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<h2 class="hd hd-2 unit-title">Part 2.1: Miller Indices</h2>
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<h3>Part 2.1: Miller Indices</h3>
<p></p>
<p>(1) Miller indices <span style="font-size: 1em;">are a type of crystallographic notation used to quantify the planes in a crystal</span></p>
<p>(2) All of the parallel planes in a crystal have the same Miller index </p>
<p>(3) The following procedure can be used to determine the Miller index of a plane:</p>
<ul>
<ul>
<ul>
<li>Read off the intercepts of the plane with axes in terms of a, b and c</li>
<li>Take the reciprocal of the intercepts</li>
<li>Reduce all fractions to integer values</li>
<li>Remove any common multiples</li>
<li>Enclose the Miller index in parenthesis. Do not use any commas between values: i.e., (hkl)</li>
<li>Negative values should be written with a bar over the negative value, i.e., (hk̄l)</li>
</ul>
</ul>
</ul>
<p></p>
<h4>Let's take a look at an example:</h4>
<p><img width="500" src="/assets/courseware/v1/ead97f0cd1f3e4b6e7bbb9c4a05cd403/asset-v1:MITx+3.012S.1x+1T2019+type@asset+block/3012_Tutorial2_Figure1.jpg" alt="Plane with intercepts at (.5, 0, 0), (0, .25,0), and (0,0,.75)" /></p>
<p>(1) We begin by writing the intercepts of the plane:</p>
<ul>
<ul>
<ul>
<li>\(\frac{1}{2} \; \frac{1}{4} \; \frac{3}{4}\)</li>
</ul>
</ul>
</ul>
<p>(2) Then we take the reciprocal of the intercepts:</p>
<ul>
<ul>
<ul>
<li>\(2 \; 4 \; \frac{4}{3}\)</li>
</ul>
</ul>
</ul>
<p>(3) We reduce the fraction to an integer value:</p>
<ul>
<ul>
<ul>
<li>\(6 \; 12 \; 4\)</li>
</ul>
</ul>
</ul>
<p>(4) Finally we remove any common multiples from our Miller index;</p>
<ul>
<ul>
<ul>
<li>\(3 \; 6 \; 2\)</li>
</ul>
</ul>
</ul>
<p>(5) This gives us a plane with the following Miller index:</p>
<ul>
<ul>
<ul>
<li>\((362)\)</li>
</ul>
</ul>
</ul>
</div>
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<h2 class="hd hd-2 unit-title">Part 2.2: Crystallographic Notation</h2>
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<h3>Part 2.2: Crystallographic Notation</h3>
<p>In the previous section, we established that we can use Miller indices to describe a plane in a crystal. </p>
<p>We can also refer to a <em>family of equivalent planes</em>. The equivalence of different planes is a function of the symmetry of a crystal. </p>
<p>Consider the primitive tetragonal cell shown below. In an tetragonal cell, the lengths \(a\) and \(b\) are equal, and they are different than the length of the side \(c\). All of the angles are equal to \(90\) degrees.</p>
<p><img width="500" src="/assets/courseware/v1/dd19e85b8b075dcbe979b2a41ac79418/asset-v1:MITx+3.012S.1x+1T2019+type@asset+block/3012_Tutorial2_Figure5.jpg" alt="Primitive Orthorhombic cells with (100) and (010) planes highlighted" /></p>
<p>Here, you can see that due to the symmetry of the crystal, the (100) plane is equivalent to the (010) plane. </p>
<p></p>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>In a body-centered cubic cell, is the \((001)\) plane a member of the \(\{100\}\) family of planes?</p>
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Question 2.2C
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<p>Now review the different ways we notate crystallographic planes, directions, and families of planes and directions. Then answer the following questions.</p>
<p>Consider the following four crystallographic references:</p>
<p>\([201]\)</p>
<p>\(\langle 201 \rangle\)</p>
<p>\((201)\)</p>
<p>\(\{201\}\)</p>
<p>The different brackets give each of these a unique meaning.</p>
<p>\([201]\) refers to:</p>
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<option value="a crystallographic direction"> a crystallographic direction</option>
<option value="the miller indices of a plane"> the miller indices of a plane</option>
