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<h2><img src="/summerschool/2005/photos/instructors/gus.jpg" hspace="10" align="right">Basics of constructing cluster expansions for alloy modeling</h2>
<p><span style="font-weight: bold">Gus Hart</span><br>
 Physics<br>
Northern Arizona University<br>
Phone: (928)-523-0426<br>
Email: Gus.Hart ---&gt; nau.edu</p>
<h3 class="stb">Lectures</h3>
<ul>
  <li><span style="font-weight: bold">Cluster Expansions: Treating the effects of strain:</span><br> 
  <a href="SS05_Hart_Coherency_Strain.pdf">pdf</a> (<span class="smaller">2.7MB</span>) | <a href="SS05_Hart_Coherency_Strain.ppt">ppt</a> (<span class="smaller">4.3MB</span>) | <a href="SS05_Hart_Coherency_Strain_Printer.ppt">ppt</a> (<span class="smaller">4.3MB, printer-friendly, reduced number of animations</span>) | <a href="SS05_Hart_Coherency_Strain.key.zip">key.zip</a> (4MB, the original Keynote presentation, zipped) </li>
  <li><span style="font-weight: bold">Building Model Hamiltonians via an Evolutionary Approach:<br>
  </span> <a href="SS05_Hart_Genetic%20Algorithms.pdf">pdf</a> (<span class="smaller">1623 K</span>) | <a href="SS05_Hart_Genetic_Algorithms.ppt">ppt</a> (<span class="smaller">3686 K</span>) | <a href="SS05_Hart_Genetic_Algorithms_Printer.ppt">ppt</a> (3392 K) 
          
| <a href="SS05_Hart_Genetic_Algorithms.key.zip">key.zip</a> (3.4 MB, the original Keynote presentation, zipped)</li>
</ul>
<h3  class="stb">Computational Laboratory</h3>
  <ul>
    <li><a href="/summerschool/2005/tutorial/AllCoherencyStrain.tar.gz">AllCoherencyStrain.tar.gz</a> (920Kb)</li>
  </ul>
  <h3 class="stb">Details about Lectures and Computer Lab </h3>
  <p><span style="font-weight: bold">Lecture abstract:</span> Basic formalism behind alloy cluster expansions, that is the generalized Ising model, will be discussed. We will learn how to calculate &quot;input data&quot; for fitting to the model and then use the resulting model Hamiltonian, a cluster expansion, to make ground state predictions, to do thermodynamic Monte Carlo simulations.</p>
  <p><span style="font-weight: bold">Lab activities:</span><br>
  (i) importance of strain in phase separating systems<br>
  (ii) how to parameterize strain for use with a cluster expansion<br>
  (iii) Thermodynamic MC for phase transitions<br>
  (iv) Constructing CEs using an evolutionary approach</p>
  <p style="font-weight: bold">Cluster Expansions: Treating the effects of strain</p>
  <ul class="inline">
    <li class="first">Generalized Ising model
    <li>An Illustration: Cluster expansion (Ising)
    <li>Cluster Expansion
    <li>Coherent semi-infinite slabs
    <li>Precipitates!
    <li>Coherent semi-infinite slabs
    <li>The resulting strain energy is &quot;long-range&quot;
      <li>What do we mean by that?
      <li>How is it that the Ising model (cluster expansion) can&rsquo;t handle this?
      <li>How can this be fixed?
      <li>Superlattices...
      <li>Consider a 2n-layer superlattice AnBn
      <li>What happens as n gets large? (Or as k gets small if we think in k-space?)
