Piezoelectricity - Wikipedia

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id="toc-Discovery_and_early_research" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Discovery_and_early_research"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Discovery and early research</span> </div> </a> <ul id="toc-Discovery_and_early_research-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-World_War_I_and_inter-war_years" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#World_War_I_and_inter-war_years"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>World War I and inter-war years</span> </div> </a> <ul id="toc-World_War_I_and_inter-war_years-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-World_War_II_and_post-war" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#World_War_II_and_post-war"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>World War II and post-war</span> </div> </a> <ul id="toc-World_War_II_and_post-war-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mechanism" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Mechanism"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Mechanism</span> </div> </a> <button aria-controls="toc-Mechanism-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 Mechanism subsection</span> </button> <ul id="toc-Mechanism-sublist" class="vector-toc-list"> <li id="toc-Mathematical_description" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mathematical_description"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Mathematical description</span> </div> </a> <ul id="toc-Mathematical_description-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Crystal_classes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Crystal_classes"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Crystal classes</span> </div> </a> <ul id="toc-Crystal_classes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Materials" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Materials"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Materials</span> </div> </a> <button aria-controls="toc-Materials-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 Materials subsection</span> </button> <ul id="toc-Materials-sublist" class="vector-toc-list"> <li id="toc-Crystalline_materials" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Crystalline_materials"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Crystalline materials</span> </div> </a> <ul id="toc-Crystalline_materials-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ceramics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ceramics"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Ceramics</span> </div> </a> <ul id="toc-Ceramics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lead-free_piezoceramics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lead-free_piezoceramics"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Lead-free piezoceramics</span> </div> </a> <ul id="toc-Lead-free_piezoceramics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-III–V_and_II–VI_semiconductors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#III–V_and_II–VI_semiconductors"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>III–V and II–VI semiconductors</span> </div> </a> <ul id="toc-III–V_and_II–VI_semiconductors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polymers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polymers"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Polymers</span> </div> </a> <ul id="toc-Polymers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_materials" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_materials"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>Other materials</span> </div> </a> <ul id="toc-Other_materials-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Application" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Application"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Application</span> </div> </a> <button aria-controls="toc-Application-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 Application subsection</span> </button> <ul id="toc-Application-sublist" class="vector-toc-list"> <li id="toc-High_voltage_and_power_sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#High_voltage_and_power_sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>High voltage and power sources</span> </div> </a> <ul id="toc-High_voltage_and_power_sources-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sensors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sensors"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Sensors</span> </div> </a> <ul id="toc-Sensors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Actuators" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Actuators"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Actuators</span> </div> </a> <ul id="toc-Actuators-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Frequency_standard" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Frequency_standard"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Frequency standard</span> </div> </a> <ul id="toc-Frequency_standard-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Piezoelectric_motors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Piezoelectric_motors"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Piezoelectric motors</span> </div> </a> <ul id="toc-Piezoelectric_motors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reduction_of_vibrations_and_noise" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Reduction_of_vibrations_and_noise"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.6</span> <span>Reduction of vibrations and noise</span> </div> </a> <ul id="toc-Reduction_of_vibrations_and_noise-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Infertility_treatment" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Infertility_treatment"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.7</span> <span>Infertility treatment</span> </div> </a> <ul id="toc-Infertility_treatment-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Surgery" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Surgery"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.8</span> <span>Surgery</span> </div> </a> <ul id="toc-Surgery-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Potential_applications" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Potential_applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.9</span> <span>Potential applications</span> </div> </a> <ul id="toc-Potential_applications-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</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"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</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">Piezoelectricity</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" 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Available in 56 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-56" 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">56 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Pi%C3%ABso-elektrisiteit" title="Piëso-elektrisiteit – Afrikaans" lang="af" hreflang="af" data-title="Piëso-elektrisiteit" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%83%D9%87%D8%B1%D8%A8%D8%A7%D8%A1_%D8%A7%D9%86%D8%B6%D8%BA%D8%A7%D8%B7%D9%8A%D8%A9" title="كهرباء انضغاطية – Arabic" lang="ar" hreflang="ar" data-title="كهرباء انضغاطية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%E2%80%99%D0%B5%D0%B7%D0%B0%D1%8D%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D1%8B%D1%87%D0%BD%D1%8B_%D1%8D%D1%84%D0%B5%D0%BA%D1%82" title="П’езаэлектрычны эфект – Belarusian" lang="be" hreflang="be" data-title="П’езаэлектрычны эфект" 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%9F%D0%B8%D0%B5%D0%B7%D0%BE%D0%B5%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%B8%D1%87%D0%B5%D0%BD_%D0%B5%D1%84%D0%B5%D0%BA%D1%82" title="Пиезоелектричен ефект – Bulgarian" lang="bg" hreflang="bg" data-title="Пиезоелектричен ефект" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Piezoelektricitet" title="Piezoelektricitet – Bosnian" lang="bs" hreflang="bs" data-title="Piezoelektricitet" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Piezoelectricitat" title="Piezoelectricitat – Catalan" lang="ca" hreflang="ca" data-title="Piezoelectricitat" 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/Piezoelektrick%C3%BD_jev" title="Piezoelektrický jev – Czech" lang="cs" hreflang="cs" data-title="Piezoelektrický