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"# Activity 1: Getting started\n",
"\n",
"MsSpec is a Fortran code with two components: Phagen (Written by R. Natoli) and Spec (written by D. Sébilleau). Phagen computes the phase shifts of the electronic wave propagating in the matter on a spherical harmonics basis. Spec uses those phase shifts to compute the multiple scattering process and simulate the intensity of different electronic spectroscopies.\n",
"\n",
"In the most recent version of MsSpec, the program is interfaced with python (https://msspec.cnrs.fr/), allowing for much more flexibility and interplay with other simulation techniques.\n",
"\n",
"## Building atomic systems\n",
"\n",
"MsSpec works in the *cluster* approach. Building such a cluster for a calculation is a fundamental step.\n",
"We use the [python Atomic Simulation Environment (ASE)](https://wiki.fysik.dtu.dk/ase/) for this.\n",
"\n",
"ASE is a set of tools and Python modules for setting up, manipulating, running, visualizing and analyzing atomistic simulations.\n",
"Building atomic systems, structures... is pretty straightforward:"
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"# To build a molecule with ASE\n",
"from ase.build import molecule\n",
"# To view\n",
"from ase.visualize import view\n",
"\n",
"# Create a water molecule\n",
"water = molecule('H2O')\n",
"# Display it\n",
"view(water)"
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"## Barebone script for MsSpec\n",
"\n",
"MsSpec can simulate different electronic spectroscopies like PED, AED, LEED, EXAFS, APECS and more will be included in the forthcoming version. However, it is really well-suited for PhotoElectron Diffraction simulation, and the python interface is only fully available for it at the moment. Since PED covers all the MsSpec features and concepts, we will focus on this technique.\n",
"\n",
"There are typically 3 steps to follow to get a result with MsSpec:\n",
"\n",
"1. Create a *cluster*\n",
"2. Create an ASE *calculator*\n",
"3. Run the simulation\n",
"\n",
"### PED polar scan for Cu(001)\n",
"\n",
"download the [cu.py](cu.py \"download\") python script and the [copper.cif](copper.cif \"download\") file. Put those files in the same folder. You can run your first MsSpec calculation by typing in a terminal:\n",
"\n",
"```shell\n",
"$ python cu.py\n",
"```\n",
"\n",
"Here is the content of the script file:"
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"```{literalinclude} cu.py\n",
":end-before: The \"emitter\"\n",
":linenos:\n",
"```"
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"source": [
"from ase.io import read\n",
"from ase.visualize import view\n",
"from msspec.calculator import MSSPEC\n",
"\n",
"\n",
"cluster = read('copper.cif')\n",
"# view the cluster\n",
"view(cluster, viewer='x3d')"
]
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"```{literalinclude} cu.py\n",
":start-at: The \"emitter\"\n",
":lineno-match:\n",
"```\n",
":::{figure-md} Cu-XPD\n",
"<img src=\"fig1.png\" width=\"600px\" align=\"center\">\n",
"\n",
"Cu(2p) polar scan for the copper cluster above\n",
":::"
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"source": [
"## Shaping a cluster"
]
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"::::{tab-set}\n",
"\n",
":::{tab-item} <i class=\"fa-solid fa-circle-question\"></i> Quiz\n",
"Based on the previous *.cif file, create a new cluster without the deepest plane and run the same calculation for the same emitter\n",
"\n",
"```{note}\n",
"Use the `cluster.edit()` method to interactively remove atoms...\n",
"\n",
"As the cluster will contain fewer atoms, the emitter index will be different\n",
"```\n",
"\n",
"What do you conclude ?\n",
":::\n",
"\n",
"::::"
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"The number of atoms used for the calculation greatly impact the calculation time and memory. Most of the time, a cluster is shaped as an hemisphere to minimize the number of atoms to take into account"
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"source": [
"from msspec.calculator import MSSPEC\n",
"from msspec.utils import hemispherical_cluster, get_atom_index\n",
"from ase.build import bulk\n",
"from ase.visualize import view\n",
"\n",
"copper = bulk('Cu', cubic=True)\n",
"cluster = hemispherical_cluster(copper, planes=3, emitter_plane=2)\n",
"cluster.emitter = get_atom_index(cluster, 0,0,0)\n",
"\n",
"view(cluster)"
]
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":::{figure-md} Cu-hemi\n",
"<img src=\"fig3.png\" width=\"600px\" align=\"center\">\n",
"\n",
"Cu(2p) polar scan for the hemispherical cluster.\n",
":::"
]
}
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"(ssc)=\n",
"# Activity 3: Adsorbates and the single scattering approach\n",
"\n",
"Photoelectron diffraction is widely used to study the adsorption of atoms or molecules on a crystalline surface. Photoelectrons from adsorbates are scattered by the underlying surface, carrying information about the adsorption site, bond length and/or molecule orientation…. Thanks to a simulation, such information becomes quantitative with a high degree of accuracy.\n",
