<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:MITx+3.012S.1x+1T2019+type@vertical+block@8df9d102df83413497c0998b7093e3bd" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+3.012S.1x+1T2019" data-block-type="vertical" data-request-token="be64bf13fe5d11ee8e26026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="VerticalStudentView"> <h2 class="hd hd-2 unit-title">Part 2.1: Miller Indices</h2><div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+3.012S.1x+1T2019+type@html+block@35c8bb9975344646baf187a6a402e237"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-usage-id="block-v1:MITx+3.012S.1x+1T2019+type@html+block@35c8bb9975344646baf187a6a402e237" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+3.012S.1x+1T2019" data-block-type="html" data-request-token="be64bf13fe5d11ee8e26026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <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> </div> <div class="vert vert-1" data-id="block-v1:MITx+3.012S.1x+1T2019+type@drag-and-drop-v2+block@eda338c343414108b537440330496f2a"> <div class="xblock xblock-public_view xblock-public_view-drag-and-drop-v2" data-usage-id="block-v1:MITx+3.012S.1x+1T2019+type@drag-and-drop-v2+block@eda338c343414108b537440330496f2a" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+3.012S.1x+1T2019" data-block-type="drag-and-drop-v2" data-request-token="be64bf13fe5d11ee8e26026cc65ec0d9" data-has-score="True" data-runtime-version="1" data-init="DragAndDropBlock"> <script type="json/xblock-args" class="xblock-json-init-args"> {"show_title": true, "max_attempts": 0, "type": "drag-and-drop-v2", "item_text_color": null, "mode": "assessment", "url_name": "eda338c343414108b537440330496f2a", "max_items_per_zone": null, "is_anonymous_user": true, "target_img_expanded_url": "/assets/courseware/v1/70b4448f59e6190ef0bb2210155e0570/asset-v1:MITx+3.012S.1x+1T2019+type@asset+block/3072Miller3.png", "weighted_max_score": 1.0, "has_deadline_passed": false, "display_zone_borders": true, "block_id": "block-v1:MITx+3.012S.1x+1T2019+type@drag-and-drop-v2+block@eda338c343414108b537440330496f2a", "item_background_color": null, "graded": false, "items": [{"imageURL": "/assets/courseware/v1/3ee6f23eab9fe0d9fea2858b91e06b09/asset-v1:MITx+3.012S.1x+1T2019+type@asset+block/3072PS2Ex4_F.png", "expandedImageURL": "/assets/courseware/v1/3ee6f23eab9fe0d9fea2858b91e06b09/asset-v1:MITx+3.012S.1x+1T2019+type@asset+block/3072PS2Ex4_F.png", "id": 0, "widthPercent": "13", "imageDescription": "(-2 -2 1)", "displayName": ""}, {"imageURL": "/assets/courseware/v1/f3e1f19dc184e45cb72c91643f8c8065/asset-v1:MITx+3.012S.1x+1T2019+type@asset+block/3072PS2Ex4_E.png", "expandedImageURL": 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crystallographic planes", "weight": 1.0, "display_name": "Tutorial Problem 2.1A: Miller Indices Practice", "title": "Tutorial Problem 2.1A: Miller Indices Practice"} </script> <div class="themed-xblock xblock--drag-and-drop"> <span class="fa fa-spin fa-spinner initial-load-spinner" aria-hidden="true"></span> Loading drag and drop problem.</div></div> </div></div></div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:MITx+3.012S.1x+1T2019+type@vertical+block@3c74cb9c9e21403293876e81d44e369c" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+3.012S.1x+1T2019" data-block-type="vertical" data-request-token="be64bf13fe5d11ee8e26026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="VerticalStudentView"> <h2 class="hd hd-2 unit-title">Part 2.2: Crystallographic Notation</h2><div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+3.012S.1x+1T2019+type@html+block@cc1e2b0b657646c0a576a404e0794894"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-usage-id="block-v1:MITx+3.012S.1x+1T2019+type@html+block@cc1e2b0b657646c0a576a404e0794894" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+3.012S.1x+1T2019" data-block-type="html" data-request-token="be64bf13fe5d11ee8e26026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <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></div> </div> <div class="vert vert-1" data-id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@74ca3c94cd3f40f39a9048d93a695a70"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@74ca3c94cd3f40f39a9048d93a695a70" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+3.012S.1x+1T2019" data-block-type="problem" data-request-token="be64bf13fe5d11ee8e26026cc65ec0d9" data-has-score="True" data-runtime-version="1" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_74ca3c94cd3f40f39a9048d93a695a70" class="problems-wrapper" role="group" aria-labelledby="74ca3c94cd3f40f39a9048d93a695a70-problem-title" data-problem-id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@74ca3c94cd3f40f39a9048d93a695a70" data-url="/courses/course-v1:MITx+3.012S.1x+1T2019/xblock/block-v1:MITx+3.012S.1x+1T2019+type@problem+block@74ca3c94cd3f40f39a9048d93a695a70/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="1" data-attempts-used="0" data-content="<h3 class="hd hd-3 problem-header" id="74ca3c94cd3f40f39a9048d93a695a70-problem-title" aria-describedby="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@74ca3c94cd3f40f39a9048d93a695a70-problem-progress" tabindex="-1"> Question 2.2A</h3><div class="problem-progress" id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@74ca3c94cd3f40f39a9048d93a695a70-problem-progress"></div><div class="problem"> <div> <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>In an primitive tetragonal cell, is the \((001)\) plane a member of the \(\{100\}\) family of planes?</p> <div class="choicegroup capa_inputtype" id="inputtype_74ca3c94cd3f40f39a9048d93a695a70_2_1"> <fieldset aria-describedby="status_74ca3c94cd3f40f39a9048d93a695a70_2_1"> <div class="field"> <input type="radio" name="input_74ca3c94cd3f40f39a9048d93a695a70_2_1" id="input_74ca3c94cd3f40f39a9048d93a695a70_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="74ca3c94cd3f40f39a9048d93a695a70_2_1-choice_0-label" for="input_74ca3c94cd3f40f39a9048d93a695a70_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_74ca3c94cd3f40f39a9048d93a695a70_2_1"> Yes, it is a member </label> </div> <div class="field"> <input type="radio" name="input_74ca3c94cd3f40f39a9048d93a695a70_2_1" id="input_74ca3c94cd3f40f39a9048d93a695a70_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="74ca3c94cd3f40f39a9048d93a695a70_2_1-choice_1-label" for="input_74ca3c94cd3f40f39a9048d93a695a70_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_74ca3c94cd3f40f39a9048d93a695a70_2_1"> No, it is not a member </label> </div> <span id="answer_74ca3c94cd3f40f39a9048d93a695a70_2_1"/> </fieldset> <div class="indicator-container"> <span class="status unanswered" id="status_74ca3c94cd3f40f39a9048d93a695a70_2_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span> </div></div><div class="solution-span"> <span id="solution_74ca3c94cd3f40f39a9048d93a695a70_solution_1"/></div></div></div> <div class="action"> <input type="hidden" name="problem_id" value="Question 2.2A" /> <div class="submit-attempt-container"> <button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_74ca3c94cd3f40f39a9048d93a695a70" > <span class="submit-label">Submit</span> </button> <div class="submission-feedback" id="submission_feedback_74ca3c94cd3f40f39a9048d93a695a70"> <span class="sr">Some problems have options such as save, reset, hints, or show answer. 