jev" 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/Piezoelektrisk_effekt" title="Piezoelektrisk effekt – Danish" lang="da" hreflang="da" data-title="Piezoelektrisk effekt" 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/Piezoelektrizit%C3%A4t" title="Piezoelektrizität – German" lang="de" hreflang="de" data-title="Piezoelektrizität" 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/Piesoelekter" title="Piesoelekter – Estonian" lang="et" hreflang="et" data-title="Piesoelekter" 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%A0%CE%B9%CE%B5%CE%B6%CE%BF%CE%B7%CE%BB%CE%B5%CE%BA%CF%84%CF%81%CE%B9%CF%83%CE%BC%CF%8C%CF%82" title="Πιεζοηλεκτρισμός – Greek" lang="el" hreflang="el" data-title="Πιεζοηλεκτρισμός" 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/Piezoelectricidad" title="Piezoelectricidad – Spanish" lang="es" hreflang="es" data-title="Piezoelectricidad" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Piezoelektro" title="Piezoelektro – Esperanto" lang="eo" hreflang="eo" data-title="Piezoelektro" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Piezoelektrizitate" title="Piezoelektrizitate – Basque" lang="eu" hreflang="eu" data-title="Piezoelektrizitate" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%BE%DB%8C%D8%B2%D9%88%D8%A7%D9%84%DA%A9%D8%AA%D8%B1%DB%8C%DA%A9%DB%8C" title="پیزوالکتریکی – Persian" lang="fa" hreflang="fa" data-title="پیزوالکتریکی" 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/Pi%C3%A9zo%C3%A9lectricit%C3%A9" title="Piézoélectricité – French" lang="fr" hreflang="fr" data-title="Piézoélectricité" 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/Iarmhairt_ph%C3%ADsileictreach" title="Iarmhairt phísileictreach – Irish" lang="ga" hreflang="ga" data-title="Iarmhairt phísileictreach" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Piezoelectricidade" title="Piezoelectricidade – Galician" lang="gl" hreflang="gl" data-title="Piezoelectricidade" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%95%95%EC%A0%84%EA%B8%B0" title="압전기 – Korean" lang="ko" hreflang="ko" data-title="압전기" 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%8A%D5%AB%D5%A5%D5%A6%D5%B8%D5%A7%D5%AC%D5%A5%D5%AF%D5%BF%D6%80%D5%A1%D5%AF%D5%A1%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Պիեզոէլեկտրականություն – Armenian" lang="hy" hreflang="hy" data-title="Պիեզոէլեկտրականություն" 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%A6%E0%A4%BE%E0%A4%AC%E0%A4%B5%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%AF%E0%A5%81%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A5%80" title="दाबविद्युतिकी – Hindi" lang="hi" hreflang="hi" data-title="दाबविद्युतिकी" 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/Piezoelektri%C4%8Dni_efekt" title="Piezoelektrični efekt – Croatian" lang="hr" hreflang="hr" data-title="Piezoelektrični efekt" 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/Piezoelektrik" title="Piezoelektrik – Indonesian" lang="id" hreflang="id" data-title="Piezoelektrik" 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/Piezoelettricit%C3%A0" title="Piezoelettricità – Italian" lang="it" hreflang="it" data-title="Piezoelettricità" 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%99%D7%90%D7%96%D7%95%D7%90%D7%9C%D7%A7%D7%98%D7%A8%D7%99%D7%95%D7%AA" title="פיאזואלקטריות – Hebrew" lang="he" hreflang="he" data-title="פיאזואלקטריות" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AA%E0%B3%80%E0%B2%9C%E0%B3%8A_%E0%B2%B5%E0%B2%BF%E0%B2%A6%E0%B3%8D%E0%B2%AF%E0%B3%81%E0%B2%A4%E0%B3%8D%E0%B2%A4%E0%B3%81" title="ಪೀಜೊ ವಿದ್ಯುತ್ತು – Kannada" lang="kn" hreflang="kn" data-title="ಪೀಜೊ ವಿದ್ಯುತ್ತು" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9F%D1%8C%D0%B5%D0%B7%D0%BE%D1%8D%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%BB%D1%96%D0%BA" title="Пьезоэлектрлік – Kazakh" lang="kk" hreflang="kk" data-title="Пьезоэлектрлік" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Pyezoelektrisite" title="Pyezoelektrisite – Haitian Creole" lang="ht" hreflang="ht" data-title="Pyezoelektrisite" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Pjezoelektriskais_efekts" title="Pjezoelektriskais efekts – Latvian" lang="lv" hreflang="lv" data-title="Pjezoelektriskais efekts" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Piezoelektromoss%C3%A1g" title="Piezoelektromosság – Hungarian" lang="hu" hreflang="hu" data-title="Piezoelektromosság" 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%9F%D0%B8%D0%B5%D0%B7%D0%BE%D0%B5%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%B8%D1%87%D0%B5%D0%BD_%D0%B5%D1%84%D0%B5%D0%BA%D1%82" title="Пиезоелектричен ефект – Macedonian" lang="mk" hreflang="mk" data-title="Пиезоелектричен ефект" 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/Piezoelektrik" title="Piezoelektrik – Malay" lang="ms" hreflang="ms" data-title="Piezoelektrik" 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/Pi%C3%ABzo-elektrisch_effect" title="Piëzo-elektrisch effect – Dutch" lang="nl" hreflang="nl" data-title="Piëzo-elektrisch effect" 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/%E5%9C%A7%E9%9B%BB%E5%8A%B9%E6%9E%9C" title="圧電効果 – Japanese" lang="ja" hreflang="ja" data-title="圧電効果" 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/Piezoelektrisitet" title="Piezoelektrisitet – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Piezoelektrisitet" 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/Piezoelektrisitet" title="Piezoelektrisitet – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Piezoelektrisitet" 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-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Piezoelectricitat" title="Piezoelectricitat – Occitan" lang="oc" hreflang="oc" data-title="Piezoelectricitat" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Piezoelektrik_effekt" title="Piezoelektrik effekt – Uzbek" lang="uz" hreflang="uz" data-title="Piezoelektrik effekt" 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/Piezoelektryk" title="Piezoelektryk – Polish" lang="pl" hreflang="pl" data-title="Piezoelektryk" 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/Piezoeletricidade" title="Piezoeletricidade – Portuguese" lang="pt" hreflang="pt" data-title="Piezoeletricidade" 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/Piezoelectricitate" title="Piezoelectricitate – Romanian" lang="ro" hreflang="ro" data-title="Piezoelectricitate" 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%9F%D1%8C%D0%B5%D0%B7%D0%BE%D1%8D%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%B8%D1%87%D0%B5%D1%81%D1%82%D0%B2%D0%BE" title="Пьезоэлектричество – Russian" lang="ru" hreflang="ru" data-title="Пьезоэлектричество" 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/Piezoelectricity" title="Piezoelectricity – Simple English" lang="en-simple" hreflang="en-simple" data-title="Piezoelectricity" 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/Piezoelektrick%C3%BD_jav" title="Piezoelektrický jav – Slovak" lang="sk" hreflang="sk" data-title="Piezoelektrický jav" 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/Piezoelektri%C4%8Dnost" title="Piezoelektričnost – Slovenian" lang="sl" hreflang="sl" data-title="Piezoelektričnost" 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%85%D8%A7%D8%AF%D8%AF%DB%95%DB%8C_%D9%BE%DB%8C%DB%8E%D8%B2%DB%86_%DA%A9%D8%A7%D8%B1%DB%95%D8%A8%D8%A7%DB%8C%DB%8C" title="ماددەی پیێزۆ کارەبایی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ماددەی پیێزۆ کارەبایی" 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%9F%D0%B8%D1%98%D0%B5%D0%B7%D0%BE%D0%B5%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%B8%D1%87%D0%BD%D0%B8_%D0%B5%D1%84%D0%B5%D0%BA%D1%82" title="Пијезоелектрични ефект – Serbian" lang="sr" hreflang="sr" data-title="Пијезоелектрични ефект" 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/Piezoelektri%C4%8Dnost" title="Piezoelektričnost – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Piezoelektričnost" 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/Pietsos%C3%A4hk%C3%B6inen_ilmi%C3%B6" title="Pietsosähköinen ilmiö – Finnish" lang="fi" hreflang="fi" data-title="Pietsosähköinen ilmiö" 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/Piezoelektricitet" title="Piezoelektricitet – Swedish" lang="sv" hreflang="sv" data-title="Piezoelektricitet" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%85%E0%AE%B4%E0%AF%81%E0%AE%A4%E0%AF%8D%E0%AE%A4%E0%AE%AE%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%B5%E0%AE%BF%E0%AE%B3%E0%AF%88%E0%AE%B5%E0%AF%81" title="அழுத்தமின் விளைவு – Tamil" lang="ta" hreflang="ta" data-title="அழுத்தமின் விளைவு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Piezoelektrik" title="Piezoelektrik – Turkish" lang="tr" hreflang="tr" data-title="Piezoelektrik" 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%9F%27%D1%94%D0%B7%D0%BE%D0%B5%D1%84%D0%B5%D0%BA%D1%82" title="П'єзоефект – Ukrainian" lang="uk" hreflang="uk" data-title="П'єзоефект" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C3%81p_%C4%91i%E1%BB%87n" title="Áp điện – Vietnamese" lang="vi" hreflang="vi" data-title="Áp điện" 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dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Electric charge generated in certain solids due to mechanical stress</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Piezoelectric_balance_presented_by_Pierre_Curie_to_Lord_Kelvin,_Hunterian_Museum,_Glasgow.