Phonon - Wikipedia

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Phonon - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"5e5230a6-f647-452d-8d91-824c1c88df30","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Phonon","wgTitle":"Phonon","wgCurRevisionId":1250846020,"wgRevisionId":1250846020,"wgArticleId":85754,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["CS1 maint: date and year","Articles with short description","Short description matches Wikidata","Articles containing Ancient Greek (to 1453)-language text","All articles with unsourced statements","Articles with unsourced statements from August 2020","Articles with unsourced statements from February 2019","Commons category link is on Wikidata","Quasiparticles","Bosons","1932 introductions"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName": "Phonon","wgRelevantArticleId":85754,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":50000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q186608","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model", "platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips", "ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Phonon - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Phonon"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Phonon&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Phonon"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Phonon rootpage-Phonon skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Phonon" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Phonon" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Phonon" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Phonon" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Definition" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definition</span> </div> </a> <ul id="toc-Definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lattice_dynamics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lattice_dynamics"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Lattice dynamics</span> </div> </a> <button aria-controls="toc-Lattice_dynamics-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Lattice dynamics subsection</span> </button> <ul id="toc-Lattice_dynamics-sublist" class="vector-toc-list"> <li id="toc-Lattice_waves" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lattice_waves"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Lattice waves</span> </div> </a> <ul id="toc-Lattice_waves-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-One-dimensional_lattice" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#One-dimensional_lattice"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>One-dimensional lattice</span> </div> </a> <ul id="toc-One-dimensional_lattice-sublist" class="vector-toc-list"> <li id="toc-Classical_treatment" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Classical_treatment"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Classical treatment</span> </div> </a> <ul id="toc-Classical_treatment-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_treatment" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Quantum_treatment"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.2</span> <span>Quantum treatment</span> </div> </a> <ul id="toc-Quantum_treatment-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Three-dimensional_lattice" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Three-dimensional_lattice"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Three-dimensional lattice</span> </div> </a> <ul id="toc-Three-dimensional_lattice-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dispersion_relation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dispersion_relation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Dispersion relation</span> </div> </a> <ul id="toc-Dispersion_relation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Interpretation_of_phonons_using_second_quantization_techniques" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Interpretation_of_phonons_using_second_quantization_techniques"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Interpretation of phonons using second quantization techniques</span> </div> </a> <ul id="toc-Interpretation_of_phonons_using_second_quantization_techniques-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Acoustic_and_optical_phonons" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Acoustic_and_optical_phonons"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Acoustic and optical phonons</span> </div> </a> <ul id="toc-Acoustic_and_optical_phonons-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Crystal_momentum" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Crystal_momentum"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Crystal momentum</span> </div> </a> <ul id="toc-Crystal_momentum-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Thermodynamics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Thermodynamics"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Thermodynamics</span> </div> </a> <button aria-controls="toc-Thermodynamics-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Thermodynamics subsection</span> </button> <ul id="toc-Thermodynamics-sublist" class="vector-toc-list"> <li id="toc-Phonon_tunneling" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Phonon_tunneling"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Phonon tunneling</span> </div> </a> <ul id="toc-Phonon_tunneling-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Operator_formalism" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Operator_formalism"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Operator formalism</span> </div> </a> <ul id="toc-Operator_formalism-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nonlinearity" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Nonlinearity"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Nonlinearity</span> </div> </a> <ul id="toc-Nonlinearity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Predicted_properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Predicted_properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Predicted properties</span> </div> </a> <ul id="toc-Predicted_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Superconductivity" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Superconductivity"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Superconductivity</span> </div> </a> <ul id="toc-Superconductivity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_research" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_research"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Other research</span> </div> </a> <ul id="toc-Other_research-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Phonon</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 50 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-50" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">50 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%81%D9%88%D9%86%D9%88%D9%86" title="فونون – Arabic" lang="ar" hreflang="ar" data-title="فونون" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%AB%27%E0%A6%A8%E0%A6%A8" title="ফ'নন – Assamese" lang="as" hreflang="as" data-title="ফ'নন" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AB%E0%A7%8B%E0%A6%A8%E0%A6%A8" title="ফোনন – Bangla" lang="bn" hreflang="bn" data-title="ফোনন" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A4%D0%B0%D0%BD%D0%BE%D0%BD" title="Фанон – Belarusian" lang="be" hreflang="be" data-title="Фанон" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A4%D0%BE%D0%BD%D0%BE%D0%BD" title="Фонон – Bulgarian" lang="bg" hreflang="bg" data-title="Фонон" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Fon%C3%B3" title="Fonó – Catalan" lang="ca" hreflang="ca" data-title="Fonó" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Fonon" title="Fonon – Czech" lang="cs" hreflang="cs" data-title="Fonon" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Fonon" title="Fonon – Danish" lang="da" hreflang="da" data-title="Fonon" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Phonon" title="Phonon – German" lang="de" hreflang="de" data-title="Phonon" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Foonon" title="Foonon – Estonian" lang="et" hreflang="et" data-title="Foonon" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A6%CF%89%CE%BD%CF%8C%CE%BD%CE%B9%CE%BF" title="Φωνόνιο – Greek" lang="el" hreflang="el" data-title="Φωνόνιο" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Fon%C3%B3n" title="Fonón – Spanish" lang="es" hreflang="es" data-title="Fonón" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D9%88%D9%86%D9%88%D9%86" title="فونون – Persian" lang="fa" hreflang="fa" data-title="فونون" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Phonon" title="Phonon – French" lang="fr" hreflang="fr" data-title="Phonon" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/F%C3%B3n%C3%B3n" title="Fónón – Irish" lang="ga" hreflang="ga" data-title="Fónón" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%8F%AC%EB%85%BC" title="포논 – Korean" lang="ko" hreflang="ko" data-title="포논" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%96%D5%B8%D5%B6%D5%B8%D5%B6" title="Ֆոնոն – Armenian" lang="hy" hreflang="hy" data-title="Ֆոնոն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AB%E0%A5%8B%E0%A4%A8%E0%A5%89%E0%A4%A8" title="फोनॉन – Hindi" lang="hi" hreflang="hi" data-title="फोनॉन" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Fonon" title="Fonon – Croatian" lang="hr" hreflang="hr" data-title="Fonon" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Fonon" title="Fonon – Indonesian" lang="id" hreflang="id" data-title="Fonon" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Fonone" title="Fonone – Italian" lang="it" hreflang="it" data-title="Fonone" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%95%D7%A0%D7%95%D7%9F" title="פונון – Hebrew" lang="he" hreflang="he" data-title="פונון" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A4%D0%BE%D0%BD%D0%BE%D0%BD" title="Фонон – Kazakh" lang="kk" hreflang="kk" data-title="Фонон" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%A4%D0%BE%D0%BD%D0%BE%D0%BD" title="Фонон – Kyrgyz" lang="ky" hreflang="ky" data-title="Фонон" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Fonon" title="Fonon – Hungarian" lang="hu" hreflang="hu" data-title="Fonon" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A4%D0%BE%D0%BD%D0%BE%D0%BD" title="Фонон – Macedonian" lang="mk" hreflang="mk" data-title="Фонон" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Fonon" title="Fonon – Malay" lang="ms" hreflang="ms" data-title="Fonon" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Fonon" title="Fonon – Dutch" lang="nl" hreflang="nl" data-title="Fonon" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%95%E3%82%A9%E3%83%8E%E3%83%B3" title="フォノン – Japanese" lang="ja" hreflang="ja" data-title="フォノン" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Fonon" title="Fonon – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Fonon" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Fonon" title="Fonon – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Fonon" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Fonon" title="Fonon – Uzbek" lang="uz" hreflang="uz" data-title="Fonon" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Fonon" title="Fonon – Polish" lang="pl" hreflang="pl" data-title="Fonon" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/F%C3%B4non" title="Fônon – Portuguese" lang="pt" hreflang="pt" data-title="Fônon" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Fonon" title="Fonon – Romanian" lang="ro" hreflang="ro" data-title="Fonon" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A4%D0%BE%D0%BD%D0%BE%D0%BD" title="Фонон – Russian" lang="ru" hreflang="ru" data-title="Фонон" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Phonon" title="Phonon – Simple English" lang="en-simple" hreflang="en-simple" data-title="Phonon" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Fon%C3%B3n" title="Fonón – Slovak" lang="sk" hreflang="sk" data-title="Fonón" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Fonon" title="Fonon – Slovenian" lang="sl" hreflang="sl" data-title="Fonon" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%81%DB%86%D9%86%DB%86%D9%86" title="فۆنۆن – Central Kurdish" lang="ckb" hreflang="ckb" data-title="فۆنۆن" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A4%D0%BE%D0%BD%D0%BE%D0%BD" title="Фонон – Serbian" lang="sr" hreflang="sr" data-title="Фонон" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Fonon" title="Fonon – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Fonon" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Fononi" title="Fononi – Finnish" lang="fi" hreflang="fi" data-title="Fononi" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Fonon" title="Fonon – Swedish" lang="sv" hreflang="sv" data-title="Fonon" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%82%E0%B8%9F%E0%B8%99%E0%B8%AD%E0%B8%99" title="โฟนอน – Thai" lang="th" hreflang="th" data-title="โฟนอน" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Fonon" title="Fonon – Turkish" lang="tr" hreflang="tr" data-title="Fonon" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A4%D0%BE%D0%BD%D0%BE%D0%BD" title="Фонон – Ukrainian" lang="uk" hreflang="uk" data-title="Фонон" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%81%D9%88%D9%86%D9%88%D9%86" title="فونون – Urdu" lang="ur" hreflang="ur" data-title="فونون" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Phonon" title="Phonon – Vietnamese" lang="vi" hreflang="vi" data-title="Phonon" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%A3%B0%E5%AD%90" title="声子 – Chinese" lang="zh" hreflang="zh" data-title="声子" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q186608#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Phonon" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Phonon" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Phonon"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Phonon&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Phonon&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Phonon"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Phonon&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Phonon&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Phonon" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Phonon" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q"><span>Special pages</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Phonon&oldid=1250846020" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Phonon&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Phonon&id=1250846020&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FPhonon"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FPhonon"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Phonon&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Phonon&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Lattice_vibrations" hreflang="en"><span>Wikimedia Commons</span></a></li><li class="wb-otherproject-link wb-otherproject-wikiquote mw-list-item"><a href="https://en.wikiquote.org/wiki/Phonon" hreflang="en"><span>Wikiquote</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q186608" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Quasiparticle of mechanical vibrations</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Photon" title="Photon">photon</a>, <a href="/wiki/Phonon_(software)" title="Phonon (software)">Phonon (software)</a>, or <a href="/wiki/Phonon_(company)" class="mw-redirect" title="Phonon (company)">Phonon (company)</a>.</div> <p>A <b>phonon</b> is a <a href="/wiki/Collective_excitation" class="mw-redirect" title="Collective excitation">collective excitation</a> in a periodic, <a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">elastic</a> arrangement of <a href="/wiki/Atom" title="Atom">atoms</a> or <a href="/wiki/Molecule" title="Molecule">molecules</a> in <a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">condensed matter</a>, specifically in <a href="/wiki/Solid" title="Solid">solids</a> and some <a href="/wiki/Liquid" title="Liquid">liquids</a>. A type of <a href="/wiki/Quasiparticle" title="Quasiparticle">quasiparticle</a> in <a href="/wiki/Physics" title="Physics">physics</a>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> a phonon is an <a href="/wiki/Excited_state" title="Excited state">excited state</a> in the <a href="/wiki/Quantum_mechanical" class="mw-redirect" title="Quantum mechanical">quantum mechanical</a> <a href="/wiki/Quantization_(physics)" title="Quantization (physics)">quantization</a> of the <a href="/wiki/Mode_of_vibration" class="mw-redirect" title="Mode of vibration">modes of vibrations</a> for elastic structures of interacting particles. Phonons can be thought of as quantized <a href="/wiki/Sound_waves" class="mw-redirect" title="Sound waves">sound waves</a>, similar to <a href="/wiki/Photons" class="mw-redirect" title="Photons">photons</a> as quantized <a href="/wiki/Light_waves" class="mw-redirect" title="Light waves">light waves</a>.<sup id="cite_ref-girvinYang_2-0" class="reference"><a href="#cite_note-girvinYang-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>The study of phonons is an important part of condensed matter physics. They play a major role in many of the physical properties of condensed matter systems, such as <a href="/wiki/Thermal_conductivity" class="mw-redirect" title="Thermal conductivity">thermal conductivity</a> and <a href="/wiki/Electrical_conductivity" class="mw-redirect" title="Electrical conductivity">electrical conductivity</a>, as well as in models of <a href="/wiki/Neutron_scattering" title="Neutron scattering">neutron scattering</a> and related effects. </p><p>The concept of phonons was introduced in 1930 by <a href="/wiki/Soviet_Union" title="Soviet Union">Soviet</a> physicist <a href="/wiki/Igor_Tamm" title="Igor Tamm">Igor Tamm</a>. The name <i>phonon</i> was suggested by <a href="/wiki/Yakov_Frenkel" title="Yakov Frenkel">Yakov Frenkel</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> It comes from the <a href="/wiki/Ancient_Greek_language" class="mw-redirect" title="Ancient Greek language">Greek</a> word <span title="Ancient Greek (to 1453)-language text"><span lang="grc">φωνή</span></span> (<span title="Ancient Greek (to 1453)-language romanization"><i lang="grc-Latn">phonē</i></span>), which translates to <i>sound</i> or <i>voice</i>, because long-wavelength phonons give rise to <a href="/wiki/Sound" title="Sound">sound</a>. The name emphasizes the analogy to the word <i>photon</i>, in that phonons represent <a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">wave-particle duality</a> for <a href="/wiki/Sound" title="Sound">sound waves</a> in the same way that photons represent wave-particle duality for <a href="/wiki/Light" title="Light">light waves</a>. Solids with more than one atom in the smallest <a href="/wiki/Unit_cell" title="Unit cell">unit cell</a> exhibit both <a href="#Acoustic_and_optical_phonons">acoustic</a> and <a href="#Acoustic_and_optical_phonons">optical</a> phonons. