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<h1>Activity 4: From single scattering to multiple scattering</h1>
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<h2> Contents </h2>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#polar-scans-of-ni-atomic-chains">Polar scans of Ni atomic chains</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="#building-a-chain-of-atoms">Building a chain of atoms</a></li>
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<section class="tex2jax_ignore mathjax_ignore" id="activity-4-from-single-scattering-to-multiple-scattering">
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<span id="ssc2ms"></span><h1>Activity 4: From single scattering to multiple scattering<a class="headerlink" href="#activity-4-from-single-scattering-to-multiple-scattering" title="Link to this heading">#</a></h1>
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<p>In the <a class="reference internal" href="../Activity03/Activity03.html#ssc"><span class="std std-ref">previous activity</span></a>, 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.
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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: <em>the defocusing effect</em>.</p>
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<p>The defocusing effect is presented in the <a class="reference internal" href="#ni-fig1"><span class="std std-ref">figure below</span></a> 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.</p>
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<figure class="align-default" id="ni-fig1">
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<a class="reference internal image-reference" href="../_images/defocusing_animation.gif"><img alt="defocusing effect" class="align-center" src="../_images/defocusing_animation.gif" style="width: 600px;" />
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</a>
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<figcaption>
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<p><span class="caption-number">Fig. 10 </span><span class="caption-text">The defocusing effect dur to multiple scattering in an atomic chain of Ni atoms.</span><a class="headerlink" href="#ni-fig1" title="Link to this image">#</a></p>
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</figcaption>
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</figure>
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<p>In 1989, M.-L Xu, J.J. Barton and M.A. Van Hove studied these multiple scattering effects on atomic chains (<a class="reference internal" href="#defocusing-paper"><span class="std std-ref">see their paper below</span></a>).
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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.</p>
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<div class="admonition seealso" id="defocusing-paper">
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<p class="admonition-title">See also</p>
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<p>based on this paper from M.-L. Xu <em>et al.</em>
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<a class="reference external" href="https://doi.org/10.1103/PhysRevB.39.8275">Phys. Rev. B <strong>39</strong> p8275 (1989)</a></p>
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</div>
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<section id="polar-scans-of-ni-atomic-chains">
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<h2>Polar scans of Ni atomic chains<a class="headerlink" href="#polar-scans-of-ni-atomic-chains" title="Link to this heading">#</a></h2>
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<section id="building-a-chain-of-atoms">
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<h3>Building a chain of atoms<a class="headerlink" href="#building-a-chain-of-atoms" title="Link to this heading">#</a></h3>
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<p>Start by creating a simple chain of 2 Ni atoms: an emitter and a scatterer in the [101] direction.</p>
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<div class="admonition tip">
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<p class="admonition-title">Tip</p>
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<p>Nickel is <em>fcc</em> with lattice parameter <span class="math notranslate nohighlight">\(a\)</span>=3.499 Å. Use the <a class="reference external" href="https://wiki.fysik.dtu.dk/ase/ase/atoms.html#ase.Atoms"><code class="docutils literal notranslate"><span class="pre">Atoms</span></code></a> class of <code class="docutils literal notranslate"><span class="pre">ase</span></code> like in the <a class="reference internal" href="../Activity03/Activity03.html#ssc"><span class="std std-ref">previous activity</span></a>…</p>
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<div class="dropdown admonition">
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<p class="admonition-title">if you need help to start…</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">msspec.calculator</span><span class="w"> </span><span class="kn">import</span> <span class="n">MSSPEC</span>
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<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>
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<span class="n">symbol</span> <span class="o">=</span> <span class="o">...</span> <span class="c1"># The kind of atom for the chain</span>
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<span class="n">a</span> <span class="o">=</span> <span class="o">...</span> <span class="c1"># The distance bewteen 2 atoms</span>
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<span class="c1"># in [101] direction</span>
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<span class="n">chain</span> <span class="o">=</span> <span class="n">Atoms</span><span class="p">(</span><span class="o">...</span><span class="p">,</span> <span class="n">positions</span><span class="o">=</span><span class="p">[</span><span class="o">...</span><span class="p">])</span>
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<span class="n">chain</span><span class="o">.</span><span class="n">rotate</span><span class="p">(</span><span class="o">...</span><span class="p">)</span>
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<span class="n">chain</span><span class="o">.</span><span class="n">edit</span><span class="p">()</span>
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</pre></div>
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</div>
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</div>
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</div>
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<div class="docutils">
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</div>
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<div class="dropdown admonition">
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<p class="admonition-title"><em>Solution…</em></p>
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<p>Building the 2-atoms chain</p>
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<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>
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<span class="linenos"> 2</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>
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<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>
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<span class="linenos"> 4</span>
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<span class="linenos"> 5</span><span class="n">symbol</span> <span class="o">=</span> <span class="s1">'Ni'</span> <span class="c1"># The kind of atom for the chain</span>
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<span class="linenos"> 6</span><span class="n">a</span> <span class="o">=</span> <span class="mf">3.499</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span> <span class="c1"># The distance bewteen 2 atoms</span>
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<span class="linenos"> 7</span> <span class="c1"># in [101] direction</span>
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<span class="linenos"> 8</span>
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<span class="linenos"> 9</span><span class="n">chain</span> <span class="o">=</span> <span class="n">Atoms</span><span class="p">(</span><span class="n">symbol</span><span class="o">*</span><span class="mi">2</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="n">a</span><span class="p">)])</span>