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Piezoelectric_balance_presented_by_Pierre_Curie_to_Lord_Kelvin%2C_Hunterian_Museum%2C_Glasgow.jpg/240px-Piezoelectric_balance_presented_by_Pierre_Curie_to_Lord_Kelvin%2C_Hunterian_Museum%2C_Glasgow.jpg" decoding="async" width="240" height="360" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Piezoelectric_balance_presented_by_Pierre_Curie_to_Lord_Kelvin%2C_Hunterian_Museum%2C_Glasgow.jpg/360px-Piezoelectric_balance_presented_by_Pierre_Curie_to_Lord_Kelvin%2C_Hunterian_Museum%2C_Glasgow.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Piezoelectric_balance_presented_by_Pierre_Curie_to_Lord_Kelvin%2C_Hunterian_Museum%2C_Glasgow.jpg/480px-Piezoelectric_balance_presented_by_Pierre_Curie_to_Lord_Kelvin%2C_Hunterian_Museum%2C_Glasgow.jpg 2x" data-file-width="2056" data-file-height="3088" /></a><figcaption>Piezoelectric balance presented by <a href="/wiki/Pierre_Curie" title="Pierre Curie">Pierre Curie</a> to <a href="/wiki/Lord_Kelvin" title="Lord Kelvin">Lord Kelvin</a>, <a href="/wiki/Hunterian_Museum_and_Art_Gallery" title="Hunterian Museum and Art Gallery">Hunterian Museum</a>, <a href="/wiki/Glasgow" title="Glasgow">Glasgow</a></figcaption></figure> <p><b>Piezoelectricity</b> (<span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="/ˌ/: secondary stress follows">ˌ</span><span title="'p' in 'pie'">p</span><span title="/iː/: 'ee' in 'fleece'">iː</span><span title="'z' in 'zoom'">z</span><span title="/oʊ/: 'o' in 'code'">oʊ</span></span>-,<span class="wrap"> </span><span style="border-bottom:1px dotted"><span title="/ˌ/: secondary stress follows">ˌ</span><span title="'p' in 'pie'">p</span><span title="/iː/: 'ee' in 'fleece'">iː</span><span title="'t' in 'tie'">t</span><span title="'s' in 'sigh'">s</span><span title="/oʊ/: 'o' in 'code'">oʊ</span></span>-,<span class="wrap"> </span><span style="border-bottom:1px dotted"><span title="'p' in 'pie'">p</span><span title="/aɪ/: 'i' in 'tide'">aɪ</span><span title="/ˌ/: secondary stress follows">ˌ</span><span title="/iː/: 'ee' in 'fleece'">iː</span><span title="'z' in 'zoom'">z</span><span title="/oʊ/: 'o' in 'code'">oʊ</span></span>-/</a></span></span>, <span class="rt-commentedText nowrap"><style data-mw-deduplicate="TemplateStyles:r1177148991">.mw-parser-output .IPA-label-small{font-size:85%}.mw-parser-output .references .IPA-label-small,.mw-parser-output .infobox .IPA-label-small,.mw-parser-output .navbox .IPA-label-small{font-size:100%}</style><span class="IPA-label IPA-label-small"><a href="/wiki/American_English" title="American English">US</a>: </span><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="'p' in 'pie'">p</span><span title="/i/: 'y' in 'happy'">i</span><span title="/ˌ/: secondary stress follows">ˌ</span><span title="/eɪ/: 'a' in 'face'">eɪ</span><span title="'z' in 'zoom'">z</span><span title="/oʊ/: 'o' in 'code'">oʊ</span></span>-,<span class="wrap"> </span><span style="border-bottom:1px dotted"><span title="'p' in 'pie'">p</span><span title="/i/: 'y' in 'happy'">i</span><span title="/ˌ/: secondary stress follows">ˌ</span><span title="/eɪ/: 'a' in 'face'">eɪ</span><span title="'t' in 'tie'">t</span><span title="'s' in 'sigh'">s</span><span title="/oʊ/: 'o' in 'code'">oʊ</span></span>-/</a></span></span>)<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> is the <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a> that accumulates in certain <a href="/wiki/Solid" title="Solid">solid</a> materials—such as <a href="/wiki/Crystal" title="Crystal">crystals</a>, certain <a href="/wiki/Ceramic" title="Ceramic">ceramics</a>, and biological matter such as <a href="/wiki/Bone" title="Bone">bone</a>, <a href="/wiki/DNA" title="DNA">DNA</a>, and various <a href="/wiki/Protein" title="Protein">proteins</a>—in response to applied <a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">mechanical stress</a>.<sup id="cite_ref-InstrumentAnalysis_2-0" class="reference"><a href="#cite_note-InstrumentAnalysis-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The word <i>piezoelectricity</i> means <a href="/wiki/Electricity" title="Electricity">electricity</a> resulting from <a href="/wiki/Pressure" title="Pressure">pressure</a> and <a href="/wiki/Latent_heat" title="Latent heat">latent heat</a>. It is derived from <a href="/wiki/Ancient_Greek_language" class="mw-redirect" title="Ancient Greek language">Ancient Greek</a> <i> </i><span lang="grc"><a href="https://en.wiktionary.org/wiki/%CF%80%CE%B9%CE%AD%CE%B6%CF%89#Ancient_Greek" class="extiw" title="wikt:πιέζω">πιέζω</a></span><i> (<span title="Ancient Greek transliteration" lang="grc-Latn"><i>piézō</i></span>)</i> 'to squeeze or press' and <i> </i><span lang="grc"><a href="https://en.wiktionary.org/wiki/%E1%BC%A4%CE%BB%CE%B5%CE%BA%CF%84%CF%81%CE%BF%CE%BD#Ancient_Greek" class="extiw" title="wikt:ἤλεκτρον">ἤλεκτρον</a></span><i> (<span title="Ancient Greek transliteration" lang="grc-Latn"><i>ḗlektron</i></span>)</i> '<a href="/wiki/Amber" title="Amber">amber</a>' (an ancient source of static electricity).<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><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> The German form of the word (<i>Piezoelektricität</i>) was coined in 1881 by the German physicist <a href="/wiki/Wilhelm_Gottlieb_Hankel" title="Wilhelm Gottlieb Hankel">Wilhelm Gottlieb Hankel</a>; the English word was coined in 1883.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>The piezoelectric effect results from the linear <a href="/wiki/Electromechanical" class="mw-redirect" title="Electromechanical">electromechanical</a> interaction between the mechanical and electrical states in crystalline materials with no <a href="/wiki/Centrosymmetry" title="Centrosymmetry">inversion symmetry</a>.<sup id="cite_ref-Piezoelectric_Sensorics_7-0" class="reference"><a href="#cite_note-Piezoelectric_Sensorics-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> The piezoelectric effect is a <a href="/wiki/Reversible_process_(thermodynamics)" title="Reversible process (thermodynamics)">reversible process</a>: <a href="/wiki/List_of_piezoelectric_materials" title="List of piezoelectric materials">materials exhibiting the piezoelectric effect</a> also exhibit the reverse piezoelectric effect, the internal generation of a mechanical strain resulting from an applied <a href="/wiki/Electric_field" title="Electric field">electric field</a>. For example, <a href="/wiki/Lead_zirconate_titanate" title="Lead zirconate titanate">lead zirconate titanate</a> crystals will generate measurable piezoelectricity when their static structure is <a href="/wiki/Deformation_(physics)" title="Deformation (physics)">deformed</a> by about 0.1% of the original dimension. Conversely, those same crystals will change about 0.1% of their static dimension when an external electric field is applied. The inverse piezoelectric effect is used in the production of <a href="/wiki/Ultrasound" title="Ultrasound">ultrasound waves</a>.<sup id="cite_ref-UT_of_Materials_8-0" class="reference"><a href="#cite_note-UT_of_Materials-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>French physicists <a href="/wiki/Jacques_Curie" class="mw-redirect" title="Jacques Curie">Jacques</a> and <a href="/wiki/Pierre_Curie" title="Pierre Curie">Pierre Curie</a> discovered piezoelectricity in 1880.