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A phonon is the <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanical</a> description of an elementary <a href="/wiki/Vibration" title="Vibration">vibrational</a> motion in which a <a href="/wiki/Lattice_model_(physics)" title="Lattice model (physics)">lattice</a> of atoms or molecules uniformly oscillates at a single <a href="/wiki/Frequency" title="Frequency">frequency</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> In <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a> this designates a <a href="/wiki/Normal_mode" title="Normal mode">normal mode</a> of vibration. Normal modes are important because any arbitrary lattice vibration can be considered to be a <a href="/wiki/Superposition_principle" title="Superposition principle">superposition</a> of these <i>elementary</i> vibration modes (cf. <a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a>). While normal modes are <a href="/wiki/Wave" title="Wave">wave-like</a> phenomena in classical mechanics, phonons have <a href="/wiki/Elementary_particle" title="Elementary particle">particle-like</a> properties too, in a way related to the <a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">wave–particle duality</a> of quantum mechanics. </p> <div class="mw-heading mw-heading2"><h2 id="Lattice_dynamics">Lattice dynamics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon&action=edit&section=2" title="Edit section: Lattice dynamics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The equations in this section do not use <a href="/wiki/Axiom" title="Axiom">axioms</a> of quantum mechanics but instead use relations for which there exists a direct <a href="/wiki/Correspondence_principle" title="Correspondence principle">correspondence</a> in classical mechanics. </p><p>For example: a rigid regular, <a href="/wiki/Crystalline" class="mw-redirect" title="Crystalline">crystalline</a> (not <a href="/wiki/Amorphous_solid" title="Amorphous solid">amorphous</a>) lattice is composed of <i>N</i> particles. These particles may be atoms or molecules. <i>N</i> is a large number, say of the order of 10<sup>23</sup>, or on the order of the <a href="/wiki/Avogadro_number" class="mw-redirect" title="Avogadro number">Avogadro number</a> for a typical sample of a solid. Since the lattice is rigid, the atoms must be exerting <a href="/wiki/Force" title="Force">forces</a> on one another to keep each atom near its equilibrium position. These forces may be <a href="/wiki/Van_der_Waals_force" title="Van der Waals force">Van der Waals forces</a>, <a href="/wiki/Covalent_bond" title="Covalent bond">covalent bonds</a>, <a href="/wiki/Electrostatic_attraction" class="mw-redirect" title="Electrostatic attraction">electrostatic attractions</a>, and others, all of which are ultimately due to the <a href="/wiki/Electric_field" title="Electric field">electric</a> force. <a href="/wiki/Magnetism" title="Magnetism">Magnetic</a> and <a href="/wiki/Gravity" title="Gravity">gravitational</a> forces are generally negligible. The forces between each pair of atoms may be characterized by a <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a> function <i>V</i> that depends on the distance of separation of the atoms. The potential energy of the entire lattice is the sum of all pairwise potential energies multiplied by a factor of 1/2 to compensate for double counting:<sup id="cite_ref-latticemechanics3_5-0" class="reference"><a href="#cite_note-latticemechanics3-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}\sum _{i\neq j}V\left(r_{i}-r_{j}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>≠</mo> <mi>j</mi> </mrow> </munder> <mi>V</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\sum _{i\neq j}V\left(r_{i}-r_{j}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ec3cbb8ef704c0ff57ad4ac1d7c777452c2d48c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:16.758ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{2}}\sum _{i\neq j}V\left(r_{i}-r_{j}\right)}"></span></dd></dl> <p>where <i>r<sub>i</sub></i> is the <a href="/wiki/Space" title="Space">position</a> of the <i>i</i>th atom, and <i>V</i> is the potential energy between two atoms. </p><p>It is difficult to solve this <a href="/wiki/Many-body_problem" title="Many-body problem">many-body problem</a> explicitly in either classical or quantum mechanics. In order to simplify the task, two important <a href="/wiki/Approximation" title="Approximation">approximations</a> are usually imposed. First, the sum is only performed over neighboring atoms. Although the electric forces in real solids extend to infinity, this approximation is still valid because the fields produced by distant atoms are effectively <a href="/wiki/Electric_field_screening" class="mw-redirect" title="Electric field screening">screened</a>. Secondly, the potentials <i>V</i> are treated as <a href="/wiki/Harmonic_oscillator" title="Harmonic oscillator">harmonic potentials</a>. This is permissible as long as the atoms remain close to their equilibrium positions. Formally, this is accomplished by <a href="/wiki/Taylor_series" title="Taylor series">Taylor expanding</a> <i>V</i> about its equilibrium value to quadratic order, giving <i>V</i> proportional to the displacement <i>x</i><sup>2</sup> and the elastic force simply proportional to <i>x</i>. The error in ignoring higher order terms remains small if <i>x</i> remains close to the equilibrium position. </p><p>The resulting lattice may be visualized as a system of balls connected by springs. The following figure shows a cubic lattice, which is a good model for many types of crystalline solid. Other lattices include a linear chain, which is a very simple lattice which we will shortly use for modeling phonons. (For other common lattices, see <a href="/wiki/Crystal_structure" title="Crystal structure">crystal structure</a>.) </p> <dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Cubic.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Cubic.svg/109px-Cubic.svg.png" decoding="async" width="109" height="127" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Cubic.svg/164px-Cubic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/55/Cubic.svg/218px-Cubic.svg.png 2x" data-file-width="109" data-file-height="127" /></a></span></dd></dl> <p>The potential energy of the lattice may now be written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{\{ij\}(\mathrm {nn} )}{\tfrac {1}{2}}m\omega ^{2}\left(R_{i}-R_{j}\right)^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mi>i</mi> <mi>j</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">n</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>m</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{\{ij\}(\mathrm {nn} )}{\tfrac {1}{2}}m\omega ^{2}\left(R_{i}-R_{j}\right)^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7443523b7f2b75bd134ab9411d37350b01bc129a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:24.17ex; height:6.009ex;" alt="{\displaystyle \sum _{\{ij\}(\mathrm {nn} )}{\tfrac {1}{2}}m\omega ^{2}\left(R_{i}-R_{j}\right)^{2}.}"></span></dd></dl> <p>Here, <i>ω</i> is the <a href="/wiki/Natural_frequency" title="Natural frequency">natural frequency</a> of the harmonic potentials, which are assumed to be the same since the lattice is regular. <i>R<sub>i</sub></i> is the position coordinate of the <i>i</i>th atom, which we now measure from its equilibrium position. The sum over nearest neighbors is denoted (nn). </p><p>It is important to mention that the mathematical treatment given here is highly simplified in order to make it accessible to non-experts. The simplification has been achieved by making two basic assumptions in the expression for the total potential energy of the crystal. These assumptions are that (i) the total potential energy can be written as a sum of pairwise interactions, and (ii) each atom interacts with only its nearest neighbors. These are used only sparingly in modern lattice dynamics.<sup id="cite_ref-:0_6-0" class="reference"><a href="#cite_note-:0-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> A more general approach is to express the potential energy in terms of force constants.<sup id="cite_ref-:0_6-1" class="reference"><a href="#cite_note-:0-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> See, for example, the Wiki article on <a href="/wiki/Multiscale_Green%27s_function" title="Multiscale Green's function">multiscale Green's functions.</a> </p> <div class="mw-heading mw-heading3"><h3 id="Lattice_waves">Lattice waves</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon&action=edit&section=3" title="Edit section: Lattice waves"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Lattice_wave.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/Lattice_wave.svg/200px-Lattice_wave.svg.png" decoding="async" width="200" height="187" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/Lattice_wave.svg/300px-Lattice_wave.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/27/Lattice_wave.svg/400px-Lattice_wave.svg.png 2x" data-file-width="506" data-file-height="472" /></a><figcaption>Phonon propagating through a square lattice (atom displacements greatly exaggerated)</figcaption></figure> <p>Due to the connections between atoms, the displacement of one or more atoms from their equilibrium positions gives rise to a set of vibration <a href="/wiki/Wave" title="Wave">waves</a> propagating through the lattice. One such wave is shown in the figure to the right. The <a href="/wiki/Amplitude" title="Amplitude">amplitude</a> of the wave is given by the displacements of the atoms from their equilibrium positions. The <a href="/wiki/Wavelength" title="Wavelength">wavelength</a> <i>λ</i> is marked. </p><p>There is a minimum possible wavelength, given by twice the equilibrium separation <i>a</i> between atoms. Any wavelength shorter than this can be mapped onto a wavelength longer than 2<i>a</i>, due to the periodicity of the lattice. This can be thought of as a consequence of the <a href="/wiki/Nyquist%E2%80%93Shannon_sampling_theorem" title="Nyquist–Shannon sampling theorem">Nyquist–Shannon sampling theorem</a>, the lattice points being viewed as the "sampling points" of a continuous wave. </p><p>Not every possible lattice vibration has a well-defined wavelength and frequency. However, the <a href="/wiki/Normal_mode" title="Normal mode">normal modes</a> do possess well-defined wavelengths and <a href="/wiki/Frequency" title="Frequency">frequencies</a>. </p> <div class="mw-heading mw-heading3"><h3 id="One-dimensional_lattice">One-dimensional lattice</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon&action=edit&section=4" title="Edit section: One-dimensional lattice"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:1D_normal_modes_(280_kB).gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/1D_normal_modes_%28280_kB%29.gif" decoding="async" width="275" height="275" class="mw-file-element" data-file-width="275" data-file-height="275" /></a><figcaption>Animation showing 6 normal modes of a one-dimensional lattice: a linear chain of particles. The shortest wavelength is at top, with progressively longer wavelengths below. In the lowest lines the motion of the waves to the right can be seen.</figcaption></figure> <p>In order to simplify the analysis needed for a 3-dimensional lattice of atoms, it is convenient to model a 1-dimensional lattice or linear chain. This model is complex enough to display the salient features of phonons. </p> <div class="mw-heading mw-heading4"><h4 id="Classical_treatment">Classical treatment</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon&action=edit&section=5" title="Edit section: Classical treatment"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The forces between the atoms are assumed to be linear and nearest-neighbour, and they are represented by an elastic spring. Each atom is assumed to be a point particle and the nucleus and electrons move in step (<a href="/wiki/Adiabatic_theorem" title="Adiabatic theorem">adiabatic theorem</a>): </p> <dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><i>n</i> − 1 <span style="padding-left:1em;"> </span> <i>n</i> <span style="padding-left:2em;"> </span> <i>n</i> + 1 <span style="padding-left:5em;"> </span> ← <span style="padding-left:1em;"> </span> <i>a</i> <span style="padding-left:1em;"> </span> →</dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl> <p>···o++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o··· </p> <dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd>→→<span style="padding-left:2em;"> </span>→<span style="padding-left:2em;"> </span>→→→</dd> <dd><i>u</i><sub><i>n</i> − 1</sub><span style="padding-left:2em;"> </span><i>u<sub>n</sub></i><span style="padding-left:2em;"> </span><i>u</i><sub><i>n</i> + 1</sub></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">n</span> labels the <span class="texhtml mvar" style="font-style:italic;">n</span>th atom out of a total of <span class="texhtml mvar" style="font-style:italic;">N</span>, <span class="texhtml mvar" style="font-style:italic;">a</span> is the distance between atoms when the chain is in equilibrium, and <span class="texhtml"><i>u<sub>n</sub></i></span> the displacement of the <span class="texhtml mvar" style="font-style:italic;">n</span>th atom from its equilibrium position. </p><p>If <i>C</i> is the elastic constant of the spring and <span class="texhtml mvar" style="font-style:italic;">m</span> the mass of the atom, then the equation of motion of the <span class="texhtml mvar" style="font-style:italic;">n</span>th atom is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -2Cu_{n}+C\left(u_{n+1}+u_{n-1}\right)=m{\frac {d^{2}u_{n}}{dt^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>2</mn> <mi>C</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mi>C</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -2Cu_{n}+C\left(u_{n+1}+u_{n-1}\right)=m{\frac {d^{2}u_{n}}{dt^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf48fc936444c49195aae1c9815cc6518a260309" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:37.668ex; height:6.009ex;" alt="{\displaystyle -2Cu_{n}+C\left(u_{n+1}+u_{n-1}\right)=m{\frac {d^{2}u_{n}}{dt^{2}}}.}"></span></dd></dl> <p>This is a set of coupled equations. </p><p>Since the solutions are expected to be oscillatory, new coordinates are defined by a <a href="/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">discrete Fourier transform</a>, in order to decouple them.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>Put </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{n}=\sum _{Nak/2\pi =1}^{N}Q_{k}e^{ikna}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mi>a</mi> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mi>π</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>n</mi> <mi>a</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{n}=\sum _{Nak/2\pi =1}^{N}Q_{k}e^{ikna}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6fc82bafe152ba110848a714bddce158b20ea49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:22.075ex; height:7.843ex;" alt="{\displaystyle u_{n}=\sum _{Nak/2\pi =1}^{N}Q_{k}e^{ikna}.}"></span></dd></dl> <p>Here, <span class="texhtml"><i>na</i></span> corresponds and devolves to the continuous variable <span class="texhtml mvar" style="font-style:italic;">x</span> of scalar field theory. The <span class="texhtml"><i>Q<sub>k</sub></i></span> are known as the <i>normal coordinates</i> for continuum field modes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{k}=e^{ikna}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>n</mi> <mi>a</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{k}=e^{ikna}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1837bff306908bdae00d37765f7990088ea8f678" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.168ex; height:3.009ex;" alt="{\displaystyle \phi _{k}=e^{ikna}}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=2\pi j/(Na)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=2\pi j/(Na)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0265c3888e2b2c6193991895d14a6231ac785466" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.027ex; height:2.843ex;" alt="{\displaystyle k=2\pi j/(Na)}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=1\dots N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>…</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j=1\dots N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0a975c69ce6619b5b3a64646cb017fdb495d8e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:10.807ex; height:2.509ex;" alt="{\displaystyle j=1\dots N}"></span>. </p><p>Substitution into the equation of motion produces the following <i>decoupled equations</i> (this requires a significant manipulation using the orthonormality and completeness relations of the discrete Fourier transform),<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2C(\cos {ka-1})Q_{k}=m{\frac {d^{2}Q_{k}}{dt^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>C</mi> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>a</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo stretchy="false">)</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2C(\cos {ka-1})Q_{k}=m{\frac {d^{2}Q_{k}}{dt^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/840858e51b48f88f5d2ff9e4076520be7544af0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:29.428ex; height:6.009ex;" alt="{\displaystyle 2C(\cos {ka-1})Q_{k}=m{\frac {d^{2}Q_{k}}{dt^{2}}}.}"></span></dd></dl> <p>These are the equations for decoupled <a href="/wiki/Harmonic_oscillators" class="mw-redirect" title="Harmonic oscillators">harmonic oscillators</a> which have the solution </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{k}=A_{k}e^{i\omega _{k}t};\qquad \omega _{k}={\sqrt {{\frac {2C}{m}}(1-\cos {ka})}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mi>t</mi> </mrow> </msup> <mo>;</mo> <mspace width="2em" /> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>C</mi> </mrow> <mi>m</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>a</mi> </mrow> <mo stretchy="false">)</mo> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{k}=A_{k}e^{i\omega _{k}t};\qquad \omega _{k}={\sqrt {{\frac {2C}{m}}(1-\cos {ka})}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa0272bc33bb1c92943a4b3cbcef2e91b0c1bdfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:43.015ex; height:6.343ex;" alt="{\displaystyle Q_{k}=A_{k}e^{i\omega _{k}t};\qquad \omega _{k}={\sqrt {{\frac {2C}{m}}(1-\cos {ka})}}.