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<h1>Activity 3: Adsorbates and the single scattering approach</h1>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#oxygen-on-rh-001">Oxygen on Rh(001)</a><ul class="nav section-nav flex-column">
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<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-scattering-factor">Computing the scattering factor</a></li>
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<section class="tex2jax_ignore mathjax_ignore" id="activity-3-adsorbates-and-the-single-scattering-approach">
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<h1>Activity 3: Adsorbates and the single scattering approach<a class="headerlink" href="#activity-3-adsorbates-and-the-single-scattering-approach" title="Link to this heading">#</a></h1>
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<p>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.</p>
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<p>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.</p>
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<section id="oxygen-on-rh-001">
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<h2>Oxygen on Rh(001)<a class="headerlink" href="#oxygen-on-rh-001" title="Link to this heading">#</a></h2>
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<p>In a paper published in 1998, T. Gerber <em>et al.</em> 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 <span class="math notranslate nohighlight">\(\pm 0.02 \mathring{A}\)</span>.</p>
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<div class="admonition seealso">
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<p class="admonition-title">See also</p>
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<p>based on this paper from T. Greber <em>et al.</em> <a class="reference external" href="https://doi.org/10.1103/PhysRevLett.81.1654">Phys. Rev. Lett. <strong>81</strong>(8) p1654 (1998)</a></p>
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<figcaption>
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<p><span class="caption-number">Fig. 6 </span><span class="caption-text">Interferences produced by the backscattering effect</span><a class="headerlink" href="#rho-fig" title="Link to this image">#</a></p>
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<section id="computing-the-scattering-factor">
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<h3>Computing the scattering factor<a class="headerlink" href="#computing-the-scattering-factor" title="Link to this heading">#</a></h3>
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<p>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.</p>
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<p>By using the <code class="docutils literal notranslate"><span class="pre">Atoms</span></code> class of the <code class="docutils literal notranslate"><span class="pre">ase</span></code> 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.</p>
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||
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span><span class="w"> </span><span class="nn">ase</span><span class="w"> </span><span class="kn">import</span> <span class="n">Atoms</span>
|
||
<span class="kn">from</span><span class="w"> </span><span class="nn">ase.visualize</span><span class="w"> </span><span class="kn">import</span> <span class="n">view</span>
|
||
|
||
<span class="c1"># Create an atomic chain O-Rh</span>
|
||
<span class="n">cluster</span> <span class="o">=</span> <span class="n">Atoms</span><span class="p">(</span><span class="o">...</span> <span class="c1"># Fill this line</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="cell tag_hide-cell docutils container">
|
||
<details class="hide above-input">
|
||
<summary aria-label="Toggle hidden content">
|
||
<span class="collapsed">Show code cell content</span>
|
||
<span class="expanded">Hide code cell content</span>
|
||
</summary>
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span><span class="w"> </span><span class="nn">ase</span><span class="w"> </span><span class="kn">import</span> <span class="n">Atoms</span>
|
||
<span class="kn">from</span><span class="w"> </span><span class="nn">ase.visualize</span><span class="w"> </span><span class="kn">import</span> <span class="n">view</span>
|
||
|
||
<span class="c1"># Create an atomic chain O-Rh</span>
|
||
<span class="n">cluster</span> <span class="o">=</span> <span class="n">Atoms</span><span class="p">([</span><span class="s1">'O'</span><span class="p">,</span> <span class="s1">'Rh'</span><span class="p">],</span> <span class="n">positions</span> <span class="o">=</span> <span class="p">[(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mf">4.</span><span class="p">)])</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</details>