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<option value="a family of planes"> a family of planes</option>
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<p>\(\langle 201 \rangle\) refers to:</p>
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<option value="a crystallographic direction"> a crystallographic direction</option>
<option value="the miller indices of a plane"> the miller indices of a plane</option>
<option value="a family of directions"> a family of directions</option>
<option value="a family of planes"> a family of planes</option>
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<p>\((201)\) refers to:</p>
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<option value="a family of directions"> a family of directions</option>
<option value="a family of planes"> a family of planes</option>
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<p>\(\{201\}\) refers to:</p>
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<option value="a crystallographic direction"> a crystallographic direction</option>
<option value="the miller indices of a plane"> the miller indices of a plane</option>
<option value="a family of directions"> a family of directions</option>
<option value="a family of planes"> a family of planes</option>
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<h2 class="hd hd-2 unit-title">Part 2.3: Reciprocal Space</h2>
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<h3>Reciprocal Space</h3>
<ul>
<ul>
<li>Consider a lattice with basis vectors \(\vec{a_1}\), \(\vec{a_2}\), and \(\vec{a_3}\).</li>
<li>How do we go about calculating the reciprocal vector for this lattice?</li>
<li>Let's consider an example. We have a unit cell with the following parameters:</li>
<ul>
<ul>
<li type="circle">\(|\vec{a_1}|\) = \(|\vec{a_2}|\) = \(|\vec{a_3}|\) = \(4 Å\)</li>
<li type="circle">\(\alpha\) = \(\beta\) = \(90^{\circ}\); \(\gamma = 120^{\circ}\)</li>
</ul>
</ul>
</ul>
</ul>
<p style="padding-left: 60px;"><img height="323" width="400" src="/assets/courseware/v1/20f5de43285c13eb12e1d0eff469b109/asset-v1:MITx+3.012S.1x+1T2019+type@asset+block/3012_Tutorial_2_3A.jpg" alt="Illustration of problem statement given in text above" /></p>
<ul>
<ul>
<li>First, let's start by reminding ourselves of the definitions of \(\alpha\), \(\beta\), and \(\gamma\):</li>
<ul>
<ul>
<li type="circle">\(\alpha = \sphericalangle(\vec{a_2},\vec{a_3})\)</li>
<li type="circle">\(\beta = \sphericalangle(\vec{a_3},\vec{a_1})\)</li>
<li type="circle">\(\gamma = \sphericalangle(\vec{a_1},\vec{a_2})\)</li>
</ul>
</ul>
<li>We also know that:</li>
<ul>
<ul>
<li type="circle">\(\vec{a}_1^* = \frac{\vec{a_2} \times \vec{a_3}}{V}\)</li>
<li type="circle">\(\vec{a}_2^* = \frac{\vec{a_3} \times \vec{a_1}}{V}\)</li>
<li type="circle">\(\vec{a}_3^* = \frac{\vec{a_1} \times \vec{a_2}}{V}\)</li>
</ul>
</ul>
<li>We will begin by expressing our lattice basis vectors in terms of unit vectors \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\)</li>
<ul>
<ul>
<li type="circle">Let's start with \(\vec{a}_1\). This vector has components in both the \(\hat{i}\) and \(\hat{j}\) directions, and no component in the \(\hat{k}\) direction. This means that we can focus on the x-y plane</li>
</ul>
</ul>
</ul>
</ul>
<p style="padding-left: 60px;"><img height="313" width="500" src="/assets/courseware/v1/6d8704c059f6069aea0fb93287614174/asset-v1:MITx+3.012S.1x+1T2019+type@asset+block/3012_Tutorial_component_vectors.jpg" alt="A view of the x-y plane with vector a_1 shown. The angle between a_1 and the y-axis is 120 degrees" /></p>
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Tutorial Problem 2.3A
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<p> Decompose the vector \(\vec{a}_1\) into its component parts, expressing \(\vec{a}_1\) in terms of unit vectors \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\).