<sup id="cite_ref-Manbachi,_A._and_Cobbold_R.S.C._2011_187–96_9-0" class="reference"><a href="#cite_note-Manbachi,_A._and_Cobbold_R.S.C._2011_187–96-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The piezoelectric effect has been exploited in many useful applications, including the production and detection of sound, piezoelectric <a href="/wiki/Inkjet_printing" title="Inkjet printing">inkjet printing</a>, generation of high voltage electricity, as a <a href="/wiki/Clock_generator" title="Clock generator">clock generator</a> in electronic devices, in <a href="/wiki/Microbalance" title="Microbalance">microbalances</a>, to drive an <a href="/wiki/Ultrasonic_nozzle" title="Ultrasonic nozzle">ultrasonic nozzle</a>, and in ultrafine focusing of optical assemblies. It forms the basis for <a href="/wiki/Scanning_probe_microscopy" title="Scanning probe microscopy">scanning probe microscopes</a> that resolve images at the scale of <a href="/wiki/Atom" title="Atom">atoms</a>. It is used in the <a href="/wiki/Pickup_(music_technology)" title="Pickup (music technology)">pickups</a> of some <a href="/wiki/Guitar_amplifier" title="Guitar amplifier">electronically amplified guitars</a> and as <a href="/wiki/Trigger_(drums)" title="Trigger (drums)">triggers</a> in most modern <a href="/wiki/Electronic_drum" title="Electronic drum">electronic drums</a>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> The piezoelectric effect also finds everyday uses, such as generating sparks to ignite gas cooking and heating devices, torches, and <a href="/wiki/Lighters" class="mw-redirect" title="Lighters">cigarette lighters</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Piezoelectricity&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Discovery_and_early_research">Discovery and early research</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Piezoelectricity&action=edit&section=2" title="Edit section: Discovery and early research"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Pyroelectricity" title="Pyroelectricity">pyroelectric effect</a>, by which a material generates an <a href="/wiki/Electric_potential" title="Electric potential">electric potential</a> in response to a temperature change, was studied by <a href="/wiki/Carl_Linnaeus" title="Carl Linnaeus">Carl Linnaeus</a> and <a href="/wiki/Franz_Aepinus" title="Franz Aepinus">Franz Aepinus</a> in the mid-18th century. Drawing on this knowledge, both <a href="/wiki/Ren%C3%A9_Just_Ha%C3%BCy" title="René Just Haüy">René Just Haüy</a> and <a href="/wiki/Antoine_C%C3%A9sar_Becquerel" title="Antoine César Becquerel">Antoine César Becquerel</a> posited a relationship between <a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">mechanical stress</a> and electric charge; however, experiments by both proved inconclusive.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Top_view_of_Curie_piezo_electric_compensator.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Top_view_of_Curie_piezo_electric_compensator.jpg/220px-Top_view_of_Curie_piezo_electric_compensator.jpg" decoding="async" width="220" height="278" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Top_view_of_Curie_piezo_electric_compensator.jpg/330px-Top_view_of_Curie_piezo_electric_compensator.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Top_view_of_Curie_piezo_electric_compensator.jpg/440px-Top_view_of_Curie_piezo_electric_compensator.jpg 2x" data-file-width="1696" data-file-height="2144" /></a><figcaption>View of piezo crystal in the top of a Curie compensator in the Museum of Scotland.</figcaption></figure> <p>The first demonstration of the direct piezoelectric effect was in 1880 by the brothers <a href="/wiki/Pierre_Curie" title="Pierre Curie">Pierre Curie</a> and <a href="/wiki/Jacques_Curie" class="mw-redirect" title="Jacques Curie">Jacques Curie</a>.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> They combined their knowledge of pyroelectricity with their understanding of the underlying crystal structures that gave rise to pyroelectricity to predict crystal behavior, and demonstrated the effect using crystals of <a href="/wiki/Tourmaline" title="Tourmaline">tourmaline</a>, <a href="/wiki/Quartz" title="Quartz">quartz</a>, <a href="/wiki/Topaz" title="Topaz">topaz</a>, <a href="/wiki/Sugar_cane" class="mw-redirect" title="Sugar cane">cane</a> <a href="/wiki/Sugar" title="Sugar">sugar</a>, and <a href="/wiki/Rochelle_salt" class="mw-redirect" title="Rochelle salt">Rochelle salt</a> (sodium potassium tartrate tetrahydrate). Quartz and <a href="/wiki/Rochelle_salt" class="mw-redirect" title="Rochelle salt">Rochelle salt</a> exhibited the most piezoelectricity. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:SchemaPiezo.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/SchemaPiezo.gif/220px-SchemaPiezo.gif" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/c/c4/SchemaPiezo.gif 1.5x" data-file-width="256" data-file-height="256" /></a><figcaption>A piezoelectric disk generates a voltage when deformed (change in shape is greatly exaggerated).</figcaption></figure> <p>The Curies, however, did not predict the converse piezoelectric effect. The converse effect was mathematically deduced from fundamental thermodynamic principles by <a href="/wiki/Gabriel_Lippmann" title="Gabriel Lippmann">Gabriel Lippmann</a> in 1881.<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> The Curies immediately confirmed the existence of the converse effect,<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> and went on to obtain quantitative proof of the complete reversibility of electro-elasto-mechanical deformations in piezoelectric crystals. </p><p>For the next few decades, piezoelectricity remained something of a laboratory curiosity, though it was a vital tool in the discovery of polonium and radium by Pierre and <a href="/wiki/Marie_Curie" title="Marie Curie">Marie Curie</a> in 1898. More work was done to explore and define the crystal structures that exhibited piezoelectricity. This culminated in 1910 with the publication of <a href="/wiki/Woldemar_Voigt" title="Woldemar Voigt">Woldemar Voigt</a>'s <i>Lehrbuch der Kristallphysik</i> (<i>Textbook on Crystal Physics</i>),<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> which described the 20 natural crystal classes capable of piezoelectricity, and rigorously defined the piezoelectric constants using <a href="/wiki/Tensor_analysis" class="mw-redirect" title="Tensor analysis">tensor analysis</a>. </p> <div class="mw-heading mw-heading3"><h3 id="World_War_I_and_inter-war_years">World War I and inter-war years</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Piezoelectricity&action=edit&section=3" title="Edit section: World War I and inter-war years"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The first practical application for piezoelectric devices was <a href="/wiki/Sonar" title="Sonar">sonar</a>, first developed during <a href="/wiki/World_War_I" title="World War I">World War I</a>. The superior performance of piezoelectric devices, operating at ultrasonic frequencies, superseded the earlier <a href="/wiki/Fessenden_oscillator" title="Fessenden oscillator">Fessenden oscillator</a>. In <a href="/wiki/France" title="France">France</a> in 1917, <a href="/wiki/Paul_Langevin" title="Paul Langevin">Paul Langevin</a> and his coworkers developed an <a href="/wiki/Ultrasound" title="Ultrasound">ultrasonic</a> <a href="/wiki/Submarine" title="Submarine">submarine</a> detector.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> The detector consisted of a <a href="/wiki/Transducer" title="Transducer">transducer</a>, made of thin quartz crystals carefully glued between two steel plates, and a <a href="/wiki/Hydrophone" title="Hydrophone">hydrophone</a> to detect the returned <a href="/wiki/Echo_(phenomenon)" class="mw-redirect" title="Echo (phenomenon)">echo</a>. By emitting a high-frequency pulse from the transducer, and measuring the amount of time it takes to hear an echo from the sound waves bouncing off an object, one can calculate the distance to that object. </p><p>The use of piezoelectricity in sonar, and the success of that project, created intense development interest in piezoelectric devices. Over the next few decades, new piezoelectric materials and new applications for those materials were explored and developed. </p><p>Piezoelectric devices found homes in many fields. Ceramic <a href="/wiki/Phonograph" title="Phonograph">phonograph</a> cartridges simplified player design, were cheap and accurate, and made record players cheaper to maintain and easier to build. The development of the ultrasonic transducer allowed for easy measurement of viscosity and elasticity in fluids and solids, resulting in huge advances in materials research. Ultrasonic <a