"\n",
"Calculations of the multiple scattering using matrix inversion have the great advantage of being exact, including all scattering paths. On the other hand, memory consumption soon becomes a problem as the kinetic energy and number of atoms to be considered increase. As an approximation, it is possible to only consider a single scattering from the emitter to any atom in the cluster. This approximation is extremely computationally fast and can give satisfactory results for adsorbates. We’ll see later that this approach is rather too simplistic for most cases.\n",
"\n",
"## Oxygen on Rh(001)\n",
"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 {math}`\\pm 0.02 \\mathring{A}`.\n",
"\n",
":::{seealso}\n",
"based on this paper from T. Greber *et al.* [Phys. Rev. Lett. **81**(8) p1654 (1998)](https://doi.org/10.1103/PhysRevLett.81.1654)\n",
":::"
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":::{figure-md} RhO-fig\n",
"<img src=\"RhO_fig0.jpg\" alt=\"RhO\" width=\"300px\" align=\"center\">\n",
"\n",
"Interferences produced by the backscattering effect\n",
":::"
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"### Computing the scattering factor\n",
"\n",
"To illustrate that photoelectrons emitted by Oxygen adsorbates towards the Rhodium surface can be backscattered, we will start by computing the scattering factor for both O and Rh atoms.\n",
"\n",
"::::{tab-set}\n",
"\n",
":::{tab-item} <i class=\"fa-solid fa-circle-question\"></i> Quiz\n",
"By using the [`Atoms`](https://wiki.fysik.dtu.dk/ase/ase/atoms.html#ase.Atoms) class of the `ase` package, try to build a O-Rh chain where atoms are 4 Å apart. Here is the begining of the script. Try to complete the line of code and view your two-atoms chain.\n",
"\n",
"```python\n",
"from ase import Atoms\n",
"from ase.visualize import view\n",
"\n",
"# Create an atomic chain O-Rh\n",
"cluster = Atoms(... # Fill this line\n",
"```\n",
":::\n",
"\n",
"::::"
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"from ase import Atoms\n",
"from ase.visualize import view\n",
"\n",
"# Create an atomic chain O-Rh\n",
"cluster = Atoms(['O', 'Rh'], positions = [(0,0,0), (0,0,4.)])"
]
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"As previously, we create a calculator, we attach our 2 atoms cluster to this calculator and we define the first atom in the chain as the emitter\n",
"\n",
":::{literalinclude} RhO_sf.py\n",
":start-at: calc =\n",
":end-at: emitter =\n",
":::"
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"::::{tab-set}\n",
"\n",
":::{tab-item} <i class=\"fa-solid fa-circle-question\"></i> Quiz\n",
"We use the `get_scattering_factors` method [(see its documentation)](https://msspec.cnrs.fr/modules/calculator.html#calculator._PED.get_scattering_factors) to compute the scattering factors at 723 eV.\n",
"\n",
"How large is the backscattering factor of Rhodium with respect to that of Oxygen ?\n",
":::\n",
"\n",
"::::"
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"### Interferences due to backscattering\n",
"\n",
"Let an Oxygen atom (in red) being adsorbed at a distance $z_0$ of an *fcc* site of the Rh(111) surface."
]
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":::{figure-md} Rho-fig2a\n",
"<img src=\"RhO_fig2a.jpg\" width=\"600px\" align=\"center\">\n",
"\n",
"Small cluster used for the calculation.\n",
":::"
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"We will compute for different values of the adsorption height $z_0$.\n",
"\n",
"::::{tab-set}\n",
"\n",
":::{tab-item} <i class=\"fa-solid fa-circle-question\"></i> Quiz\n",
"Complete the [script below](RhO_tofill.py \"download\") to compute the ($\\theta,\\phi$) scan of the photodiffraction of O(1s) adsorbed on a *fcc* site on Rh(111) surface.\n",
"\n",
"```{literalinclude} RhO_tofill.py\n",
":lineno-match:\n",
":emphasize-lines: 6,8,12,13\n",
"```\n",
"\n",
"As proposed in the comments, add a loop to vary the adsorption height of Oxygen between 1.10 and 1.65 Å.\n",
"What is the bond length difference between to intensity maxima ?\n",
":::\n",
"\n",
"::::"
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"(ssc2ms)=\n",
"# Activity 4: From single scattering to multiple scattering\n",
"\n",
"In the [previous activity](ssc), we saw that simple single scattering calculations (SSC) can be used to simulate photodiffraction diagrams with good accuracy. The approximation works fine when the emitting atom is very close to the surface.\n",
"However, the SSC approach is no longer suitable for deeper emitter atoms, where multiple scattering effects come into play. In this activity, we will focus on a major consequence of multiple scattering: *the defocusing effect*.\n",
"\n",
"The defocusing effect is presented in the [figure below](Ni-fig1) for a chain of nickel atoms. Although purely illustrative, understanding multiple scattering in atomic chains is fundamental because they are found in many situations, such as in particular directions of a crystal or in molecules of various lengths."