}"></span></dd></dl> <p>Each normal coordinate <i>Q<sub>k</sub></i> represents an independent vibrational mode of the lattice with wavenumber <span class="texhtml mvar" style="font-style:italic;">k</span>, which is known as a <a href="/wiki/Normal_mode" title="Normal mode">normal mode</a>. </p><p>The second equation, for <span class="texhtml"><i>ω<sub>k</sub></i></span>, is known as the <a href="/wiki/Dispersion_relation" title="Dispersion relation">dispersion relation</a> between the <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a> and the <a href="/wiki/Wavenumber" title="Wavenumber">wavenumber</a>. </p><p>In the <a href="/wiki/Continuum_limit" title="Continuum limit">continuum limit</a>, <span class="texhtml mvar" style="font-style:italic;">a</span>→0, <span class="texhtml mvar" style="font-style:italic;">N</span>→∞, with <span class="texhtml"><i>Na</i></span> held fixed, <span class="texhtml"><i>u<sub>n</sub></i></span> → <span class="texhtml"><i>φ</i>(<i>x</i>)</span>, a scalar field, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega (k)\propto ka}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>∝</mo> <mi>k</mi> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega (k)\propto ka}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9200245b4f280f7f4157f3c7375eb3b985e638a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.006ex; height:2.843ex;" alt="{\displaystyle \omega (k)\propto ka}"></span>. This amounts to classical free <a href="/wiki/Scalar_field_theory" title="Scalar field theory">scalar field theory</a>, an assembly of independent oscillators. </p> <div class="mw-heading mw-heading4"><h4 id="Quantum_treatment">Quantum treatment</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon&action=edit&section=6" title="Edit section: Quantum treatment"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A one-dimensional quantum mechanical harmonic chain consists of <i>N</i> identical atoms. This is the simplest quantum mechanical model of a lattice that allows phonons to arise from it. The formalism for this model is readily generalizable to two and three dimensions. </p><p>In contrast to the previous section, the positions of the masses are not denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14f13cb025ff2e136dcbd2fc81ddf965b728e3d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle u_{i}}"></span>, but instead by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},x_{2},\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},x_{2},\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e71a0bff46c1d8ad13a271387dcf9c11fb30cee9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.559ex; height:2.009ex;" alt="{\displaystyle x_{1},x_{2},\dots }"></span> as measured from their equilibrium positions. (I.e. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/491e215fe273840aab63acab32e73cb14f2fc8e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.39ex; height:2.509ex;" alt="{\displaystyle x_{i}=0}"></span> if particle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> is at its equilibrium position.) In two or more dimensions, the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}"></span> are vector quantities. The <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a> for this system is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}=\sum _{i=1}^{N}{\frac {p_{i}^{2}}{2m}}+{\frac {1}{2}}m\omega ^{2}\sum _{\{ij\}(\mathrm {nn} )}\left(x_{i}-x_{j}\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mi>i</mi> <mi>j</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">n</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </munder> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}=\sum _{i=1}^{N}{\frac {p_{i}^{2}}{2m}}+{\frac {1}{2}}m\omega ^{2}\sum _{\{ij\}(\mathrm {nn} )}\left(x_{i}-x_{j}\right)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bf24d4e3ac682daa870eb28c47025d78ead0fee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:39.066ex; height:7.843ex;" alt="{\displaystyle {\mathcal {H}}=\sum _{i=1}^{N}{\frac {p_{i}^{2}}{2m}}+{\frac {1}{2}}m\omega ^{2}\sum _{\{ij\}(\mathrm {nn} )}\left(x_{i}-x_{j}\right)^{2}}"></span></dd></dl> <p>where <i>m</i> is the mass of each atom (assuming it is equal for all), and <i>x<sub>i</sub></i> and <i>p<sub>i</sub></i> are the position and <a href="/wiki/Momentum" title="Momentum">momentum</a> operators, respectively, for the <i>i</i>th atom and the sum is made over the nearest neighbors (nn). However one expects that in a lattice there could also appear waves that behave like particles. It is customary to deal with <a href="/wiki/Wave" title="Wave">waves</a> in <a href="/wiki/Fourier_space" class="mw-redirect" title="Fourier space">Fourier space</a> which uses <a href="/wiki/Normal_modes" class="mw-redirect" title="Normal modes">normal modes</a> of the <a href="/wiki/Wavevector" class="mw-redirect" title="Wavevector">wavevector</a> as variables instead of coordinates of particles. The number of normal modes is the same as the number of particles. Still, the Fourier space is very useful given the <a href="/wiki/Fourier_series" title="Fourier series">periodicity</a> of the system. </p><p>A set of <i>N</i> "normal coordinates" <i>Q<sub>k</sub></i> may be introduced, defined as the <a href="/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">discrete Fourier transforms</a> of the <i>x<sub>k</sub></i> and <i>N</i> "conjugate momenta" <i>Π<sub>k</sub></i> defined as the Fourier transforms of the <i>p<sub>k</sub></i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}Q_{k}&={\frac {1}{\sqrt {N}}}\sum _{l}e^{ikal}x_{l}\\\Pi _{k}&={\frac {1}{\sqrt {N}}}\sum _{l}e^{-ikal}p_{l}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>N</mi> </msqrt> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>a</mi> <mi>l</mi> </mrow> </msup> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>N</mi> </msqrt> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>k</mi> <mi>a</mi> <mi>l</mi> </mrow> </msup> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}Q_{k}&={\frac {1}{\sqrt {N}}}\sum _{l}e^{ikal}x_{l}\\\Pi _{k}&={\frac {1}{\sqrt {N}}}\sum _{l}e^{-ikal}p_{l}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fc51bf982e53d0819fa2063057719fbfcb9eebd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:23.659ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}Q_{k}&={\frac {1}{\sqrt {N}}}\sum _{l}e^{ikal}x_{l}\\\Pi _{k}&={\frac {1}{\sqrt {N}}}\sum _{l}e^{-ikal}p_{l}.\end{aligned}}}"></span></dd></dl> <p>The quantity <i>k</i> turns out to be the <a href="/wiki/Wavenumber" title="Wavenumber">wavenumber</a> of the phonon, i.e. 2<span class="texhtml mvar" style="font-style:italic;">π</span> divided by the <a href="/wiki/Wavelength" title="Wavelength">wavelength</a>. </p><p>This choice retains the desired commutation relations in either real space or wavevector space </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\left[x_{l},p_{m}\right]&=i\hbar \delta _{l,m}\\\left[Q_{k},\Pi _{k'}\right]&={\frac {1}{N}}\sum _{l,m}e^{ikal}e^{-ik'am}\left[x_{l},p_{m}\right]\\&={\frac {i\hbar }{N}}\sum _{l}e^{ial\left(k-k'\right)}=i\hbar \delta _{k,k'}\\\left[Q_{k},Q_{k'}\right]&=\left[\Pi _{k},\Pi _{k'}\right]=0\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow> <mo>[</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>[</mo> <mrow> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>k</mi> <mo>′</mo> </msup> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mo>,</mo> <mi>m</mi> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>a</mi> <mi>l</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <msup> <mi>k</mi> <mo>′</mo> </msup> <mi>a</mi> <mi>m</mi> </mrow> </msup> <mrow> <mo>[</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mrow> <mi>N</mi> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>a</mi> <mi>l</mi> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>−</mo> <msup> <mi>k</mi> <mo>′</mo> </msup> </mrow> <mo>)</mo> </mrow> </mrow> </msup> <mo>=</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <msup> <mi>k</mi> <mo>′</mo> </msup> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>[</mo> <mrow> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>k</mi> <mo>′</mo> </msup> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>k</mi> <mo>′</mo> </msup> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left[x_{l},p_{m}\right]&=i\hbar \delta _{l,m}\\\left[Q_{k},\Pi _{k'}\right]&={\frac {1}{N}}\sum _{l,m}e^{ikal}e^{-ik'am}\left[x_{l},p_{m}\right]\\&={\frac {i\hbar }{N}}\sum _{l}e^{ial\left(k-k'\right)}=i\hbar \delta _{k,k'}\\\left[Q_{k},Q_{k'}\right]&=\left[\Pi _{k},\Pi _{k'}\right]=0\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1829b425a1c6cb72b77a83ed8f279de7279fa4db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.09ex; margin-bottom: -0.247ex; width:38.165ex; height:19.843ex;" alt="{\displaystyle {\begin{aligned}\left[x_{l},p_{m}\right]&=i\hbar \delta _{l,m}\\\left[Q_{k},\Pi _{k'}\right]&={\frac {1}{N}}\sum _{l,m}e^{ikal}e^{-ik'am}\left[x_{l},p_{m}\right]\\&={\frac {i\hbar }{N}}\sum _{l}e^{ial\left(k-k'\right)}=i\hbar \delta _{k,k'}\\\left[Q_{k},Q_{k'}\right]&=\left[\Pi _{k},\Pi _{k'}\right]=0\end{aligned}}}"></span></dd></dl> <p>From the general result </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sum _{l}x_{l}x_{l+m}&={\frac {1}{N}}\sum _{kk'}Q_{k}Q_{k'}\sum _{l}e^{ial\left(k+k'\right)}e^{iamk'}=\sum _{k}Q_{k}Q_{-k}e^{iamk}\\\sum _{l}{p_{l}}^{2}&=\sum _{k}\Pi _{k}\Pi _{-k}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </munder> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mo>+</mo> <mi>m</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <msup> <mi>k</mi> <mo>′</mo> </msup> </mrow> </munder> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>k</mi> <mo>′</mo> </msup> </mrow> </msub> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>a</mi> <mi>l</mi> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>+</mo> <msup> <mi>k</mi> <mo>′</mo> </msup> </mrow> <mo>)</mo> </mrow> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>a</mi> <mi>m</mi> <msup> <mi>k</mi> <mo>′</mo> </msup> </mrow> </msup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>a</mi> <mi>m</mi> <mi>k</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </munder> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sum _{l}x_{l}x_{l+m}&={\frac {1}{N}}\sum _{kk'}Q_{k}Q_{k'}\sum _{l}e^{ial\left(k+k'\right)}e^{iamk'}=\sum _{k}Q_{k}Q_{-k}e^{iamk}\\\sum _{l}{p_{l}}^{2}&=\sum _{k}\Pi _{k}\Pi _{-k}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6182e4d24585975b5ccb24f52ca6a16ffc6e149d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:64.842ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}\sum _{l}x_{l}x_{l+m}&={\frac {1}{N}}\sum _{kk'}Q_{k}Q_{k'}\sum _{l}e^{ial\left(k+k'\right)}e^{iamk'}=\sum _{k}Q_{k}Q_{-k}e^{iamk}\\\sum _{l}{p_{l}}^{2}&=\sum _{k}\Pi _{k}\Pi _{-k}\end{aligned}}}"></span></dd></dl> <p>The potential energy term is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}m\omega ^{2}\sum _{j}\left(x_{j}-x_{j+1}\right)^{2}={\tfrac {1}{2}}m\omega ^{2}\sum _{k}Q_{k}Q_{-k}(2-e^{ika}-e^{-ika})={\tfrac {1}{2}}\sum _{k}m{\omega _{k}}^{2}Q_{k}Q_{-k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>m</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>m</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>a</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>k</mi> <mi>a</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mi>m</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}m\omega ^{2}\sum _{j}\left(x_{j}-x_{j+1}\right)^{2}={\tfrac {1}{2}}m\omega ^{2}\sum _{k}Q_{k}Q_{-k}(2-e^{ika}-e^{-ika})={\tfrac {1}{2}}\sum _{k}m{\omega _{k}}^{2}Q_{k}Q_{-k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89bfeda3fd49c8cad819c4f7f7ab14117e1e77e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:81.967ex; height:5.843ex;" alt="{\displaystyle {\tfrac {1}{2}}m\omega ^{2}\sum _{j}\left(x_{j}-x_{j+1}\right)^{2}={\tfrac {1}{2}}m\omega ^{2}\sum _{k}Q_{k}Q_{-k}(2-e^{ika}-e^{-ika})={\tfrac {1}{2}}\sum _{k}m{\omega _{k}}^{2}Q_{k}Q_{-k}}"></span></dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{k}={\sqrt {2\omega ^{2}\left(1-\cos {ka}\right)}}=2\omega \left|\sin {\frac {ka}{2}}\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>a</mi> </mrow> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> <mo>=</mo> <mn>2</mn> <mi>ω</mi> <mrow> <mo>|</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mi>a</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{k}={\sqrt {2\omega ^{2}\left(1-\cos {ka}\right)}}=2\omega \left|\sin {\frac {ka}{2}}\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c89073199ecf8f20acc9234c7585568a85991f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:37.665ex; height:5.843ex;" alt="{\displaystyle \omega _{k}={\sqrt {2\omega ^{2}\left(1-\cos {ka}\right)}}=2\omega \left|\sin {\frac {ka}{2}}\right|}"></span></dd></dl> <p>The Hamiltonian may be written in wavevector space as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}={\frac {1}{2m}}\sum _{k}\left(\Pi _{k}\Pi _{-k}+m^{2}\omega _{k}^{2}Q_{k}Q_{-k}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}={\frac {1}{2m}}\sum _{k}\left(\Pi _{k}\Pi _{-k}+m^{2}\omega _{k}^{2}Q_{k}Q_{-k}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2c571c3784fdc567a61458bb3e120f496ae95a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.905ex; height:6.343ex;" alt="{\displaystyle {\mathcal {H}}={\frac {1}{2m}}\sum _{k}\left(\Pi _{k}\Pi _{-k}+m^{2}\omega _{k}^{2}Q_{k}Q_{-k}\right)}"></span></dd></dl> <p>The couplings between the position variables have been transformed away; if the <i>Q</i> and <i>Π</i> were <a href="/wiki/Hermitian_operator" class="mw-redirect" title="Hermitian operator">Hermitian</a> (which they are not), the transformed Hamiltonian would describe <i>N</i> uncoupled harmonic oscillators. </p><p>The form of the quantization depends on the choice of boundary conditions; for simplicity, <i>periodic</i> boundary conditions are imposed, defining the (<i>N</i> + 1)th atom as equivalent to the first atom. Physically, this corresponds to joining the chain at its ends. The resulting quantization is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=k_{n}={\frac {2\pi n}{Na}}\quad {\mbox{for }}n=0,\pm 1,\pm 2,\ldots \pm {\frac {N}{2}}.\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>n</mi> </mrow> <mrow> <mi>N</mi> <mi>a</mi> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>for </mtext> </mstyle> </mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo>±</mo> <mn>1</mn> <mo>,</mo> <mo>±</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>N</mi> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=k_{n}={\frac {2\pi n}{Na}}\quad {\mbox{for }}n=0,\pm 1,\pm 2,\ldots \pm {\frac {N}{2}}.\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0348b25b00ad3156853a6f8727f44be3e77151e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:45.029ex; height:5.176ex;" alt="{\displaystyle k=k_{n}={\frac {2\pi n}{Na}}\quad {\mbox{for }}n=0,\pm 1,\pm 2,\ldots \pm {\frac {N}{2}}.\ }"></span></dd></dl> <p>The upper bound to <i>n</i> comes from the minimum wavelength, which is twice the lattice spacing <i>a</i>, as discussed above. </p><p>The harmonic oscillator eigenvalues or energy levels for the mode <i>ω<sub>k</sub></i> are: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{n}=\left({\tfrac {1}{2}}+n\right)\hbar \omega _{k}\qquad n=0,1,2,3\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> <mi class="MJX-variant">ℏ</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mspace width="2em" /> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{n}=\left({\tfrac {1}{2}}+n\right)\hbar \omega _{k}\qquad n=0,1,2,3\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c93610bae700c6c54e3237ad365f0caac8c5640e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:38.284ex; height:3.509ex;" alt="{\displaystyle E_{n}=\left({\tfrac {1}{2}}+n\right)\hbar \omega _{k}\qquad n=0,1,2,3\ldots }"></span></dd></dl> <p>The levels are evenly spaced at: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}\hbar \omega ,\ {\tfrac {3}{2}}\hbar \omega ,\ {\tfrac {5}{2}}\hbar \omega \ \cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi class="MJX-variant">ℏ</mi> <mi>ω</mi> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi class="MJX-variant">ℏ</mi> <mi>ω</mi> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi class="MJX-variant">ℏ</mi> <mi>ω</mi> <mtext> </mtext> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}\hbar \omega ,\ {\tfrac {3}{2}}\hbar \omega ,\ {\tfrac {5}{2}}\hbar \omega \ \cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21df907c0eaf709b89e8ed2d946cf9770b4ae024" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.151ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}\hbar \omega ,\ {\tfrac {3}{2}}\hbar \omega ,\ {\tfrac {5}{2}}\hbar \omega \ \cdots }"></span></dd></dl> <p>where <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span><i>ħω</i> is the <a href="/wiki/Zero-point_energy" title="Zero-point energy">zero-point energy</a> of a <a href="/wiki/Quantum_harmonic_oscillator" title="Quantum harmonic oscillator">quantum harmonic oscillator</a>. </p><p>An <b>exact</b> amount of <a href="/wiki/Energy" title="Energy">energy</a> <i>ħω</i> must be supplied to the harmonic oscillator lattice to push it to the next energy level. By analogy to the <a href="/wiki/Photon" title="Photon">photon</a> case when the <a href="/wiki/Electromagnetic_field" title="Electromagnetic field">electromagnetic field</a> is quantized, the quantum of vibrational energy is called a phonon. </p><p>All quantum systems show wavelike and particlelike properties simultaneously. The particle-like properties of the phonon are best understood using the methods of <a href="/wiki/Second_quantization" title="Second quantization">second quantization</a> and operator techniques described later.<sup id="cite_ref-Mahan_9-0" class="reference"><a href="#cite_note-Mahan-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Canonical_quantization#Real_scalar_field" title="Canonical quantization">Canonical quantization § Real scalar field</a></div> <div class="mw-heading mw-heading3"><h3 id="Three-dimensional_lattice">Three-dimensional lattice</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon&action=edit&section=7" title="Edit section: Three-dimensional lattice"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This may be generalized to a three-dimensional lattice. The wavenumber <i>k</i> is replaced by a three-dimensional <a href="/wiki/Wavevector" class="mw-redirect" title="Wavevector">wavevector</a> <b>k</b>. Furthermore, each <b>k</b> is now associated with three normal coordinates. </p><p>The new indices <i>s</i> = 1, 2, 3 label the <a href="/wiki/Polarization_(waves)" title="Polarization (waves)">polarization</a> of the phonons. In the one-dimensional model, the atoms were restricted to moving along the line, so the phonons corresponded to <a href="/wiki/Longitudinal_wave" title="Longitudinal wave">longitudinal waves</a>. In three dimensions, vibration is not restricted to the direction of propagation, and can also occur in the perpendicular planes, like <a href="/wiki/Transverse_wave" title="Transverse wave">transverse waves</a>. This gives rise to the additional normal coordinates, which, as the form of the Hamiltonian indicates, we may view as independent species of phonons. </p> <div class="mw-heading mw-heading3"><h3 id="Dispersion_relation">Dispersion relation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon&action=edit&section=8" title="Edit section: Dispersion relation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Diatomic_phonons.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/Diatomic_phonons.png/220px-Diatomic_phonons.png" decoding="async" width="220" height="202" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/Diatomic_phonons.png/330px-Diatomic_phonons.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/04/Diatomic_phonons.png/440px-Diatomic_phonons.png 2x" data-file-width="674" data-file-height="619" /></a><figcaption>Dispersion curves in linear diatomic chain</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Optical_%26_acoustic_vibrations-en.