href="/wiki/Time-domain_reflectometer" title="Time-domain reflectometer">time-domain reflectometers</a> (which send an ultrasonic pulse through a material and measure reflections from discontinuities) could find flaws inside cast metal and stone objects, improving structural safety. </p> <div class="mw-heading mw-heading3"><h3 id="World_War_II_and_post-war">World War II and post-war</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Piezoelectricity&action=edit&section=4" title="Edit section: World War II and post-war"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>During <a href="/wiki/World_War_II" title="World War II">World War II</a>, independent research groups in the <a href="/wiki/United_States" title="United States">United States</a>, <a href="/wiki/USSR" class="mw-redirect" title="USSR">USSR</a>, and <a href="/wiki/Japan" title="Japan">Japan</a> discovered a new class of synthetic materials, called <a href="/wiki/Ferroelectricity" title="Ferroelectricity">ferroelectrics</a>, which exhibited piezoelectric constants many times higher than natural materials. This led to intense research to develop <a href="/wiki/Barium_titanate" title="Barium titanate">barium titanate</a> and later lead zirconate titanate materials with specific properties for particular applications. </p><p>One significant example of the use of piezoelectric crystals was developed by <a href="/wiki/Bell_Telephone_Laboratories" class="mw-redirect" title="Bell Telephone Laboratories">Bell Telephone Laboratories</a>. Following World War I, Frederick R. Lack, working in radio telephony in the engineering department, developed the "AT cut" crystal, a crystal that operated through a wide range of temperatures. Lack's crystal did not need the heavy accessories previous crystal used, facilitating its use on the aircraft. This development allowed Allied air forces to engage in coordinated mass attacks through the use of aviation radio. </p><p>Development of piezoelectric devices and materials in the United States was kept within the companies doing the development, mostly due to the wartime beginnings of the field, and in the interests of securing profitable patents. New materials were the first to be developed—quartz crystals were the first commercially exploited piezoelectric material, but scientists searched for higher-performance materials. Despite the advances in materials and the maturation of manufacturing processes, the United States market did not grow as quickly as Japan's did. Without many new applications, the growth of the United States' piezoelectric industry suffered. </p><p>In contrast, Japanese manufacturers shared their information, quickly overcoming technical and manufacturing challenges and creating new markets. In Japan, a temperature stable crystal cut was developed by <a href="/wiki/Issac_Koga" title="Issac Koga">Issac Koga</a>. Japanese efforts in materials research created piezoceramic materials competitive to the United States materials but free of expensive patent restrictions. Major Japanese piezoelectric developments included new designs of piezoceramic filters for radios and televisions, piezo buzzers and audio transducers that can connect directly to electronic circuits, and the <a href="/wiki/Piezo_ignition" title="Piezo ignition">piezoelectric igniter</a>, which generates sparks for small engine ignition systems and gas-grill lighters, by compressing a ceramic disc. Ultrasonic transducers that transmit sound waves through air had existed for quite some time but first saw major commercial use in early television remote controls. These transducers now are mounted on several <a href="/wiki/Automobile" class="mw-redirect" title="Automobile">car</a> models as an <a href="/wiki/Acoustic_location" title="Acoustic location">echolocation</a> device, helping the driver determine the distance from the car to any objects that may be in its path. </p> <div class="mw-heading mw-heading2"><h2 id="Mechanism">Mechanism</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Piezoelectricity&action=edit&section=5" title="Edit section: Mechanism"><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:Piezo_bending_principle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Piezo_bending_principle.svg/220px-Piezo_bending_principle.svg.png" decoding="async" width="220" height="275" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Piezo_bending_principle.svg/330px-Piezo_bending_principle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Piezo_bending_principle.svg/440px-Piezo_bending_principle.svg.png 2x" data-file-width="320" data-file-height="400" /></a><figcaption>Piezoelectric plate used to convert <a href="/wiki/Audio_signal" title="Audio signal">audio signal</a> to sound waves</figcaption></figure> <p>The nature of the piezoelectric effect is closely related to the occurrence of <a href="/wiki/Electric_dipole_moment" title="Electric dipole moment">electric dipole moments</a> in solids. The latter may either be induced for <a href="/wiki/Ions" class="mw-redirect" title="Ions">ions</a> on <a href="/wiki/Crystal_lattice" class="mw-redirect" title="Crystal lattice">crystal lattice</a> sites with asymmetric charge surroundings (as in <a href="/wiki/BaTiO3" class="mw-redirect" title="BaTiO3">BaTiO<sub>3</sub></a> and <a href="/wiki/PZT" class="mw-redirect" title="PZT">PZTs</a>) or may directly be carried by molecular groups (as in <a href="/wiki/Cane_sugar" class="mw-redirect" title="Cane sugar">cane sugar</a>). The dipole density or <a href="/wiki/Polarization_density" title="Polarization density">polarization</a> (dimensionality [C·m/m<sup>3</sup>] ) may easily be calculated for <a href="/wiki/Crystals" class="mw-redirect" title="Crystals">crystals</a> by summing up the dipole moments per volume of the crystallographic <a href="/wiki/Unit_cell" title="Unit cell">unit cell</a>.<sup id="cite_ref-ZPB1995a_18-0" class="reference"><a href="#cite_note-ZPB1995a-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> As every dipole is a vector, the dipole density <i><b>P</b></i> is a <a href="/wiki/Vector_field" title="Vector field">vector field</a>. Dipoles near each other tend to be aligned in regions called Weiss domains. The domains are usually randomly oriented, but can be aligned using the process of <i>poling</i> (not the same as <a href="/wiki/Magnet#Magnetizing_ferromagnets" title="Magnet">magnetic poling</a>), a process by which a strong electric field is applied across the material, usually at elevated temperatures. Not all piezoelectric materials can be poled.<sup id="cite_ref-PAMTA_19-0" class="reference"><a href="#cite_note-PAMTA-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>Of decisive importance for the piezoelectric effect is the change of polarization <i><b>P</b></i> when applying a <a href="/wiki/Mechanical_stress" class="mw-redirect" title="Mechanical stress">mechanical stress</a>. This might either be caused by a reconfiguration of the dipole-inducing surrounding or by re-orientation of molecular dipole moments under the influence of the external stress. Piezoelectricity may then manifest in a variation of the polarization strength, its direction or both, with the details depending on: 1. the orientation of <i><b>P</b></i> within the crystal; 2. <a href="/wiki/Crystal_symmetry" class="mw-redirect" title="Crystal symmetry">crystal symmetry</a>; and 3. the applied mechanical stress. The change in <i><b>P</b></i> appears as a variation of surface <a href="/wiki/Charge_density" title="Charge density">charge density</a> upon the crystal faces, i.e. as a variation of the <a href="/wiki/Electric_field" title="Electric field">electric field</a> extending between the faces caused by a change in dipole density in the bulk. For example, a 1 cm<sup>3</sup> cube of quartz with 2 kN (500 lbf) of correctly applied force can produce a voltage of 12500 <a href="/wiki/Volt" title="Volt">V</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><p>Piezoelectric materials also show the opposite effect, called the <b>converse piezoelectric effect</b>, where the application of an electrical field creates mechanical deformation in the crystal. </p> <div class="mw-heading mw-heading3"><h3 id="Mathematical_description">Mathematical description</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Piezoelectricity&action=edit&section=6" title="Edit section: Mathematical description"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Linear piezoelectricity is the combined effect of </p> <ul><li>The linear electrical