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":::{figure-md} Ni-fig1\n",
"<img src=\"defocusing_animation.gif\" alt=\"defocusing effect\" width=\"600px\" align=\"center\">\n",
"\n",
"The defocusing effect dur to multiple scattering in an atomic chain of Ni atoms.\n",
":::"
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"In 1989, M.-L Xu, J.J. Barton and M.A. Van Hove studied these multiple scattering effects on atomic chains ([see their paper below](defocusing-paper)).\n",
"In the spirit of figure 3 of their paper, we will create 3 atomic chains of Ni atoms (2, 3 and 5 atoms) tilted by 45° and we will compare the intensity of the forward scattering peak for single scattering and for full multiple scattering."
]
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"(defocusing-paper)=\n",
":::{seealso}\n",
"based on this paper from M.-L. Xu *et al.*\n",
"[Phys. Rev. B **39** p8275 (1989)](https://doi.org/10.1103/PhysRevB.39.8275) \n",
":::"
]
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"## Polar scans of Ni atomic chains\n",
"\n",
"### Building a chain of atoms\n",
"\n",
"Start by creating a simple chain of 2 Ni atoms: an emitter and a scatterer in the [101] direction.\n",
"\n",
":::{tip}\n",
"Nickel is *fcc* with lattice parameter $a$=3.499 Å. Use the [`Atoms`](https://wiki.fysik.dtu.dk/ase/ase/atoms.html#ase.Atoms) class of `ase` like in the [previous activity](ssc)...\n",
"\n",
":::{admonition} if you need help to start...\n",
":class: dropdown\n",
"\n",
":::{code} python\n",
"from msspec.calculator import MSSPEC\n",
"from ase import Atoms\n",
"\n",
"symbol = ... # The kind of atom for the chain\n",
"a = ... # The distance bewteen 2 atoms\n",
" # in [101] direction\n",
"\n",
"chain = Atoms(..., positions=[...])\n",
"chain.rotate(...)\n",
"chain.edit()\n",
":::\n",
"\n",
":::\n",
"\n",
":::"
]
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"::::{tab-set}\n",
"\n",
":::{tab-item} <i class=\"fa-solid fa-circle-question\"></i> Quiz\n",
"Create an `MSSPEC` calculator with `expansion` algortithm and set the `scattering_order`=1 to compute a polar scan of the Ni(3s) in single scattering. How is varying the height of the peak at 45° (along the chain) if you increase the number of atoms in the chain ?\n",
"\n",
"Repeat the same experiment with `inversion` algorithm for having the full multiple scattering result. What do you observe ?\n",
":::\n",
"\n",
"::::"
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"(forward-scattering)=\n",
"# Activity 5: Multiple scattering in the forward scattering regime"
]
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"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.\n",
"\n",
"Peaks of intentisity are then usually observed for dense atomic directions of the sample. This is the **forward scattering approximation**.\n",
"\n",
"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",
"$(N \\times (L_{max}+1)^2)^2$ 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."
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"::::{tab-set}\n",
"\n",
":::{tab-item} <i class=\"fa-solid fa-circle-question\"></i> Quiz\n",
"Try to evaluate how much memory you would need for this matrix for a hemispherical cluster of copper 15 angströms thick (1 MFP) for $L_{max} = 25$ ?\n",
":::\n",
"\n",
"::::"
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"606 atoms, 2.685 TB\n"
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"source": [
"import numpy as np\n",
"\n",
"# lattice constant of fcc copper\n",
"a = 3.6\n",
"# radius of the cluster\n",
"r = 15\n",
"# volume of the cluster\n",
"V = .5 * (4/3) * np.pi * r**3\n",
"# volume of the cell\n",
"v = a**3\n",
"# number of atoms in the unit cell\n",
"n = 4\n",
"# number of atoms in the cluster\n",
"N = int(V/v * n)\n",
"\n",
"Lmax = 25\n",
"M = (N * (Lmax+1)**2 )**2 * 2 * 64 / 8\n",
"print(f\"{N:d} atoms, {M/1e12:.3f} TB\")"
]
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"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"
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"## PED of the 1T-TiSe<sub>2</sub> surface"
]
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"Let us try to model the Ti2p XPD pattern of the transition metal dichalcogenide 1T-TiSe<sub>2</sub>.\n",
"\n",
":::{seealso}\n",
"based on this paper from M.V. Kuznetsov *et al.*\n",
"[Surf. Sci. **606** p1760–70 (2012)](https://doi.org/10.1016/j.susc.2012.06.008)\n",
":::\n",
"\n",
"### Creating the TiSe{sub}`2` cluster\n",
"\n",
"Start by creating a small cluster of 1T-TiSe<sub>2</sub> using the `mx2` function of `ase.build` and the `hemispherical_cluster` function of `msspec.utils`."
