266
14 Principles: Bond-Band-Barrier Correlation
As the energy of the beam increases, the perpendicular momentum of the diffracted
beam increases, so the beam travels further from the substrate, increasing the phase
change. When the beam energy is close to the emergence energy, a small increase
causes a large change in phase, so the peaks cluster together towards the fixed energy.
The same physical process is responsible for the image-like states seen in the inverse
PES spectra [6, 7]. Each pre-emergent beam contributes to the observed fine structure. The interference will produce visible modulations in the observed intensity.
The interference between the pre-emergent and the specular beams must meet the
following conditions:
• Same direction. The incident beam is partially diffracted into other beams that are
then diffracted by the substrate and the SPB repeatedly. These diffracted beams
are partially diffracted into the (00) direction being responsible for the detection.
• Compatible amplitudes. The pre-emergent beam and the specular beam must have
similar amplitudes for them to be any observable modulation and interfere in the
intensity.
• Fixed phase difference. Same wavelengths of the diffracted beams are vital to the
interference and the difference in phase must be an integer multiple of 2π.
The emergence energy of the pre-emergent beam meets Bragg-diffraction condition. The emergence energy depends on the two-dimensional crystal geometry only.
The dependence of the emergence energy on the two-dimensional crystal geometry
can be assessed from the Ewald construction. The emergence energy corresponds
to the point where the sphere first touches the rod in the reciprocal space. The finestructure features converge in a Rydberg-like series and the height and shape of the
SPB modulates the beam energies slightly.
14.1.3 Scattering Matrices and Phase Shift
The barrier-scattering matrices were calculated by integrating the complex
Schrödinger’s equation from the position of the top atomic layer to a point some
distance away from the surface where the influence of the potential has become
insignificant. To create all the required reflection and transmission coefficients, the
integration was started from inside the crystal to large distances in the negative
z-direction pointing outward of the surface. The integration range depends on the
energy of the electron compared to the inner potential. The proper reflection (r) and
transmission (t) coefficients can be found from the value of the wave function and its
derivative following this integration, using the expressions given in Eq. (14.1). The
choice of the integration limits depends on which part of coefficients is being evaluated. If r
−+ and t
− are being evaluated, then the integration of the wave function is
started at some large distance from the surface and continued into the point z, which
is the terminal of the surface z = 0. The choice of where the integration starting from
depends on the energy of the electron compared to the potential.
14 Principles: Bond-Band-Barrier Correlation
As the energy of the beam increases, the perpendicular momentum of the diffracted
beam increases, so the beam travels further from the substrate, increasing the phase
change. When the beam energy is close to the emergence energy, a small increase
causes a large change in phase, so the peaks cluster together towards the fixed energy.
The same physical process is responsible for the image-like states seen in the inverse
PES spectra [6, 7]. Each pre-emergent beam contributes to the observed fine structure. The interference will produce visible modulations in the observed intensity.
The interference between the pre-emergent and the specular beams must meet the
following conditions:
• Same direction. The incident beam is partially diffracted into other beams that are
then diffracted by the substrate and the SPB repeatedly. These diffracted beams
are partially diffracted into the (00) direction being responsible for the detection.
• Compatible amplitudes. The pre-emergent beam and the specular beam must have
similar amplitudes for them to be any observable modulation and interfere in the
intensity.
• Fixed phase difference. Same wavelengths of the diffracted beams are vital to the
interference and the difference in phase must be an integer multiple of 2π.
The emergence energy of the pre-emergent beam meets Bragg-diffraction condition. The emergence energy depends on the two-dimensional crystal geometry only.
The dependence of the emergence energy on the two-dimensional crystal geometry
can be assessed from the Ewald construction. The emergence energy corresponds
to the point where the sphere first touches the rod in the reciprocal space. The finestructure features converge in a Rydberg-like series and the height and shape of the
SPB modulates the beam energies slightly.
14.1.3 Scattering Matrices and Phase Shift
The barrier-scattering matrices were calculated by integrating the complex
Schrödinger’s equation from the position of the top atomic layer to a point some
distance away from the surface where the influence of the potential has become
insignificant. To create all the required reflection and transmission coefficients, the
integration was started from inside the crystal to large distances in the negative
z-direction pointing outward of the surface. The integration range depends on the
energy of the electron compared to the inner potential. The proper reflection (r) and
transmission (t) coefficients can be found from the value of the wave function and its
derivative following this integration, using the expressions given in Eq. (14.1). The
choice of the integration limits depends on which part of coefficients is being evaluated. If r
−+ and t
− are being evaluated, then the integration of the wave function is
started at some large distance from the surface and continued into the point z, which
is the terminal of the surface z = 0. The choice of where the integration starting from
depends on the energy of the electron compared to the potential.