|
||
</div>
|
||
<div class="cell tag_remove-input docutils container">
|
||
<div class="cell_output docutils container">
|
||
<div class="output text_html"><html>
|
||
<head>
|
||
<title>ASE atomic visualization</title>
|
||
<link rel="stylesheet" type="text/css" href="https://www.x3dom.org/release/x3dom.css"></link>
|
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<script type="text/javascript" src="https://www.x3dom.org/release/x3dom.js"></script>
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<X3D width="400px"; height="300px";>
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|
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|
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|
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|
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|
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|
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|
||
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|
||
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|
||
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|
||
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|
||
<!--End of Inserted Scene-->
|
||
|
||
</X3D>
|
||
</body>
|
||
</html>
|
||
|
||
</div></div>
|
||
</div>
|
||
<p>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</p>
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">calc</span> <span class="o">=</span> <span class="n">MSSPEC</span><span class="p">(</span><span class="n">spectroscopy</span><span class="o">=</span><span class="s1">'PED'</span><span class="p">)</span>
|
||
<span class="n">calc</span><span class="o">.</span><span class="n">set_atoms</span><span class="p">(</span><span class="n">cluster</span><span class="p">)</span>
|
||
<span class="n">cluster</span><span class="o">.</span><span class="n">emitter</span> <span class="o">=</span> <span class="mi">0</span>
|
||
</pre></div>
|
||
</div>
|
||
<div class="sd-tab-set docutils">
|
||
<input checked="checked" id="sd-tab-item-1" name="sd-tab-set-1" type="radio">
|
||
<label class="sd-tab-label" for="sd-tab-item-1">
|
||
<i class="fa-solid fa-circle-question"></i> Quiz</label><div class="sd-tab-content docutils">
|
||
<p>We use the <code class="docutils literal notranslate"><span class="pre">get_scattering_factors</span></code> method <a class="reference external" href="https://msspec.cnrs.fr/modules/calculator.html#calculator._PED.get_scattering_factors">(see its documentation)</a> to compute the scattering factors at 723 eV.</p>
|
||
<p>How large is the backscattering factor of Rhodium with respect to that of Oxygen ?</p>
|
||
</div>
|
||
</div>
|
||
<div class="toggle docutils container">
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="c1"># Compute the scattering factor</span>
|
||
<span class="n">data</span> <span class="o">=</span> <span class="n">calc</span><span class="o">.</span><span class="n">get_scattering_factors</span><span class="p">(</span><span class="n">kinetic_energy</span><span class="o">=</span><span class="mi">723</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Popup the results</span>
|
||
<span class="n">data</span><span class="o">.</span><span class="n">view</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
<figure class="align-default" id="sf-fig">
|
||
<a class="reference internal image-reference" href="../_images/RhO_fig1.jpg"><img alt="Scattering factors" class="align-center" src="../_images/RhO_fig1.jpg" style="width: 600px;" />
|
||
</a>
|
||
<figcaption>
|
||
<p><span class="caption-number">Fig. 7 </span><span class="caption-text">Polar representation of the scattering factor</span><a class="headerlink" href="#sf-fig" title="Link to this image">#</a></p>
|
||
</figcaption>
|
||
</figure>
|
||
</div>
|
||
</section>
|
||
<section id="interferences-due-to-backscattering">
|
||
<h3>Interferences due to backscattering<a class="headerlink" href="#interferences-due-to-backscattering" title="Link to this heading">#</a></h3>
|
||
<p>Let an Oxygen atom (in red) being adsorbed at a distance <span class="math notranslate nohighlight">\(z_0\)</span> of an <em>fcc</em> site of the Rh(111) surface.</p>
|
||
<figure class="align-default" id="rho-fig2a">
|
||
<a class="reference internal image-reference" href="../_images/RhO_fig2a.jpg"><img alt="../_images/RhO_fig2a.jpg" class="align-center" src="../_images/RhO_fig2a.jpg" style="width: 600px;" />
|
||
</a>
|
||
<figcaption>
|
||
<p><span class="caption-number">Fig. 8 </span><span class="caption-text">Small cluster used for the calculation.</span><a class="headerlink" href="#rho-fig2a" title="Link to this image">#</a></p>
|
||
</figcaption>
|
||