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data-course-id="course-v1:MITx+3.012S.1x+1T2019" data-block-type="problem" data-request-token="be64bf13fe5d11ee8e26026cc65ec0d9" data-has-score="True" data-runtime-version="1" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_7f8cadc8c1164048bf80bbd587e00023" class="problems-wrapper" role="group" aria-labelledby="7f8cadc8c1164048bf80bbd587e00023-problem-title" data-problem-id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@7f8cadc8c1164048bf80bbd587e00023" data-url="/courses/course-v1:MITx+3.012S.1x+1T2019/xblock/block-v1:MITx+3.012S.1x+1T2019+type@problem+block@7f8cadc8c1164048bf80bbd587e00023/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="1" data-attempts-used="0" data-content="<h3 class="hd hd-3 problem-header" id="7f8cadc8c1164048bf80bbd587e00023-problem-title" aria-describedby="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@7f8cadc8c1164048bf80bbd587e00023-problem-progress" tabindex="-1"> Question 2.2B</h3><div class="problem-progress" id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@7f8cadc8c1164048bf80bbd587e00023-problem-progress"></div><div class="problem"> <div> <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> <div class="choicegroup capa_inputtype" id="inputtype_7f8cadc8c1164048bf80bbd587e00023_2_1"> <fieldset aria-describedby="status_7f8cadc8c1164048bf80bbd587e00023_2_1"> <div class="field"> <input type="radio" name="input_7f8cadc8c1164048bf80bbd587e00023_2_1" id="input_7f8cadc8c1164048bf80bbd587e00023_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="7f8cadc8c1164048bf80bbd587e00023_2_1-choice_0-label" for="input_7f8cadc8c1164048bf80bbd587e00023_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_7f8cadc8c1164048bf80bbd587e00023_2_1"> Yes, it is a member </label> </div> <div class="field"> <input type="radio" name="input_7f8cadc8c1164048bf80bbd587e00023_2_1" id="input_7f8cadc8c1164048bf80bbd587e00023_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="7f8cadc8c1164048bf80bbd587e00023_2_1-choice_1-label" for="input_7f8cadc8c1164048bf80bbd587e00023_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_7f8cadc8c1164048bf80bbd587e00023_2_1"> No, it is not a member </label> </div> <span id="answer_7f8cadc8c1164048bf80bbd587e00023_2_1"/> </fieldset> <div class="indicator-container"> <span class="status unanswered" id="status_7f8cadc8c1164048bf80bbd587e00023_2_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span> </div></div><div class="solution-span"> <span id="solution_7f8cadc8c1164048bf80bbd587e00023_solution_1"/></div></div></div> <div class="action"> <input type="hidden" name="problem_id" value="Question 2.2B" /> <div class="submit-attempt-container"> <button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_7f8cadc8c1164048bf80bbd587e00023" > <span class="submit-label">Submit</span> </button> <div class="submission-feedback" id="submission_feedback_7f8cadc8c1164048bf80bbd587e00023"> <span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span> </div> </div> <div class="problem-action-buttons-wrapper"> </div> </div> <div class="notification warning notification-gentle-alert is-hidden" tabindex="-1"> <span class="icon fa fa-exclamation-circle" aria-hidden="true"></span> <span class="notification-message" aria-describedby="7f8cadc8c1164048bf80bbd587e00023-problem-title"> </span> <div class="notification-btn-wrapper"> <button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button> </div></div> <div class="notification warning notification-save is-hidden" tabindex="-1"> <span class="icon fa fa-save" aria-hidden="true"></span> <span class="notification-message" aria-describedby="7f8cadc8c1164048bf80bbd587e00023-problem-title">None </span> <div class="notification-btn-wrapper"> <button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button> </div></div> <div class="notification general notification-show-answer is-hidden" tabindex="-1"> <span class="icon fa fa-info-circle" aria-hidden="true"></span> <span class="notification-message" aria-describedby="7f8cadc8c1164048bf80bbd587e00023-problem-title">Answers are displayed within the problem </span> <div class="notification-btn-wrapper"> <button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button> </div></div></div>" data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading…</span> </p></div></div> </div> <div class="vert vert-3" data-id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@9823cae3b9c44b1a9444abb302287101"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@9823cae3b9c44b1a9444abb302287101" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+3.012S.1x+1T2019" data-block-type="problem" data-request-token="be64bf13fe5d11ee8e26026cc65ec0d9" data-has-score="True" data-runtime-version="1" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_9823cae3b9c44b1a9444abb302287101" class="problems-wrapper" role="group" aria-labelledby="9823cae3b9c44b1a9444abb302287101-problem-title" data-problem-id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@9823cae3b9c44b1a9444abb302287101" data-url="/courses/course-v1:MITx+3.012S.1x+1T2019/xblock/block-v1:MITx+3.012S.1x+1T2019+type@problem+block@9823cae3b9c44b1a9444abb302287101/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="4" data-attempts-used="0" data-content="<h3 class="hd hd-3 problem-header" id="9823cae3b9c44b1a9444abb302287101-problem-title" aria-describedby="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@9823cae3b9c44b1a9444abb302287101-problem-progress" tabindex="-1"> Question 2.2C</h3><div class="problem-progress" id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@9823cae3b9c44b1a9444abb302287101-problem-progress"></div><div class="problem"> <div> <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> <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="inputtype option-input "> <select name="input_9823cae3b9c44b1a9444abb302287101_2_1" id="input_9823cae3b9c44b1a9444abb302287101_2_1" aria-describedby="status_9823cae3b9c44b1a9444abb302287101_2_1"> <option value="option_9823cae3b9c44b1a9444abb302287101_2_1_dummy_default">Select an option</option> <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> </select> <div class="indicator-container"> <span class="status unanswered" id="status_9823cae3b9c44b1a9444abb302287101_2_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span> </div> <p class="answer" id="answer_9823cae3b9c44b1a9444abb302287101_2_1"/></div></div> <p>\(\langle 201 \rangle\) refers to:</p> <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="inputtype option-input "> <select name="input_9823cae3b9c44b1a9444abb302287101_3_1" id="input_9823cae3b9c44b1a9444abb302287101_3_1" aria-describedby="status_9823cae3b9c44b1a9444abb302287101_3_1"> <option value="option_9823cae3b9c44b1a9444abb302287101_3_1_dummy_default">Select an option</option> <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> </select> <div class="indicator-container"> <span class="status unanswered" id="status_9823cae3b9c44b1a9444abb302287101_3_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span> </div> <p class="answer" id="answer_9823cae3b9c44b1a9444abb302287101_3_1"/></div></div> <p>\((201)\) refers to:</p> <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="inputtype option-input "> <select name="input_9823cae3b9c44b1a9444abb302287101_4_1" id="input_9823cae3b9c44b1a9444abb302287101_4_1" aria-describedby="status_9823cae3b9c44b1a9444abb302287101_4_1"> <option value="option_9823cae3b9c44b1a9444abb302287101_4_1_dummy_default">Select an option</option> <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> </select> <div class="indicator-container"> <span class="status unanswered" id="status_9823cae3b9c44b1a9444abb302287101_4_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span> </div> <p class="answer" id="answer_9823cae3b9c44b1a9444abb302287101_4_1"/></div></div> <p>\(\{201\}\) refers to:</p> <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div class="inputtype option-input "> <select name="input_9823cae3b9c44b1a9444abb302287101_5_1" id="input_9823cae3b9c44b1a9444abb302287101_5_1" aria-describedby="status_9823cae3b9c44b1a9444abb302287101_5_1"> <option value="option_9823cae3b9c44b1a9444abb302287101_5_1_dummy_default">Select an option</option> <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> </select> <div class="indicator-container"> <span class="status unanswered" id="status_9823cae3b9c44b1a9444abb302287101_5_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span> </div> <p class="answer" id="answer_9823cae3b9c44b1a9444abb302287101_5_1"/></div></div></div> <div class="action"> <input type="hidden" name="problem_id" value="Question 2.2C" /> <div class="submit-attempt-container"> <button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_9823cae3b9c44b1a9444abb302287101" > <span class="submit-label">Submit</span> </button> <div class="submission-feedback" 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<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:MITx+3.012S.1x+1T2019+type@vertical+block@9790909da7144c33abf39f5248fc720b" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+3.012S.1x+1T2019" data-block-type="vertical" data-request-token="be64bf13fe5d11ee8e26026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="VerticalStudentView"> <h2 class="hd hd-2 unit-title">Part 2.3: Reciprocal Space</h2><div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+3.012S.1x+1T2019+type@html+block@36001dab68db46418d0f8ee1149b7e6c"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-usage-id="block-v1:MITx+3.012S.1x+1T2019+type@html+block@36001dab68db46418d0f8ee1149b7e6c" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+3.012S.1x+1T2019" data-block-type="html" data-request-token="be64bf13fe5d11ee8e26026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <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></div> </div> <div class="vert