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":::{figure-md} TiSe2-fig\n",
"<img src=\"TiSe2_cell.jpg\" alt=\"TiSe2\" width=\"300px\" align=\"center\">\n",
"\n",
"Structure of 1T-TiSe<sub>2</sub> ($a_0=b_0=3.535$ Å, $c_0=6.004$ Å, $d=3.450$ Å, $D=2.554$ Å)\n",
":::"
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"::::{tab-set}\n",
"\n",
":::{tab-item} <i class=\"fa-solid fa-circle-question\"></i> Quiz\n",
"Complete the code snipet provided below (or [here](TiSe2_1_tofill.py \"download\")) to create a small TiSe{sub}`2` cluster with Ti emitter in the 2{sup}`nd` plane:\n",
"\n",
"```{literalinclude} TiSe2_1_tofill.py\n",
":lines: 1-29\n",
":linenos:\n",
":emphasize-lines: 7,11,15,26 \n",
"```\n",
"\n",
":::\n",
"\n",
"::::"
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"source": [
"from ase.build import mx2\n",
"from ase.visualize import view\n",
"from msspec.calculator import MSSPEC\n",
"from msspec.utils import hemispherical_cluster, get_atom_index\n",
"\n",
"# Some usefull constants (a, c, d, D) for defining the structure\n",
"a=3.535;c=6.004;d=3.450;D=2.554\n",
"\n",
"# Create the TiSe2 trilayer\n",
"# use ase help for this function\n",
"TiSe2 = mx2(formula='TiSe2', kind='1T', a=a, thickness=d, size=(1, 1, 1), vacuum=None)\n",
"\n",
"# The preious cell is 2D, let's define the c-axis to take into account \n",
"# the Van der Waals gap between trilayers\n",
"TiSe2.cell[2] = [0, 0, c]\n",
"\n",
"# To be aligned like in the paper\n",
"TiSe2.rotate(60, 'z', rotate_cell=True)\n",
"\n",
"# Since the material is multi-elements, \"tag\" each inequivalent atom \n",
"# of the unit cell with a number. The \"Ti\" atom is tagged 0 and \"Se\" \n",
"# atoms are 1 and 2.\n",
"for i in range(3): \n",
" TiSe2[i].tag = i\n",
"\n",
"cluster = hemispherical_cluster(TiSe2, emitter_tag=0, emitter_plane=1, planes=5)\n",
"cluster.emitter = get_atom_index(cluster, 0, 0, 0)\n",
"view(cluster, viewer='x3d')"
]
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"### Effect of the scattering order\n",
"\n",
"Use the line belows to create a calculator and compute a $\\theta$-$\\phi$ scan of the Ti(2p)\n",
"\n",
"```{literalinclude} TiSe2_1_tofill.py\n",
" :start-at: Create a\n",
"```"
]
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"::::{tab-set}\n",
"\n",
":::{tab-item} <i class=\"fa-solid fa-circle-question\"></i> Quiz\n",
"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.\n",
"What do you conclude about the value of the `calc.calculation_parameters.scattering_order` ?\n",
"\n",
"```{figure-md} results-fig\n",
"<img src=\"results.jpg\" width=\"400px\" align=\"center\">\n",
"\n",
"$\\theta$-$\\phi$ 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).\n",
"```\n",
"\n",
":::\n",
"\n",
"::::"
]
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"source": [
"# Activity 6: Effect of the temperature\n",
"\n",
"\n",
"## Surface and bulk effects of the temperature\n",
"\n",
"In this example, we will look at the effects of the temperature on the X-Ray photoelectron diffraction from a copper substrate. We will base our python script on a paper published in 1986 by R. Trehan and S. Fadley. In their work, they performed azimutal scans of a copper(001) surface at 2 different polar angles: one at grazing incidence and one at 45° for incresing temperatures from 298K to roughly 1000K.\n",
"\n",
"For each azimutal scan, they looked at the anisotropy of the signal, that is:\n",
"\n",
"```{math}\n",
":label: eq-anisotropy\n",
"\\frac{I_{max} - I_{min}}{I_{max}}=\\frac{\\Delta I}{I_{max}}\n",
"```\n",
"\n",
"This value is representative of how clear are the *modulations* of the signal. They also proposed single scattering calculations that reproduced well these results.\n",
"\n",
"We will reproduce this kind of calculations to introduce the parameters that control the vibrational damping.\n",
"\n",
"(msd-paper)=\n",
":::{seealso}\n",
"based on this paper from R. Trehan and C.S. Fadley\n",
"[Phys. Rev. B **34** p6784–98 (2012)](https://doi.org/10.1103/PhysRevB.34.6784)\n",
":::\n",
"\n",
"### The script\n",
"\n",
"Since we want to distinguish between bulk (low polar angles, $\\theta = 45°$ in the paper) and surface effects (large polar angles, $\\theta = 83°$ in the paper), we need to compute scans for different emitters and different depths in the cluster.\n",
"\n",
"The script contains 3 functions:\n",
"1. The `create_clusters` function will build clusters with increasing number of planes, with the emitter being in the deepest plane for each cluster.\n",
"2. The function `compute` will compute the azimuthal scan for a given cluster.\n",
"3. The `analysis` function will sum the intensity from each plane for a given temperature. This will be the *substrate total signal* (more on this in the {ref}`path-filtering` section). The function will also compute the anisotropy (equation {eq}`eq-anisotropy`).\n",