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<span class="linenos">10</span><span class="n">chain</span><span class="o">.</span><span class="n">rotate</span><span class="p">(</span><span class="mi">45</span><span class="p">,</span> <span class="s1">'y'</span><span class="p">)</span>
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</pre></div>
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</div>
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<figure class="align-default" id="ni-fig2">
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<a class="reference internal image-reference" href="../_images/Ni_2atomsSSC.png"><img alt="Ni 2 atoms" class="align-center" src="../_images/Ni_2atomsSSC.png" style="width: 600px;" />
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</a>
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<figcaption>
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<p><span class="caption-number">Fig. 11 </span><span class="caption-text">Polar scan of a Ni(3s) chain of 2 atoms for single scattering.</span><a class="headerlink" href="#ni-fig2" title="Link to this image">#</a></p>
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</figcaption>
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</figure>
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</div>
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<div class="sd-tab-set docutils">
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<input checked="checked" id="sd-tab-item-0" name="sd-tab-set-0" type="radio">
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<label class="sd-tab-label" for="sd-tab-item-0">
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<i class="fa-solid fa-circle-question"></i> Quiz</label><div class="sd-tab-content docutils">
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<p>Create an <code class="docutils literal notranslate"><span class="pre">MSSPEC</span></code> calculator with <code class="docutils literal notranslate"><span class="pre">expansion</span></code> algortithm and set the <code class="docutils literal notranslate"><span class="pre">scattering_order</span></code>=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 ?</p>
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<p>Repeat the same experiment with <code class="docutils literal notranslate"><span class="pre">inversion</span></code> algorithm for having the full multiple scattering result. What do you observe ?</p>
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</div>
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</div>
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<div class="dropdown admonition">
|
|
<p class="admonition-title"><em>Solution…</em></p>
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|
<p>The peak at 45° is increasing with the number of atoms in the chain for SSC. We observe defocusing of this peak for full multiple scattering calculations (MSC). For the 2-atoms chain, both SSC and MSC give the same result owing to the fact that the emitter has only one scatterer atom and that kinetic energy is high enough to cancel out all backscattering.</p>
|
|
<figure class="align-default" id="ni-figx">
|
|
<a class="reference internal image-reference" href="../_images/Ni_SSCvsMI.jpg"><img alt="Ni chain" class="align-center" src="../_images/Ni_SSCvsMI.jpg" style="width: 600px;" />
|
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</a>
|
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<figcaption>
|
|
<p><span class="caption-number">Fig. 12 </span><span class="caption-text">Polar scan of a Ni chain of 2-5 atoms for single and full mutliple scattering.</span><a class="headerlink" href="#ni-figx" title="Link to this image">#</a></p>
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</figcaption>
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</figure>
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<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>
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<span class="linenos"> 2</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>
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<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>
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<span class="linenos"> 4</span>
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<span class="linenos"> 5</span><span class="n">symbol</span> <span class="o">=</span> <span class="s1">'Ni'</span> <span class="c1"># The kind of atom for the chain</span>
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<span class="linenos"> 6</span><span class="n">a</span> <span class="o">=</span> <span class="mf">3.499</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span> <span class="c1"># The distance bewteen 2 atoms</span>
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<span class="linenos"> 7</span> <span class="c1"># in [101] direction</span>
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<span class="linenos"> 8</span>
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|
<span class="linenos"> 9</span><span class="n">chain</span> <span class="o">=</span> <span class="n">Atoms</span><span class="p">(</span><span class="n">symbol</span><span class="p">)</span>
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<span class="linenos">10</span><span class="n">data</span> <span class="o">=</span> <span class="kc">None</span>
|
|
<span class="linenos">11</span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">5</span><span class="p">):</span>
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<span class="linenos">12</span> <span class="n">atom</span> <span class="o">=</span> <span class="n">Atoms</span><span class="p">(</span><span class="n">symbol</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="n">i</span><span class="o">*</span><span class="n">a</span><span class="p">]])</span>
|
|
<span class="linenos">13</span> <span class="n">atom</span><span class="o">.</span><span class="n">rotate</span><span class="p">(</span><span class="s1">'y'</span><span class="p">,</span> <span class="mi">45</span><span class="p">)</span>
|
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<span class="linenos">14</span> <span class="n">chain</span> <span class="o">+=</span> <span class="n">atom</span>
|
|
<span class="linenos">15</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">'inversion'</span><span class="p">)</span>
|
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<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="linenos">19</span> <span class="n">chain</span><span class="o">.</span><span class="n">emitter</span> <span class="o">=</span> <span class="mi">0</span>
|
|
<span class="linenos">20</span> <span class="n">calc</span><span class="o">.</span><span class="n">set_atoms</span><span class="p">(</span><span class="n">chain</span><span class="p">)</span>
|
|
<span class="linenos">21</span>
|
|
<span class="linenos">22</span> <span class="n">data</span> <span class="o">=</span> <span class="n">calc</span><span class="o">.</span><span class="n">get_theta_scan</span><span class="p">(</span><span class="n">level</span><span class="o">=</span><span class="s1">'3s'</span><span class="p">,</span> <span class="n">theta</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="mi">0</span><span class="p">,</span><span class="mi">80</span><span class="p">,</span><span class="mf">0.5</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">23</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="o">.</span><span class="n">views</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_plot_options</span><span class="p">(</span><span class="n">ylim</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mf">0.045</span><span class="p">])</span>
|
|
<span class="linenos">24</span><span class="n">data</span><span class="o">.</span><span class="n">view</span><span class="p">()</span>
|
|
</pre></div>
|
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</div>
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</section>
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<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#building-a-chain-of-atoms">Building a chain of atoms</a></li>
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