vert-1" data-id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@7cd1297a6e5547b7b5839a28f889a8a4"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@7cd1297a6e5547b7b5839a28f889a8a4" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+3.012S.1x+1T2019" data-block-type="problem" data-request-token="be64bf13fe5d11ee8e26026cc65ec0d9" data-has-score="True" data-runtime-version="1" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_7cd1297a6e5547b7b5839a28f889a8a4" class="problems-wrapper" role="group" aria-labelledby="7cd1297a6e5547b7b5839a28f889a8a4-problem-title" data-problem-id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@7cd1297a6e5547b7b5839a28f889a8a4" data-url="/courses/course-v1:MITx+3.012S.1x+1T2019/xblock/block-v1:MITx+3.012S.1x+1T2019+type@problem+block@7cd1297a6e5547b7b5839a28f889a8a4/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="3" data-attempts-used="0" data-content="<h3 class="hd hd-3 problem-header" id="7cd1297a6e5547b7b5839a28f889a8a4-problem-title" aria-describedby="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@7cd1297a6e5547b7b5839a28f889a8a4-problem-progress" tabindex="-1"> Tutorial Problem 2.3A</h3><div class="problem-progress" id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@7cd1297a6e5547b7b5839a28f889a8a4-problem-progress"></div><div class="problem"> <div> <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> <div class="table"> <table cellspacing="0" class="tabular" style="table-layout:auto"> <tr> <td style="text-align:center"> <p> \(\mathbf{a_1}\)=</p> </td> <td style="text-align:center"> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_7cd1297a6e5547b7b5839a28f889a8a4_2_1" class=" capa_inputtype inline textline"> <div class="unanswered inline"><input type="text" name="input_7cd1297a6e5547b7b5839a28f889a8a4_2_1" id="input_7cd1297a6e5547b7b5839a28f889a8a4_2_1" aria-describedby="trailing_text_7cd1297a6e5547b7b5839a28f889a8a4_2_1 status_7cd1297a6e5547b7b5839a28f889a8a4_2_1" value=""/><span class="trailing_text" id="trailing_text_7cd1297a6e5547b7b5839a28f889a8a4_2_1">\(\hat{i}\)+</span><span class="status unanswered" id="status_7cd1297a6e5547b7b5839a28f889a8a4_2_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span><p id="answer_7cd1297a6e5547b7b5839a28f889a8a4_2_1" class="answer"/> </div></div></div> </td> <td style="text-align:center"> <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_7cd1297a6e5547b7b5839a28f889a8a4_3_1" class=" capa_inputtype inline textline"> <div class="unanswered inline"><input type="text" name="input_7cd1297a6e5547b7b5839a28f889a8a4_3_1" id="input_7cd1297a6e5547b7b5839a28f889a8a4_3_1" aria-describedby="trailing_text_7cd1297a6e5547b7b5839a28f889a8a4_3_1 status_7cd1297a6e5547b7b5839a28f889a8a4_3_1" value=""/><span class="trailing_text" id="trailing_text_7cd1297a6e5547b7b5839a28f889a8a4_3_1">\(\hat{j}\)+</span><span class="status unanswered" id="status_7cd1297a6e5547b7b5839a28f889a8a4_3_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span><p id="answer_7cd1297a6e5547b7b5839a28f889a8a4_3_1" class="answer"/> </div></div></div> </td> <td style="text-align:center"> <div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_7cd1297a6e5547b7b5839a28f889a8a4_4_1" class=" capa_inputtype inline textline"> <div class="unanswered inline"><input type="text" name="input_7cd1297a6e5547b7b5839a28f889a8a4_4_1" id="input_7cd1297a6e5547b7b5839a28f889a8a4_4_1" aria-describedby="trailing_text_7cd1297a6e5547b7b5839a28f889a8a4_4_1 status_7cd1297a6e5547b7b5839a28f889a8a4_4_1" value=""/><span class="trailing_text" id="trailing_text_7cd1297a6e5547b7b5839a28f889a8a4_4_1">\(\hat{k}\)</span><span class="status unanswered" id="status_7cd1297a6e5547b7b5839a28f889a8a4_4_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span><p id="answer_7cd1297a6e5547b7b5839a28f889a8a4_4_1" class="answer"/> </div></div></div> </td> </tr> </table> </div> <div class="solution-span"> <span id="solution_7cd1297a6e5547b7b5839a28f889a8a4_solution_1"/></div></div> <div class="action"> <input type="hidden" name="problem_id" value="Tutorial Problem 2.3A" 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These options follow the Submit button.</span> </div> </div> <div class="problem-action-buttons-wrapper"> </div> </div> <div class="notification warning notification-gentle-alert is-hidden" tabindex="-1"> <span class="icon fa fa-exclamation-circle" aria-hidden="true"></span> <span class="notification-message" aria-describedby="7cd1297a6e5547b7b5839a28f889a8a4-problem-title"> </span> <div class="notification-btn-wrapper"> <button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button> </div></div> <div class="notification warning notification-save is-hidden" tabindex="-1"> <span class="icon fa fa-save" aria-hidden="true"></span> <span class="notification-message" aria-describedby="7cd1297a6e5547b7b5839a28f889a8a4-problem-title">None </span> <div class="notification-btn-wrapper"> <button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button> </div></div> <div class="notification general 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data-course-id="course-v1:MITx+3.012S.1x+1T2019" data-block-type="problem" data-request-token="be64bf13fe5d11ee8e26026cc65ec0d9" data-has-score="True" data-runtime-version="1" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_a505b2c8331842dababadde29128d588" class="problems-wrapper" role="group" aria-labelledby="a505b2c8331842dababadde29128d588-problem-title" data-problem-id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@a505b2c8331842dababadde29128d588" data-url="/courses/course-v1:MITx+3.012S.1x+1T2019/xblock/block-v1:MITx+3.012S.1x+1T2019+type@problem+block@a505b2c8331842dababadde29128d588/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="6" data-attempts-used="0" data-content="<h3 class="hd hd-3 problem-header" id="a505b2c8331842dababadde29128d588-problem-title" aria-describedby="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@a505b2c8331842dababadde29128d588-problem-progress" tabindex="-1"> Tutorial Problem 2.3B</h3><div class="problem-progress" id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@a505b2c8331842dababadde29128d588-problem-progress"></div><div class="problem"> <div> <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> <div class="table"> <table cellspacing="0" class="tabular" style="table-layout:auto"> <tr> <td style="text-align:center"> <p> \(\mathbf{a_2}\)=</p> </td> <td style="text-align:center"> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_a505b2c8331842dababadde29128d588_2_1" class=" capa_inputtype inline textline"> <div class="unanswered inline"><input type="text" name="input_a505b2c8331842dababadde29128d588_2_1" id="input_a505b2c8331842dababadde29128d588_2_1" aria-describedby="trailing_text_a505b2c8331842dababadde29128d588_2_1 status_a505b2c8331842dababadde29128d588_2_1" value=""/><span class="trailing_text" id="trailing_text_a505b2c8331842dababadde29128d588_2_1">\(\hat{i}\)+</span><span class="status unanswered" id="status_a505b2c8331842dababadde29128d588_2_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span><p id="answer_a505b2c8331842dababadde29128d588_2_1" class="answer"/> </div></div></div> </td> <td style="text-align:center"> <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_a505b2c8331842dababadde29128d588_3_1" class=" capa_inputtype inline textline"> <div class="unanswered inline"><input type="text" name="input_a505b2c8331842dababadde29128d588_3_1" id="input_a505b2c8331842dababadde29128d588_3_1" aria-describedby="trailing_text_a505b2c8331842dababadde29128d588_3_1 status_a505b2c8331842dababadde29128d588_3_1" value=""/><span class="trailing_text" id="trailing_text_a505b2c8331842dababadde29128d588_3_1">\(\hat{j}\)+</span><span class="status unanswered" id="status_a505b2c8331842dababadde29128d588_3_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span><p id="answer_a505b2c8331842dababadde29128d588_3_1" class="answer"/> </div></div></div> </td> <td style="text-align:center"> <div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_a505b2c8331842dababadde29128d588_4_1" class=" capa_inputtype inline textline"> <div class="unanswered inline"><input type="text" name="input_a505b2c8331842dababadde29128d588_4_1" id="input_a505b2c8331842dababadde29128d588_4_1" aria-describedby="trailing_text_a505b2c8331842dababadde29128d588_4_1 status_a505b2c8331842dababadde29128d588_4_1" value=""/><span class="trailing_text" id="trailing_text_a505b2c8331842dababadde29128d588_4_1">\(\hat{k}\)</span><span class="status unanswered" id="status_a505b2c8331842dababadde29128d588_4_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span><p