</p>
<p> Lattice vector \(\vec{a}_1\) (in \(&#197;\)):</p>
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<p> \(\mathbf{a_1}\)=</p>
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<span class="trailing_text" id="trailing_text_7cd1297a6e5547b7b5839a28f889a8a4_2_1">\(\hat{i}\)+</span>
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<span class="trailing_text" id="trailing_text_7cd1297a6e5547b7b5839a28f889a8a4_3_1">\(\hat{j}\)+</span>
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<span class="trailing_text" id="trailing_text_7cd1297a6e5547b7b5839a28f889a8a4_4_1">\(\hat{k}\)</span>
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Tutorial Problem 2.3B
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<p> Because \(\vec{a}_2\) and \(\vec{a}_3\) lie in the \(\hat{j}\) and \(\hat{k}\) directions, respectly, we can write them directly:
</p>
<p> Lattice vector \(\vec{a}_2\) (in \(&#197;\)):</p>
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<p> \(\mathbf{a_2}\)=</p>
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<span class="trailing_text" id="trailing_text_a505b2c8331842dababadde29128d588_2_1">\(\hat{i}\)+</span>
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<span class="trailing_text" id="trailing_text_a505b2c8331842dababadde29128d588_3_1">\(\hat{j}\)+</span>
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<span class="trailing_text" id="trailing_text_a505b2c8331842dababadde29128d588_4_1">\(\hat{k}\)</span>
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<p> Lattice vector \(\vec{a}_3\) (in \(&#197;\)):</p>
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<p> \(\mathbf{a_3}\)=</p>
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<span class="trailing_text" id="trailing_text_a505b2c8331842dababadde29128d588_5_1">\(\hat{i}\)+</span>
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<span class="trailing_text" id="trailing_text_a505b2c8331842dababadde29128d588_6_1">\(\hat{j}\)+</span>
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<span class="trailing_text" id="trailing_text_a505b2c8331842dababadde29128d588_7_1">\(\hat{k}\)</span>
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<p>Now, let's calculate the volume of the cell, \(V\).</p>
<p>The volume of the cell is given by the triple product:</p>
<ul>
<ul>
<ul>
<ul>
<li>\(V = |\vec{a}_1 \cdot \vec{a}_2 \times \vec{a}_3| = |\vec{a}_2 \cdot \vec{a}_3 \times \vec{a}_1| = |\vec{a}_3 \cdot \vec{a}_1 \times \vec{a}_2| \)</li>
</ul>
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<p>Let's calculate \(|\vec{a}_2 \cdot \vec{a}_3 \times \vec{a}_1|\).</p>
<p>Remember, you can write the cross-product of \((a \times b)\) as:</p>
<ul>
<ul>
<ul>
<ul>
<li>\( \begin{vmatrix} a_2 & a_3 \\ b_2 & b_3 \end{vmatrix} \hat{i} - \begin{vmatrix} a_1 & a_3 \\ b_1 & b_3 \end{vmatrix} \hat{j} + \begin{vmatrix} a_1 & a_2 \\ b_1 & b_2 \end{vmatrix} \hat{k}\)</li>
</ul>
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</ul>
<p>From our work above, we know the following:</p>
<ul>
<ul>
<ul>
<ul>
<li>\(\vec{a}_2 = 0 \hat{i} + 4 \hat{j} + 0 \hat{k}\)</li>
<li>\(\vec{a}_3 = 0 \hat{i} + 0 \hat{j} + 4 \hat{k}\)</li>
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Tutorial Problem 2.3C
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<p> Start by calculating \(\vec{a}_2 \times \vec{a}_3\).