"\n",
"MsSpec can introduce temperature effects in two ways: either by using the Debye Waller model or by doing an average over the phase shifts. We will explore the latter in this example.\n",
"\n",
"Regardless of the option chosen to model temperature effects, we need to define mean square displacement (MSD) values of the atoms in our cluster. We will use the Debye model for this. The MSD is defined as follows:\n",
"\n",
":::{math}\n",
":label: eq-debye\n",
"<U_j^2(T)> = \\frac{3\\hbar^2 T}{M K_B \\Theta_D^2}\n",
":::\n",
"\n",
"where $\\hbar$ is the reduce Planck's constant, $T$ is the temperature, $M$ is the mass of the atom, $k_B$ is the Boltzmann's constant and $\\Theta_D$ is the Debye temperature of the sample.\n",
"\n",
"To get an idea of the typical order of magnitude for MSD, the figure below plots the values of MSD for copper for temperatures ranging from 300 K to 1000 K.\n",
"\n",
":::{figure-md} msd-fig\n",
"<img src=\"msd.jpg\" alt=\"Cu MSD\" width=\"600px\" align=\"center\">\n",
"\n",
"Variation of MSD for copper versus temperature using equation {eq}`eq-debye`\n",
":::"
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"::::{tab-set}\n",
"\n",
":::{tab-item} <i class=\"fa-solid fa-circle-question\"></i> Quiz\n",
"With the help of the [MsSpec documentation](https://msspec.cnrs.fr/parameters.html) and the second paragraph p6791 of the [article cited above](#msd-paper),\n",
"complete the hilighted lines in the [following script](Cu_temperature.py \"download\") to compute the anisotropy of Cu(2p) $\\phi$-scans for polar angle $\\theta$=45° and 83°.\n",
"\n",
"How is varying the anisotropy versus the temperature. How can you qualitatively explain this variation ?\n",
"\n",
"```{literalinclude} Cu_temperature.py\n",
":lineno-match:\n",
":emphasize-lines: 40-41,44-48\n",
"```\n",
"\n",
":::\n",
"\n",
"::::"
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"(path-filtering)=\n",
"# Activity 7: Large clusters and path filtering"
]
},
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"So far you have seen that multiple scattering calculations with MsSpec can use two different algorithms: *matrix inversion* or *Rehr Albers series expansion*. When matrix inversion becomes impossible because the kinetic energy is too high and the number of atoms is too large, serial expansion is the alternative. This algorithm requires very little memory but it processes the scattering paths sequentially, which can lead to very long calculation times.\n",
"\n",
"In this activity, we will explore how to configure MsSpec to reduce this calculation time to compute the signal from a deep emitter in Si(001).\n",
"\n",
"\n",
"\n",
":::"
]
},
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"## The number of scattering paths\n",
"\n",
"To fix the idea, we will first evaluate how many scattering paths we need to compute for an emitter in the subsurface (2{sup}`nd` plane) or in the bulk (7{sup}`th` plane) of a Si(001) cluster."
]
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"::::{tab-set}\n",
"\n",
":::{tab-item} <i class=\"fa-solid fa-circle-question\"></i> Quiz\n",
"Create a cluster of Si(001) with the emitter in the subsurface, 40 Å diameter with 2 planes (since atoms below the emitter can be ignored at high kinetic energies).\n",
"\n",
"1. For an emitter in the subsurface, we can use single scattering (see {ref}`forward-scattering`). How many paths would be generated for this calculation ?\n",
"\n",
"2. Same question for an emitter in the 7{sup}`th` plane. If we were able to treat each scattering path within only 1 µs. How long would be such calculation ?\n",
"\n",
"```{note}\n",
"Remember that \n",
"1. for an emitter in plane $p$, the scattering order has to be at least the number of planes `above` the emitter\n",
"2. The number of scattering paths of order $n$ corresponds to the number of possibilities of arranging up to $n$ atoms (taking order into account).\n",
"```\n",
"\n",
"\n",
"::::"
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"## Paths filtering in MsSpec\n",
"\n",
"As you may expect, not all paths contribute significantly to the total intensity. This is why we can filter out some scattering paths and drastically reduce the computation time. MsSpec offers several filters for this. The 3 most common filters are:\n",
"1. the `forward_scattering` filter which allows all paths where each scattering angle is within a cone of defined aperture\n",
"2. the `backward_scattering` filter which is similar to the previous one but for backscattering direction\n",
"3. the `distance` filter which rejects all paths longer than a threshold distance\n",
"\n",
"The following figure illustrate the effect of theses filters on scattering paths\n",
"\n",