id="answer_a505b2c8331842dababadde29128d588_4_1" class="answer"/> </div></div></div> </td> </tr> </table> </div> <p> Lattice vector \(\vec{a}_3\) (in \(&#197;\)):</p> <div class="table"> <table cellspacing="0" class="tabular" style="table-layout:auto"> <tr> <td style="text-align:center"> <p> \(\mathbf{a_3}\)=</p> </td> <td style="text-align:center"> <div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_a505b2c8331842dababadde29128d588_5_1" class=" capa_inputtype inline textline"> <div class="unanswered inline"><input type="text" name="input_a505b2c8331842dababadde29128d588_5_1" id="input_a505b2c8331842dababadde29128d588_5_1" aria-describedby="trailing_text_a505b2c8331842dababadde29128d588_5_1 status_a505b2c8331842dababadde29128d588_5_1" value=""/><span class="trailing_text" id="trailing_text_a505b2c8331842dababadde29128d588_5_1">\(\hat{i}\)+</span><span class="status unanswered" id="status_a505b2c8331842dababadde29128d588_5_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span><p id="answer_a505b2c8331842dababadde29128d588_5_1" class="answer"/> </div></div></div> </td> <td style="text-align:center"> <div class="inline" tabindex="-1" aria-label="Question 5" role="group"><div id="inputtype_a505b2c8331842dababadde29128d588_6_1" class=" capa_inputtype inline textline"> <div class="unanswered inline"><input type="text" name="input_a505b2c8331842dababadde29128d588_6_1" id="input_a505b2c8331842dababadde29128d588_6_1" aria-describedby="trailing_text_a505b2c8331842dababadde29128d588_6_1 status_a505b2c8331842dababadde29128d588_6_1" value=""/><span class="trailing_text" id="trailing_text_a505b2c8331842dababadde29128d588_6_1">\(\hat{j}\)+</span><span class="status unanswered" id="status_a505b2c8331842dababadde29128d588_6_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span><p id="answer_a505b2c8331842dababadde29128d588_6_1" class="answer"/> </div></div></div> </td> <td style="text-align:center"> <div class="inline" tabindex="-1" aria-label="Question 6" role="group"><div id="inputtype_a505b2c8331842dababadde29128d588_7_1" class=" capa_inputtype inline textline"> <div class="unanswered inline"><input type="text" name="input_a505b2c8331842dababadde29128d588_7_1" id="input_a505b2c8331842dababadde29128d588_7_1" aria-describedby="trailing_text_a505b2c8331842dababadde29128d588_7_1 status_a505b2c8331842dababadde29128d588_7_1" value=""/><span class="trailing_text" id="trailing_text_a505b2c8331842dababadde29128d588_7_1">\(\hat{k}\)</span><span class="status unanswered" id="status_a505b2c8331842dababadde29128d588_7_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span><p id="answer_a505b2c8331842dababadde29128d588_7_1" class="answer"/> </div></div></div> </td> </tr> </table> </div></div> <div class="action"> <input type="hidden" name="problem_id" value="Tutorial Problem 2.3B" /> <div class="submit-attempt-container"> <button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_a505b2c8331842dababadde29128d588" > <span class="submit-label">Submit</span> </button> <div class="submission-feedback" id="submission_feedback_a505b2c8331842dababadde29128d588"> <span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span> </div> </div> <div class="problem-action-buttons-wrapper"> </div> </div> <div class="notification warning notification-gentle-alert is-hidden" tabindex="-1"> <span class="icon fa fa-exclamation-circle" aria-hidden="true"></span> <span class="notification-message" aria-describedby="a505b2c8331842dababadde29128d588-problem-title"> </span> <div class="notification-btn-wrapper"> <button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button> </div></div> <div class="notification warning notification-save is-hidden" tabindex="-1"> <span class="icon fa fa-save" aria-hidden="true"></span> <span class="notification-message" aria-describedby="a505b2c8331842dababadde29128d588-problem-title">None </span> <div class="notification-btn-wrapper"> <button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button> </div></div> <div class="notification general notification-show-answer is-hidden" tabindex="-1"> <span class="icon fa fa-info-circle" aria-hidden="true"></span> <span class="notification-message" aria-describedby="a505b2c8331842dababadde29128d588-problem-title">Answers are displayed within the problem </span> <div class="notification-btn-wrapper"> <button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button> </div></div></div>" data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading…</span> </p></div></div> </div> <div class="vert vert-3" data-id="block-v1:MITx+3.012S.1x+1T2019+type@html+block@0f5499cfa52f414db49f111ae7ed6b41"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-usage-id="block-v1:MITx+3.012S.1x+1T2019+type@html+block@0f5499cfa52f414db49f111ae7ed6b41" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+3.012S.1x+1T2019" data-block-type="html" data-request-token="be64bf13fe5d11ee8e26026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <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></ul></ul></ul><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></ul></ul></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></ul></ul></ul></ul><p></p></div> </div> <div class="vert vert-4" data-id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@0fc7e3e599c64015a02af810de82914f"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@0fc7e3e599c64015a02af810de82914f" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+3.012S.1x+1T2019" data-block-type="problem" data-request-token="be64bf13fe5d11ee8e26026cc65ec0d9" data-has-score="True" data-runtime-version="1" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_0fc7e3e599c64015a02af810de82914f" class="problems-wrapper" role="group" aria-labelledby="0fc7e3e599c64015a02af810de82914f-problem-title" data-problem-id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@0fc7e3e599c64015a02af810de82914f" data-url="/courses/course-v1:MITx+3.012S.1x+1T2019/xblock/block-v1:MITx+3.012S.1x+1T2019+type@problem+block@0fc7e3e599c64015a02af810de82914f/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="3" data-attempts-used="0" data-content="<h3 class="hd hd-3 problem-header" id="0fc7e3e599c64015a02af810de82914f-problem-title" aria-describedby="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@0fc7e3e599c64015a02af810de82914f-problem-progress" tabindex="-1"> Tutorial Problem 2.3C</h3><div class="problem-progress" id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@0fc7e3e599c64015a02af810de82914f-problem-progress"></div><div class="problem"> <div> <p> Start by calculating \(\vec{a}_2 \times \vec{a}_3\). </p> <p> \(\vec{a}_2 \times \vec{a}_3\) (in \(&#197;\)):</p> <div class="table"> <table cellspacing="0" class="tabular" style="table-layout:auto"> <tr> <td style="text-align:center"> <p> \(\mathbf{a_1}\)=</p> </td> <td style="text-align:center"> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_0fc7e3e599c64015a02af810de82914f_2_1" class=" capa_inputtype inline textline"> <div class="unanswered inline"><input type="text" name="input_0fc7e3e599c64015a02af810de82914f_2_1" id="input_0fc7e3e599c64015a02af810de82914f_2_1" aria-describedby="trailing_text_0fc7e3e599c64015a02af810de82914f_2_1 status_0fc7e3e599c64015a02af810de82914f_2_1" value=""/><span class="trailing_text" id="trailing_text_0fc7e3e599c64015a02af810de82914f_2_1">\(\hat{i}\)+</span><span class="status unanswered" id="status_0fc7e3e599c64015a02af810de82914f_2_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span><p id="answer_0fc7e3e599c64015a02af810de82914f_2_1" class="answer"/> </div></div></div> </td> <td style="text-align:center"> <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_0fc7e3e599c64015a02af810de82914f_3_1" class=" capa_inputtype inline textline"> <div class="unanswered inline"><input type="text" name="input_0fc7e3e599c64015a02af810de82914f_3_1" id="input_0fc7e3e599c64015a02af810de82914f_3_1" aria-describedby="trailing_text_0fc7e3e599c64015a02af810de82914f_3_1 status_0fc7e3e599c64015a02af810de82914f_3_1" value=""/><span class="trailing_text" id="trailing_text_0fc7e3e599c64015a02af810de82914f_3_1">\(\hat{j}\)+</span><span class="status unanswered" id="status_0fc7e3e599c64015a02af810de82914f_3_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span><p id="answer_0fc7e3e599c64015a02af810de82914f_3_1" class="answer"/> </div></div></div> </td> <td style="text-align:center"> <div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_0fc7e3e599c64015a02af810de82914f_4_1" class=" capa_inputtype inline textline"> <div class="unanswered inline"><input type="text" name="input_0fc7e3e599c64015a02af810de82914f_4_1" id="input_0fc7e3e599c64015a02af810de82914f_4_1" aria-describedby="trailing_text_0fc7e3e599c64015a02af810de82914f_4_1 status_0fc7e3e599c64015a02af810de82914f_4_1" value=""/><span class="trailing_text" id="trailing_text_0fc7e3e599c64015a02af810de82914f_4_1">\(\hat{k}\)</span><span class="status unanswered" id="status_0fc7e3e599c64015a02af810de82914f_4_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span><p id="answer_0fc7e3e599c64015a02af810de82914f_4_1" class="answer"/> </div></div></div> </td> </tr> </table> </div> <div class="solution-span"> <span id="solution_0fc7e3e599c64015a02af810de82914f_solution_1"/></div></div> <div class="action"> <input type="hidden" name="problem_id" value="Tutorial Problem 2.3C" /> <div 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These options follow the Submit button.