      <li>Non-analyticity 
        
</ul>
  <ul>
    <li><a href="SS05_Hart_Coherency_Strain.pdf">pdf</a> (<span class="smaller">2.7MB</span>) | <a href="SS05_Hart_Coherency_Strain.ppt">ppt</a> (<span class="smaller">4.3MB</span>) | <a href="SS05_Hart_Coherency_Strain_Printer.ppt">ppt</a> (<span class="smaller">4.3MB, printer-friendly, reduced number of animations</span>) | <a href="SS05_Hart_Coherency_Strain.key.zip">key.zip</a> (4MB, the original Keynote presentation, zipped)
  </ul>
  <p style="font-weight: bold">Building Model Hamiltonians via an Evolutionary Approach</p>
  <ul class="inline">
    <li class="first">The argument for &quot;fancy&quot;
    <li>Goal: A fast, but quantitatively accurate, Hamiltonian
    <li>Such models are complex, almost by definition generally not even the &quot;shape&quot;of the model is simple
    <li>That is, the value of parameters is not the issue, which parameters to use in the model (or even how many!) is essentially unguessable 
    <li>Fancy or not, we need an approach that &quot;works&quot;
    <li>An Illustration: Cluster expansion (Ising)
    <li>How do we truncate? Determine the model's &quot;shape?&quot;
    <li>Selecting the multi-body interaction types (MBITs)
    <li>Optimization (search) problem:
    <li>(Actually, the problem is even worse for surfaces...)
    <li>How to pick the terms
    <li>By physics &quot;intuition&quot; dangerous (and physics shouldn't be an &quot;art&quot;)
    <li>Formal hierarchy (converges way too slow to be practical)
    <li>Simulated annealing (didn't work highly correlated problem)
    <li>Systematic &quot;guessing&quot;hierarchal approach
    <li>Something &quot;fancier&quot;(anything that works!)
    <li>Formal hierarchy
    <li>Evolutionary optimization
    <li>Good approach for some highly correlated problems
    <li>Not &quot;too fancy&quot;if it's the simplest thing that works
    <li>Make a cool figure: on the left, b/w vs. color genomes
    <li>Fade in on right 0010011111001111
    <li>But does it work?
    <li>Still have problems with local minima...
    <li>&quot;Lock-out&quot;makes the approach completely robust
    <li>Do we really need all this accuracy in the fitting/construction? Does it really make a difference?!
    <li>Now for the fun part... (Today's lab)
    <li>A simple view of the problem
    <li>The specific problem for today
    <li>Computing the energy
    <li>
</ul>
  <ul>
    <li><a href="SS05_Hart_Genetic%20Algorithms.pdf">pdf</a> (<span class="smaller">1623 K</span>) | <a href="SS05_Hart_Genetic_Algorithms.ppt">ppt</a> (<span class="smaller">3686 K</span>) | <a href="SS05_Hart_Genetic_Algorithms_Printer.ppt">ppt</a> (3392 K) | <a href="SS05_Hart_Genetic_Algorithms.key.zip">key.zip</a> (3.4 MB, the original Keynote presentation, zipped)
  </ul>
  <h3>Computational Laboratory</h3>
  <p>Download the collection of tutorial materials:</p>
  <ul>
    <li><a href="/summerschool/2005/tutorial/AllCoherencyStrain.tar.gz">AllCoherencyStrain.tar.gz</a> (920Kb)</li>
  </ul>
  <p>Or the individual files: </p>
  <ul>
    <li><a href="coherency_strain_lab.zip">coherency_strain_lab.zip</a> (104 Kb)
        <ul class="inline">
          <li class="first">strain_lab.nb
          <li>strain_lab.pdf
        </ul>
    <li><a href="MatLabGA_Solution/">MatLabGA_Solution.zip</a> (12Kb)
        <ul class="inline">
          <li class="first">compute_energy.m 
          <li> enumerate.m
          <li>simulated_annealing.m
          <li>ga_minimize.m
          <li>ga_driver_sweep_r_mut.m
          <li>ga_driver_sweep_pop_size.m
          <li>ga_driver.m
        </ul>
    <li> <a href="ga_lab.pdf">ga_lab.pdf</a> (<span class="smaller"><i>81 K</i></span>) 
    <li><a href="ga_lab_solutions.pdf">ga_lab_solutions.pdf</a> (<span class="smaller"><i>846 K</i></span>) 
    <li><a href="compute_energy.m">compute_energy.m</a> (<span class="smaller"><i>0 K</i></span>) 
    <li><a href="enumerate.m">enumerate.m</a> (<span class="smaller"><i>1 K</i></span>)
    <li><a href="initialize_hamiltonian.m">initialize_hamiltonian.m</a> (<span class="smaller"><i>0 K</i></span>) 
    <li><a href="simulated_annealing.m">simulated_annealing.m</a> (<span class="smaller"><i>2 K</i></span>) 
  </ul>
  <p>&nbsp;</p>
  <p>&nbsp;</p>
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