</figure>
|
||
<p>We will compute for different values of the adsorption height <span class="math notranslate nohighlight">\(z_0\)</span>.</p>
|
||
<div class="sd-tab-set docutils">
|
||
<input checked="checked" id="sd-tab-item-2" name="sd-tab-set-2" type="radio">
|
||
<label class="sd-tab-label" for="sd-tab-item-2">
|
||
<i class="fa-solid fa-circle-question"></i> Quiz</label><div class="sd-tab-content docutils">
|
||
<p>Complete the script below to compute the (<span class="math notranslate nohighlight">\(\theta,\phi\)</span>) scan of the photodiffraction of O(1s) adsorbed on a <em>fcc</em> site on Rh(111) surface.</p>
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="linenos"> 1</span><span class="kn">from</span><span class="w"> </span><span class="nn">msspec.calculator</span><span class="w"> </span><span class="kn">import</span> <span class="n">MSSPEC</span>
|
||
<span class="linenos"> 2</span><span class="kn">from</span><span class="w"> </span><span class="nn">ase.build</span><span class="w"> </span><span class="kn">import</span> <span class="n">fcc111</span><span class="p">,</span> <span class="n">add_adsorbate</span>
|
||
<span class="linenos"> 3</span><span class="kn">import</span><span class="w"> </span><span class="nn">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
|
||
<span class="linenos"> 4</span>
|
||
<span class="linenos"> 5</span><span class="n">data</span> <span class="o">=</span> <span class="kc">None</span>
|
||
<span class="hll"><span class="linenos"> 6</span><span class="n">all_z</span> <span class="o">=</span> <span class="o">...</span> <span class="c1"># -> Define a list of z values for the adsorbate</span>
|
||
</span><span class="linenos"> 7</span>
|
||
<span class="hll"><span class="linenos"> 8</span><span class="k">for</span> <span class="o">...</span> <span class="c1"># -> Complete this for-loop over z values</span>
|
||
</span><span class="linenos"> 9</span> <span class="c1"># construct the cluster</span>
|
||
<span class="linenos">10</span> <span class="n">cluster</span> <span class="o">=</span> <span class="n">fcc111</span><span class="p">(</span><span class="s1">'Rh'</span><span class="p">,</span> <span class="n">size</span> <span class="o">=</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span>
|
||
<span class="linenos">11</span> <span class="n">cluster</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
|
||
<span class="hll"><span class="linenos">12</span> <span class="n">add_adsorbate</span><span class="p">(</span><span class="o">...</span> <span class="c1"># -> Put the oxygen atom on the fcc site</span>
|
||
</span><span class="hll"><span class="linenos">13</span> <span class="n">cluster</span><span class="o">.</span><span class="n">emitter</span> <span class="o">=</span> <span class="o">...</span> <span class="c1"># -> Oxygen is the last atom we added, so the indice is...</span>
|
||
</span><span class="linenos">14</span>
|
||
<span class="linenos">15</span> <span class="c1"># Define a calculator for single scattering calculations</span>
|
||
<span class="linenos">16</span> <span class="n">calc</span> <span class="o">=</span> <span class="n">MSSPEC</span><span class="p">(</span><span class="n">spectroscopy</span><span class="o">=</span><span class="s1">'PED'</span><span class="p">,</span> <span class="n">algorithm</span><span class="o">=</span><span class="s1">'expansion'</span><span class="p">)</span>
|
||
<span class="linenos">17</span> <span class="n">calc</span><span class="o">.</span><span class="n">calculation_parameters</span><span class="o">.</span><span class="n">scattering_order</span> <span class="o">=</span> <span class="mi">1</span>
|
||
<span class="linenos">18</span> <span class="n">calc</span><span class="o">.</span><span class="n">set_atoms</span><span class="p">(</span><span class="n">cluster</span><span class="p">)</span>
|
||
<span class="linenos">19</span>
|
||
<span class="linenos">20</span> <span class="c1"># Compute</span>
|
||
<span class="linenos">21</span> <span class="n">data</span> <span class="o">=</span> <span class="n">calc</span><span class="o">.</span><span class="n">get_theta_phi_scan</span><span class="p">(</span><span class="n">level</span><span class="o">=</span><span class="s1">'1s'</span><span class="p">,</span> <span class="n">kinetic_energy</span><span class="o">=</span><span class="mi">723</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">data</span><span class="p">)</span>
|
||
<span class="linenos">22</span>
|
||
<span class="linenos">23</span><span class="n">data</span><span class="o">.</span><span class="n">view</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
<p>As proposed in the comments, add a loop to vary the adsorption height of Oxygen between 1.10 and 1.65 Å.