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Optical_%26_acoustic_vibrations-en.svg/250px-Optical_%26_acoustic_vibrations-en.svg.png" decoding="async" width="250" height="113" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Optical_%26_acoustic_vibrations-en.svg/375px-Optical_%26_acoustic_vibrations-en.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Optical_%26_acoustic_vibrations-en.svg/500px-Optical_%26_acoustic_vibrations-en.svg.png 2x" data-file-width="780" data-file-height="351" /></a><figcaption>Optical and acoustic vibrations in a linear diatomic chain.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Diatomic_chain.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Diatomic_chain.gif/220px-Diatomic_chain.gif" decoding="async" width="220" height="144" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Diatomic_chain.gif/330px-Diatomic_chain.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Diatomic_chain.gif/440px-Diatomic_chain.gif 2x" data-file-width="768" data-file-height="502" /></a><figcaption>Vibrations of the diatomic chain at different frequencies.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Phonon_dispersion_relations_in_GaAs.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/Phonon_dispersion_relations_in_GaAs.png/250px-Phonon_dispersion_relations_in_GaAs.png" decoding="async" width="250" height="135" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/Phonon_dispersion_relations_in_GaAs.png/375px-Phonon_dispersion_relations_in_GaAs.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/45/Phonon_dispersion_relations_in_GaAs.png/500px-Phonon_dispersion_relations_in_GaAs.png 2x" data-file-width="801" data-file-height="433" /></a><figcaption>Dispersion relation <i>ω</i> = <i>ω</i>(<b>k</b>) for some waves corresponding to lattice vibrations in GaAs.<sup id="cite_ref-Cardona_10-0" class="reference"><a href="#cite_note-Cardona-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>For a one-dimensional alternating array of two types of ion or atom of mass <i>m</i><sub>1</sub>, <i>m</i><sub>2</sub> repeated periodically at a distance <i>a</i>, connected by springs of spring constant <i>K</i>, two modes of vibration result:<sup id="cite_ref-Misra_11-0" class="reference"><a href="#cite_note-Misra-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{\pm }^{2}=K\left({\frac {1}{m_{1}}}+{\frac {1}{m_{2}}}\right)\pm K{\sqrt {\left({\frac {1}{m_{1}}}+{\frac {1}{m_{2}}}\right)^{2}-{\frac {4\sin ^{2}{\frac {ka}{2}}}{m_{1}m_{2}}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mi>K</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>±</mo> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mi>a</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\pm }^{2}=K\left({\frac {1}{m_{1}}}+{\frac {1}{m_{2}}}\right)\pm K{\sqrt {\left({\frac {1}{m_{1}}}+{\frac {1}{m_{2}}}\right)^{2}-{\frac {4\sin ^{2}{\frac {ka}{2}}}{m_{1}m_{2}}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/419d1c8ffe378ac3cf077e30ad3b4ff3847bd760" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:57.901ex; height:7.843ex;" alt="{\displaystyle \omega _{\pm }^{2}=K\left({\frac {1}{m_{1}}}+{\frac {1}{m_{2}}}\right)\pm K{\sqrt {\left({\frac {1}{m_{1}}}+{\frac {1}{m_{2}}}\right)^{2}-{\frac {4\sin ^{2}{\frac {ka}{2}}}{m_{1}m_{2}}}}},}"></span></dd></dl> <p>where <i>k</i> is the wavevector of the vibration related to its wavelength by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k={\tfrac {2\pi }{\lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>λ</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={\tfrac {2\pi }{\lambda }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bb0a1ba22db801ee2cbe8bb4183de2304d81cd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.91ex; height:3.676ex;" alt="{\displaystyle k={\tfrac {2\pi }{\lambda }}}"></span>. </p><p>The connection between frequency and wavevector, <i>ω</i> = <i>ω</i>(<i>k</i>), is known as a <a href="/wiki/Dispersion_relation" title="Dispersion relation">dispersion relation</a>. The plus sign results in the so-called <i>optical</i> mode, and the minus sign to the <i>acoustic</i> mode. In the optical mode two adjacent different atoms move against each other, while in the acoustic mode they move together. </p><p>The speed of propagation of an acoustic phonon, which is also the <a href="/wiki/Speed_of_sound" title="Speed of sound">speed of sound</a> in the lattice, is given by the slope of the acoustic dispersion relation, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">∂<i>ω<sub>k</sub></i></span><span class="sr-only">/</span><span class="den">∂<i>k</i></span></span>⁠</span> (see <a href="/wiki/Group_velocity" title="Group velocity">group velocity</a>.) At low values of <i>k</i> (i.e. long wavelengths), the dispersion relation is almost linear, and the speed of sound is approximately <i>ωa</i>, independent of the phonon frequency. As a result, packets of phonons with different (but long) wavelengths can propagate for large distances across the lattice without breaking apart. This is the reason that sound propagates through solids without significant distortion. This behavior fails at large values of <i>k</i>, i.e. short wavelengths, due to the microscopic details of the lattice. </p><p>For a crystal that has at least two atoms in its <a href="/wiki/Wigner-Seitz_cell#Primitive_cell" class="mw-redirect" title="Wigner-Seitz cell">primitive cell</a>, the dispersion relations exhibit two types of phonons, namely, optical and acoustic modes corresponding to the upper blue and lower red curve in the diagram, respectively. The vertical axis is the energy or frequency of phonon, while the horizontal axis is the <a href="/wiki/Wavevector" class="mw-redirect" title="Wavevector">wavevector</a>. The boundaries at −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><span class="texhtml mvar" style="font-style:italic;">π</span></span><span class="sr-only">/</span><span class="den"><i>a</i></span></span>⁠</span> and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><span class="texhtml mvar" style="font-style:italic;">π</span></span><span class="sr-only">/</span><span class="den"><i>a</i></span></span>⁠</span> are those of the first <a href="/wiki/Brillouin_zone" title="Brillouin zone">Brillouin zone</a>.<sup id="cite_ref-Misra_11-1" class="reference"><a href="#cite_note-Misra-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> A crystal with <i>N</i> ≥ 2 different atoms in the <a href="/wiki/Primitive_cell" class="mw-redirect" title="Primitive cell">primitive cell</a> exhibits three acoustic modes: one <a href="/wiki/Longitudinal_wave" title="Longitudinal wave">longitudinal acoustic mode</a> and two <a href="/wiki/Transverse_wave" title="Transverse wave">transverse acoustic modes</a>. The number of optical modes is 3<i>N</i> – 3. The lower figure shows the dispersion relations for several phonon modes in <a href="/wiki/GaAs" class="mw-redirect" title="GaAs">GaAs</a> as a function of wavevector <b>k</b> in the <a href="/wiki/Brillouin_zone#Critical_points" title="Brillouin zone">principal directions</a> of its Brillouin zone.<sup id="cite_ref-Cardona_10-1" class="reference"><a href="#cite_note-Cardona-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>The modes are also referred to as the branches of phonon dispersion. In general, if there are p atoms (denoted by N earlier) in the primitive unit cell, there will be 3p branches of phonon dispersion in a 3-dimensional crystal. Out of these, 3 branches correspond to acoustic modes and the remaining 3p-3 branches will correspond to optical modes. In some special directions, some branches coincide due to symmetry. These branches are called degenerate. In acoustic modes, all the p atoms vibrate in phase. So there is no change in the relative displacements of these atoms during the wave propagation. </p><p>Study of phonon dispersion is useful for modeling propagation of sound waves in solids, which is characterized by phonons. The energy of each phonon, as given earlier, is <i>ħω.</i> The velocity of the wave also is given in terms of <i>ω</i> and k <i>.</i> The direction of the wave vector is the direction of the wave propagation and the phonon polarization vector gives the direction in which the atoms vibrate. Actually, in general, the wave velocity in a crystal is different for different directions of k. In other words, most crystals are anisotropic for phonon propagation. </p><p>A wave is longitudinal if the atoms vibrate in the same direction as the wave propagation. In a transverse wave, the atoms vibrate perpendicular to the wave propagation. However, except for isotropic crystals, waves in a crystal are not exactly longitudinal or transverse. For general anisotropic crystals, the phonon waves are longitudinal or transverse only in certain special symmetry directions. In other directions, they can be nearly longitudinal or nearly transverse. It is only for labeling convenience, that they are often called longitudinal or transverse but are actually quasi-longitudinal or quasi-transverse. Note that in the three-dimensional case, there are two directions perpendicular to a straight line at each point on the line. Hence, there are always two (quasi) transverse waves for each (quasi) longitudinal wave. </p><p>Many phonon dispersion curves have been measured by <a href="/wiki/Inelastic_neutron_scattering" class="mw-redirect" title="Inelastic neutron scattering">inelastic neutron scattering</a>. </p><p>The physics of sound in <a href="/wiki/Fluid" title="Fluid">fluids</a> differs from the physics of sound in solids, although both are density waves: sound waves in fluids only have longitudinal components, whereas sound waves in solids have longitudinal and transverse components. This is because fluids cannot support <a href="/wiki/Shear_stress" title="Shear stress">shear stresses</a> (but see <a href="/wiki/Viscoelastic" class="mw-redirect" title="Viscoelastic">viscoelastic</a> fluids, which only apply to high frequencies). </p> <div class="mw-heading mw-heading3"><h3 id="Interpretation_of_phonons_using_second_quantization_techniques">Interpretation of phonons using second quantization techniques</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon&action=edit&section=9" title="Edit section: Interpretation of phonons using second quantization techniques"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The above-derived Hamiltonian may look like a classical Hamiltonian function, but if it is interpreted as an <a href="/wiki/Operator_(physics)" title="Operator (physics)">operator</a>, then it describes a <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a> of non-interacting <a href="/wiki/Boson" title="Boson">bosons</a>.<sup id="cite_ref-girvinYang_2-1" class="reference"><a href="#cite_note-girvinYang-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Second_quantization" title="Second quantization">second quantization</a> technique, similar to the <a href="/wiki/Ladder_operator" title="Ladder operator">ladder operator</a> method used for <a href="/wiki/Quantum_harmonic_oscillator#Ladder_operator_method" title="Quantum harmonic oscillator">quantum harmonic oscillators</a>, is a means of extracting energy <a href="/wiki/Eigenvalues" class="mw-redirect" title="Eigenvalues">eigenvalues</a> without directly solving the differential equations. Given the Hamiltonian, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19ef4c7b923a5125ac91aa491838a95ee15b804f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\mathcal {H}}}"></span>, as well as the conjugate position, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fa6cb6cf7ffc157202e52dd5711e755892d1015" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.927ex; height:2.509ex;" alt="{\displaystyle Q_{k}}"></span>, and conjugate momentum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35cef58c6187d4932357154d686d829167f30381" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.832ex; height:2.509ex;" alt="{\displaystyle \Pi _{k}}"></span> defined in the quantum treatment section above, we can define <a href="/wiki/Creation_and_annihilation_operators" title="Creation and annihilation operators">creation and annihilation operators</a>:<sup id="cite_ref-ashcrroftMermin_12-0" class="reference"><a href="#cite_note-ashcrroftMermin-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{k}={\sqrt {\frac {m\omega _{k}}{2\hbar }}}\left(Q_{k}+{\frac {i}{m\omega _{k}}}\Pi _{-k}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>m</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <mi class="MJX-variant">ℏ</mi> </mrow> </mfrac> </msqrt> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mrow> <mi>m</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{k}={\sqrt {\frac {m\omega _{k}}{2\hbar }}}\left(Q_{k}+{\frac {i}{m\omega _{k}}}\Pi _{-k}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23464c0e4930fe5b2c35394e61429a965f201058" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:32.017ex; height:6.343ex;" alt="{\displaystyle b_{k}={\sqrt {\frac {m\omega _{k}}{2\hbar }}}\left(Q_{k}+{\frac {i}{m\omega _{k}}}\Pi _{-k}\right)}"></span>   and   <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {b_{k}}^{\dagger }={\sqrt {\frac {m\omega _{k}}{2\hbar }}}\left(Q_{-k}-{\frac {i}{m\omega _{k}}}\Pi _{k}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>m</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <mi class="MJX-variant">ℏ</mi> </mrow> </mfrac> </msqrt> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mrow> <mi>m</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {b_{k}}^{\dagger }={\sqrt {\frac {m\omega _{k}}{2\hbar }}}\left(Q_{-k}-{\frac {i}{m\omega _{k}}}\Pi _{k}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55fa0326f17c9b2bb51f0763b90ef026f7b28a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:32.979ex; height:6.343ex;" alt="{\displaystyle {b_{k}}^{\dagger }={\sqrt {\frac {m\omega _{k}}{2\hbar }}}\left(Q_{-k}-{\frac {i}{m\omega _{k}}}\Pi _{k}\right)}"></span></dd></dl> <p>The following commutators can be easily obtained by substituting in the <a href="/wiki/Canonical_commutation_relation" title="Canonical commutation relation">canonical commutation relation</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[b_{k},{b_{k'}}^{\dagger }\right]=\delta _{k,k'},\quad {\Big [}b_{k},b_{k'}{\Big ]}=\left[{b_{k}}^{\dagger },{b_{k'}}^{\dagger }\right]=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>k</mi> <mo>′</mo> </msup> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <msup> <mi>k</mi> <mo>′</mo> </msup> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>k</mi> <mo>′</mo> </msup> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>k</mi> <mo>′</mo> </msup> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[b_{k},{b_{k'}}^{\dagger }\right]=\delta _{k,k'},\quad {\Big [}b_{k},b_{k'}{\Big ]}=\left[{b_{k}}^{\dagger },{b_{k'}}^{\dagger }\right]=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b4bb0435990871df574fdc8fdbd1826e347f8ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:43.955ex; height:4.843ex;" alt="{\displaystyle \left[b_{k},{b_{k'}}^{\dagger }\right]=\delta _{k,k'},\quad {\Big [}b_{k},b_{k'}{\Big ]}=\left[{b_{k}}^{\dagger },{b_{k'}}^{\dagger }\right]=0}"></span></dd></dl> <p>Using this, the operators <i>b<sub>k</sub></i><sup>†</sup> and <i>b<sub>k</sub></i> can be inverted to redefine the conjugate position and momentum as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{k}={\sqrt {\frac {\hbar }{2m\omega _{k}}}}\left({b_{k}}^{\dagger }+b_{-k}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi class="MJX-variant">ℏ</mi> <mrow> <mn>2</mn> <mi>m</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </msqrt> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{k}={\sqrt {\frac {\hbar }{2m\omega _{k}}}}\left({b_{k}}^{\dagger }+b_{-k}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a0fd7d7c27c4946816855bd7f445bba8fcc84ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.693ex; height:7.676ex;" alt="{\displaystyle Q_{k}={\sqrt {\frac {\hbar }{2m\omega _{k}}}}\left({b_{k}}^{\dagger }+b_{-k}\right)}"></span>   and   <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{k}=i{\sqrt {\frac {\hbar m\omega _{k}}{2}}}\left({b_{k}}^{\dagger }-b_{-k}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi class="MJX-variant">ℏ</mi> <mi>m</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </msqrt> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{k}=i{\sqrt {\frac {\hbar m\omega _{k}}{2}}}\left({b_{k}}^{\dagger }-b_{-k}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/881e4cb4e830ca9a523800765ca5f5ca467341ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.545ex; height:6.343ex;" alt="{\displaystyle \Pi _{k}=i{\sqrt {\frac {\hbar m\omega _{k}}{2}}}\left({b_{k}}^{\dagger }-b_{-k}\right)}"></span></dd></dl> <p>Directly substituting these definitions for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fa6cb6cf7ffc157202e52dd5711e755892d1015" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.927ex; height:2.509ex;" alt="{\displaystyle Q_{k}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35cef58c6187d4932357154d686d829167f30381" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.832ex; height:2.509ex;" alt="{\displaystyle \Pi _{k}}"></span> into the wavevector space Hamiltonian, as it is defined above, and simplifying then results in the Hamiltonian taking the form:<sup id="cite_ref-girvinYang_2-2" class="reference"><a href="#cite_note-girvinYang-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}=\sum _{k}\hbar \omega _{k}\left({b_{k}}^{\dagger }b_{k}+{\tfrac {1}{2}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mi class="MJX-variant">ℏ</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}=\sum _{k}\hbar \omega _{k}\left({b_{k}}^{\dagger }b_{k}+{\tfrac {1}{2}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04d084fb7c16b0af7a10906a91415d5d43e42243" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.796ex; height:5.509ex;" alt="{\displaystyle {\mathcal {H}}=\sum _{k}\hbar \omega _{k}\left({b_{k}}^{\dagger }b_{k}+{\tfrac {1}{2}}\right)}"></span></dd></dl> <p>This is known as the second quantization technique, also known as the occupation number formulation, where <i>n<sub>k</sub></i> = <i>b<sub>k</sub></i><sup>†</sup><i>b<sub>k</sub></i> is the occupation number. This can be seen to be a sum of N independent oscillator Hamiltonians, each with a unique wave vector, and compatible with the methods used for the quantum harmonic oscillator (note that <i>n<sub>k</sub></i> is <a href="/wiki/Hermitian_matrix" title="Hermitian matrix">hermitian</a>).<sup id="cite_ref-ashcrroftMermin_12-1" class="reference"><a href="#cite_note-ashcrroftMermin-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> When a Hamiltonian can be written as a sum of commuting sub-Hamiltonians, the energy eigenstates will be given by the products of eigenstates of each of the separate sub-Hamiltonians. The corresponding <a href="/wiki/Energy" title="Energy">energy</a> <a href="/wiki/Spectrum_of_an_operator" class="mw-redirect" title="Spectrum of an operator">spectrum</a> is then given by the sum of the individual eigenvalues of the sub-Hamiltonians.