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<p> \(\vec{a}_2 \times \vec{a}_3\) (in \(&#197;\)):</p>
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Tutorial Problem 2.3D
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>Now, we can take the dot product with \(\vec{a}_1\).</p>
<p>\((\vec{a}_2 \times \vec{a}_3) \cdot \vec{a}_1\) in \(&#197;^3\):</p>
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Tutorial Problem 2.3E
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<p> Now that we know \(\vec{a}_1\), \(\vec{a}_2\),\(\vec{a}_3\), and \(V\), we can calculate the reciprocal lattice vectors.</p>
<p>Please express your answer in inverse angstroms.</p>
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<p> \(\vec{a}_1^{*}\)=</p>
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<span class="trailing_text" id="trailing_text_8c4a353a4c64494b97c4c10d58187ee9_2_1">\(\hat{i}\)+</span>
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<span class="trailing_text" id="trailing_text_8c4a353a4c64494b97c4c10d58187ee9_3_1">\(\hat{j}\)+</span>
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<span class="trailing_text" id="trailing_text_8c4a353a4c64494b97c4c10d58187ee9_4_1">\(\hat{k}\)</span>
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<p> \(\vec{a}_2^{*}\) =</p>
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<span class="trailing_text" id="trailing_text_8c4a353a4c64494b97c4c10d58187ee9_5_1">\(\hat{i}\)+</span>
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<span class="trailing_text" id="trailing_text_8c4a353a4c64494b97c4c10d58187ee9_6_1">\(\hat{j}\)+</span>
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<span class="trailing_text" id="trailing_text_8c4a353a4c64494b97c4c10d58187ee9_7_1">\(\hat{k}\)</span>
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<p> \(\vec{a}_3^{*}\)=</p>
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<span class="trailing_text" id="trailing_text_8c4a353a4c64494b97c4c10d58187ee9_8_1">\(\hat{i}\)+</span>
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<span class="trailing_text" id="trailing_text_8c4a353a4c64494b97c4c10d58187ee9_9_1">\(\hat{j}\)+</span>
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<h2 class="hd hd-2 unit-title">Part 2.4: Bragg's Law & Diffraction</h2>
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<h3>Bragg's Law and Diffraction</h3>
<ul>
<ul>
<li>Bragg's Law describes a necessary condition for constructive interference to occur</li>
<li>It is given by \(2d sin(\theta) = n \lambda\), where</li>
<ul>
<ul>
<li type="circle">\(d\) is the distance between planes</li>
<li type="circle">\(\theta\) is the angle of the incident x-rays</li>
<li type="circle">\(\lambda\) is the wavelength of the incident x-rays</li>
<li type="circle">\(n\) is an integer</li>
</ul>
</ul>
<li>Bragg's law is a necessary, but not sufficient, condition for constructive interference. </li>
</ul>
</ul>
<p></p>
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Tutorial Problem 2.4A
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>An x-ray is incident on a plane of atoms with an interplanar spacing of \(5 &#197;\). The wavelength of the incedent light is \(0.3 \; nm\).</p>
<div class="choicegroup capa_inputtype" id="inputtype_3f593b19c12f4c0f9303065d7020392b_2_1">
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<legend id="3f593b19c12f4c0f9303065d7020392b_2_1-legend" class="response-fieldset-legend field-group-hd">At what incident angles could we possibly see diffraction peaks</legend>
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<input type="checkbox" name="input_3f593b19c12f4c0f9303065d7020392b_2_1[]" id="input_3f593b19c12f4c0f9303065d7020392b_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="3f593b19c12f4c0f9303065d7020392b_2_1-choice_0-label" for="input_3f593b19c12f4c0f9303065d7020392b_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_3f593b19c12f4c0f9303065d7020392b_2_1"> \(17.46^{\circ}\)
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<input type="checkbox" name="input_3f593b19c12f4c0f9303065d7020392b_2_1[]" id="input_3f593b19c12f4c0f9303065d7020392b_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="3f593b19c12f4c0f9303065d7020392b_2_1-choice_1-label" for="input_3f593b19c12f4c0f9303065d7020392b_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_3f593b19c12f4c0f9303065d7020392b_2_1"> \(24.39^{\circ}\)