|
||
What is the bond length difference between to intensity maxima ?</p>
|
||
</div>
|
||
</div>
|
||
<div class="toggle docutils container">
|
||
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="linenos"> 1</span><span class="kn">from</span><span class="w"> </span><span class="nn">msspec.calculator</span><span class="w"> </span><span class="kn">import</span> <span class="n">MSSPEC</span>
|
||
<span class="linenos"> 2</span><span class="kn">from</span><span class="w"> </span><span class="nn">ase.build</span><span class="w"> </span><span class="kn">import</span> <span class="n">fcc111</span><span class="p">,</span> <span class="n">add_adsorbate</span>
|
||
<span class="linenos"> 3</span><span class="kn">import</span><span class="w"> </span><span class="nn">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
|
||
<span class="linenos"> 4</span>
|
||
<span class="linenos"> 5</span><span class="n">data</span> <span class="o">=</span> <span class="kc">None</span>
|
||
<span class="hll"><span class="linenos"> 6</span><span class="n">all_z</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mf">1.10</span><span class="p">,</span> <span class="mf">1.65</span><span class="p">,</span> <span class="mf">0.05</span><span class="p">)</span>
|
||
</span><span class="linenos"> 7</span>
|
||
<span class="hll"><span class="linenos"> 8</span><span class="k">for</span> <span class="n">zi</span><span class="p">,</span> <span class="n">z0</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">all_z</span><span class="p">):</span>
|
||
</span><span class="linenos"> 9</span> <span class="c1"># construct the cluster</span>
|
||
<span class="linenos">10</span> <span class="n">cluster</span> <span class="o">=</span> <span class="n">fcc111</span><span class="p">(</span><span class="s1">'Rh'</span><span class="p">,</span> <span class="n">size</span> <span class="o">=</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span>
|
||
<span class="linenos">11</span> <span class="n">cluster</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
|
||
<span class="hll"><span class="linenos">12</span> <span class="n">add_adsorbate</span><span class="p">(</span><span class="n">cluster</span><span class="p">,</span> <span class="s1">'O'</span><span class="p">,</span> <span class="n">z0</span><span class="p">,</span> <span class="n">position</span><span class="o">=</span><span class="s1">'fcc'</span><span class="p">)</span>
|
||
</span><span class="hll"><span class="linenos">13</span> <span class="n">cluster</span><span class="o">.</span><span class="n">emitter</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">cluster</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
|
||
</span><span class="linenos">14</span>
|
||
<span class="linenos">15</span> <span class="c1"># Define a calculator for single scattering calculations</span>
|
||
<span class="linenos">16</span> <span class="n">calc</span> <span class="o">=</span> <span class="n">MSSPEC</span><span class="p">(</span><span class="n">spectroscopy</span><span class="o">=</span><span class="s1">'PED'</span><span class="p">,</span> <span class="n">algorithm</span><span class="o">=</span><span class="s1">'expansion'</span><span class="p">)</span>
|
||
<span class="linenos">17</span> <span class="n">calc</span><span class="o">.</span><span class="n">calculation_parameters</span><span class="o">.</span><span class="n">scattering_order</span> <span class="o">=</span> <span class="mi">1</span>
|
||
<span class="linenos">18</span> <span class="n">calc</span><span class="o">.</span><span class="n">set_atoms</span><span class="p">(</span><span class="n">cluster</span><span class="p">)</span>
|
||
<span class="linenos">19</span>
|
||
<span class="linenos">20</span> <span class="c1"># Compute</span>
|
||
<span class="linenos">21</span> <span class="n">data</span> <span class="o">=</span> <span class="n">calc</span><span class="o">.</span><span class="n">get_theta_phi_scan</span><span class="p">(</span><span class="n">level</span><span class="o">=</span><span class="s1">'1s'</span><span class="p">,</span> <span class="n">kinetic_energy</span><span class="o">=</span><span class="mi">723</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">data</span><span class="p">)</span>
|
||
<span class="linenos">22</span> <span class="n">dset</span> <span class="o">=</span> <span class="n">data</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
|
||
<span class="linenos">23</span> <span class="n">dset</span><span class="o">.</span><span class="n">title</span> <span class="o">=</span> <span class="s2">"</span><span class="si">{:d}</span><span class="s2">) z = </span><span class="si">{:.2f}</span><span class="s2"> angstroms"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">zi</span><span class="p">,</span> <span class="n">z0</span><span class="p">)</span>
|
||
<span class="linenos">24</span>
|
||
<span class="linenos">25</span><span class="n">data</span><span class="o">.</span><span class="n">view</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
<figure class="align-default" id="stereo-fig">
|
||
<a class="reference internal image-reference" href="../_images/RhO_fig2b.gif"><img alt="Scattering factors" class="align-center" src="../_images/RhO_fig2b.gif" style="width: 600px;" />
|
||
</a>
|
||
<figcaption>
|
||
<p><span class="caption-number">Fig. 9 </span><span class="caption-text">Stereographic projections of O(1s) emission at <span class="math notranslate nohighlight">\(E_0\)</span> = 723 eV for an oxygen atom
|
||
on top of a fcc site of 3 Rh atoms at various altitudes <span class="math notranslate nohighlight">\(z_0\)</span></span><a class="headerlink" href="#stereo-fig" title="Link to this image">#</a></p>
|
||
</figcaption>
|
||
</figure>
|
||
<p>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.</p>
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