<sup id="cite_ref-ashcrroftMermin_12-2" class="reference"><a href="#cite_note-ashcrroftMermin-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>As with the quantum harmonic oscillator, one can show that <i>b<sub>k</sub></i><sup>†</sup> and <i>b<sub>k</sub></i> respectively create and destroy a single field excitation, a phonon, with an energy of <i>ħω<sub>k</sub></i>.<sup id="cite_ref-ashcrroftMermin_12-3" class="reference"><a href="#cite_note-ashcrroftMermin-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-girvinYang_2-3" class="reference"><a href="#cite_note-girvinYang-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Three important properties of phonons may be deduced from this technique. First, phonons are <a href="/wiki/Boson" title="Boson">bosons</a>, since any number of identical excitations can be created by repeated application of the creation operator <i>b<sub>k</sub></i><sup>†</sup>. Second, each phonon is a "collective mode" caused by the motion of every atom in the lattice. This may be seen from the fact that the creation and annihilation operators, defined here in momentum space, contain sums over the position and momentum operators of every atom when written in position space. (See <a href="/wiki/Position_and_momentum_space" class="mw-redirect" title="Position and momentum space">position and momentum space</a>.)<sup id="cite_ref-ashcrroftMermin_12-4" class="reference"><a href="#cite_note-ashcrroftMermin-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Finally, using the <i>position–position <a href="/wiki/Correlation_function" title="Correlation function">correlation function</a></i>, it can be shown that phonons act as waves of lattice displacement.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (August 2020)">citation needed</span></a></i>]</sup> </p><p>This technique is readily generalized to three dimensions, where the Hamiltonian takes the form:<sup id="cite_ref-ashcrroftMermin_12-5" class="reference"><a href="#cite_note-ashcrroftMermin-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-girvinYang_2-4" class="reference"><a href="#cite_note-girvinYang-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}=\sum _{k}\sum _{s=1}^{3}\hbar \,\omega _{k,s}\left({b_{k,s}}^{\dagger }b_{k,s}+{\tfrac {1}{2}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <mi class="MJX-variant">ℏ</mi> <mspace width="thinmathspace" /> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}=\sum _{k}\sum _{s=1}^{3}\hbar \,\omega _{k,s}\left({b_{k,s}}^{\dagger }b_{k,s}+{\tfrac {1}{2}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3f2ad8ea02d7fb5160b94ede3c0f6ef3869ac43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.644ex; height:7.176ex;" alt="{\displaystyle {\mathcal {H}}=\sum _{k}\sum _{s=1}^{3}\hbar \,\omega _{k,s}\left({b_{k,s}}^{\dagger }b_{k,s}+{\tfrac {1}{2}}\right).}"></span></dd></dl> <p>This can be interpreted as the sum of 3N independent oscillator Hamiltonians, one for each wave vector and polarization.<sup id="cite_ref-ashcrroftMermin_12-6" class="reference"><a href="#cite_note-ashcrroftMermin-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Acoustic_and_optical_phonons">Acoustic and optical phonons</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon&action=edit&section=10" title="Edit section: Acoustic and optical phonons"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Solids with more than one atom in the smallest <a href="/wiki/Unit_cell" title="Unit cell">unit cell</a> exhibit two types of phonons: acoustic phonons and optical phonons. </p><p><b>Acoustic phonons</b> are coherent movements of atoms of the lattice out of their equilibrium positions. If the displacement is in the direction of propagation, then in some areas the atoms will be closer, in others farther apart, as in a sound wave in air (hence the name acoustic). Displacement perpendicular to the propagation direction is comparable to waves on a string. If the wavelength of acoustic phonons goes to infinity, this corresponds to a simple displacement of the whole crystal, and this costs zero deformation energy. Acoustic phonons exhibit a linear relationship between frequency and phonon wave-vector for long wavelengths. The frequencies of acoustic phonons tend to zero with longer wavelength. Longitudinal and transverse acoustic phonons are often abbreviated as LA and TA phonons, respectively. </p><p><b>Optical phonons</b> are out-of-phase movements of the atoms in the lattice, one atom moving to the left, and its neighbor to the right. This occurs if the lattice basis consists of two or more atoms. They are called <i>optical</i> because in ionic crystals, such as <a href="/wiki/Sodium_chloride" title="Sodium chloride">sodium chloride</a>, fluctuations in displacement create an electrical polarization that couples to the electromagnetic field.<sup id="cite_ref-girvinYang_2-5" class="reference"><a href="#cite_note-girvinYang-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Hence, they can be excited by <a href="/wiki/Infrared_radiation" class="mw-redirect" title="Infrared radiation">infrared radiation</a>, the electric field of the light will move every positive sodium ion in the direction of the field, and every negative chloride ion in the other direction, causing the crystal to vibrate. </p><p>Optical phonons have a non-zero frequency at the <a href="/wiki/Brillouin_zone" title="Brillouin zone">Brillouin zone</a> center and show no dispersion near that long wavelength limit. This is because they correspond to a mode of vibration where positive and negative ions at adjacent lattice sites swing against each other, creating a time-varying <a href="/wiki/Electrical_dipole_moment" class="mw-redirect" title="Electrical dipole moment">electrical dipole moment</a>. Optical phonons that interact in this way with light are called <i>infrared active</i>. Optical phonons that are <i>Raman active</i> can also interact indirectly with light, through <a href="/wiki/Raman_scattering" title="Raman scattering">Raman scattering</a>. Optical phonons are often abbreviated as LO and TO phonons, for the longitudinal and transverse modes respectively; the splitting between LO and TO frequencies is often described accurately by the <a href="/wiki/Lyddane%E2%80%93Sachs%E2%80%93Teller_relation" title="Lyddane–Sachs–Teller relation">Lyddane–Sachs–Teller relation</a>. </p><p>When measuring optical phonon energy experimentally, optical phonon frequencies are sometimes given in spectroscopic <a href="/wiki/Wavenumber" title="Wavenumber">wavenumber</a> notation, where the symbol <i>ω</i> represents ordinary frequency (not angular frequency), and is expressed in units of <a href="/wiki/Centimetre" title="Centimetre">cm</a><sup>−1</sup>. The value is obtained by dividing the frequency by the <a href="/wiki/Speed_of_light_in_vacuum" class="mw-redirect" title="Speed of light in vacuum">speed of light in vacuum</a>. In other words, the wave-number in cm<sup>−1</sup> units corresponds to the inverse of the <a href="/wiki/Wavelength" title="Wavelength">wavelength</a> of a <a href="/wiki/Photon" title="Photon">photon</a> in vacuum that has the same frequency as the measured phonon.<sup id="cite_ref-cmian_13-0" class="reference"><a href="#cite_note-cmian-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Crystal_momentum">Crystal momentum</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon&action=edit&section=11" title="Edit section: Crystal momentum"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Crystal_momentum" title="Crystal momentum">Crystal momentum</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Phonon_k_3k.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Phonon_k_3k.gif/250px-Phonon_k_3k.gif" decoding="async" width="250" height="81" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Phonon_k_3k.gif/375px-Phonon_k_3k.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Phonon_k_3k.gif/500px-Phonon_k_3k.gif 2x" data-file-width="616" data-file-height="200" /></a><figcaption>k-vectors exceeding the first <a href="/wiki/Brillouin_zone" title="Brillouin zone">Brillouin zone</a> (red) do not carry any more information than their counterparts (black) in the first Brillouin zone.</figcaption></figure> <p>By analogy to <a href="/wiki/Photon" title="Photon">photons</a> and <a href="/wiki/De_Broglie_wavelength" class="mw-redirect" title="De Broglie wavelength">matter waves</a>, phonons have been treated with <a href="/wiki/Wavevector" class="mw-redirect" title="Wavevector">wavevector</a> <i>k</i> as though it has a <a href="/wiki/Momentum" title="Momentum">momentum</a> <i>ħk</i>;<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> however, this is not strictly correct, because <i>ħk</i> is not actually a physical momentum; it is called the <i>crystal momentum</i> or <i>pseudomomentum</i>. This is because <i>k</i> is only determined up to addition of constant vectors (the <a href="/wiki/Reciprocal_lattice" title="Reciprocal lattice">reciprocal lattice vectors</a> and integer multiples thereof). For example, in the one-dimensional model, the normal coordinates <i>Q</i> and <i>Π</i> are defined so that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{k}{\stackrel {\mathrm {def} }{=}}Q_{k+K};\quad \Pi _{k}{\stackrel {\mathrm {def} }{=}}\Pi _{k+K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mi>K</mi> </mrow> </msub> <mo>;</mo> <mspace width="1em" /> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{k}{\stackrel {\mathrm {def} }{=}}Q_{k+K};\quad \Pi _{k}{\stackrel {\mathrm {def} }{=}}\Pi _{k+K}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60bb081ff71de00eb0f60628b2964f7041649779" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.865ex; height:3.676ex;" alt="{\displaystyle Q_{k}{\stackrel {\mathrm {def} }{=}}Q_{k+K};\quad \Pi _{k}{\stackrel {\mathrm {def} }{=}}\Pi _{k+K}}"></span></dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\frac {2n\pi }{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> <mi>π</mi> </mrow> <mi>a</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\frac {2n\pi }{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75219b63f54cea8747eb6e5e1408dfa5c7c90602" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.89ex; height:5.176ex;" alt="{\displaystyle K={\frac {2n\pi }{a}}}"></span></dd></dl> <p>for any integer <i>n</i>. A phonon with wavenumber <i>k</i> is thus equivalent to an infinite family of phonons with wavenumbers <i>k</i> ± <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">2<span class="texhtml mvar" style="font-style:italic;">π</span></span><span class="sr-only">/</span><span class="den"><i>a</i></span></span>⁠</span>, <i>k</i> ± <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">4<span class="texhtml mvar" style="font-style:italic;">π</span></span><span class="sr-only">/</span><span class="den"><i>a</i></span></span>⁠</span>, and so forth. Physically, the reciprocal lattice vectors act as additional chunks of momentum which the lattice can impart to the phonon. <a href="/wiki/Bloch_electron" class="mw-redirect" title="Bloch electron">Bloch electrons</a> obey a similar set of restrictions. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Brillouin_zone.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Brillouin_zone.svg/220px-Brillouin_zone.svg.png" decoding="async" width="220" height="243" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Brillouin_zone.svg/330px-Brillouin_zone.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Brillouin_zone.svg/440px-Brillouin_zone.svg.png 2x" data-file-width="825" data-file-height="913" /></a><figcaption>Brillouin zones, (a) in a square lattice, and (b) in a hexagonal lattice</figcaption></figure> <p>It is usually convenient to consider phonon wavevectors <i>k</i> which have the smallest magnitude |<i>k</i>| in their "family". The set of all such wavevectors defines the <i>first <a href="/wiki/Brillouin_zone" title="Brillouin zone">Brillouin zone</a></i>. Additional Brillouin zones may be defined as copies of the first zone, shifted by some reciprocal lattice vector. </p> <div class="mw-heading mw-heading2"><h2 id="Thermodynamics">Thermodynamics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon&action=edit&section=12" title="Edit section: Thermodynamics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamic</a> properties of a solid are directly related to its phonon structure. The entire set of all possible phonons that are described by the phonon dispersion relations combine in what is known as the phonon <a href="/wiki/Density_of_states" title="Density of states">density of states</a> which determines the <a href="/wiki/Heat_capacity" title="Heat capacity">heat capacity</a> of a crystal. By the nature of this distribution, the heat capacity is dominated by the high-frequency part of the distribution, while thermal conductivity is primarily the result of the low-frequency region.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (February 2019)">citation needed</span></a></i>]</sup> </p><p>At <a href="/wiki/Absolute_zero" title="Absolute zero">absolute zero</a> temperature, a crystal lattice lies in its <a href="/wiki/Ground_state" title="Ground state">ground state</a>, and contains no phonons. A lattice at a nonzero <a href="/wiki/Temperature" title="Temperature">temperature</a> has an energy that is not constant, but fluctuates <a href="/wiki/Random" class="mw-redirect" title="Random">randomly</a> about some <a href="/wiki/Arithmetic_mean" title="Arithmetic mean">mean value</a>. These energy fluctuations are caused by random lattice vibrations, which can be viewed as a gas of phonons. Because these phonons are generated by the temperature of the lattice, they are sometimes designated thermal phonons.<sup id="cite_ref-nonMetalsThermalPhonons_15-0" class="reference"><a href="#cite_note-nonMetalsThermalPhonons-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>Thermal phonons can be created and destroyed by random energy fluctuations. In the language of statistical mechanics this means that the chemical potential for adding a phonon is zero.<sup id="cite_ref-nonMetalsThermalPhonons_15-1" class="reference"><a href="#cite_note-nonMetalsThermalPhonons-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> This behavior is an extension of the harmonic potential into the anharmonic regime. The behavior of thermal phonons is similar to the <a href="/wiki/Photon_gas" title="Photon gas">photon gas</a> produced by an <a href="/wiki/Electromagnetic_cavity" title="Electromagnetic cavity">electromagnetic cavity</a>, wherein photons may be emitted or absorbed by the cavity walls. This similarity is not coincidental, for it turns out that the electromagnetic field behaves like a set of harmonic oscillators, giving rise to <a href="/wiki/Black-body_radiation" title="Black-body radiation">black-body radiation</a>. Both gases obey the <a href="/wiki/Bose%E2%80%93Einstein_statistics" title="Bose–Einstein statistics">Bose–Einstein statistics</a>: in thermal equilibrium and within the harmonic regime, the probability of finding phonons or photons in a given state with a given angular frequency is:<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\left(\omega _{k,s}\right)={\frac {1}{\exp \left({\dfrac {\hbar \omega _{k,s}}{k_{\mathrm {B} }T}}\right)-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mrow> <mo>(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi class="MJX-variant">ℏ</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\left(\omega _{k,s}\right)={\frac {1}{\exp \left({\dfrac {\hbar \omega _{k,s}}{k_{\mathrm {B} }T}}\right)-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4c0e8a119fcc72bcf6c03947a35788915827a99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:28.171ex; height:9.343ex;" alt="{\displaystyle n\left(\omega _{k,s}\right)={\frac {1}{\exp \left({\dfrac {\hbar \omega _{k,s}}{k_{\mathrm {B} }T}}\right)-1}}}"></span></dd></dl> <p>where <i>ω</i><sub><i>k</i>,<i>s</i></sub> is the frequency of the phonons (or photons) in the state, <i>k</i><sub>B</sub> is the <a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a>, and <i>T</i> is the temperature. </p> <div class="mw-heading mw-heading3"><h3 id="Phonon_tunneling">Phonon tunneling</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon&action=edit&section=13" title="Edit section: Phonon tunneling"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Phonons have been shown to exhibit <a href="/wiki/Quantum_tunneling" class="mw-redirect" title="Quantum tunneling">quantum tunneling</a> behavior (or <i>phonon tunneling</i>) where, across gaps up to a nanometer wide, heat can flow via phonons that "tunnel" between two materials.<sup id="cite_ref-mitPhononTunneling_17-0" class="reference"><a href="#cite_note-mitPhononTunneling-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> This type of heat transfer works between distances too large for <a href="/wiki/Thermal_conduction" title="Thermal conduction">conduction</a> to occur but too small for <a href="/wiki/Thermal_radiation" title="Thermal radiation">radiation</a> to occur and therefore cannot be explained by classical <a href="/wiki/Heat_transfer" title="Heat transfer">heat transfer</a> models.