</span> </div> </div> <div class="problem-action-buttons-wrapper"> </div> </div> <div class="notification warning notification-gentle-alert is-hidden" tabindex="-1"> <span class="icon fa fa-exclamation-circle" aria-hidden="true"></span> <span class="notification-message" aria-describedby="0fc7e3e599c64015a02af810de82914f-problem-title"> </span> <div class="notification-btn-wrapper"> <button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button> </div></div> <div class="notification warning notification-save is-hidden" tabindex="-1"> <span class="icon fa fa-save" aria-hidden="true"></span> <span class="notification-message" aria-describedby="0fc7e3e599c64015a02af810de82914f-problem-title">None </span> <div class="notification-btn-wrapper"> <button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button> </div></div> <div class="notification general 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data-course-id="course-v1:MITx+3.012S.1x+1T2019" data-block-type="problem" data-request-token="be64bf13fe5d11ee8e26026cc65ec0d9" data-has-score="True" data-runtime-version="1" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_128ac32a05424a738d2a9c009b334e28" class="problems-wrapper" role="group" aria-labelledby="128ac32a05424a738d2a9c009b334e28-problem-title" data-problem-id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@128ac32a05424a738d2a9c009b334e28" data-url="/courses/course-v1:MITx+3.012S.1x+1T2019/xblock/block-v1:MITx+3.012S.1x+1T2019+type@problem+block@128ac32a05424a738d2a9c009b334e28/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="1" data-attempts-used="0" data-content="<h3 class="hd hd-3 problem-header" id="128ac32a05424a738d2a9c009b334e28-problem-title" aria-describedby="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@128ac32a05424a738d2a9c009b334e28-problem-progress" tabindex="-1"> Tutorial Problem 2.3D</h3><div class="problem-progress" id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@128ac32a05424a738d2a9c009b334e28-problem-progress"></div><div class="problem"> <div> <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> <div id="formulaequationinput_128ac32a05424a738d2a9c009b334e28_2_1" class="inputtype formulaequationinput"> <div class="unanswered"> <input type="text" name="input_128ac32a05424a738d2a9c009b334e28_2_1" id="input_128ac32a05424a738d2a9c009b334e28_2_1" data-input-id="128ac32a05424a738d2a9c009b334e28_2_1" value="" aria-describedby="status_128ac32a05424a738d2a9c009b334e28_2_1" size="20"/> <span class="trailing_text" id="trailing_text_128ac32a05424a738d2a9c009b334e28_2_1"/> <span class="status unanswered" id="status_128ac32a05424a738d2a9c009b334e28_2_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span> <p id="answer_128ac32a05424a738d2a9c009b334e28_2_1" class="answer"/> <div id="input_128ac32a05424a738d2a9c009b334e28_2_1_preview" class="equation"> \(\) <img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/> </div> </div> <div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/></div><div class="solution-span"> <span id="solution_128ac32a05424a738d2a9c009b334e28_solution_1"/></div></div></div> <div class="action"> <input type="hidden" name="problem_id" value="Tutorial Problem 2.3D" /> <div class="submit-attempt-container"> <button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" 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data-course-id="course-v1:MITx+3.012S.1x+1T2019" data-block-type="problem" data-request-token="be64bf13fe5d11ee8e26026cc65ec0d9" data-has-score="True" data-runtime-version="1" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_8c4a353a4c64494b97c4c10d58187ee9" class="problems-wrapper" role="group" aria-labelledby="8c4a353a4c64494b97c4c10d58187ee9-problem-title" data-problem-id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@8c4a353a4c64494b97c4c10d58187ee9" data-url="/courses/course-v1:MITx+3.012S.1x+1T2019/xblock/block-v1:MITx+3.012S.1x+1T2019+type@problem+block@8c4a353a4c64494b97c4c10d58187ee9/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="9" data-attempts-used="0" data-content="<h3 class="hd hd-3 problem-header" id="8c4a353a4c64494b97c4c10d58187ee9-problem-title" aria-describedby="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@8c4a353a4c64494b97c4c10d58187ee9-problem-progress" tabindex="-1"> Tutorial Problem 2.3E</h3><div class="problem-progress" id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@8c4a353a4c64494b97c4c10d58187ee9-problem-progress"></div><div class="problem"> <div> <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> <div class="table"> <table cellspacing="0" class="tabular" style="table-layout:auto"> <tr> <td style="text-align:center"> <p> \(\vec{a}_1^{*}\)=</p> </td> <td style="text-align:center"> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_8c4a353a4c64494b97c4c10d58187ee9_2_1" class=" capa_inputtype inline textline"> <div class="unanswered inline"><input type="text" name="input_8c4a353a4c64494b97c4c10d58187ee9_2_1" id="input_8c4a353a4c64494b97c4c10d58187ee9_2_1" aria-describedby="trailing_text_8c4a353a4c64494b97c4c10d58187ee9_2_1 status_8c4a353a4c64494b97c4c10d58187ee9_2_1" value=""/><span class="trailing_text" id="trailing_text_8c4a353a4c64494b97c4c10d58187ee9_2_1">\(\hat{i}\)+</span><span class="status unanswered" id="status_8c4a353a4c64494b97c4c10d58187ee9_2_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span><p id="answer_8c4a353a4c64494b97c4c10d58187ee9_2_1" class="answer"/> </div></div></div> </td> <td style="text-align:center"> <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_8c4a353a4c64494b97c4c10d58187ee9_3_1" class=" capa_inputtype inline textline"> <div class="unanswered inline"><input type="text" name="input_8c4a353a4c64494b97c4c10d58187ee9_3_1" id="input_8c4a353a4c64494b97c4c10d58187ee9_3_1" aria-describedby="trailing_text_8c4a353a4c64494b97c4c10d58187ee9_3_1 status_8c4a353a4c64494b97c4c10d58187ee9_3_1" value=""/><span class="trailing_text" id="trailing_text_8c4a353a4c64494b97c4c10d58187ee9_3_1">\(\hat{j}\)+</span><span class="status unanswered" id="status_8c4a353a4c64494b97c4c10d58187ee9_3_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span><p id="answer_8c4a353a4c64494b97c4c10d58187ee9_3_1" class="answer"/> </div></div></div> </td> <td style="text-align:center"> <div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_8c4a353a4c64494b97c4c10d58187ee9_4_1" class=" capa_inputtype inline textline"> <div class="unanswered inline"><input type="text" name="input_8c4a353a4c64494b97c4c10d58187ee9_4_1" id="input_8c4a353a4c64494b97c4c10d58187ee9_4_1" aria-describedby="trailing_text_8c4a353a4c64494b97c4c10d58187ee9_4_1 status_8c4a353a4c64494b97c4c10d58187ee9_4_1" value=""/><span class="trailing_text" id="trailing_text_8c4a353a4c64494b97c4c10d58187ee9_4_1">\(\hat{k}\)</span><span class="status unanswered" id="status_8c4a353a4c64494b97c4c10d58187ee9_4_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span><p id="answer_8c4a353a4c64494b97c4c10d58187ee9_4_1" class="answer"/> </div></div></div> </td> </tr> </table> </div> <div class="table"> <table cellspacing="0" class="tabular" style="table-layout:auto"> <tr> <td style="text-align:center"> <p> \(\vec{a}_2^{*}\) =</p> </td> <td style="text-align:center"> <div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_8c4a353a4c64494b97c4c10d58187ee9_5_1" class=" capa_inputtype inline textline"> <div class="unanswered inline"><input type="text" name="input_8c4a353a4c64494b97c4c10d58187ee9_5_1" id="input_8c4a353a4c64494b97c4c10d58187ee9_5_1" aria-describedby="trailing_text_8c4a353a4c64494b97c4c10d58187ee9_5_1 status_8c4a353a4c64494b97c4c10d58187ee9_5_1" value=""/><span class="trailing_text" id="trailing_text_8c4a353a4c64494b97c4c10d58187ee9_5_1">\(\hat{i}\)+</span><span class="status unanswered" id="status_8c4a353a4c64494b97c4c10d58187ee9_5_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span><p