behavior of the material:</li></ul> <dl><dd><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 \mathbf {D} ={\boldsymbol {\varepsilon }}\,\mathbf {E} \quad \implies }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mspace width="1em" /> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {D} ={\boldsymbol {\varepsilon }}\,\mathbf {E} \quad \implies }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae376be1e3f1d354b926e9c1f2e49bfacd018484" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.586ex; height:2.176ex;" alt="{\displaystyle \mathbf {D} ={\boldsymbol {\varepsilon }}\,\mathbf {E} \quad \implies }"></span> <a href="/wiki/Summation" title="Summation"><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 \quad D_{i}=\sum _{j}\varepsilon _{ij}\,E_{j}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad D_{i}=\sum _{j}\varepsilon _{ij}\,E_{j}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7becd50a4e2117af8ac266e930c386cb8ea6c6df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:18.105ex; height:5.843ex;" alt="{\displaystyle \quad D_{i}=\sum _{j}\varepsilon _{ij}\,E_{j}\;}"></span></a></dd></dl></dd> <dd>where <b>D</b> is the electric flux density<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><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> (<a href="/wiki/Electric_displacement" class="mw-redirect" title="Electric displacement">electric displacement</a>), <b>ε</b> is the <a href="/wiki/Permittivity" title="Permittivity">permittivity</a> (free-body dielectric constant), <b>E</b> is the <a href="/wiki/Electric_field_strength" class="mw-redirect" title="Electric field strength">electric field strength</a>, and <a href="/wiki/Divergence" title="Divergence"><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 \nabla \cdot \mathbf {D} =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {D} =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9090472e24557b8c02000e3ed9e20afe3ff7f681" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.926ex; height:2.176ex;" alt="{\displaystyle \nabla \cdot \mathbf {D} =0}"></span></a> , <a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)"><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 \nabla \times \mathbf {E} =\mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \times \mathbf {E} =\mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad7e78fa2586a1ddcf90559d8317826d82a18995" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.968ex; height:2.176ex;" alt="{\displaystyle \nabla \times \mathbf {E} =\mathbf {0} }"></span></a>.</dd></dl> <ul><li><a href="/wiki/Hooke%27s_law" title="Hooke's law">Hooke's law</a> for linear elastic materials:</li></ul> <dl><dd><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 {\boldsymbol {S}}={\mathsf {s}}\,{\boldsymbol {T}}\quad \implies \quad S_{ij}=\sum _{k,\ell }s_{ijk\ell }\,T_{k\ell }\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">S</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">s</mi> </mrow> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">T</mi> </mrow> <mspace width="1em" /> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mspace width="1em" /> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>ℓ</mi> </mrow> </munder> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> <mi>ℓ</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>ℓ</mi> </mrow> </msub> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {S}}={\mathsf {s}}\,{\boldsymbol {T}}\quad \implies \quad S_{ij}=\sum _{k,\ell }s_{ijk\ell }\,T_{k\ell }\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11d25255a7c9675c0792f20a40175c77e341c689" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:36.878ex; height:5.843ex;" alt="{\displaystyle {\boldsymbol {S}}={\mathsf {s}}\,{\boldsymbol {T}}\quad \implies \quad S_{ij}=\sum _{k,\ell }s_{ijk\ell }\,T_{k\ell }\;}"></span></dd></dl></dd> <dd>where <b>S</b> is the linearized <a href="/wiki/Strain_(materials_science)" class="mw-redirect" title="Strain (materials science)">strain</a>, <b>s</b> is <a href="/wiki/Compliance_(mechanics)" class="mw-redirect" title="Compliance (mechanics)">compliance</a> under short-circuit conditions, <b>T</b> is <a href="/wiki/Stress_(physics)" class="mw-redirect" title="Stress (physics)">stress</a>, and <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 \nabla \cdot {\boldsymbol {T}}=\mathbf {0} \,\,,\,{\boldsymbol {S}}={\frac {\nabla \mathbf {u} +\mathbf {u} \nabla }{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">T</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">S</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mi mathvariant="normal">∇</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot {\boldsymbol {T}}=\mathbf {0} \,\,,\,{\boldsymbol {S}}={\frac {\nabla \mathbf {u} +\mathbf {u} \nabla }{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e2673f1d4cbe14db08061dc48ebf5fb29e99273" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.963ex; height:5.176ex;" alt="{\displaystyle \nabla \cdot {\boldsymbol {T}}=\mathbf {0} \,\,,\,{\boldsymbol {S}}={\frac {\nabla \mathbf {u} +\mathbf {u} \nabla }{2}},}"></span></dd></dl></dd> <dd>where <b>u</b> is the <i>displacement vector</i>.</dd></dl> <p>These may be combined into so-called <i>coupled equations</i>, of which the <b>strain-charge form</b> is:<sup id="cite_ref-ikeda_23-0" class="reference"><a href="#cite_note-ikeda-23"><span class="cite-bracket">[</span>23<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 {\begin{aligned}{\boldsymbol {S}}&={\mathsf {s}}\,{\boldsymbol {T}}+{\mathfrak {d}}^{t}\,\mathbf {E} \ &&\implies \quad S_{ij}=\sum _{k,\ell }s_{ijk\ell }\,T_{k\ell }+\sum _{k}d_{ijk}^{t}\,E_{k},\\[6pt]\mathbf {D} &={\mathfrak {d}}\,{\boldsymbol {T}}+{\boldsymbol {\varepsilon }}\,\mathbf {E} &&\implies \quad D_{i}=\sum _{j,k}d_{ijk}\,T_{jk}+\sum _{j}\varepsilon _{ij}\,E_{j},\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="0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">S</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">s</mi> </mrow> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">T</mi> </mrow> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">d</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mtext> </mtext> </mtd> <mtd /> <mtd> <mi></mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mspace width="1em" /> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>ℓ</mi> </mrow> </munder> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> <mi>ℓ</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>ℓ</mi> </mrow> </msub> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">d</mi> </mrow> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">T</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mspace width="1em" /> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </munder> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>k</mi> </mrow> </msub> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\boldsymbol {S}}&={\mathsf {s}}\,{\boldsymbol {T}}+{\mathfrak {d}}^{t}\,\mathbf {E} \ &&\implies \quad S_{ij}=\sum _{k,\ell }s_{ijk\ell }\,T_{k\ell }+\sum _{k}d_{ijk}^{t}\,E_{k},\\[6pt]\mathbf {D} &={\mathfrak {d}}\,{\boldsymbol {T}}+{\boldsymbol {\varepsilon }}\,\mathbf {E} &&\implies \quad D_{i}=\sum _{j,k}d_{ijk}\,T_{jk}+\sum _{j}\varepsilon _{ij}\,E_{j},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6704b020e31a346b2db7c110f71830692ccf8774" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.821ex; margin-bottom: -0.184ex; width:61.211ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}{\boldsymbol {S}}&={\mathsf {s}}\,{\boldsymbol {T}}+{\mathfrak {d}}^{t}\,\mathbf {E} \ &&\implies \quad S_{ij}=\sum _{k,\ell }s_{ijk\ell }\,T_{k\ell }+\sum _{k}d_{ijk}^{t}\,E_{k},\\[6pt]\mathbf {D} &={\mathfrak {d}}\,{\boldsymbol {T}}+{\boldsymbol {\varepsilon }}\,\mathbf {E} &&\implies \quad D_{i}=\sum _{j,k}d_{ijk}\,T_{jk}+\sum _{j}\varepsilon _{ij}\,E_{j},\end{aligned}}}"></span></dd></dl> <p>where <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 {\mathfrak {d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">d</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f205c4184ddb813ff6d4cdbdefbb7f8f04bbabb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.16ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {d}}}"></span> is the piezoelectric tensor and the superscript t stands for its transpose. Due to the symmetry of <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 {\mathfrak {d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">d</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f205c4184ddb813ff6d4cdbdefbb7f8f04bbabb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.16ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {d}}}"></span>, <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 d_{ijk}^{t}=d_{kji}=d_{kij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>j</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{ijk}^{t}=d_{kji}=d_{kij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d54489db2ef7c9376fe074fef720f3977fcc7ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:16.824ex; height:3.509ex;" alt="{\displaystyle d_{ijk}^{t}=d_{kji}=d_{kij}}"></span>. </p><p>In matrix form, </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}\{S\}&=\left[s^{E}\right]\{T\}+[d^{\mathrm {t} }]\{E\},\\[6pt]\{D\}&=[d]\{T\}+\left[\varepsilon ^{T}\right]\{E\},\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="0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo fence="false" stretchy="false">{</mo> <mi>S</mi> <mo fence="false" stretchy="false">}</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>[</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msup> <mo>]</mo> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>T</mi> <mo fence="false" stretchy="false">}</mo> <mo>+</mo> <mo stretchy="false">[</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msup> <mo stretchy="false">]</mo> <mo fence="false" stretchy="false">{</mo> <mi>E</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo fence="false" stretchy="false">{</mo> <mi>D</mi> <mo fence="false" stretchy="false">}</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">[</mo> <mi>d</mi> <mo stretchy="false">]</mo> <mo fence="false" stretchy="false">{</mo> <mi>T</mi> <mo fence="false" stretchy="false">}</mo> <mo>+</mo> <mrow> <mo>[</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>]</mo> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>E</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\{S\}&=\left[s^{E}\right]\{T\}+[d^{\mathrm {t} }]\{E\},\\[6pt]\{D\}&=[d]\{T\}+\left[\varepsilon ^{T}\right]\{E\},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b7cb5ba4d1d2a459c0419b385f752a627a25687" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:27.936ex; height:8.509ex;" alt="{\displaystyle {\begin{aligned}\{S\}&=\left[s^{E}\right]\{T\}+[d^{\mathrm {t} }]\{E\},\\[6pt]\{D\}&=[d]\{T\}+\left[\varepsilon ^{T}\right]\{E\},\end{aligned}}}"></span></dd></dl> <p>where [<i>d</i>] is the matrix for the direct piezoelectric effect and [<i>d</i><span style="padding-left:0.12em;"><sup>t</sup></span>] is the matrix for the converse piezoelectric effect. The superscript <i>E</i> indicates a zero, or constant, electric field; the superscript <i>T</i> indicates a zero, or constant, stress field; and the superscript t stands for <a href="/wiki/Transpose" title="Transpose">transposition</a> of a <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a>. </p><p>Notice that the third order tensor <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 {\mathfrak {d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">d</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f205c4184ddb813ff6d4cdbdefbb7f8f04bbabb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.16ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {d}}}"></span> maps vectors into symmetric matrices. There are no non-trivial rotation-invariant tensors that have this property, which is why there are no isotropic piezoelectric materials. </p><p>The strain-charge for a material of the <a href="/wiki/Tetragonal_crystal_system" title="Tetragonal crystal system">4mm</a> (C<sub>4v</sub>) <a href="/wiki/Crystal_system" title="Crystal system">crystal class</a> (such as a poled piezoelectric ceramic such as tetragonal PZT or BaTiO<sub>3</sub>) as well as the <a href="/wiki/Hexagonal_crystal_system#Hexagonal_crystal_system" class="mw-redirect" title="Hexagonal crystal system">6mm</a> crystal class may also be written as (ANSI IEEE 176): </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}&{\begin{bmatrix}S_{1}\\S_{2}\\S_{3}\\S_{4}\\S_{5}\\S_{6}\end{bmatrix}}={\begin{bmatrix}s_{11}^{E}&s_{12}^{E}&s_{13}^{E}&0&0&0\\s_{21}^{E}&s_{22}^{E}&s_{23}^{E}&0&0&0\\s_{31}^{E}&s_{32}^{E}&s_{33}^{E}&0&0&0\\0&0&0&s_{44}^{E}&0&0\\0&0&0&0&s_{55}^{E}&0\\0&0&0&0&0&s_{66}^{E}=2\left(s_{11}^{E}-s_{12}^{E}\right)\end{bmatrix}}{\begin{bmatrix}T_{1}\\T_{2}\\T_{3}\\T_{4}\\T_{5}\\T_{6}\end{bmatrix}}+{\begin{bmatrix}0&0&d_{31}\\0&0&d_{32}\\0&0&d_{33}\\0&d_{24}&0\\d_{15}&0&0\\0&0&0\end{bmatrix}}{\begin{bmatrix}E_{1}\\E_{2}\\E_{3}\end{bmatrix}}\\[8pt]&{\begin{bmatrix}D_{1}\\D_{2}\\D_{3}\end{bmatrix}}={\begin{bmatrix}0&0&0&0&d_{15}&0\\0&0&0&d_{24}&0&0\\d_{31}&d_{32}&d_{33}&0&0&0\end{bmatrix}}{\begin{bmatrix}T_{1}\\T_{2}\\T_{3}\\T_{4}\\T_{5}\\T_{6}\end{bmatrix}}+{\begin{bmatrix}{\varepsilon }_{11}&0&0\\0&{\varepsilon }_{22}&0\\0&0&{\varepsilon }_{33}\end{bmatrix}}{\begin{bmatrix}E_{1}\\E_{2}\\E_{3}\end{bmatrix}}\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="1.1em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>31</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>44</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>55</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>66</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msubsup> <mo>=</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>31</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> 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<mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>31</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\begin{bmatrix}S_{1}\\S_{2}\\S_{3}\\S_{4}\\S_{5}\\S_{6}\end{bmatrix}}={\begin{bmatrix}s_{11}^{E}&s_{12}^{E}&s_{13}^{E}&0&0&0\\s_{21}^{E}&s_{22}^{E}&s_{23}^{E}&0&0&0\\s_{31}^{E}&s_{32}^{E}&s_{33}^{E}&0&0&0\\0&0&0&s_{44}^{E}&0&0\\0&0&0&0&s_{55}^{E}&0\\0&0&0&0&0&s_{66}^{E}=2\left(s_{11}^{E}-s_{12}^{E}\right)\end{bmatrix}}{\begin{bmatrix}T_{1}\\T_{2}\\T_{3}\\T_{4}\\T_{5}\\T_{6}\end{bmatrix}}+{\begin{bmatrix}0&0&d_{31}\\0&0&d_{32}\\0&0&d_{33}\\0&d_{24}&0\\d_{15}&0&0\\0&0&0\end{bmatrix}}{\begin{bmatrix}E_{1}\\E_{2}\\E_{3}\end{bmatrix}}\\[8pt]&{\begin{bmatrix}D_{1}\\D_{2}\\D_{3}\end{bmatrix}}={\begin{bmatrix}0&0&0&0&d_{15}&0\\0&0&0&d_{24}&0&0\\d_{31}&d_{32}&d_{33}&0&0&0\end{bmatrix}}{\begin{bmatrix}T_{1}\\T_{2}\\T_{3}\\T_{4}\\T_{5}\\T_{6}\end{bmatrix}}+{\begin{bmatrix}{\varepsilon }_{11}&0&0\\0&{\varepsilon }_{22}&0\\0&0&{\varepsilon }_{33}\end{bmatrix}}{\begin{bmatrix}E_{1}\\E_{2}\\E_{3}\end{bmatrix}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/274067106e58b659dd52cbc526e4476eec76f231" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -20.671ex; width:92.477ex; height:42.509ex;" alt="{\displaystyle {\begin{aligned}&{\begin{bmatrix}S_{1}\\S_{2}\\S_{3}\\S_{4}\\S_{5}\\S_{6}\end{bmatrix}}={\begin{bmatrix}s_{11}^{E}&s_{12}^{E}&s_{13}^{E}&0&0&0\\s_{21}^{E}&s_{22}^{E}&s_{23}^{E}&0&0&0\\s_{31}^{E}&s_{32}^{E}&s_{33}^{E}&0&0&0\\0&0&0&s_{44}^{E}&0&0\\0&0&0&0&s_{55}^{E}&0\\0&0&0&0&0&s_{66}^{E}=2\left(s_{11}^{E}-s_{12}^{E}\right)\end{bmatrix}}{\begin{bmatrix}T_{1}\\T_{2}\\T_{3}\\T_{4}\\T_{5}\\T_{6}\end{bmatrix}}+{\begin{bmatrix}0&0&d_{31}\\0&0&d_{32}\\0&0&d_{33}\\0&d_{24}&0\\d_{15}&0&0\\0&0&0\end{bmatrix}}{\begin{bmatrix}E_{1}\\E_{2}\\E_{3}\end{bmatrix}}\\[8pt]&{\begin{bmatrix}D_{1}\\D_{2}\\D_{3}\end{bmatrix}}={\begin{bmatrix}0&0&0&0&d_{15}&0\\0&0&0&d_{24}&0&0\\d_{31}&d_{32}&d_{33}&0&0&0\end{bmatrix}}{\begin{bmatrix}T_{1}\\T_{2}\\T_{3}\\T_{4}\\T_{5}\\T_{6}\end{bmatrix}}+{\begin{bmatrix}{\varepsilon }_{11}&0&0\\0&{\varepsilon }_{22}&0\\0&0&{\varepsilon }_{33}\end{bmatrix}}{\begin{bmatrix}E_{1}\\E_{2}\\E_{3}\end{bmatrix}}\end{aligned}}}"></span></dd></dl> <p>where the first equation represents the relationship for the converse piezoelectric effect and the latter for the direct piezoelectric effect.