energy, the path is about 150 atomic unit long.
However, if the energy of electron is small or large compared with the barrier height,
then the path is shortened to around 50 atomic unit. The other pair of reflection
and transmission coefficients was found by reversing the direction of integration.
The Runga-Kutta technique was used to integrate through the barrier. This approach
provides both ψ and ψ
. The initial wave function was taken to be a plane wave.
The partial-wave phase shifts were determined by numerically integrating the radial
Schrödinger’s equation for the ion cores and matching the solutions at the muffin-tin
radius.
14.1.4 Multi-Atom Calculation Code
The code used previously to do VLEED calculation was developed for a single atom
per unit cell and so it was unsuitable for systems where the surface had reconstructed
or where there was more than one atom per unit cell. To extend the analysis to such
complex systems, a multi-atom code is necessary with the following modifications
to the previous one-atom code [3]:
• Based on the code of Malmstrom and Rundgren [8], Lindroos had developed a code
to calculate the effect of the SPB on the I-E curves by counting in the reflection
and transmission coefficients of the SPB. A modification of the code of Lindroos
et al. [9] by dealing with the layer-scattering matrices to make it suitable for such
low energies. The Kambe’s method was used to sum the scattering from atoms in
the layers.
• The Runga-Kutta method was used to integrate the Schrödinger’s equation from
the surface layer out to distance from the substrate. The package by Malmstrom
and Rundgren calculated the reflection and transmission coefficients of a SPB by
integrating the image-like part of the potential to large distances from the substrate.
All these attempts to find the effect of the SPB assumed that the SPB could be
represented by a one-dimensional function of distance from the substrate.
• A subroutine sums the effect of the SPB interference to the substrate lattice scattering. Once the reflection and transmission coefficients are calculated, reflection
and transmission matrices for the SPB layer are produced. The scattering of a single layer from all incident beams into all exciting beams is represented by a (n ×
n) matrix, where n is the number of active beams. In this convention, the barrier
matrices must be diagonal, as the barrier is one-dimensional so a beam incident in
one direction cannot be scattered into a beam in other directions. To save space,
Thurgate represented the reflection and transmission matrices as vectors and wrote
a subroutine to add the SPB effect to the substrate scattering. This was done by
summing to infinity all multiple-scattering effects between the substrate and the
SPB.