id="answer_8c4a353a4c64494b97c4c10d58187ee9_5_1" class="answer"/> </div></div></div> </td> <td style="text-align:center"> <div class="inline" tabindex="-1" aria-label="Question 5" role="group"><div id="inputtype_8c4a353a4c64494b97c4c10d58187ee9_6_1" class=" capa_inputtype inline textline"> <div class="unanswered inline"><input type="text" name="input_8c4a353a4c64494b97c4c10d58187ee9_6_1" id="input_8c4a353a4c64494b97c4c10d58187ee9_6_1" aria-describedby="trailing_text_8c4a353a4c64494b97c4c10d58187ee9_6_1 status_8c4a353a4c64494b97c4c10d58187ee9_6_1" value=""/><span class="trailing_text" id="trailing_text_8c4a353a4c64494b97c4c10d58187ee9_6_1">\(\hat{j}\)+</span><span class="status unanswered" id="status_8c4a353a4c64494b97c4c10d58187ee9_6_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span><p id="answer_8c4a353a4c64494b97c4c10d58187ee9_6_1" class="answer"/> </div></div></div> </td> <td style="text-align:center"> <div class="inline" tabindex="-1" aria-label="Question 6" role="group"><div id="inputtype_8c4a353a4c64494b97c4c10d58187ee9_7_1" class=" capa_inputtype inline textline"> <div class="unanswered inline"><input type="text" name="input_8c4a353a4c64494b97c4c10d58187ee9_7_1" id="input_8c4a353a4c64494b97c4c10d58187ee9_7_1" aria-describedby="trailing_text_8c4a353a4c64494b97c4c10d58187ee9_7_1 status_8c4a353a4c64494b97c4c10d58187ee9_7_1" value=""/><span class="trailing_text" id="trailing_text_8c4a353a4c64494b97c4c10d58187ee9_7_1">\(\hat{k}\)</span><span class="status unanswered" id="status_8c4a353a4c64494b97c4c10d58187ee9_7_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span><p id="answer_8c4a353a4c64494b97c4c10d58187ee9_7_1" class="answer"/> </div></div></div> </td> </tr> </table> </div> <div class="table"> <table cellspacing="0" class="tabular" style="table-layout:auto"> <tr> <td style="text-align:center"> <p> \(\vec{a}_3^{*}\)=</p> </td> <td style="text-align:center"> <div class="inline" tabindex="-1" aria-label="Question 7" role="group"><div id="inputtype_8c4a353a4c64494b97c4c10d58187ee9_8_1" class=" capa_inputtype inline textline"> <div class="unanswered inline"><input type="text" name="input_8c4a353a4c64494b97c4c10d58187ee9_8_1" id="input_8c4a353a4c64494b97c4c10d58187ee9_8_1" aria-describedby="trailing_text_8c4a353a4c64494b97c4c10d58187ee9_8_1 status_8c4a353a4c64494b97c4c10d58187ee9_8_1" value=""/><span class="trailing_text" id="trailing_text_8c4a353a4c64494b97c4c10d58187ee9_8_1">\(\hat{i}\)+</span><span class="status unanswered" id="status_8c4a353a4c64494b97c4c10d58187ee9_8_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span><p id="answer_8c4a353a4c64494b97c4c10d58187ee9_8_1" class="answer"/> </div></div></div> </td> <td style="text-align:center"> <div class="inline" tabindex="-1" aria-label="Question 8" role="group"><div id="inputtype_8c4a353a4c64494b97c4c10d58187ee9_9_1" class=" capa_inputtype inline textline"> <div class="unanswered inline"><input type="text" name="input_8c4a353a4c64494b97c4c10d58187ee9_9_1" id="input_8c4a353a4c64494b97c4c10d58187ee9_9_1" aria-describedby="trailing_text_8c4a353a4c64494b97c4c10d58187ee9_9_1 status_8c4a353a4c64494b97c4c10d58187ee9_9_1" value=""/><span class="trailing_text" id="trailing_text_8c4a353a4c64494b97c4c10d58187ee9_9_1">\(\hat{j}\)+</span><span class="status unanswered" id="status_8c4a353a4c64494b97c4c10d58187ee9_9_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span><p id="answer_8c4a353a4c64494b97c4c10d58187ee9_9_1" class="answer"/> </div></div></div> </td> <td style="text-align:center"> <div class="inline" tabindex="-1" aria-label="Question 9" role="group"><div id="inputtype_8c4a353a4c64494b97c4c10d58187ee9_10_1" class=" capa_inputtype inline textline"> <div class="unanswered inline"><input type="text" name="input_8c4a353a4c64494b97c4c10d58187ee9_10_1" id="input_8c4a353a4c64494b97c4c10d58187ee9_10_1" aria-describedby="trailing_text_8c4a353a4c64494b97c4c10d58187ee9_10_1 status_8c4a353a4c64494b97c4c10d58187ee9_10_1" value=""/><span class="trailing_text" id="trailing_text_8c4a353a4c64494b97c4c10d58187ee9_10_1">\(\hat{k}\)</span><span class="status unanswered" id="status_8c4a353a4c64494b97c4c10d58187ee9_10_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span><p id="answer_8c4a353a4c64494b97c4c10d58187ee9_10_1" class="answer"/> </div></div></div> </td> </tr> </table> </div> <div class="solution-span"> <span id="solution_8c4a353a4c64494b97c4c10d58187ee9_solution_1"/></div></div> <div class="action"> <input type="hidden" name="problem_id" value="Tutorial Problem 2.3E" /> <div class="submit-attempt-container"> <button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_8c4a353a4c64494b97c4c10d58187ee9" > <span class="submit-label">Submit</span> </button> <div class="submission-feedback" id="submission_feedback_8c4a353a4c64494b97c4c10d58187ee9"> <span class="sr">Some problems have options such as save, reset, hints, or show answer. 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<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:MITx+3.012S.1x+1T2019+type@vertical+block@c3e46e6968f64523a10f661086c1e45b" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+3.012S.1x+1T2019" data-block-type="vertical" data-request-token="be64bf13fe5d11ee8e26026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="VerticalStudentView"> <h2 class="hd hd-2 unit-title">Part 2.4: Bragg's Law & Diffraction</h2><div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+3.012S.1x+1T2019+type@html+block@ccd069bef45b4268be4df5637bbf7b92"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-usage-id="block-v1:MITx+3.012S.1x+1T2019+type@html+block@ccd069bef45b4268be4df5637bbf7b92" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+3.012S.1x+1T2019" data-block-type="html" data-request-token="be64bf13fe5d11ee8e26026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <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></div> </div> <div class="vert vert-1" data-id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@3f593b19c12f4c0f9303065d7020392b"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@3f593b19c12f4c0f9303065d7020392b" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+3.012S.1x+1T2019" data-block-type="problem" data-request-token="be64bf13fe5d11ee8e26026cc65ec0d9" data-has-score="True" data-runtime-version="1" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_3f593b19c12f4c0f9303065d7020392b" class="problems-wrapper" role="group" aria-labelledby="3f593b19c12f4c0f9303065d7020392b-problem-title" data-problem-id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@3f593b19c12f4c0f9303065d7020392b" data-url="/courses/course-v1:MITx+3.012S.1x+1T2019/xblock/block-v1:MITx+3.012S.1x+1T2019+type@problem+block@3f593b19c12f4c0f9303065d7020392b/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="1" data-attempts-used="0" data-content="<h3 class="hd hd-3 problem-header" id="3f593b19c12f4c0f9303065d7020392b-problem-title" aria-describedby="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@3f593b19c12f4c0f9303065d7020392b-problem-progress" tabindex="-1"> Tutorial Problem 2.4A</h3><div class="problem-progress" id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@3f593b19c12f4c0f9303065d7020392b-problem-progress"></div><div class="problem"> <div> <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"> <fieldset aria-describedby="status_3f593b19c12f4c0f9303065d7020392b_2_1"> <legend id="3f593b19c12f4c0f9303065d7020392b_2_1-legend" class="response-fieldset-legend field-group-hd">At what incident angles could we possibly see diffraction peaks</legend> <div class="field"> <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"> \(17.46^{\circ}\) </label> </div> <div class="field"> <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"> \(24.39^{\circ}\) </label> </div> <div class="field"> <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"> \(31.29^{\circ}\) </label> </div> <div class="field"> <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> </div> <div class="field"> <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"> \(64.16^{\circ}\) </label> </div> <div class="field"> <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 </label> </div> <span id="answer_3f593b19c12f4c0f9303065d7020392b_2_1"/> </fieldset> <div class="indicator-container"> <span class="status unanswered" id="status_3f593b19c12f4c0f9303065d7020392b_2_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span> </div></div><div class="solution-span"> <span id="solution_3f593b19c12f4c0f9303065d7020392b_solution_1"/></div></div></div> <div class="action"> <input type="hidden" name="problem_id" value="Tutorial Problem 2.4A" /> <div class="submit-attempt-container"> <button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_3f593b19c12f4c0f9303065d7020392b" > <span class="submit-label">Submit</span> </button> <div class="submission-feedback" id="submission_feedback_3f593b19c12f4c0f9303065d7020392b"> <span class="sr">Some problems have options such as save, reset, hints, or show answer. 