":::{figure-md} filters-fig\n",
"<img src=\"filters.jpg\" alt=\"path filtering\" width=\"600px\" align=\"center\">\n",
"\n",
"Some examples of scattering paths with `forward_scattering`, `backward_scattering` and `distance` filters selected. The accepted forward angle is 45°, the accepted backscattering angle is 20° and the threshold distance is $6a_0$ where $a_0$ is the lattice parameter. Note that the yellow path is rejected but if the `off_cone_events` option is set to a value > 1, then it could have been accepted.\n",
":::"
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"## Application to a deep plane in a Si(001) sample\n",
"\n",
"The following script will compute the contribution of a Si(2p) atom in the 4{sup}`th` plane of a Si(001) cluster at scattering order 3.\n",
"\n",
"Taking into account all scattering paths took 15 minutes to compute.\n",
"\n",
"(msd-paper)=\n",
":::{seealso}\n",
"based on this paper from S. Tricot *et al.*\n",
"[J. Electron. Spectrosc. Relat. Phenom. **256** 147176 (2022)](https://doi.org/10.1016/j.elspec.2022.147176)\n",
":::\n",
"\n",
"\n",
"::::{tab-set}\n",
"\n",
":::{tab-item} <i class=\"fa-solid fa-circle-question\"></i> Quiz\n",
"\n",
"The [following script](Si001.py \"download\") is almost completed, try to define path filtering options (no backscattering, accept all paths with forward angles < 40° and reject paths longer than the diameter of the cluster).\n",
"\n",
"```{literalinclude} Si001.py\n",
":lineno-match:\n",
":emphasize-lines: 37-41\n",
"```\n",
"\n",
"1. How long was your calculation ?\n",
"2. How does it compare to the calculation with **all** scattering paths up to order 3 ?\n",
"3. What is the proportion of scattering paths of order 3 that were actually taken into account ?\n",
"\n",
":::\n",
"\n",
"::::"
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"# Activity 8: Inequivalent emitters and the XPD of a substrate"
]
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"XPD can be used to study the adsorption of atoms or molecules on surfaces ([Activity 3](#ssc)), or atomic substitutions on surfaces ([Activity 2](#exp-setup)). In this case, modeling is relatively straightforward, since only one emitter atom is involved.\n",
"\n",
"We have seen from previous examples that, for kinetic energies $\\gtrsim$ 500 eV, the use of Rehr-Albers series expansion and scattering path filtering give access to the intensity of deeper emitter atoms ([Activity 7](#path-filtering)).\n",
"This is the key to computing the total photodiffraction signal of a *substrate*. As emitted photoelectrons originate from highly localized core levels around the atoms, the total signal corresponds to the (incoherent) sum of the intensities of all inequivalent emitters in the probed volume.\n",
"\n",
"Let's take a look at how this is done on the following example. \n",
"\n",
"\n",
"## The Aluminium Nitride (AlN) polarity\n",
"\n",
"In this example, we will compute polar diagrams of an aluminum nitride substrate.\n",
"\n",
"In a work published in 1999, Lebedev *et al.* demonstrated that Photoelectron diffraction can be used as a non invasive tool to unambiguously state the polarity of an AlN surface. Aluminium nitride cristallizes in an hexagonal cell and the authors experimentally showed that the polarity of the surface can be controlled by the annealing temperature during the growth. Both polarities are sketched in the [figure](#AlN-fig1) below.\n"
]
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"(aln-paper)=\n",
":::{seealso}\n",
"based on this paper from V. Lebedev *et al.*\n",
"[J. Cryst. Growth. **207(4)** p266-72 (1999)](https://doi.org/10.1016/S0022-0248(99)00375-9)\n",
":::"
]
},
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":::{figure-md} AlN-fig1\n",
"<img src=\"AlN-fig1.jpg\" alt=\"AlN crystal direction\" width=\"600px\" align=\"center\">\n",
"\n",
"AlN hexagonal lattice. Left) N polarity with nitrogen terminated surface and AlN{sub}`4` tetrahedrons pointing downward. Right) Al polarity with aluminium terminated surface and AlN{sub}`4` tetrahedrons pointing upward\n",
":::"
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"The AlN(0001) and (00.-1) faces share the same crystallograpphic symmetry and the Al and N atoms have the same geometrical surrounding differing only in the exchange of Al and N atoms ({numref}`Fig. %s <AlN-fig2>`).\n",
"\n",
"It is thus expected that Al(2p) and N(1s) XPD patterns exhibit almost the same features with only small differences due to the contrast between Al and N scattering amplitudes.\n",
"\n",
":::{figure-md} AlN-fig2\n",
"<img src=\"AlN-fig2.jpg\" alt=\"AlN crystal direction\" width=\"600px\" align=\"center\">\n",
"\n",