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<input type="checkbox" name="input_3f593b19c12f4c0f9303065d7020392b_2_1[]" id="input_3f593b19c12f4c0f9303065d7020392b_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="3f593b19c12f4c0f9303065d7020392b_2_1-choice_2-label" for="input_3f593b19c12f4c0f9303065d7020392b_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_3f593b19c12f4c0f9303065d7020392b_2_1"> \(31.29^{\circ}\)
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<input type="checkbox" name="input_3f593b19c12f4c0f9303065d7020392b_2_1[]" id="input_3f593b19c12f4c0f9303065d7020392b_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="3f593b19c12f4c0f9303065d7020392b_2_1-choice_3-label" for="input_3f593b19c12f4c0f9303065d7020392b_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_3f593b19c12f4c0f9303065d7020392b_2_1"> \(36.87^{\circ}\)
</label>
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<input type="checkbox" name="input_3f593b19c12f4c0f9303065d7020392b_2_1[]" id="input_3f593b19c12f4c0f9303065d7020392b_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="3f593b19c12f4c0f9303065d7020392b_2_1-choice_4-label" for="input_3f593b19c12f4c0f9303065d7020392b_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_3f593b19c12f4c0f9303065d7020392b_2_1"> \(64.16^{\circ}\)
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<input type="checkbox" name="input_3f593b19c12f4c0f9303065d7020392b_2_1[]" id="input_3f593b19c12f4c0f9303065d7020392b_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="3f593b19c12f4c0f9303065d7020392b_2_1-choice_5-label" for="input_3f593b19c12f4c0f9303065d7020392b_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_3f593b19c12f4c0f9303065d7020392b_2_1"> None of these
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Tutorial Problem 2.4B
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>An x-ray is incident on a plane of atoms with an interplanar spacing of \(5 &#197;\). The wavelength of the incedent light is \(0.3 \; nm\).</p>
<div class="choicegroup capa_inputtype" id="inputtype_fc7e4f82ef6b437a8b990418e90f5aca_2_1">
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<legend id="fc7e4f82ef6b437a8b990418e90f5aca_2_1-legend" class="response-fieldset-legend field-group-hd">At which angles will we definitely observe diffraction peaks?</legend>
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<input type="checkbox" name="input_fc7e4f82ef6b437a8b990418e90f5aca_2_1[]" id="input_fc7e4f82ef6b437a8b990418e90f5aca_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="fc7e4f82ef6b437a8b990418e90f5aca_2_1-choice_0-label" for="input_fc7e4f82ef6b437a8b990418e90f5aca_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_fc7e4f82ef6b437a8b990418e90f5aca_2_1"> \(17.46^{\circ}\)
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<input type="checkbox" name="input_fc7e4f82ef6b437a8b990418e90f5aca_2_1[]" id="input_fc7e4f82ef6b437a8b990418e90f5aca_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="fc7e4f82ef6b437a8b990418e90f5aca_2_1-choice_1-label" for="input_fc7e4f82ef6b437a8b990418e90f5aca_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_fc7e4f82ef6b437a8b990418e90f5aca_2_1"> \(24.39^{\circ}\)
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<input type="checkbox" name="input_fc7e4f82ef6b437a8b990418e90f5aca_2_1[]" id="input_fc7e4f82ef6b437a8b990418e90f5aca_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="fc7e4f82ef6b437a8b990418e90f5aca_2_1-choice_2-label" for="input_fc7e4f82ef6b437a8b990418e90f5aca_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_fc7e4f82ef6b437a8b990418e90f5aca_2_1"> \(31.29^{\circ}\)
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<input type="checkbox" name="input_fc7e4f82ef6b437a8b990418e90f5aca_2_1[]" id="input_fc7e4f82ef6b437a8b990418e90f5aca_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="fc7e4f82ef6b437a8b990418e90f5aca_2_1-choice_3-label" for="input_fc7e4f82ef6b437a8b990418e90f5aca_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_fc7e4f82ef6b437a8b990418e90f5aca_2_1"> \(36.87^{\circ}\)
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<input type="checkbox" name="input_fc7e4f82ef6b437a8b990418e90f5aca_2_1[]" id="input_fc7e4f82ef6b437a8b990418e90f5aca_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="fc7e4f82ef6b437a8b990418e90f5aca_2_1-choice_4-label" for="input_fc7e4f82ef6b437a8b990418e90f5aca_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_fc7e4f82ef6b437a8b990418e90f5aca_2_1"> \(64.16^{\circ}\)