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The wavelength of the incedent light is \(0.3 \; nm\).</p> <div class="choicegroup capa_inputtype" id="inputtype_fc7e4f82ef6b437a8b990418e90f5aca_2_1"> <fieldset aria-describedby="status_fc7e4f82ef6b437a8b990418e90f5aca_2_1"> <legend id="fc7e4f82ef6b437a8b990418e90f5aca_2_1-legend" class="response-fieldset-legend field-group-hd">At which angles will we definitely observe diffraction peaks?</legend> <div class="field"> <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"> \(17.46^{\circ}\) </label> </div> <div class="field"> <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"> \(24.39^{\circ}\) </label> </div> <div class="field"> <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"> \(31.29^{\circ}\) </label> </div> <div class="field"> <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"> \(36.87^{\circ}\) </label> </div> <div class="field"> <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"> \(64.16^{\circ}\) </label> </div> <div class="field"> <input type="checkbox" name="input_fc7e4f82ef6b437a8b990418e90f5aca_2_1[]" id="input_fc7e4f82ef6b437a8b990418e90f5aca_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="fc7e4f82ef6b437a8b990418e90f5aca_2_1-choice_5-label" for="input_fc7e4f82ef6b437a8b990418e90f5aca_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_fc7e4f82ef6b437a8b990418e90f5aca_2_1"> None of these </label> </div> <span id="answer_fc7e4f82ef6b437a8b990418e90f5aca_2_1"/> </fieldset> <div class="indicator-container"> <span class="status unanswered" id="status_fc7e4f82ef6b437a8b990418e90f5aca_2_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span> </div></div><div class="solution-span"> <span id="solution_fc7e4f82ef6b437a8b990418e90f5aca_solution_1"/></div></div></div> <div class="action"> <input type="hidden" name="problem_id" value="Tutorial Problem 2.4B" /> <div class="submit-attempt-container"> <button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_fc7e4f82ef6b437a8b990418e90f5aca" > <span class="submit-label">Submit</span> </button> <div class="submission-feedback" id="submission_feedback_fc7e4f82ef6b437a8b990418e90f5aca"> <span class="sr">Some problems have options such as save, reset, hints, or show answer. 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<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:MITx+3.012S.1x+1T2019+type@vertical+block@5a15cf7eb48c404780d23e3753c7ca07" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+3.012S.1x+1T2019" data-block-type="vertical" data-request-token="be64bf13fe5d11ee8e26026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="VerticalStudentView"> <h2 class="hd hd-2 unit-title">Part 2.5: Interplanar Spacing</h2><div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+3.012S.1x+1T2019+type@html+block@a5bccd01aaa24c2eaff081b0f6c166f1"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-usage-id="block-v1:MITx+3.012S.1x+1T2019+type@html+block@a5bccd01aaa24c2eaff081b0f6c166f1" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+3.012S.1x+1T2019" data-block-type="html" data-request-token="be64bf13fe5d11ee8e26026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <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></ul></ul></ul></ul></div> </div> <div class="vert vert-1" data-id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@b48b15b3792f4565a7b849744222b4f7"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@b48b15b3792f4565a7b849744222b4f7" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+3.012S.1x+1T2019" data-block-type="problem" data-request-token="be64bf13fe5d11ee8e26026cc65ec0d9" data-has-score="True" data-runtime-version="1" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_b48b15b3792f4565a7b849744222b4f7" class="problems-wrapper" role="group" aria-labelledby="b48b15b3792f4565a7b849744222b4f7-problem-title" data-problem-id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@b48b15b3792f4565a7b849744222b4f7" data-url="/courses/course-v1:MITx+3.012S.1x+1T2019/xblock/block-v1:MITx+3.012S.1x+1T2019+type@problem+block@b48b15b3792f4565a7b849744222b4f7/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="3" data-attempts-used="0" data-content="<h3 class="hd hd-3 problem-header" id="b48b15b3792f4565a7b849744222b4f7-problem-title" aria-describedby="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@b48b15b3792f4565a7b849744222b4f7-problem-progress" tabindex="-1"> Tutorial Problem 2.5A</h3><div class="problem-progress" id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@b48b15b3792f4565a7b849744222b4f7-problem-progress"></div><div class="problem"> <div> <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> <div class="table"> <table cellspacing="0" class="tabular" style="table-layout:auto"> <tr> <td style="text-align:center"> <p> \(\vec{g}_{110}\)=</p> </td> <td style="text-align:center"> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_b48b15b3792f4565a7b849744222b4f7_2_1" class=" capa_inputtype inline textline"> <div class="unanswered inline"><input type="text" name="input_b48b15b3792f4565a7b849744222b4f7_2_1" id="input_b48b15b3792f4565a7b849744222b4f7_2_1" aria-describedby="trailing_text_b48b15b3792f4565a7b849744222b4f7_2_1 status_b48b15b3792f4565a7b849744222b4f7_2_1" value=""/><span class="trailing_text" id="trailing_text_b48b15b3792f4565a7b849744222b4f7_2_1">\(\vec{a}_1^*\)+</span><span class="status unanswered" id="status_b48b15b3792f4565a7b849744222b4f7_2_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span><p id="answer_b48b15b3792f4565a7b849744222b4f7_2_1" class="answer"/> </div></div></div> </td> <td style="text-align:center"> <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_b48b15b3792f4565a7b849744222b4f7_3_1" class=" capa_inputtype inline textline"> <div class="unanswered inline"><input type="text" name="input_b48b15b3792f4565a7b849744222b4f7_3_1" id="input_b48b15b3792f4565a7b849744222b4f7_3_1" aria-describedby="trailing_text_b48b15b3792f4565a7b849744222b4f7_3_1 status_b48b15b3792f4565a7b849744222b4f7_3_1" value=""/><span class="trailing_text" id="trailing_text_b48b15b3792f4565a7b849744222b4f7_3_1">\(\vec{a}_2^*\)+</span><span class="status unanswered" id="status_b48b15b3792f4565a7b849744222b4f7_3_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span><p id="answer_b48b15b3792f4565a7b849744222b4f7_3_1" class="answer"/> </div></div></div> </td> <td style="text-align:center"> <div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_b48b15b3792f4565a7b849744222b4f7_4_1" class=" capa_inputtype inline textline"> <div class="unanswered inline"><input type="text" name="input_b48b15b3792f4565a7b849744222b4f7_4_1" id="input_b48b15b3792f4565a7b849744222b4f7_4_1" aria-describedby="trailing_text_b48b15b3792f4565a7b849744222b4f7_4_1 status_b48b15b3792f4565a7b849744222b4f7_4_1" value=""/><span class="trailing_text" id="trailing_text_b48b15b3792f4565a7b849744222b4f7_4_1">\(\vec{a}_3^*\)</span><span class="status unanswered" id="status_b48b15b3792f4565a7b849744222b4f7_4_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span><p id="answer_b48b15b3792f4565a7b849744222b4f7_4_1" class="answer"/> </div></div></div> </td> </tr> </table> </div></div> <div class="action"> <input type="hidden" name="problem_id" value="Tutorial Problem 2.5A" /> <div class="submit-attempt-container"> <button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" 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Please give the coefficients in terms of \(&#197;^{-1}\).