"Side views of N- or Al- terminated surfaces showing nearest neighbours main polar crystallographic directions. The inset shows the experimental Al(2p)/N(1s) ratio versus polar angle for both AlN polarities (taken from [Lebedev *et al.*](#aln-paper)).\n",
":::\n",
"\n",
"The strongest differences in photoemission intensities suitable for a quick and unambiguous determination of polarity were found in the (10-10) azimuthal plane at **32°** and **59°** (polar scans in the inset of {numref}`Fig. %s <AlN-fig2>`).\n",
"\n",
"These are the directions of short neighbor distances between the atoms of the same element (32°) and between Al and N atoms (58.5°), respectively.\n"
]
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"::::{tab-set}\n",
"\n",
":::{tab-item} <i class=\"fa-solid fa-circle-question\"></i> Quiz\n",
"Using the crystal view in {numref}`Fig. %s <AlN-fig1>` and assuming that we want to compute Al(2p) and N(1s) intensities for emitters located in 3 different planes to get a *substrate* signal. How many clusters do we need to build ? \n",
"\n",
":::\n",
"\n",
"::::"
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"::::{tab-set}\n",
"\n",
":::{tab-item} <i class=\"fa-solid fa-circle-question\"></i> Quiz\n",
"Download [this script](AlN.py) and fill in the lines indicated by the comments “FILL HERE”. Run the calculation and check that you are reproducing polar scan of {numref}`Fig. %s <AlN-fig2>`.\n",
"\n",
":::\n",
"\n",
"::::"
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"(RFactor)=\n",
"# Activity 9: Comparing simulation and experiment with R-factors\n",
"In order to extract precise crystallographic information from electronic spectroscopy, we need to compare MsSpec calculations with experimental results and adjust the modelling parameters to simulate the experiment as accurately as possible.\n",
"\n",
"*R-factors* (reliability factors) are commonly used for this task. In the following example, we will see how MsSpec can extract the adsorption geometry of a molecule.\n",
"\n",
"## The unusual tilt of CO molecule on Fe(001)\n",
"The carbon monoxide molecule can be adsorbed onto a Fe(001) surface in the hollow site. It was experimentally demonstrated that the CO molecule is tilted by 55$\\pm$2° in <100> azimuthal directions. The molecule is bonded to the Fe surface by the carbon atom and the adsorption height was estimated to be $\\sim$ 0.6 Å.\n",
"\n",
"(COFe-paper)=\n",
":::{seealso}\n",
"based on this paper from R. S. Saiki *et al.*\n",
"[Phys. Rev. Lett. **63(3)** p283-6 (1989)](https://doi.org/10.1103/PhysRevLett.63.283)\n",
":::\n",
"\n",
"We will try to reproduce the polar scan of the figure below of CO adsorbed in the hollow site of the Fe(001) surface with simple single scattering calculations with MsSpec and by using R-Factors to find the best adsorption geometry. We will use the simple cluster displayed on the left hand side of the figure.\n",
"\n",
":::{figure-md} COFe-fig1\n",
"<img src=\"COFe_fig1.jpg\" alt=\"\" width=\"600px\" align=\"center\">\n",
"\n",
"Small cluster used for SSC calculations (left) and (right) Normalized polar scan of the C(1s) at 1202 eV for [100] and [1-10] azimths.\n",
":::"
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"::::{tab-set}\n",
"\n",
":::{tab-item} <i class=\"fa-solid fa-circle-question\"></i> Quiz\n",
"Download [this script](./COFe.py) and write the body of the function `create_cluster`. The function should return a \n",
"small cluster of 5 Fe atoms with the CO molecule adsorbed like in figure {numref}`Fig. %s <COFe-fig2>` below. The function\n",
"will accept 4 keyword arguments to control the adsorption geometry.\n",
"\n",
"```{figure-md} COFe-fig2\n",
"<img src=\"COFe_fig2.jpg\" alt=\"CO-Fe adsorption geometry\" width=\"600px\" align=\"center\">\n",
"\n",
"Adsorption geometry of carbon monoxide on Fe(001).\n",
"```\n",
"\n",
"```{note}\n",
"Remember to use the [`ase.build.bulk`](https://wiki.fysik.dtu.dk/ase/ase/build/build.html#ase.build.bulk), the [`msspec.utils.hemispherical_cluster`](https://msspec.cnrs.fr/faq/hemispherical_cluster/hemispherical_cluster.html#hemispherical-cluster-faq) and the [`ase.build.add_adsorbate`](https://wiki.fysik.dtu.dk/ase/ase/build/surface.html#ase.build.add_adsorbate) \n",
"functions.\n",
"```\n",
"\n",
":::\n",
"\n",
"::::"
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"Now that the `create_cluster` function is done, look at the rest of the script. The next function is called `compute_polar_scan` and will obviously be used to compute the $\\theta$-scan of the C1s for a given cluster geometry.\n",
"Finally there is the *Main part* that is built in two sections:\n",
"\n",
"1. A section containing nested *for loops*\n",
"2. A final section for *R-Factor analysis*\n",
"\n",
"::::{tab-set}\n",
"\n",
":::{tab-item} <i class=\"fa-solid fa-circle-question\"></i> Quiz\n",
