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<img src="/summerschool/2005/photos/instructors/dane.jpg" width="127" height="154" hspace="10" align="right">
<h2>Thermodynamics, Phase Diagrams, and the Cluster Expansion</h2>
<p><span style="font-weight: bold">Dane Morgan</span><br>
  Materials Science<br>
University of Wisconsin-Madison<br>
Phone: (608) 265-5879<br>
Email: ddmorgan --&gt; wisc.edu </p>
<h3>Lectures</h3>
<ul><li>Thermodynamics and Phase Diagrams from Cluster Expansions  | <a href="/summerschool/2005/week_two_lectures/dane_morgan/SS05_Morgan_Lectures.pdf">PDF</a> (408 Kb) | <a href="/summerschool/2005/week_two_lectures/dane_morgan/SS05_Morgan_Lectures.ppt">PPT</a> (1.3 MB) </li>

</ul>

<h3 class="stb">Computer Lab </h3>
<ul>
  <li><a href="http://www.mcc.uiuc.edu/summerschool/2005/tutorial/Lab_MonteCarlo.pdf"> Lab_MonteCarlo.pdf</a> (77Kb)</li>
  <li><a href="/summerschool/2005/tutorial/AMTB1.31_Tutorial.tar.gz">AMTB1.31_Tutorial.tar.gz</a> (21Kb) </ul>
<h3 class="stb">References</h3>
<ul class="plain">
  <li>Curtarolo, S., Morgan, D., et al. &quot;Predicting Crystal Structures with Data Mining of Quantum Calculations,&quot; Phys. Rev. Lett. 91, 135503 (2003).</li>
</ul>
<h3 class="stb">Details about Lectures and Computer Lab </h3>
<p>General aspects of thermodynamic simulation based on the CE and Monte Carlo will be extended to include how to identify phase transition, use of thermodynamic integration to get free-energy, how to obtain the phase diagram.<b> </b></p>
<h5>Thermodynamics and Phase Diagrams from Cluster Expansions | <a href="/summerschool/2005/week_two_lectures/dane_morgan/SS05_Morgan_Lectures.pdf">PDF</a> (408 Kb) | <a href="/summerschool/2005/week_two_lectures/dane_morgan/SS05_Morgan_Lectures.ppt">PPT</a> (1.3 MB) </h5>
<ul class="inline">
  <li class="first">Thermodynamics and Phase Diagrams from Cluster Expansions
  <li>The Cluster Expansion and Phase Diagrams
  <li>Outline
  <li>Phase Diagram Basics
  <li>What is A Phase Diagram?
  <li>Thermodynamics of Phase Stability
  <li>Stable Phases from Cluster Expansion
  <li> the Ground State Problem
  <li>Determining Possible Phases
  <li>The Convex Hull
  <li>Getting the Convex Hull of a Cluster Expansion Hamiltonian
  <li>Phase Diagrams from Cluster Expansion: 
  <li> Semi-Analytical Approximations
  <li>Semi-Analytic Expressions for F (F)
  <li>High-Temperature Expansion
  <li> (NN Cluster Expansion)
  <li>Low-Temperature Expansion
  <li> (NN Cluster Expansion)
  <li>Transition Temperature from LT and HT Expansion
  <li>Mean-Field Theory - The Idea
  <li>Implementing Mean-Field Theory
  <li> The Cluster Variation Method
  <li>Writing f[{r(s)}].
  <li>Factoring the Probability to Simplify r(s)
  <li>Truncating the Probability Factorization
  <li>&nbsp;= Mean Field
  <li>The Mean-Field Potential
  <li>The Modern Formalism
  <li>The CVM Potential
  <li>Simplest CVM Approximation - The Point
  <li> (Bragg-Williams, Weiss Molecular Field)
  <li>CVM Point Approximation - Bragg-Williams
  <li> (NN Cluster Expansion)
  <li>Bragg-Williams Approximation 
  <li> (NN Cluster Expansion)
  <li>Comparison of Bragg-Williams and High-Temperature Expansion
  <li>Critical Temperatures
  <li>Limitations of the CVM (Mean-Field), High- and Low-Temperature Expansions
  <li>Phase Diagrams from Cluster Expansion: Simulation with Monte Carlo
  <li>What Is MC and What is it for?
  <li>MC Sampling
  <li>Problem with Simple MC Sampling
  <li> r(s) is Very Sharply Peaked
  <li>Better MC Sampling
  <li>Detailed Balance and The Metropolis Algorithm
  <li>The Metropolis Algorithm (General)
  <li>Metropolis Algorithm for Cluster Expansion Model (Real Space)
  <li>Obtaining Thermal Averages From MC
  <li>Energy Vs. MC Step
  <li>Measuring Accuracy of Averages
  <li>Example of Autocorrelation Function
  <li>Semiquantitative Understanding of Role of Correlation in Averaging Errors
  <li>Methods to Test Convergence Efficiently
  <li>Finding Phases With MC
  <li>Thermodynamic Potentials in MC
  <li>Thermodynamic Integration
  <li>Example of Thermodynamic Integration
  <li>Summary
    
</ul>
<h3>Computer Lab </h3>
<p>A 2D Monte Carlo based on simplified CE, and possibly (time permitting) ATAT toolkit, will be used to investigate phase transitions, hysteresis, thermodynamic integration, and phase diagram.</p>
<ul>
  <li><a href="http://www.mcc.uiuc.edu/summerschool/2005/tutorial/Lab_MonteCarlo.pdf"> Lab_MonteCarlo.pdf</a> (77Kb)</li>
  <ul class="inline">
    <LI class="first">Computational Laboratory: Monte Carlo for Phase Stability Calculations </LI>
    <LI> Atomistic Modeling Toolbox (AMTB) </LI>
    <LI> Useful Matlab tricks </LI>
    <LI> Introduction </LI>
    <LI> Doing the labs </LI>
    <LI> Exercise 0: Exploring the Code </LI>
    <LI> Exercise 1: Convergence </LI>
    <LI> Exercise 2: Phase stability from direct MC: concentration, susceptibility </LI>
    <LI> Exercise 3: Phase stability from varying &micro;, thermodynamic integration and free energies </LI>
    <LI> Exercise 4: Varying &micro; for an ordering system </LI>
  </ul>
  <li><a href="/summerschool/2005/tutorial/AMTB1.31_Tutorial.tar.gz">AMTB1.31_Tutorial.tar.gz</a> (21Kb) 
</ul>
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