</p> <div class="table"> <table cellspacing="0" class="tabular" style="table-layout:auto"> <tr> <td style="text-align:center"> <p> \(\vec{g}_{110}\)=</p> </td> <td style="text-align:center"> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ce1c944fe6f44446bb264d7b1601a1e2_2_1" class=" capa_inputtype inline textline"> <div class="unanswered inline"><input type="text" name="input_ce1c944fe6f44446bb264d7b1601a1e2_2_1" id="input_ce1c944fe6f44446bb264d7b1601a1e2_2_1" aria-describedby="trailing_text_ce1c944fe6f44446bb264d7b1601a1e2_2_1 status_ce1c944fe6f44446bb264d7b1601a1e2_2_1" value=""/><span class="trailing_text" id="trailing_text_ce1c944fe6f44446bb264d7b1601a1e2_2_1">\(\hat{i}\)+</span><span class="status unanswered" id="status_ce1c944fe6f44446bb264d7b1601a1e2_2_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span><p id="answer_ce1c944fe6f44446bb264d7b1601a1e2_2_1" class="answer"/> </div></div></div> </td> <td style="text-align:center"> <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ce1c944fe6f44446bb264d7b1601a1e2_3_1" class=" capa_inputtype inline textline"> <div class="unanswered inline"><input type="text" name="input_ce1c944fe6f44446bb264d7b1601a1e2_3_1" id="input_ce1c944fe6f44446bb264d7b1601a1e2_3_1" aria-describedby="trailing_text_ce1c944fe6f44446bb264d7b1601a1e2_3_1 status_ce1c944fe6f44446bb264d7b1601a1e2_3_1" value=""/><span class="trailing_text" id="trailing_text_ce1c944fe6f44446bb264d7b1601a1e2_3_1">\(\hat{j}\)+</span><span class="status unanswered" id="status_ce1c944fe6f44446bb264d7b1601a1e2_3_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span><p id="answer_ce1c944fe6f44446bb264d7b1601a1e2_3_1" class="answer"/> </div></div></div> </td> <td style="text-align:center"> <div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_ce1c944fe6f44446bb264d7b1601a1e2_4_1" class=" capa_inputtype inline textline"> <div class="unanswered inline"><input type="text" name="input_ce1c944fe6f44446bb264d7b1601a1e2_4_1" id="input_ce1c944fe6f44446bb264d7b1601a1e2_4_1" aria-describedby="trailing_text_ce1c944fe6f44446bb264d7b1601a1e2_4_1 status_ce1c944fe6f44446bb264d7b1601a1e2_4_1" value=""/><span class="trailing_text" id="trailing_text_ce1c944fe6f44446bb264d7b1601a1e2_4_1">\(\hat{k}\)</span><span class="status unanswered" id="status_ce1c944fe6f44446bb264d7b1601a1e2_4_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span><p id="answer_ce1c944fe6f44446bb264d7b1601a1e2_4_1" class="answer"/> </div></div></div> </td> </tr> </table> </div> <div class="solution-span"> <span id="solution_ce1c944fe6f44446bb264d7b1601a1e2_solution_1"/></div></div> <div class="action"> <input type="hidden" name="problem_id" value="Tutorial Problem 2.5B" /> <div class="submit-attempt-container"> <button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_ce1c944fe6f44446bb264d7b1601a1e2" > <span class="submit-label">Submit</span> </button> <div class="submission-feedback" id="submission_feedback_ce1c944fe6f44446bb264d7b1601a1e2"> <span class="sr">Some problems have options such as save, reset, hints, or show answer. 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aria-describedby="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@a1edfbde1ea9404888af98b458c82c5e-problem-progress" tabindex="-1"> Tutorial Problem 2.5C</h3><div class="problem-progress" id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@a1edfbde1ea9404888af98b458c82c5e-problem-progress"></div><div class="problem"> <div> <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>Next, we need to calculate \(|\vec{g}_{110}|\).</p> <div id="formulaequationinput_a1edfbde1ea9404888af98b458c82c5e_2_1" class="inputtype formulaequationinput"> <div class="unanswered"> <label class="problem-group-label" for="input_a1edfbde1ea9404888af98b458c82c5e_2_1" id="label_a1edfbde1ea9404888af98b458c82c5e_2_1">\(|\vec{g}_{110}|\), in (\(&#197;^{-1} \)):</label> <input type="text" name="input_a1edfbde1ea9404888af98b458c82c5e_2_1" id="input_a1edfbde1ea9404888af98b458c82c5e_2_1" data-input-id="a1edfbde1ea9404888af98b458c82c5e_2_1" value="" 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aria-describedby="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@314593b3ddf34ad2a7fe0c431c8d380b-problem-progress" tabindex="-1"> Tutorial Problem 2.5D</h3><div class="problem-progress" id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@314593b3ddf34ad2a7fe0c431c8d380b-problem-progress"></div><div class="problem"> <div> <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> <div id="formulaequationinput_314593b3ddf34ad2a7fe0c431c8d380b_2_1" class="inputtype formulaequationinput"> <div class="unanswered"> <input type="text" name="input_314593b3ddf34ad2a7fe0c431c8d380b_2_1" id="input_314593b3ddf34ad2a7fe0c431c8d380b_2_1" data-input-id="314593b3ddf34ad2a7fe0c431c8d380b_2_1" value="" aria-describedby="status_314593b3ddf34ad2a7fe0c431c8d380b_2_1" size="20"/> <span class="trailing_text" 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<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:MITx+3.012S.1x+1T2019+type@vertical+block@9bd5249ca0aa4e598a1d6e9b3f2267ec" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+3.012S.1x+1T2019" data-block-type="vertical" data-request-token="be64bf13fe5d11ee8e26026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="VerticalStudentView"> <h2 class="hd hd-2 unit-title">Part 2.6: Selection Rules</h2><div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+3.012S.1x+1T2019+type@html+block@dde6d845c59c4bafbfff7b8da16cb63f"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-usage-id="block-v1:MITx+3.012S.1x+1T2019+type@html+block@dde6d845c59c4bafbfff7b8da16cb63f" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+3.012S.1x+1T2019" data-block-type="html" data-request-token="be64bf13fe5d11ee8e26026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <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><p></p></div> </div> <div class="vert vert-1" data-id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@372bc5463ec3423b93697a6e7cc31fb8"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@372bc5463ec3423b93697a6e7cc31fb8" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+3.012S.1x+1T2019" 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Tutorial Problem 2.6A</h3><div class="problem-progress" id="block-v1:MITx+3.012S.1x+1T2019+type@problem+block@372bc5463ec3423b93697a6e7cc31fb8-problem-progress"></div><div class="problem"> <div> <p>Would you expect to observe diffraction peaks off the \((111)\) plane in the following cells?</p> <p>1) Simple cubic cell</p> <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_372bc5463ec3423b93697a6e7cc31fb8_2_1"> <fieldset aria-describedby="status_372bc5463ec3423b93697a6e7cc31fb8_2_1"> <div class="field"> <input type="radio" name="input_372bc5463ec3423b93697a6e7cc31fb8_2_1" id="input_372bc5463ec3423b93697a6e7cc31fb8_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="372bc5463ec3423b93697a6e7cc31fb8_2_1-choice_0-label" for="input_372bc5463ec3423b93697a6e7cc31fb8_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_372bc5463ec3423b93697a6e7cc31fb8_2_1"> Yes </label> </div> <div class="field"> <input type="radio" name="input_372bc5463ec3423b93697a6e7cc31fb8_2_1" id="input_372bc5463ec3423b93697a6e7cc31fb8_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="372bc5463ec3423b93697a6e7cc31fb8_2_1-choice_1-label" for="input_372bc5463ec3423b93697a6e7cc31fb8_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_372bc5463ec3423b93697a6e7cc31fb8_2_1"> No </label> </div> <span id="answer_372bc5463ec3423b93697a6e7cc31fb8_2_1"/> </fieldset> <div class="indicator-container"> <span class="status unanswered" id="status_372bc5463ec3423b93697a6e7cc31fb8_2_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span> </div></div></div> <p>2) Body-centered cubic cell</p> <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_372bc5463ec3423b93697a6e7cc31fb8_3_1"> <fieldset aria-describedby="status_372bc5463ec3423b93697a6e7cc31fb8_3_1"> <div class="field"> <input type="radio" name="input_372bc5463ec3423b93697a6e7cc31fb8_3_1" id="input_372bc5463ec3423b93697a6e7cc31fb8_3_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="372bc5463ec3423b93697a6e7cc31fb8_3_1-choice_0-label" for="input_372bc5463ec3423b93697a6e7cc31fb8_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_372bc5463ec3423b93697a6e7cc31fb8_3_1"> Yes </label> </div> <div class="field"> <input type="radio" name="input_372bc5463ec3423b93697a6e7cc31fb8_3_1" id="input_372bc5463ec3423b93697a6e7cc31fb8_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="372bc5463ec3423b93697a6e7cc31fb8_3_1-choice_1-label" for="input_372bc5463ec3423b93697a6e7cc31fb8_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_372bc5463ec3423b93697a6e7cc31fb8_3_1"> No </label> </div> <span id="answer_372bc5463ec3423b93697a6e7cc31fb8_3_1"/> </fieldset> <div class="indicator-container"> <span class="status unanswered" id="status_372bc5463ec3423b93697a6e7cc31fb8_3_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/></span> </div></div></div> <p>3) Face-centered cubic cell</p> <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_372bc5463ec3423b93697a6e7cc31fb8_4_1"> <fieldset aria-describedby="status_372bc5463ec3423b93697a6e7cc31fb8_4_1"> <div class="field"> <input type="radio" name="input_372bc5463ec3423b93697a6e7cc31fb8_4_1" id="input_372bc5463ec3423b93697a6e7cc31fb8_4_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="372bc5463ec3423b93697a6e7cc31fb8_4_1-choice_0-label" 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