"1. Complete the nested for loops in section 1) in order to compute polar scans for CO molecule tilted from 45° to 60° with a 1° step and aligned either along the [100] direction of Fe, 30° from this direction or along [110]. The adsorption height is 0.6 Å and the bond length is 1.157 Å.\n",
"2. What is the best $(\\theta, \\phi)$ according to the R-Factor analysis\n",
":::\n",
"\n",
"::::"
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"MsSpec uses 12 different R-Factors formula to compare the calculted curve to the experimental data. The set of parameters giving the best agreement according to the majority of those 12 R-factors is assumed to be the best solution. Here are the R-factors formula used in MsSpec\n",
"\n",
":::{figure-md} rfactors-formula\n",
"<img src=\"formula.jpg\" alt=\"R-Factor result\" width=\"800px\" align=\"center\">\n",
"\n",
"The 12 R-Factors used in MsSpec. The Pendry's R-Factor is n°11.\n",
":::\n",
"\n",
"::::{tab-set}\n",
"\n",
":::{tab-item} <i class=\"fa-solid fa-circle-question\"></i> Quiz\n",
"How many R-Factors do agree that $(\\theta,\\phi)=(55,0)$ gives the best agreement ?\n",
":::\n",
"\n",
"::::"
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"# Activity 10: Parallelization and multi-processing in MsSpec"
]
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"As you can see from the previous examples, a complete simulation may require several multiple scattering calculations, for instance to calculate the total intensity of a substrate or to optimize the geometry of a system. As the calculations are often time consuming, it can be useful to distribute these tasks over several processors to make the most of hardware resources.\n",
"Although MsSpec is not fully parallelized, the code does offer a number of features, which we will explore here.\n",
"\n",
"## Matrix inversion parallelization\n",
"\n",
"When available during installation, MsSpec will link with the system lapack library. It will be used to invert the matrix in the `inversion` option of the MsSpec `calculator`. To allow MsSpec to use this shared memory parallelism, you need to set the number of cores to be used in the `OMP_NUM_THREADS` environment variable.\n",
"\n",
"You can set this variable just for the execution of your script. For example:\n",
"\n",
"```sh\n",
"$ OMP_NUM_THREADS=12 python my_script.py\n",
"```\n",
"\n",
"will use 12 cores for inverting the matrix in your script.\n",
"\n",
"It is also possible to set environment variable inside your python script.\n",
"\n",
"```python\n",
"import os\n",
"\n",
"os.environ['OMP_NUM_THREADS'] = 12\n",
"```\n",
"\n",
"It may be useful for technical reasons or to use different number of cores in some parts of your script.\n",
"\n",
"## Process-based parallelism\n",
"\n",
"Another kind of parallelization used in MsSpec is multiprocessing. Quite often, you need to run different *independent* calculations. MsSpec provides a simple *looper* that can be useful for multiprocessing. Let's demonstrate it with the previous example CO/Fe(001).\n",
"\n",
"[This script](COFe_mp.py) is the multiprocessed version of the previous one. You can see that the previous nested for loops are now replaced by some declarative content (lines 63-67) and the definition of a `process` function (whose name \n",
"can be changed).\n",
"\n",
"With the `msspec.looper` package, the user define `Sweep` objects that are parameters of the calculation or of the cluster. The `process` function must accept as many arguments as parameters to sweep (+ the `**kwargs`).\n",
"\n",
"A `Looper` object is created (line 76) and the `process` function is set to its `pipeline` attribute (line 77). When MsSpec will run the `looper`, it will combine all parameters values to unique individual sets and MsSpec will distribute the calculations over the number of processors specified in the `ncpu` option.\n",
"\n",
":::{literalinclude} COFe_mp.py\n",
":lineno-start: 63\n",
":linenos: true\n",
":lines: 63-84\n",
":::\n",
"\n",
"\n",
"::::{tab-set}\n",
"\n",
":::{tab-item} <i class=\"fa-solid fa-circle-question\"></i> Quiz\n",
"In the paper discussed in {ref}`RFactor`, experimental values of the anisotropy suggest an adsorption height between 0.2 and 0.6 Å. Modify the script to add another sweep for variying the adsorption height of the CO molecule.\n",
":::\n",
"\n",
"::::"
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# Table of contents
# Learn more at https://jupyterbook.org/customize/toc.html
format: jb-book
root: intro
chapters:
- file: Activity01/Activity01_light
- file: Activity02/Activity02_light
- file: Activity03/Activity03_light
- file: Activity04/Activity04_light
- file: Activity05/Activity05_light
- file: Activity06/Activity06_light
- file: Activity07/Activity07_light
- file: Activity08/Activity08_light
- file: Activity09/Activity09_light
- file: Activity10/Activity10_light