<sup id="cite_ref-mitPhononTunneling_17-1" class="reference"><a href="#cite_note-mitPhononTunneling-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Operator_formalism">Operator formalism</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon&action=edit&section=14" title="Edit section: Operator formalism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The phonon Hamiltonian is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}={\tfrac {1}{2}}\sum _{\alpha }\left(p_{\alpha }^{2}+\omega _{\alpha }^{2}q_{\alpha }^{2}-\hbar \omega _{\alpha }\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mi class="MJX-variant">ℏ</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}={\tfrac {1}{2}}\sum _{\alpha }\left(p_{\alpha }^{2}+\omega _{\alpha }^{2}q_{\alpha }^{2}-\hbar \omega _{\alpha }\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb93c548ed634d3eb005fc64ed7cd256f596c7e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.201ex; height:5.509ex;" alt="{\displaystyle {\mathcal {H}}={\tfrac {1}{2}}\sum _{\alpha }\left(p_{\alpha }^{2}+\omega _{\alpha }^{2}q_{\alpha }^{2}-\hbar \omega _{\alpha }\right)}"></span></dd></dl> <p>In terms of the <a href="/wiki/Creation_and_annihilation_operators" title="Creation and annihilation operators">creation and annihilation operators</a>, these are given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}=\sum _{\alpha }\hbar \omega _{\alpha }{a_{\alpha }}^{\dagger }a_{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </munder> <mi class="MJX-variant">ℏ</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}=\sum _{\alpha }\hbar \omega _{\alpha }{a_{\alpha }}^{\dagger }a_{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17eff3cc28042b1254362ee75ba3c89f86331189" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.831ex; height:5.509ex;" alt="{\displaystyle {\mathcal {H}}=\sum _{\alpha }\hbar \omega _{\alpha }{a_{\alpha }}^{\dagger }a_{\alpha }}"></span></dd></dl> <p>Here, in expressing the <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a> in operator formalism, we have not taken into account the <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span><i>ħω<sub>q</sub></i> term as, given a <a href="/wiki/Linear_continuum" title="Linear continuum">continuum</a> or <a href="/wiki/Bravais_lattice" title="Bravais lattice">infinite lattice</a>, the <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span><i>ħω<sub>q</sub></i> terms will add up yielding an <a href="/wiki/Singularity_(mathematics)" title="Singularity (mathematics)">infinite term</a>. Because the difference in energy is what we measure and not the absolute value of it, the constant term <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span><i>ħω<sub>q</sub></i> can be ignored without changing the equations of motion. Hence, the <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span><i>ħω<sub>q</sub></i> factor is absent in the operator formalized expression for the <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a>. </p><p>The ground state, also called the "<a href="/wiki/Vacuum_state" class="mw-redirect" title="Vacuum state">vacuum state</a>", is the state composed of no phonons. Hence, the energy of the ground state is 0. When a system is in the state <span class="nowrap">|<i>n</i><sub>1</sub><i>n</i><sub>2</sub><i>n</i><sub>3</sub>…⟩</span>, we say there are <i>n<sub>α</sub></i> phonons of type <i>α</i>, where <i>n<sub>α</sub></i> is the occupation number of the phonons. The energy of a single phonon of type <i>α</i> is given by <i>ħω<sub>q</sub></i> and the total energy of a general phonon system is given by <i>n</i><sub>1</sub><i>ħω</i><sub>1</sub> + <i>n</i><sub>2</sub><i>ħω</i><sub>2</sub> +.... As there are no cross terms (e.g. <i>n</i><sub>1</sub><i>ħω</i><sub>2</sub>), the phonons are said to be non-interacting. The action of the creation and annihilation operators is given by: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {a_{\alpha }}^{\dagger }{\Big |}n_{1}\ldots n_{\alpha -1}n_{\alpha }n_{\alpha +1}\ldots {\Big \rangle }={\sqrt {n_{\alpha }+1}}{\Big |}n_{1}\ldots ,n_{\alpha -1},(n_{\alpha }+1),n_{\alpha +1}\ldots {\Big \rangle }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">|</mo> </mrow> </mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>…</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">⟩</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">|</mo> </mrow> </mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>…</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">⟩</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {a_{\alpha }}^{\dagger }{\Big |}n_{1}\ldots n_{\alpha -1}n_{\alpha }n_{\alpha +1}\ldots {\Big \rangle }={\sqrt {n_{\alpha }+1}}{\Big |}n_{1}\ldots ,n_{\alpha -1},(n_{\alpha }+1),n_{\alpha +1}\ldots {\Big \rangle }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1a845b302d62c8ac69e1c694c1fa1b0673d0aa3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:71.991ex; height:4.843ex;" alt="{\displaystyle {a_{\alpha }}^{\dagger }{\Big |}n_{1}\ldots n_{\alpha -1}n_{\alpha }n_{\alpha +1}\ldots {\Big \rangle }={\sqrt {n_{\alpha }+1}}{\Big |}n_{1}\ldots ,n_{\alpha -1},(n_{\alpha }+1),n_{\alpha +1}\ldots {\Big \rangle }}"></span></dd></dl> <p>and, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{\alpha }{\Big |}n_{1}\ldots n_{\alpha -1}n_{\alpha }n_{\alpha +1}\ldots {\Big \rangle }={\sqrt {n_{\alpha }}}{\Big |}n_{1}\ldots ,n_{\alpha -1},(n_{\alpha }-1),n_{\alpha +1},\ldots {\Big \rangle }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">|</mo> </mrow> </mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>…</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">⟩</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">|</mo> </mrow> </mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">⟩</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{\alpha }{\Big |}n_{1}\ldots n_{\alpha -1}n_{\alpha }n_{\alpha +1}\ldots {\Big \rangle }={\sqrt {n_{\alpha }}}{\Big |}n_{1}\ldots ,n_{\alpha -1},(n_{\alpha }-1),n_{\alpha +1},\ldots {\Big \rangle }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/039e12a6795277d8d1c46bd776a8cbf20114eb62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:67.285ex; height:4.843ex;" alt="{\displaystyle a_{\alpha }{\Big |}n_{1}\ldots n_{\alpha -1}n_{\alpha }n_{\alpha +1}\ldots {\Big \rangle }={\sqrt {n_{\alpha }}}{\Big |}n_{1}\ldots ,n_{\alpha -1},(n_{\alpha }-1),n_{\alpha +1},\ldots {\Big \rangle }}"></span></dd></dl> <p>The creation operator, <i>a<sub>α</sub></i><sup>†</sup> creates a phonon of type <i>α</i> while <i>a<sub>α</sub></i> annihilates one. Hence, they are respectively the creation and annihilation operators for phonons. Analogous to the <a href="/wiki/Quantum_harmonic_oscillator" title="Quantum harmonic oscillator">quantum harmonic oscillator</a> case, we can define <a href="/wiki/Particle_number_operator" title="Particle number operator">particle number operator</a> as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=\sum _{\alpha }{a_{\alpha }}^{\dagger }a_{\alpha }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </munder> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=\sum _{\alpha }{a_{\alpha }}^{\dagger }a_{\alpha }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b70d2d0db9e10b227fde21b0afb945c6521398b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.541ex; height:5.509ex;" alt="{\displaystyle N=\sum _{\alpha }{a_{\alpha }}^{\dagger }a_{\alpha }.}"></span></dd></dl> <p>The number operator commutes with a string of products of the creation and annihilation operators if and only if the number of creation operators is equal to number of annihilation operators. </p><p>It can be shown that phonons are symmetric under exchange (i.e. <span class="nowrap">|<i>α</i>,<i>β</i>⟩</span> = <span class="nowrap">|<i>β</i>,<i>α</i>⟩</span>), so therefore they are considered <a href="/wiki/Bosons" class="mw-redirect" title="Bosons">bosons</a>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Nonlinearity">Nonlinearity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon&action=edit&section=15" title="Edit section: Nonlinearity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As well as <a href="/wiki/Photons" class="mw-redirect" title="Photons">photons</a>, phonons can interact via <a href="/wiki/Parametric_down_conversion" class="mw-redirect" title="Parametric down conversion">parametric down conversion</a><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> and form <a href="/wiki/Squeezed_coherent_state" title="Squeezed coherent state">squeezed coherent states</a>.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Predicted_properties">Predicted properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon&action=edit&section=16" title="Edit section: Predicted properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Recent research has shown that phonons and <a href="/wiki/Roton" title="Roton">rotons</a> may have a non-negligible mass and be affected by gravity just as standard particles are.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> In particular, phonons are predicted to have a kind of <a href="/wiki/Negative_mass" title="Negative mass">negative mass</a> and negative gravity.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> This can be explained by how phonons are known to travel faster in denser materials. Because the part of a material pointing towards a gravitational source is closer to the object, it becomes denser on that end. From this, it is predicted that phonons would deflect away as it detects the difference in densities, exhibiting the qualities of a negative gravitational field.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> Although the effect would be too small to measure, it is possible that future equipment could lead to successful results. </p> <div class="mw-heading mw-heading2"><h2 id="Superconductivity">Superconductivity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon&action=edit&section=17" title="Edit section: Superconductivity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Superconductivity" title="Superconductivity">Superconductivity</a> is a state of electronic matter in which <a href="/wiki/Electrical_resistance_and_conductance" title="Electrical resistance and conductance">electrical resistance</a> vanishes and <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic fields</a> are expelled from the material. In a superconductor, electrons are bound together into <a href="/wiki/Cooper_pair" title="Cooper pair">Cooper pairs</a> by a weak attractive force. In a conventional superconductor, this attraction is caused by an <a href="/wiki/Force_carrier" title="Force carrier">exchange</a> of phonons between the electrons.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> The evidence that phonons, the vibrations of the ionic lattice, are relevant for superconductivity is provided by the <a href="/wiki/BCS_theory#Underlying_evidence" title="BCS theory">isotope effect</a>, the dependence of the superconducting critical temperature on the mass of the ions. </p> <div class="mw-heading mw-heading2"><h2 id="Other_research">Other research</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon&action=edit&section=18" title="Edit section: Other research"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 2019, researchers were able to isolate individual phonons without destroying them for the first time.<sup id="cite_ref-nature_25-0" class="reference"><a href="#cite_note-nature-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p><p>They have been also shown to form “phonon winds” where an electric current in a graphene surface is generated by a liquid flow above it due to the viscous forces at the liquid–solid interface.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon&action=edit&section=19" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/25px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png" decoding="async" width="25" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/37px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/49px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 2x" data-file-width="530" data-file-height="600" /></a></span></span><span class="portalbox-link"><a href="/wiki/Portal:Physics" title="Portal:Physics">Physics portal</a></span></li></ul> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 22em;"> <ul><li><a href="/wiki/Boson" title="Boson">Boson</a></li> <li><a href="/wiki/Brillouin_scattering" title="Brillouin scattering">Brillouin scattering</a></li> <li><a href="/wiki/Fracton" title="Fracton">Fracton</a></li> <li><a href="/wiki/Linear_elasticity" title="Linear elasticity">Linear elasticity</a></li> <li><a href="/wiki/Mechanical_wave" title="Mechanical wave">Mechanical wave</a></li> <li><a href="/wiki/Phonon_scattering" title="Phonon scattering">Phonon scattering</a></li> <li><a href="/wiki/Carrier_scattering" title="Carrier scattering">Carrier scattering</a></li> <li><a href="/wiki/Acoustic_metamaterials#Phononic_crystal" class="mw-redirect" title="Acoustic metamaterials">Phononic crystal</a></li> <li><a href="/wiki/Rayleigh_wave" title="Rayleigh wave">Rayleigh wave</a></li> <li><a href="/wiki/Relativistic_heat_conduction" title="Relativistic heat conduction">Relativistic heat conduction</a></li> <li><a href="/wiki/Rigid_unit_modes" title="Rigid unit modes">Rigid unit modes</a></li> <li><a href="/wiki/SASER" class="mw-redirect" title="SASER">SASER</a></li> <li><a href="/wiki/Second_sound" title="Second sound">Second sound</a></li> <li><a href="/wiki/Surface_acoustic_wave" title="Surface acoustic wave">Surface acoustic wave</a></li> <li><a href="/wiki/Surface_phonon" title="Surface phonon">Surface phonon</a></li> <li><a href="/wiki/Thermal_conductivity" class="mw-redirect" title="Thermal conductivity">Thermal conductivity</a></li> <li><a href="/wiki/Vibration" title="Vibration">Vibration</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon&action=edit&section=20" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFSchwabl2008" class="citation book cs1">Schwabl, Franz (2008). <i>Advanced Quantum Mechanics</i> (4th ed.). Springer. p. 253. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-85062-5" title="Special:BookSources/978-3-540-85062-5"><bdi>978-3-540-85062-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+Quantum+Mechanics&rft.pages=253&rft.edition=4th&rft.pub=Springer&rft.date=2008&rft.isbn=978-3-540-85062-5&rft.aulast=Schwabl&rft.aufirst=Franz&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhonon" class="Z3988"></span></span> </li> <li id="cite_note-girvinYang-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-girvinYang_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-girvinYang_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-girvinYang_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-girvinYang_2-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-girvinYang_2-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-girvinYang_2-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGirvinYang2019" class="citation book cs1">Girvin, Steven M.; Yang, Kun (2019). <i>Modern Condensed Matter Physics</i>. Cambridge University Press. pp. 78–96. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-13739-4" title="Special:BookSources/978-1-107-13739-4"><bdi>978-1-107-13739-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Condensed+Matter+Physics&rft.pages=78-96&rft.pub=Cambridge+University+Press&rft.date=2019&rft.isbn=978-1-107-13739-4&rft.aulast=Girvin&rft.aufirst=Steven+M.&rft.au=Yang%2C+Kun&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhonon" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKozhevnikov2004" class="citation book cs1">Kozhevnikov, A. B. (2004). <a rel="nofollow" class="external text" href="https://archive.org/details/stalinsgreatscie0000kozh/mode/1up?q=tamm+phonon"><i>Stalin's great science : the times and adventures of Soviet physicists</i></a>. London: Imperial College Press. pp. 64–69. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-86094-419-2" title="Special:BookSources/978-1-86094-419-2"><bdi>978-1-86094-419-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Stalin%27s+great+science+%3A+the+times+and+adventures+of+Soviet+physicists&rft.place=London&rft.pages=64-69&rft.pub=Imperial+College+Press&rft.date=2004&rft.isbn=978-1-86094-419-2&rft.aulast=Kozhevnikov&rft.aufirst=A.+B.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fstalinsgreatscie0000kozh%2Fmode%2F1up%3Fq%3Dtamm%2Bphonon&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhonon" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: date and year (<a href="/wiki/Category:CS1_maint:_date_and_year" title="Category:CS1 maint: date and year">link</a>)</span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSimon2013" class="citation book cs1">Simon, Steven H. (2013). <i>The Oxford solid state basics</i> (1st ed.). Oxford: Oxford University Press. p. 82. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-968077-1" title="Special:BookSources/978-0-19-968077-1"><bdi>978-0-19-968077-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Oxford+solid+state+basics&rft.place=Oxford&rft.pages=82&rft.edition=1st&rft.pub=Oxford+University+Press&rft.date=2013&rft.isbn=978-0-19-968077-1&rft.aulast=Simon&rft.aufirst=Steven+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhonon" class="Z3988"></span></span> </li> <li id="cite_note-latticemechanics3-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-latticemechanics3_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKrauth2006" class="citation book cs1">Krauth, Werner (April 2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EnabPPmmS4sC&q=Mechanics+of+particles+on+a+lattice&pg=RA1-PA231"><i>Statistical mechanics: algorithms and computations</i></a>. International publishing locations: Oxford University Press. pp. 231–232. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-851536-4" title="Special:BookSources/978-0-19-851536-4"><bdi>978-0-19-851536-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistical+mechanics%3A+algorithms+and+computations&rft.place=International+publishing+locations&rft.pages=231-232&rft.pub=Oxford+University+Press&rft.date=2006-04&rft.isbn=978-0-19-851536-4&rft.aulast=Krauth&rft.aufirst=Werner&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DEnabPPmmS4sC%26q%3DMechanics%2Bof%2Bparticles%2Bon%2Ba%2Blattice%26pg%3DRA1-PA231&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhonon" class="Z3988"></span></span> </li> <li id="cite_note-:0-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Maradudin, A.; Montroll, E.; Weiss, G.; Ipatova, I. (1971). <i>Theory of lattice dynamics in the harmonic approximation</i>. Solid State Physics. Vol. Supplement 3 (Second ed.). New York: Academic Press.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMattuck1976" class="citation book cs1">Mattuck, R. (1976). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/guidetofeynmandi0000matt"><i>A guide to Feynman Diagrams in the many-body problem</i></a></span>. McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780070409545" title="Special:BookSources/9780070409545"><bdi>9780070409545</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+guide+to+Feynman+Diagrams+in+the+many-body+problem&rft.pub=McGraw-Hill&rft.date=1976&rft.isbn=9780070409545&rft.aulast=Mattuck&rft.aufirst=R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fguidetofeynmandi0000matt&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhonon" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFetterWalecka2003" class="citation book cs1">Fetter, Alexander; Walecka, John (2003-12-16). <i>Theoretical Mechanics of Particles and Continua</i>. Dover Books on Physics. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0486432618" title="Special:BookSources/978-0486432618"><bdi>978-0486432618</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theoretical+Mechanics+of+Particles+and+Continua&rft.pub=Dover+Books+on+Physics&rft.date=2003-12-16&rft.isbn=978-0486432618&rft.aulast=Fetter&rft.aufirst=Alexander&rft.au=Walecka%2C+John&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhonon" class="Z3988"></span></span> </li> <li id="cite_note-Mahan-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-Mahan_9-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMahan1981" class="citation book cs1">Mahan, G. D. (1981). <i>Many-Particle Physics</i>. New York: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-306-46338-9" title="Special:BookSources/978-0-306-46338-9"><bdi>978-0-306-46338-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Many-Particle+Physics&rft.place=New+York&rft.pub=Springer&rft.date=1981&rft.isbn=978-0-306-46338-9&rft.aulast=Mahan&rft.aufirst=G.+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhonon" class="Z3988"></span></span> </li> <li id="cite_note-Cardona-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-Cardona_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Cardona_10-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYuCardona2010" class="citation book cs1">Yu, Peter Y.; Cardona, Manuel (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5aBuKYBT_hsC&pg=PA111">"Fig. 3.2: Phonon dispersion curves in GaAs along high-symmetry axes"</a>. <i>Fundamentals of Semiconductors</i>. Physics and Materials Properties (4th ed.). Springer. p. 111. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-00709-5" title="Special:BookSources/978-3-642-00709-5"><bdi>978-3-642-00709-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Fig.+3.2%3A+Phonon+dispersion+curves+in+GaAs+along+high-symmetry+axes&rft.btitle=Fundamentals+of+Semiconductors&rft.series=Physics+and+Materials+Properties&rft.pages=111&rft.edition=4th&rft.pub=Springer&rft.date=2010&rft.isbn=978-3-642-00709-5&rft.aulast=Yu&rft.aufirst=Peter+Y.&rft.au=Cardona%2C+Manuel&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D5aBuKYBT_hsC%26pg%3DPA111&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhonon" class="Z3988"></span></span> </li> <li id="cite_note-Misra-11"><span class="mw-cite-backlink">^ <a href="#cite_ref-Misra_11-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Misra_11-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMisra2010" class="citation book cs1">Misra, Prasanta Kumar (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=J6rMISLVCmcC&pg=PA44">"§2.1.3 Normal modes of a one-dimensional chain with a basis"</a>. <i>Physics of Condensed Matter</i>. Academic Press. p. 44. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-384954-0" title="Special:BookSources/978-0-12-384954-0"><bdi>978-0-12-384954-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=%C2%A72.1.3+Normal+modes+of+a+one-dimensional+chain+with+a+basis&rft.btitle=Physics+of+Condensed+Matter&rft.pages=44&rft.pub=Academic+Press&rft.date=2010&rft.isbn=978-0-12-384954-0&rft.aulast=Misra&rft.aufirst=Prasanta+Kumar&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJ6rMISLVCmcC%26pg%3DPA44&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhonon" class="Z3988"></span></span> </li> <li id="cite_note-ashcrroftMermin-12"><span class="mw-cite-backlink">^ <a href="#cite_ref-ashcrroftMermin_12-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ashcrroftMermin_12-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-ashcrroftMermin_12-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-ashcrroftMermin_12-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-ashcrroftMermin_12-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-ashcrroftMermin_12-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-ashcrroftMermin_12-6"><sup><i><b>g</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAshcroftMermin1976" class="citation book cs1">Ashcroft, Neil W.; Mermin, N. David (1976). <i>Solid State Physics</i>. Saunders College Publishing. pp. 780–783. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-03-083993-9" title="Special:BookSources/0-03-083993-9"><bdi>0-03-083993-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Solid+State+Physics&rft.pages=780-783&rft.pub=Saunders+College+Publishing&rft.date=1976&rft.isbn=0-03-083993-9&rft.aulast=Ashcroft&rft.aufirst=Neil+W.&rft.au=Mermin%2C+N.+David&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhonon" class="Z3988"></span></span> </li> <li id="cite_note-cmian-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-cmian_13-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMahan2010" class="citation book cs1">Mahan, Gerald (2010). <i>Condensed Matter in a Nutshell</i>. Princeton: Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-14016-2" title="Special:BookSources/978-0-691-14016-2"><bdi>978-0-691-14016-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Condensed+Matter+in+a+Nutshell&rft.place=Princeton&rft.pub=Princeton+University+Press&rft.date=2010&rft.isbn=978-0-691-14016-2&rft.aulast=Mahan&rft.aufirst=Gerald&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhonon" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKittel2004" class="citation book cs1">Kittel, Charles (2004). <i><a href="/wiki/Introduction_to_Solid_State_Physics" title="Introduction to Solid State Physics">Introduction to Solid State Physics</a>, 8th Edition</i>. Wiley. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/isbn_9780471415268/page/100">100</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-41526-8" title="Special:BookSources/978-0-471-41526-8"><bdi>978-0-471-41526-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Solid+State+Physics%2C+8th+Edition&rft.pages=100&rft.pub=Wiley&rft.date=2004&rft.isbn=978-0-471-41526-8&rft.aulast=Kittel&rft.aufirst=Charles&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhonon" class="Z3988"></span></span> </li> <li id="cite_note-nonMetalsThermalPhonons-15"><span class="mw-cite-backlink">^ <a href="#cite_ref-nonMetalsThermalPhonons_15-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-nonMetalsThermalPhonons_15-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.doitpoms.ac.uk/tlplib/thermal_electrical/nonmetal_thermal.php#:~:text=When%20the%20lattice%20is%20heated,as%20a%20gas%20of%20phonons.">"Non-metals: thermal phonons"</a>. <i>University of Cambridge Teaching and Learning Packages Library</i><span class="reference-accessdate">. Retrieved <span class="nowrap">15 August</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=University+of+Cambridge+Teaching+and+Learning+Packages+Library&rft.atitle=Non-metals%3A+thermal+phonons&rft_id=https%3A%2F%2Fwww.doitpoms.ac.uk%2Ftlplib%2Fthermal_electrical%2Fnonmetal_thermal.php%23%3A~%3Atext%3DWhen%2520the%2520lattice%2520is%2520heated%2Cas%2520a%2520gas%2520of%2520phonons.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhonon" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPathriaBeale2011" class="citation book cs1">Pathria; Beale (2011). <i>Statistical Mechanics</i> (3 ed.). India: Elsevier. p. 201. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-93-80931-89-0" title="Special:BookSources/978-93-80931-89-0"><bdi>978-93-80931-89-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistical+Mechanics&rft.place=India&rft.pages=201&rft.edition=3&rft.pub=Elsevier&rft.date=2011&rft.isbn=978-93-80931-89-0&rft.au=Pathria&rft.au=Beale&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhonon" class="Z3988"></span></span> </li> <li id="cite_note-mitPhononTunneling-17"><span class="mw-cite-backlink">^ <a href="#cite_ref-mitPhononTunneling_17-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-mitPhononTunneling_17-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://news.mit.edu/2015/phonon-tunneling-heat-flow-nanometer-gaps-0407">"Tunneling across a tiny gap"</a>. <i>News.mit.edu</i>. 7 April 2015<span class="reference-accessdate">. Retrieved <span class="nowrap">13 August</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=News.mit.edu&rft.atitle=Tunneling+across+a+tiny+gap&rft.date=2015-04-07&rft_id=https%3A%2F%2Fnews.mit.edu%2F2015%2Fphonon-tunneling-heat-flow-nanometer-gaps-0407&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhonon" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeynman1982" class="citation book cs1">Feynman, Richard P. (1982). <a rel="nofollow" class="external text" href="https://archive.org/details/statisticalmecha00rich/page/159"><i>Statistical Mechanics, A Set of Lectures</i></a>. Reading, MA: Benjamin-Cummings. p. <a rel="nofollow" class="external text" href="https://archive.org/details/statisticalmecha00rich/page/159">159</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8053-2508-9" title="Special:BookSources/978-0-8053-2508-9"><bdi>978-0-8053-2508-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistical+Mechanics%2C+A+Set+of+Lectures&rft.place=Reading%2C+MA&rft.pages=159&rft.pub=Benjamin-Cummings&rft.date=1982&rft.isbn=978-0-8053-2508-9&rft.aulast=Feynman&rft.aufirst=Richard+P.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fstatisticalmecha00rich%2Fpage%2F159&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhonon" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarquetSchmidt-KalerJames2003" class="citation journal cs1">Marquet, C.; Schmidt-Kaler, F.; James, D. F. V. (2003). <a rel="nofollow" class="external text" href="http://www.quantumoptics.at/images/publications/papers/apb03_marquet.pdf">"Phonon–phonon interactions due to non-linear effects in a linear ion trap"</a> <span class="cs1-format">(PDF)</span>. <i>Applied Physics B</i>. <b>76</b> (3): 199–208. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0211079">quant-ph/0211079</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2003ApPhB..76..199M">2003ApPhB..76..199M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00340-003-1097-7">10.1007/s00340-003-1097-7</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17019967">17019967</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Applied+Physics+B&rft.atitle=Phonon%E2%80%93phonon+interactions+due+to+non-linear+effects+in+a+linear+ion+trap&rft.volume=76&rft.issue=3&rft.pages=199-208&rft.date=2003&rft_id=info%3Aarxiv%2Fquant-ph%2F0211079&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17019967%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs00340-003-1097-7&rft_id=info%3Abibcode%2F2003ApPhB..76..199M&rft.aulast=Marquet&rft.aufirst=C.&rft.au=Schmidt-Kaler%2C+F.&rft.au=James%2C+D.+F.+V.&rft_id=http%3A%2F%2Fwww.quantumoptics.at%2Fimages%2Fpublications%2Fpapers%2Fapb03_marquet.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhonon" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReiterSauerHunekePapenkort2009" class="citation journal cs1">Reiter, D. E.; Sauer, S.; Huneke, J.; Papenkort, T.; Kuhn, T.; Vagov, A.; Axt, V. M. (2009). <a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1742-6596%2F193%2F1%2F012121">"Generation of squeezed phonon states by optical excitation of a quantum dot"</a>. <i>Journal of Physics: Conference Series</i>. <b>193</b> (1). <a href="/wiki/Institute_of_Physics" title="Institute of Physics">Institute of Physics</a>: 012121. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009JPhCS.193a2121R">2009JPhCS.193a2121R</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1742-6596%2F193%2F1%2F012121">10.1088/1742-6596/193/1/012121</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Physics%3A+Conference+Series&rft.atitle=Generation+of+squeezed+phonon+states+by+optical+excitation+of+a+quantum+dot&rft.volume=193&rft.issue=1&rft.pages=012121&rft.date=2009&rft_id=info%3Adoi%2F10.1088%2F1742-6596%2F193%2F1%2F012121&rft_id=info%3Abibcode%2F2009JPhCS.193a2121R&rft.aulast=Reiter&rft.aufirst=D.+E.&rft.au=Sauer%2C+S.&rft.au=Huneke%2C+J.&rft.au=Papenkort%2C+T.&rft.au=Kuhn%2C+T.&rft.au=Vagov%2C+A.&rft.au=Axt%2C+V.+M.&rft_id=https%3A%2F%2Fdoi.org%2F10.1088%252F1742-6596%252F193%252F1%252F012121&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhonon" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text">Alberto Nicolis and Riccardo Penco. (2017). <a rel="nofollow" class="external text" href="https://arxiv.org/abs/1705.08914">Mutual Interactions of Phonons, Rotons, and Gravity</a>, Arxiv.org, Retrieved November 27, 2018</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text">Angelo Esposito, Rafael Krichevsky, and Alberto Nicolis. (2018). <a rel="nofollow" class="external text" href="https://arxiv.org/abs/1807.08771">The mass of sound</a> Retrieved November 11, 2018</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://phys.org/news/2018-08-phonons-mass-negative-gravity.html">"Researchers suggest phonons may have mass and perhaps negative gravity"</a>. <i>Phys.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">13 August</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Phys.org&rft.atitle=Researchers+suggest+phonons+may+have+mass+and+perhaps+negative+gravity&rft_id=https%3A%2F%2Fphys.org%2Fnews%2F2018-08-phonons-mass-negative-gravity.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhonon" class="Z3988"></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTinkham1996" class="citation book cs1">Tinkham, Michael (1996). <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontosu0000mich/page/8/mode/2up?view=theater"><i>Introduction to Superconductivity</i></a>. Mineola, New York: Dover Publications, Inc. p. 9. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0486435032" title="Special:BookSources/0486435032"><bdi>0486435032</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Superconductivity&rft.place=Mineola%2C+New+York&rft.pages=9&rft.pub=Dover+Publications%2C+Inc.&rft.date=1996&rft.isbn=0486435032&rft.aulast=Tinkham&rft.aufirst=Michael&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontosu0000mich%2Fpage%2F8%2Fmode%2F2up%3Fview%3Dtheater&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhonon" class="Z3988"></span></span> </li> <li id="cite_note-nature-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-nature_25-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation journal cs1"><a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fd41586-019-02009-5">"Detecting the softest sounds in the Universe"</a>. <i><a href="/wiki/Nature_(journal)" title="Nature (journal)">Nature</a></i>. <b>571</b> (7763): 8–9. July 1, 2019. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2019Natur.571....8.">2019Natur.571....8.</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fd41586-019-02009-5">10.1038/d41586-019-02009-5</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:195774243">195774243</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Detecting+the+softest+sounds+in+the+Universe&rft.volume=571&rft.issue=7763&rft.pages=8-9&rft.date=2019-07-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A195774243%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1038%2Fd41586-019-02009-5&rft_id=info%3Abibcode%2F2019Natur.571....8.&rft_id=https%3A%2F%2Fdoi.org%2F10.1038%252Fd41586-019-02009-5&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhonon" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLizéeMarcotteCoquinotKavokine2023" class="citation journal cs1">Lizée, Mathieu; Marcotte, Alice; Coquinot, Baptiste; Kavokine, Nikita; Sobnath, Karen; Barraud, Clément; Bhardwaj, Ankit; Radha, Boya; Niguès, Antoine; Bocquet, Lydéric; Siria, Alessandro (2023-02-17). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/PhysRevX.13.011020">"Strong Electronic Winds Blowing under Liquid Flows on Carbon Surfaces"</a>. <i>Physical Review X</i>. <b>13</b> (1): 011020. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2205.05037">2205.05037</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2023PhRvX..13a1020L">2023PhRvX..13a1020L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevX.13.011020">10.1103/PhysRevX.13.011020</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:248665478">248665478</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+X&rft.atitle=Strong+Electronic+Winds+Blowing+under+Liquid+Flows+on+Carbon+Surfaces&rft.volume=13&rft.issue=1&rft.pages=011020&rft.date=2023-02-17&rft_id=info%3Aarxiv%2F2205.05037&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A248665478%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevX.13.011020&rft_id=info%3Abibcode%2F2023PhRvX..13a1020L&rft.aulast=Liz%C3%A9e&rft.aufirst=Mathieu&rft.au=Marcotte%2C+Alice&rft.au=Coquinot%2C+Baptiste&rft.au=Kavokine%2C+Nikita&rft.au=Sobnath%2C+Karen&rft.au=Barraud%2C+Cl%C3%A9ment&rft.au=Bhardwaj%2C+Ankit&rft.au=Radha%2C+Boya&rft.au=Nigu%C3%A8s%2C+Antoine&rft.au=Bocquet%2C+Lyd%C3%A9ric&rft.au=Siria%2C+Alessandro&rft_id=https%3A%2F%2Flink.aps.org%2Fdoi%2F10.1103%2FPhysRevX.13.011020&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhonon" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchirber2023" class="citation journal cs1">Schirber, Michael (2023-02-17). <a rel="nofollow" class="external text" href="https://physics.aps.org/articles/v16/26">"Secret of Flow-Induced Electric Currents Revealed"</a>. <i>Physics</i>. <b>16</b> (1): 26. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2205.05037">2205.05037</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2023PhRvX..13a1020L">2023PhRvX..13a1020L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevX.13.011020">10.1103/PhysRevX.13.011020</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:248665478">248665478</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics&rft.atitle=Secret+of+Flow-Induced+Electric+Currents+Revealed&rft.volume=16&rft.issue=1&rft.pages=26&rft.date=2023-02-17&rft_id=info%3Aarxiv%2F2205.05037&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A248665478%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevX.13.011020&rft_id=info%3Abibcode%2F2023PhRvX..13a1020L&rft.aulast=Schirber&rft.aufirst=Michael&rft_id=https%3A%2F%2Fphysics.aps.org%2Farticles%2Fv16%2F26&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhonon" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon&action=edit&section=21" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Lattice_vibrations" class="extiw" title="commons:Category:Lattice vibrations">Lattice vibrations</a></span>.</div></div> </div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wikiquote-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/13px-Wikiquote-logo.svg.png" decoding="async" width="13" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/20px-Wikiquote-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/27px-Wikiquote-logo.svg.png 2x" data-file-width="300" data-file-height="355" /></a></span> Quotations related to <a href="https://en.wikiquote.org/wiki/Special:Search/Phonon" class="extiw" title="wikiquote:Special:Search/Phonon">Phonon</a> at Wikiquote</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://news.mit.edu/2010/explained-phonons-0706">"Explained: Phonons"</a>. <i>MIT News | Massachusetts Institute of Technology</i>. 2010-07-08<span class="reference-accessdate">. Retrieved <span class="nowrap">2024-10-12</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MIT+News+%7C+Massachusetts+Institute+of+Technology&rft.atitle=Explained%3A+Phonons&rft.date=2010-07-08&rft_id=https%3A%2F%2Fnews.mit.edu%2F2010%2Fexplained-phonons-0706&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhonon" class="Z3988"></span>* <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080318142623/http://dept.kent.edu/projects/ksuviz/leeviz/phonon/phonon.html">Optical and acoustic modes</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeatusTlustyBar-Ziv2006" class="citation journal cs1">Beatus, Tsevi; Tlusty, Tsvi; Bar-Ziv, Roy (November 2006). <a rel="nofollow" class="external text" href="https://www.nature.com/articles/nphys432">"Phonons in a one-dimensional microfluidic crystal"</a>. <i>Nature Physics</i>. <b>2</b> (11): 743–748. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnphys432">10.1038/nphys432</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1745-2481">1745-2481</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature+Physics&rft.atitle=Phonons+in+a+one-dimensional+microfluidic+crystal&rft.volume=2&rft.issue=11&rft.pages=743-748&rft.date=2006-11&rft_id=info%3Adoi%2F10.1038%2Fnphys432&rft.issn=1745-2481&rft.aulast=Beatus&rft.aufirst=Tsevi&rft.au=Tlusty%2C+Tsvi&rft.au=Bar-Ziv%2C+Roy&rft_id=https%3A%2F%2Fwww.nature.com%2Farticles%2Fnphys432&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhonon" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeatusTlustyBar-Ziv2006" class="citation journal cs1">Beatus, Tsevi; Tlusty, Tsvi; Bar-Ziv, Roy (November 2006). <a rel="nofollow" class="external text" href="https://arxiv.org/abs/1008.1155">"Phonons in a one-dimensional microfluidic crystal"</a>. <i>Nature Physics</i>. <b>2</b> (11): 743–748. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnphys432">10.1038/nphys432</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1745-2473">1745-2473</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature+Physics&rft.atitle=Phonons+in+a+one-dimensional+microfluidic+crystal&rft.volume=2&rft.issue=11&rft.pages=743-748&rft.date=2006-11&rft_id=info%3Adoi%2F10.1038%2Fnphys432&rft.issn=1745-2473&rft.aulast=Beatus&rft.aufirst=Tsevi&rft.au=Tlusty%2C+Tsvi&rft.au=Bar-Ziv%2C+Roy&rft_id=https%3A%2F%2Farxiv.org%2Fabs%2F1008.1155&rfr_id=info%3Asid%2Fen.wikipedia.org%3APhonon" class="Z3988"></span> with movies in <a rel="nofollow" class="external text" href="http://www.weizmann.ac.il/materials/barziv/project_1.htm">Bar-Ziv Lab</a>.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Particles_in_physics" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="text-align: center;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Particles" title="Template:Particles"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Particles" title="Template talk:Particles"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Particles" title="Special:EditPage/Template:Particles"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Particles_in_physics" style="font-size:114%;margin:0 4em"><a href="/wiki/Particle_physics" title="Particle physics">Particles in physics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="text-align: center;;width:1%"><a href="/wiki/Elementary_particle" title="Elementary particle">Elementary</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;"><a href="/wiki/Fermion" title="Fermion">Fermions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Quark" title="Quark">Quarks</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Up_quark" title="Up quark">Up (quark</a></li> <li><a href="/wiki/Up_antiquark" class="mw-redirect" title="Up antiquark">antiquark)</a></li> <li><a href="/wiki/Down_quark" title="Down quark">Down (quark</a></li> <li><a href="/wiki/Down_antiquark" class="mw-redirect" title="Down antiquark">antiquark)</a></li> <li><a href="/wiki/Charm_quark" title="Charm quark">Charm (quark</a></li> <li><a href="/wiki/Charm_antiquark" class="mw-redirect" title="Charm antiquark">antiquark)</a></li> <li><a href="/wiki/Strange_quark" title="Strange quark">Strange (quark</a></li> <li><a href="/wiki/Strange_antiquark" class="mw-redirect" title="Strange antiquark">antiquark)</a></li> <li><a href="/wiki/Top_quark" title="Top quark">Top (quark</a></li> <li><a href="/wiki/Top_antiquark" class="mw-redirect" title="Top antiquark">antiquark)</a></li> <li><a href="/wiki/Bottom_quark" title="Bottom quark">Bottom (quark</a></li> <li><a href="/wiki/Bottom_antiquark" class="mw-redirect" title="Bottom antiquark">antiquark)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Lepton" title="Lepton">Leptons</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Electron" title="Electron">Electron</a></li> <li><a href="/wiki/Positron" title="Positron">Positron</a></li> <li><a href="/wiki/Muon" title="Muon">Muon</a></li> <li><a href="/wiki/Muon" title="Muon">Antimuon</a></li> <li><a href="/wiki/Tau_(particle)" title="Tau (particle)">Tau</a></li> <li><a href="/wiki/Tau_(particle)" title="Tau (particle)">Antitau</a></li> <li><a href="/wiki/Neutrino" title="Neutrino">Neutrino</a> <ul><li><a href="/wiki/Electron_neutrino" title="Electron neutrino">Electron neutrino</a></li> <li><a href="/wiki/Neutrino#Antineutrinos" title="Neutrino">Electron antineutrino</a></li> <li><a href="/wiki/Muon_neutrino" title="Muon neutrino">Muon neutrino</a></li> <li><a href="/wiki/Neutrino#Antineutrinos" title="Neutrino">Muon antineutrino</a></li> <li><a href="/wiki/Tau_neutrino" title="Tau neutrino">Tau neutrino</a></li> <li><a href="/wiki/Neutrino#Antineutrinos" title="Neutrino">Tau antineutrino</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;"><a href="/wiki/Boson" title="Boson">Bosons</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:4em;font-weight:normal; text-align: center;"><a href="/wiki/Gauge_boson" title="Gauge boson">Gauge</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Photon" title="Photon">Photon</a></li> <li><a href="/wiki/Gluon" title="Gluon">Gluon</a></li> <li><a href="/wiki/W_and_Z_bosons" title="W and Z bosons">W and Z bosons</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:4em;font-weight:normal; text-align: center;"><a href="/wiki/Scalar_boson" title="Scalar boson">Scalar</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Higgs_boson" title="Higgs boson">Higgs boson </a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;;font-weight:normal; text-align: center;"><a href="/wiki/Ghost_(physics)" title="Ghost (physics)">Ghost fields</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Faddeev%E2%80%93Popov_ghost" title="Faddeev–Popov ghost">Faddeev–Popov ghosts</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;"><a href="/wiki/Hypothetical_particles" class="mw-redirect" title="Hypothetical particles">Hypothetical</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal; text-align: center;"><a href="/wiki/Superpartner" title="Superpartner">Superpartners</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal; text-align: center;"><a href="/wiki/Gaugino" title="Gaugino">Gauginos</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Gluino" title="Gluino">Gluino</a></li> <li><a href="/wiki/Gravitino" title="Gravitino">Gravitino</a></li> <li><a href="/wiki/Photino" title="Photino">Photino</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal; text-align: center;">Others</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Axino" title="Axino">Axino</a></li> <li><a href="/wiki/Chargino" title="Chargino">Chargino</a></li> <li><a href="/wiki/Higgsino" title="Higgsino">Higgsino</a></li> <li><a href="/wiki/Neutralino" title="Neutralino">Neutralino</a></li> <li><a href="/wiki/Sfermion" title="Sfermion">Sfermion</a> (<a href="/wiki/Stop_squark" title="Stop squark">Stop squark</a>)</li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal; text-align: center;">Others</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Axion" title="Axion">Axion</a></li> <li><a href="/wiki/Curvaton" title="Curvaton">Curvaton</a></li> <li><a href="/wiki/Dilaton" title="Dilaton">Dilaton</a></li> <li><a href="/wiki/Dual_graviton" title="Dual graviton">Dual graviton</a></li> <li><a href="/wiki/Graviphoton" title="Graviphoton">Graviphoton</a></li> <li><a href="/wiki/Graviton" title="Graviton">Graviton</a></li> <li><a href="/wiki/Inflaton" title="Inflaton">Inflaton</a></li> <li><a href="/wiki/Leptoquark" title="Leptoquark">Leptoquark</a></li> <li><a href="/wiki/Magnetic_monopole" title="Magnetic monopole">Magnetic monopole</a></li> <li><a href="/wiki/Majoron" title="Majoron">Majoron</a></li> <li><a href="/wiki/Majorana_fermion" title="Majorana fermion">Majorana fermion</a></li> <li><a href="/wiki/Dark_photon" title="Dark photon">Dark photon</a></li> <li><a href="/wiki/Preon" title="Preon">Preon</a></li> <li><a href="/wiki/Sterile_neutrino" title="Sterile neutrino">Sterile neutrino</a></li> <li><a href="/wiki/Tachyon" title="Tachyon">Tachyon</a></li> <li><a href="/wiki/W%E2%80%B2_and_Z%E2%80%B2_bosons" title="W′ and Z′ bosons">W′ and Z′ bosons</a></li> <li><a href="/wiki/X_and_Y_bosons" title="X and Y bosons">X and Y bosons</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align: center;;width:1%"><a href="/wiki/Bound_state" title="Bound state">Composite</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Hadrons" scope="row" class="navbox-group" style="width:1%;text-align: center;"><a href="/wiki/Hadron" title="Hadron">Hadrons</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal; text-align: center;"><a href="/wiki/Baryon" title="Baryon">Baryons</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nucleon" title="Nucleon">Nucleon</a> <ul><li><a href="/wiki/Proton" title="Proton">Proton</a></li> <li><a href="/wiki/Antiproton" title="Antiproton">Antiproton</a></li> <li><a href="/wiki/Neutron" title="Neutron">Neutron</a></li> <li><a href="/wiki/Antineutron" title="Antineutron">Antineutron</a></li></ul></li> <li><a href="/wiki/Delta_baryon" title="Delta baryon">Delta baryon</a></li> <li><a href="/wiki/Lambda_baryon" title="Lambda baryon">Lambda baryon</a></li> <li><a href="/wiki/Sigma_baryon" title="Sigma baryon">Sigma baryon</a></li> <li><a href="/wiki/Xi_baryon" title="Xi baryon">Xi baryon</a></li> <li><a href="/wiki/Omega_baryon" title="Omega baryon">Omega baryon</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal; text-align: center;"><a href="/wiki/Meson" title="Meson">Mesons</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pion" title="Pion">Pion</a></li> <li><a href="/wiki/Rho_meson" title="Rho meson">Rho meson</a></li> <li><a href="/wiki/Eta_meson" class="mw-redirect" title="Eta meson">Eta and eta prime mesons</a></li> <li><a href="/wiki/Bottom_eta_meson" title="Bottom eta meson">Bottom eta meson</a></li> <li><a href="/wiki/Phi_meson" title="Phi meson">Phi meson</a></li> <li><a href="/wiki/J/psi_meson" title="J/psi meson">J/psi meson</a></li> <li><a href="/wiki/Omega_meson" title="Omega meson">Omega meson</a></li> <li><a href="/wiki/Upsilon_meson" title="Upsilon meson">Upsilon meson</a></li> <li><a href="/wiki/Kaon" title="Kaon">Kaon</a></li> <li><a href="/wiki/B_meson" title="B meson">B meson</a></li> <li><a href="/wiki/D_meson" title="D meson">D meson</a></li> <li><a href="/wiki/Quarkonium" title="Quarkonium">Quarkonium</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal; text-align: center;"><a href="/wiki/Exotic_hadron" title="Exotic hadron">Exotic hadrons</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Tetraquark" title="Tetraquark">Tetraquark</a> (<a href="/wiki/Double-charm_tetraquark" title="Double-charm tetraquark">Double-charm tetraquark</a>)</li> <li><a href="/wiki/Pentaquark" title="Pentaquark">Pentaquark</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;;font-weight:normal; text-align: center;">Others</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Atomic_nucleus" title="Atomic nucleus">Atomic nuclei</a></li> <li><a href="/wiki/Atom" title="Atom">Atoms</a></li> <li><a href="/wiki/Exotic_atom" title="Exotic atom">Exotic atoms</a> <ul><li><a href="/wiki/Positronium" title="Positronium">Positronium</a></li> <li><a href="/wiki/Muonium" title="Muonium">Muonium</a></li> <li><a href="/wiki/Tauonium" class="mw-redirect" title="Tauonium">Tauonium</a></li> <li><a href="/wiki/Onium" title="Onium">Onia</a></li> <li><a href="/wiki/Pionium" title="Pionium">Pionium</a></li> <li><a href="/wiki/Protonium" title="Protonium">Protonium</a></li></ul></li> <li><a href="/wiki/Superatom" title="Superatom">Superatoms</a></li> <li><a href="/wiki/Molecule" title="Molecule">Molecules</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;"><a href="/wiki/Category:Hypothetical_composite_particles" title="Category:Hypothetical composite particles">Hypothetical</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Baryons</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hexaquark" title="Hexaquark">Hexaquark</a></li> <li><a href="/wiki/Heptaquark" title="Heptaquark">Heptaquark</a></li> <li><a href="/wiki/Skyrmion" title="Skyrmion">Skyrmion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Mesons</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Glueball" title="Glueball">Glueball</a></li> <li><a href="/wiki/Theta_meson" title="Theta meson">Theta meson</a></li> <li><a href="/wiki/T_meson" title="T meson">T meson</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal; text-align: center;;font-weight:normal; text-align: center;">Others</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mesonic_molecule" title="Mesonic molecule">Mesonic molecule</a></li> <li><a href="/wiki/Pomeron" title="Pomeron">Pomeron</a></li> <li><a href="/wiki/Diquark" title="Diquark">Diquark</a></li> <li><a href="/wiki/R-hadron" title="R-hadron">R-hadron</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align: center;;width:1%"><a href="/wiki/Quasiparticle" title="Quasiparticle">Quasiparticles</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Anyon" title="Anyon">Anyon</a></li> <li><a href="/wiki/Davydov_soliton" title="Davydov soliton">Davydov soliton</a></li> <li><a href="/wiki/Dropleton" title="Dropleton">Dropleton</a></li> <li><a href="/wiki/Exciton" title="Exciton">Exciton</a></li> <li><a href="/wiki/Fracton_(subdimensional_particle)" title="Fracton (subdimensional particle)">Fracton</a></li> <li><a href="/wiki/Electron_hole" title="Electron hole">Hole</a></li> <li><a href="/wiki/Magnon" title="Magnon">Magnon</a></li> <li><a class="mw-selflink selflink">Phonon</a></li> <li><a href="/wiki/Plasmaron" title="Plasmaron">Plasmaron</a></li> <li><a href="/wiki/Plasmon" title="Plasmon">Plasmon</a></li> <li><a href="/wiki/Polariton" title="Polariton">Polariton</a></li> <li><a href="/wiki/Polaron" title="Polaron">Polaron</a></li> <li><a href="/wiki/Roton" title="Roton">Roton</a></li> <li><a href="/wiki/Trion_(physics)" title="Trion (physics)">Trion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align: center;;width:1%">Lists</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_baryons" title="List of baryons">Baryons</a></li> <li><a href="/wiki/List_of_mesons" title="List of mesons">Mesons</a></li> <li><a href="/wiki/List_of_particles" title="List of particles">Particles</a></li> <li><a href="/wiki/List_of_quasiparticles" title="List of quasiparticles">Quasiparticles</a></li> <li><a href="/wiki/Timeline_of_particle_discoveries" title="Timeline of particle discoveries">Timeline of particle discoveries</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align: center;;width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_subatomic_physics" title="History of subatomic physics">History of subatomic physics</a> <ul><li><a href="/wiki/Timeline_of_atomic_and_subatomic_physics" title="Timeline of atomic and subatomic physics">timeline</a></li></ul></li> <li><a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> <ul><li><a href="/wiki/Mathematical_formulation_of_the_Standard_Model" title="Mathematical formulation of the Standard Model">mathematical formulation</a></li></ul></li> <li><a href="/wiki/Subatomic_particle" title="Subatomic particle">Subatomic particles</a></li> <li><a href="/wiki/Particle" title="Particle">Particles</a></li> <li><a href="/wiki/Antiparticle" title="Antiparticle">Antiparticles</a></li> <li><a href="/wiki/Nuclear_physics" title="Nuclear physics">Nuclear physics</a></li> <li><a href="/wiki/Eightfold_way_(physics)" title="Eightfold way (physics)">Eightfold way</a> <ul><li><a href="/wiki/Quark_model" title="Quark model">Quark model</a></li></ul></li> <li><a href="/wiki/Exotic_matter" title="Exotic matter">Exotic matter</a></li> <li><a href="/wiki/Massless_particle" title="Massless particle">Massless particle</a></li> <li><a href="/wiki/Relativistic_particle" title="Relativistic particle">Relativistic particle</a></li> <li><a href="/wiki/Virtual_particle" title="Virtual particle">Virtual particle</a></li> <li><a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">Wave–particle duality</a></li> <li><a href="/wiki/Particle_chauvinism" title="Particle chauvinism">Particle chauvinism</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2" style="text-align: center;"><div><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Symbol_portal_class.svg" class="mw-file-description" title="Portal"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/16px-Symbol_portal_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/23px-Symbol_portal_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/31px-Symbol_portal_class.svg.png 2x" data-file-width="180" data-file-height="185" /></a></span> <b><a href="/wiki/Portal:Physics" title="Portal:Physics">Physics portal</a></b></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q186608#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4129373-3">Germany</a></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Phonons"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85101066">United States</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00568885">Japan</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007543518005171">Israel</a></span></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Phonon&oldid=1250846020">https://en.wikipedia.org/w/index.php?title=Phonon&oldid=1250846020</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Quasiparticles" title="Category:Quasiparticles">Quasiparticles</a></li><li><a href="/wiki/Category:Bosons" title="Category:Bosons">Bosons</a></li><li><a href="/wiki/Category:1932_introductions" title="Category:1932 introductions">1932 introductions</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_maint:_date_and_year" title="Category:CS1 maint: date and year">CS1 maint: date and year</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Articles_containing_Ancient_Greek_(to_1453)-language_text" title="Category:Articles containing Ancient Greek (to 1453)-language text">Articles containing Ancient Greek (to 1453)-language text</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_August_2020" title="Category:Articles with unsourced statements from August 2020">Articles with unsourced statements from August 2020</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_February_2019" title="Category:Articles with unsourced statements from February 2019">Articles with unsourced statements from February 2019</a></li><li><a href="/wiki/Category:Commons_category_link_is_on_Wikidata" title="Category:Commons category link is on Wikidata">Commons category link is on Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 12 October 2024, at 21:30<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Phonon&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-f69cdc8f6-7g25x","wgBackendResponseTime":157,"wgPageParseReport":{"limitreport":{"cputime":"0.806","walltime":"1.090","ppvisitednodes":{"value":4136,"limit":1000000},"postexpandincludesize":{"value":140173,"limit":2097152},"templateargumentsize":{"value":3414,"limit":2097152},"expansiondepth":{"value":13,"limit":100},"expensivefunctioncount":{"value":9,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":124866,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 767.094 1 -total"," 27.86% 213.683 1 Template:Reflist"," 18.61% 142.781 10 Template:Navbox"," 17.38% 133.343 16 Template:Cite_book"," 15.95% 122.368 1 Template:Particles"," 12.75% 97.839 1 Template:Lang"," 10.34% 79.341 1 Template:Short_description"," 7.59% 58.193 2 Template:Pagetype"," 6.16% 47.241 1 Template:Authority_control"," 5.84% 44.827 7 Template:Cite_journal"]},"scribunto":{"limitreport-timeusage":{"value":"0.469","limit":"10.000"},"limitreport-memusage":{"value":14439631,"limit":52428800}},"cachereport":{"origin":"mw-web.codfw.main-f69cdc8f6-x6txf","timestamp":"20241122140659","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Phonon","url":"https:\/\/en.wikipedia.org\/wiki\/Phonon","sameAs":"http:\/\/www.wikidata.org\/entity\/Q186608","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q186608","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2002-09-15T10:18:34Z","dateModified":"2024-10-12T21:30:09Z","headline":"quasiparticle of mechanical vibrations"}</script> </body> </html>

CINXE.COM

Phonon - Wikipedia