Activity 5: Multiple scattering in the forward scattering regime

Activity 5: Multiple scattering in the forward scattering regime#

In photoelectron diffraction, it is well known that for high photoelectron kinetic energy (typically > 900 eV), the scattering factor is strongly peaked in the forward direction. It means that photoelectrons are almost not deviated after a scattering event.

Peaks of intentisity are then usually observed for dense atomic directions of the sample. This is the forward scattering approximation.

For such high kinetic energy, multiple scattering is needed to accurately describe the measured intensity, but the matrix inversion algorithm cannot be used since the memory needed for storing the matrix itself would be generally too large. The matrix will contain \((N \times (L_{max}+1)^2)^2\) elements of complex type with double precision (64 bits) where \(N\) is the number of atoms and \(L_{max}\) is the number of spherical harmonics used to expand the electron wave around each atomic center. As the kinetic energy increases, the mean free path (MFP) of the photoelectron is larger and the number of atoms in the cluster has to be greater. Lmax also increases with the kinetic energy.

Try to evaluate how much memory you would need for this matrix for a hemispherical cluster of copper 15 angströms thick (1 MFP) for \(L_{max} = 25\) ?

Hide code cell content 
import numpy as np

# lattice constant of fcc copper
a = 3.6
# radius of the cluster
r = 15
# volume of the cluster
V = .5 * (4/3) * np.pi * r**3
# volume of the cell
v = a**3
# number of atoms in the unit cell
n = 4
# number of atoms in the cluster
N = int(V/v * n)

Lmax = 25
M = (N * (Lmax+1)**2 )**2 * 2 * 64 / 8
print(f"{N:d} atoms, {M/1e12:.3f} TB")

606 atoms, 2.685 TB

This is too much memory. We will use another algorithm available in MsSpec: The Rehr-Albers series expansion. We already used that algorithm in activity 3 for the single scattering approach. But this time, we will explore a bit more the effect of the scattering order > 1

PED of the 1T-TiSe2 surface#

Let us try to model the Ti2p XPD pattern of the transition metal dichalcogenide 1T-TiSe2.

See also

based on this paper from M.V. Kuznetsov et al. Surf. Sci. 606 p1760–70 (2012)

Creating the TiSe2 cluster#

Start by creating a small cluster of 1T-TiSe2 using the mx2 function of ase.build and the hemispherical_cluster function of msspec.utils.

TiSe2

Structure of 1T-TiSe2 (\(a_0=b_0=3.535\) Å, \(c_0=6.004\) Å, \(d=3.450\) Å, \(D=2.554\) Å)#

Complete the code snipet provided below (or here) to create a small TiSe2 cluster with Ti emitter in the 2nd plane:

ASE atomic visualization

Effect of the scattering order#

Use the line belows to create a calculator and compute a \(\theta\)-\(\phi\) scan of the Ti(2p)

Compute a scan for an emitter in the first trilayer and in the second trilayer for scattering orders from 1 (single scattering) to 3 in order to complete the figure below. What do you conclude about the value of the calc.calculation_parameters.scattering_order ?

light/results.jpg

\(\theta\)-\(\phi\) scan of Ti(2p) at 1030 eV kinetic energy for an emitter in the first trilayer (left column) and in the second trilayer (right column). Each row correspond to a growing value for the calc.calculation_parameters.scattering_order parameter (from 1 to 5).#

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