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<h2 class="hd hd-2 unit-title">Part 2.5: Interplanar Spacing</h2>
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<h3>Interplanar Spacing</h3>
<ul>
<ul>
<li>How can we determine the interplanar distance \(d\) between two planes?</li>
<li>Consider an arbitrary vector \(\vec{g}\) expressed in reciprocal space</li>
<ul>
<ul>
<li type="circle">\(\vec{g}_{hkl} = h\vec{a}_1^* +k\vec{a}_2^* + l\vec{a}_3^*\)</li>
</ul>
</ul>
<li>\(\vec{g}_{hkl} \) is perpendicular to plane \((hkl)\)</li>
<ul>
<ul>
<li type="circle">For example, \(\vec{g}_{111} \perp\) \((111)\)</li>
</ul>
</ul>
<li>\(|\vec{g}_{hkl}| = \frac{1}{d_{hkl}}\)</li>
<ul>
<ul>
<li type="circle">where \(d\) is the distance between the \((hkl)\) planes</li>
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Tutorial Problem 2.5A
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<p> Consider a unit cell with the following reciprocal lattice vectors:
</p>
<p>\(\vec{a}_1^* = 0.29 \hat{i}\)</p>
<p> \(\vec{a}_2^* = 0.145 \hat{i} + 0.251 \hat{j}\)</p>
<p> \(\vec{a}_3^* = 0.251 \hat{k}\)</p>
<p> Let's determine the distance between the \(110\) planes in this crystal.</p>
<p>Start by writing an expression for \(\vec{g}_{hkl}\) in terms of \(\vec{a}_1^*\), \(\vec{a}_2^*\), and \(\vec{a}_3^*\).</p>
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<p> \(\vec{g}_{110}\)=</p>
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<span class="trailing_text" id="trailing_text_b48b15b3792f4565a7b849744222b4f7_2_1">\(\vec{a}_1^*\)+</span>
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<span class="trailing_text" id="trailing_text_b48b15b3792f4565a7b849744222b4f7_4_1">\(\vec{a}_3^*\)</span>
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<p> Now express \(\vec{g}_{110}\) in terms of \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\). Please give the coefficients in terms of \(&#197;^{-1}\).</p>
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<p> \(\vec{g}_{110}\)=</p>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>Next, we need to calculate \(|\vec{g}_{110}|\).</p>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>Finally, knowing that \(|\vec{g}_{hkl}|\) = \(\frac{1}{d_{hkl}}\), what is the interplanar spacing \(d_{110}\)?</p>
<p>\(d_{110}\) (in \(&#197;\)):</p>
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\(\)
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<h2 class="hd hd-2 unit-title">Part 2.6: Selection Rules</h2>
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<h3>Selection Rules</h3>
<ul>
<ul>
<li>The Structure Factor \(F(\Delta k)\) of a unit cell describes the intensity of diffracted beams</li>
<li>\(F(\Delta k) = \sum\limits_{i=1}^{N} f(r_1)e^{i 2 \pi \Delta k r_i}\) \( = \sum\limits_{n= 1}^{N} f(\vec{r}_i) e^{i2 \pi(hx_i + ky_i + lz_i)}\)</li>
<ul>
<ul>
<li type="circle">\(N\) is equal to the number of atoms in the unit cell</li>
<li type="circle">\(\vec{r_1} = x_i \vec{a}_1+ y_i \vec{a}_2 + z_i \vec{a}_3\)</li>
<li type="circle">\(f\) is the atomic scattering factor of the atom</li>
</ul>
</ul>
<li>Knowledge of the Structure Factor allows us to determine selection rules for the crystal</li>
<li>Selection Rules tell us which planes of a crystal will demonstrate reflections</li>
</ul>
</ul>
<p></p>
<table style="width: 100%;">
<tbody>
<tr><th>Crystal Type</th><th>Reflection present for</th><th>F</th><th>Number of lattice <br />points per cell</th></tr>
<tr>
<td>Primitive</td>
<td>Any h,k,l</td>
<td>f</td>
<td>1</td>
</tr>
<tr>
<td>Body centered</td>
<td>(h + k + l) = 2n</td>
<td>2f</td>
<td>2</td>
</tr>
<tr>
<td>Face centered</td>
<td>h, k, and l all odd or even</td>
<td>4f</td>
<td>4</td>
</tr>
<tr>
<td>Diamond</td>
<td>Like fcc, but if all even and h + k + l is not equal to 4n<br /> then absent anyway</td>
<td></td>
<td></td>
</tr>
<tr>
<td>Base centered</td>
<td>h,k, and l all odd or even</td>
<td>2f</td>
<td>2</td>
</tr>
</tbody>
</table>
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Tutorial Problem 2.6A
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<p>Would you expect to observe diffraction peaks off the \((111)\) plane in the following cells?</p>
<p>1) Simple cubic cell</p>
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<p>2) Body-centered cubic cell</p>
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<p>3) Face-centered cubic cell</p>
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