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@DallasTrinkle
DallasTrinkle committed May 12, 2016
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137 changes: 137 additions & 0 deletions MSE304/LJ_byteshelf.xml
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<?xml version="1.0" encoding="utf-8"?>
<stdhw weight="1" tags="">
<title>Lennard-Jones Potential</title>
<code>boltzmann = 8.617e-5
epsi = rRound(boltzmann * rand(900, 2000, 45)/10, 3)
sig = rand(100, 500, 63)/100
mindist = 2^(1/6)*sig
innerdist = 0.9*mindist
outerdist = 1.1*mindist
innerpot = 4*epsi*((sig/innerdist)^12-(sig/innerdist)^6)
outerpot = 4*epsi*((sig/outerdist)^12-(sig/outerdist)^6)
innerforce = -4*epsi*(-12*sig^12/innerdist^13 + 6*sig^6/innerdist^7)*1E10*1.602E-19
outerforce = -4*epsi*(-12*sig^12/outerdist^13 + 6*sig^6/outerdist^7)*1E10*1.602E-19
</code>
<problemsetup>
<p>As you have learned in class, the potential that an atom sees when it encounters another atom is defined by a combination of attractive forces and repulsive forces from the repulsion of the electron clouds due to the Pauli exclusion principle. Rather than explicitly solving the Schrödinger Equation, the <em>Lennard-Jones</em> potential provides a reasonable approximation for the case of neutral atoms or molecules that experience only van der Waals forces (or dispersion forces) for attraction.</p>
<p>The Lennard-Jones potential is given by $V(r) = 4 \epsilon \left[ ( \frac{\sigma}{r} )^{12} - (\frac{\sigma}{r})^6 \right]$. Here, $r$ is the distance between atoms/molecules, $\epsilon$ is related to the depth of the potential, and $\sigma$ is related to the equilibrium bond length.</p>
<p>This homework problem will illustrate the Lennard-Jones potential for you. Let $\epsilon = ${epsi} eV and $\sigma = ${sig} å.</p>
</problemsetup>
<question type="radio" feedbacktype="afterdeadline" weight="1">
<questionprompt>
<p>Visualize the potential. Which of the two terms is due to the Pauli exclusion principle?</p>
</questionprompt>
<correctFeedback>Right answer !</correctFeedback>
<solution>
<p>The Pauli exclusion principle governs the potential at small $r$, i.e., when the electron clouds overlap. When $r$ is small, $r^{12}$ is much smaller than $r^{6}$. Hence, $1/r^{12}$ is much larger than $1/r^6$ and dominates for small $r$.</p>
</solution>
<answerchoice iscorrect="true">
<text>The one proportional to $1/r^{12}$.</text>
</answerchoice>
<answerchoice iscorrect="false">
<text>The one proportional to $1/r^{6}$.</text>
</answerchoice>
<answerchoice iscorrect="false">
<text>No individual term. It is their combination.</text>
</answerchoice>
</question>
<question type="freeresponse" feedbacktype="immediate" weight="1">
<questionprompt>
<p>At what point $r$ (besides infinity) does the potential vanish?</p>
</questionprompt>
<answer>{sig}</answer>
<units>å</units>
<correctFeedback>
<p>Right answer !</p>
</correctFeedback>
<solution>
<p>If you set $r = \sigma$, then you find that both terms inside become 1, so the attractive and repulsive forces cancel out.</p>
</solution>
</question>
<question type="freeresponse" feedbacktype="immediate" weight="1">
<questionprompt>
<p>At what position $r_{\rm min}$ is $V(r)$ globally minimal?</p>
</questionprompt>
<answer>{mindist}</answer>
<units>å</units>
<correctFeedback>Right answer !</correctFeedback>
<help>
<p>How do you find the minimum of a mathematical function?</p>
</help>
<solution>
<p>$\frac{dV(r)}{dr} =4\epsilon\left[-12\frac{\sigma^{12}}{r^{13}} + 6\frac{\sigma^6}{r^7}\right]$</p>
<p>Setting this to 0 gives you $r = 2^{1/6} \sigma$.</p>
</solution>
</question>
<question type="freeresponse" feedbacktype="immediate" weight="1">
<questionprompt>
<p>What is the value of $V(r)$ at this minimum?</p>
</questionprompt>
<answer>-{epsi}</answer>
<units>eV</units>
<correctFeedback>Right answer !</correctFeedback>
<solution>
<p>Just put $r$=$2^{1/6} \sigma$=$2^{1/6}\cdot $ {sigma} into $V(r)$, this gives $-\epsilon$.</p>
</solution>
</question>
<question type="freeresponse" feedbacktype="immediate" weight="1">
<questionprompt>
<p>Now assume that an external force pushes two atoms closer together, such that $r$=$0.9\cdot r_{\rm min}$ ($r_{\rm min}$ as in Question 3). By how much does the potential energy increase?</p>
</questionprompt>
<answer>{innerpot}+{epsi}</answer>
<units>eV</units>
<correctFeedback>Right answer !</correctFeedback>
</question>
<question type="freeresponse" feedbacktype="immediate" weight="1">
<questionprompt>
<p>How large is the increase in potential energy if the external force pulls the two atoms apart such that $r=1.1\cdot r_{\rm min}$?</p>
</questionprompt>
<answer>{outerpot}+{epsi}</answer>
<units>eV</units>
<correctFeedback>Right answer !</correctFeedback>
</question>
<question type="freeresponse" feedbacktype="immediate" weight="1">
<questionprompt>
<p>
<span style="line-height: 20.7999992370605px;">What is the value (and sign) of the force acting on the system at $r = 0.9 \cdot r_{\rm min}$?</span>
</p>
</questionprompt>
<answer>{innerforce}</answer>
<units>N</units>
<correctFeedback>Right answer !</correctFeedback>
<solution>
<p>
<span style="line-height: 20.7999992370605px;">The force follows from $F(r)=-\frac{d}{dr}V(r)$.</span>
</p>
</solution>
<feedback answer="-{innerforce}">
<message>
<p>
<span style="line-height: 20.7999992370605px;">Is the force equal to $\frac{dV}{dr}$? Draw some potentials out and think about what the sign of the force must be at different areas of the potential.</span>
</p>
</message>
</feedback>
</question>
<question type="freeresponse" feedbacktype="immediate" weight="1">
<questionprompt>
<p>
<span style="line-height: 20.7999992370605px;">What is the value (and sign) of that force at $r = 1.1 \cdot r_{\rm min}$?</span>
</p>
</questionprompt>
<answer>{outerforce}</answer>
<units>N</units>
<correctFeedback>Right answer !</correctFeedback>
<solution>
<p>
<span style="line-height: 20.7999992370605px;">The force follows from $F(r)=-\frac{d}{dr}V(r)$.</span>
</p>
</solution>
<feedback answer="-{outerforce}">
<message>
<p>
<span style="line-height: 20.7999992370605px;">Is the force equal to $\frac{dV}{dr}$? Draw some potentials out and think about what the sign of the force must be at different areas of the potential.</span>
</p>
</message>
</feedback>
</question>
</stdhw>
3 changes: 2 additions & 1 deletion README.md
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Expand Up @@ -26,5 +26,6 @@ The `xml` files are inputs for ByteShelf. These have been converted by hand into
The primary files that need to be constructed are `server.js` (which will create the variables for the question and answers); `info.json` contains information about files that are needed to be accessed. The two HTML files, `question.html` and `answer.html` include references to the variables.

**Example to be converted:**
* ??
* LJ_byteshelf.xml

*This can be done by hand, but you may want to use a script to do some initial conversion.*

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