<sup id="cite_ref-DD1998_24-0" class="reference"><a href="#cite_note-DD1998-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p><p>Although the above equations are the most used form in literature, some comments about the notation are necessary. Generally, <i>D</i> and <i>E</i> are <a href="/wiki/Vector_(geometric)" class="mw-redirect" title="Vector (geometric)">vectors</a>, that is, <a href="/wiki/Cartesian_tensor" title="Cartesian tensor">Cartesian tensors</a> of rank 1; and permittivity <i>ε</i> is a Cartesian tensor of rank 2. Strain and stress are, in principle, also rank-2 <a href="/wiki/Tensors" class="mw-redirect" title="Tensors">tensors</a>. But conventionally, because strain and stress are all symmetric tensors, the subscript of strain and stress can be relabeled in the following fashion: 11 → 1; 22 → 2; 33 → 3; 23 → 4; 13 → 5; 12 → 6. (Different conventions may be used by different authors in literature. For example, some use 12 → 4; 23 → 5; 31 → 6 instead.) That is why <i>S</i> and <i>T</i> appear to have the "vector form" of six components. Consequently, <i>s</i> appears to be a 6-by-6 matrix instead of a rank-3 tensor. Such a relabeled notation is often called <a href="/wiki/Voigt_notation" title="Voigt notation">Voigt notation</a>. Whether the shear strain components <i>S</i><sub>4</sub>, <i>S</i><sub>5</sub>, <i>S</i><sub>6</sub> are tensor components or engineering strains is another question. In the equation above, they must be engineering strains for the 6,6 coefficient of the compliance matrix to be written as shown, i.e., 2(<i>s</i><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>E</i></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">11</sub></span></span> − <i>s</i><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>E</i></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">12</sub></span></span>). Engineering shear strains are double the value of the corresponding tensor shear, such as <i>S</i><sub>6</sub> = 2<i>S</i><sub>12</sub> and so on. This also means that <i>s</i><sub>66</sub> = <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"><i>G</i><sub>12</sub></span></span>⁠</span>, where <i>G</i><sub>12</sub> is the shear modulus. </p><p>In total, there are four piezoelectric coefficients, <i>d<sub>ij</sub></i>, <i>e<sub>ij</sub></i>, <i>g<sub>ij</sub></i>, and <i>h<sub>ij</sub></i> defined as follows: </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}d_{ij}&={\phantom {+}}\left({\frac {\partial D_{i}}{\partial T_{j}}}\right)^{E}&&={\phantom {+}}\left({\frac {\partial S_{j}}{\partial E_{i}}}\right)^{T}\\[6pt]e_{ij}&={\phantom {+}}\left({\frac {\partial D_{i}}{\partial S_{j}}}\right)^{E}&&=-\left({\frac {\partial T_{j}}{\partial E_{i}}}\right)^{S}\\[6pt]g_{ij}&=-\left({\frac {\partial E_{i}}{\partial T_{j}}}\right)^{D}&&={\phantom {+}}\left({\frac {\partial S_{j}}{\partial D_{i}}}\right)^{T}\\[6pt]h_{ij}&=-\left({\frac {\partial E_{i}}{\partial S_{j}}}\right)^{D}&&=-\left({\frac {\partial T_{j}}{\partial D_{i}}}\right)^{S}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right 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<mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>+</mo> </mphantom> </mrow> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msup> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msup> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>+</mo> </mphantom> </mrow> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msup> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}d_{ij}&={\phantom {+}}\left({\frac {\partial D_{i}}{\partial T_{j}}}\right)^{E}&&={\phantom {+}}\left({\frac {\partial S_{j}}{\partial E_{i}}}\right)^{T}\\[6pt]e_{ij}&={\phantom {+}}\left({\frac {\partial D_{i}}{\partial S_{j}}}\right)^{E}&&=-\left({\frac {\partial T_{j}}{\partial E_{i}}}\right)^{S}\\[6pt]g_{ij}&=-\left({\frac {\partial E_{i}}{\partial T_{j}}}\right)^{D}&&={\phantom {+}}\left({\frac {\partial S_{j}}{\partial D_{i}}}\right)^{T}\\[6pt]h_{ij}&=-\left({\frac {\partial E_{i}}{\partial S_{j}}}\right)^{D}&&=-\left({\frac {\partial T_{j}}{\partial D_{i}}}\right)^{S}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1898273adf66239c0ffea05c8a1752752defc04d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -15.455ex; margin-bottom: -0.217ex; width:37.502ex; height:32.509ex;" alt="{\displaystyle {\begin{aligned}d_{ij}&={\phantom {+}}\left({\frac {\partial D_{i}}{\partial T_{j}}}\right)^{E}&&={\phantom {+}}\left({\frac {\partial S_{j}}{\partial E_{i}}}\right)^{T}\\[6pt]e_{ij}&={\phantom {+}}\left({\frac {\partial D_{i}}{\partial S_{j}}}\right)^{E}&&=-\left({\frac {\partial T_{j}}{\partial E_{i}}}\right)^{S}\\[6pt]g_{ij}&=-\left({\frac {\partial E_{i}}{\partial T_{j}}}\right)^{D}&&={\phantom {+}}\left({\frac {\partial S_{j}}{\partial D_{i}}}\right)^{T}\\[6pt]h_{ij}&=-\left({\frac {\partial E_{i}}{\partial S_{j}}}\right)^{D}&&=-\left({\frac {\partial T_{j}}{\partial D_{i}}}\right)^{S}\end{aligned}}}"></span></dd></dl> <p>where the first set of four terms corresponds to the direct piezoelectric effect and the second set of four terms corresponds to the converse piezoelectric effect. The equality between the direct piezoelectric tensor and the transpose of the converse piezoelectric tensor originates from the <a href="/wiki/Maxwell_relations" title="Maxwell relations">Maxwell relations</a> of thermodynamics.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> For those piezoelectric crystals for which the polarization is of the crystal-field induced type, a formalism has been worked out that allows for the calculation of piezoelectrical coefficients <i>d<sub>ij</sub></i> from electrostatic lattice constants or higher-order <a href="/wiki/Madelung_constant" title="Madelung constant">Madelung constants</a>.<sup id="cite_ref-ZPB1995a_18-1" class="reference"><a href="#cite_note-ZPB1995a-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Crystal_classes">Crystal classes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Piezoelectricity&action=edit&section=7" title="Edit section: Crystal classes"><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:Capacitor_schematic_with_dielectric.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Capacitor_schematic_with_dielectric.svg/220px-Capacitor_schematic_with_dielectric.svg.png" decoding="async" width="220" height="242" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Capacitor_schematic_with_dielectric.svg/330px-Capacitor_schematic_with_dielectric.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Capacitor_schematic_with_dielectric.svg/440px-Capacitor_schematic_with_dielectric.svg.png 2x" data-file-width="430" data-file-height="473" /></a><figcaption>Any spatially separated charge will result in an <a href="/wiki/Electric_field" title="Electric field">electric field</a>, and therefore an <a href="/wiki/Electric_potential" title="Electric potential">electric potential</a>. Shown here is a standard dielectric in a <a href="/wiki/Capacitor" title="Capacitor">capacitor</a>. In a piezoelectric device, mechanical stress, instead of an externally applied voltage, causes the charge separation in the individual atoms of the material.</figcaption></figure> <p>Of the 32 <a href="/wiki/Crystal_class" class="mw-redirect" title="Crystal class">crystal classes</a>, 21 are non-<a href="/wiki/Centrosymmetric" class="mw-redirect" title="Centrosymmetric">centrosymmetric</a> (not having a centre of symmetry), and of these, 20 exhibit direct piezoelectricity<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> (the 21st is the cubic class 432). Ten of these represent the polar crystal classes,<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> which show a spontaneous polarization without mechanical stress due to a non-vanishing electric dipole moment associated with their unit cell, and which exhibit <a href="/wiki/Pyroelectricity" title="Pyroelectricity">pyroelectricity</a>. If the dipole moment can be reversed by applying an external electric field, the material is said to be <a href="/wiki/Ferroelectric" class="mw-redirect" title="Ferroelectric">ferroelectric</a>. </p> <ul><li>The 10 polar (pyroelectric) crystal classes: 1, 2, m, mm2, 4, 4mm, 3, 3m, 6, 6mm.</li> <li>The other 10 piezoelectric crystal classes: 222, <span style="text-decoration:overline;">4</span>, 422, <span style="text-decoration:overline;">4</span>2m, 32, <span style="text-decoration:overline;">6</span>, 622, <span style="text-decoration:overline;">6</span>2m, 23, <span style="text-decoration:overline;">4</span>3m.</li></ul> <p>For polar crystals, for which <i><b>P</b></i> ≠ 0 holds without applying a mechanical load, the piezoelectric effect manifests itself by changing the magnitude or the direction of <i><b>P</b></i> or both. </p><p>For the nonpolar but piezoelectric crystals, on the other hand, a polarization <i><b>P</b></i> different from zero is only elicited by applying a mechanical load. For them the stress can be imagined to transform the material from a nonpolar crystal class (<i><b>P</b></i> = 0) to a polar one,<sup id="cite_ref-ZPB1995a_18-2" class="reference"><a href="#cite_note-ZPB1995a-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> having <i><b>P</b></i> ≠ 0. </p> <div class="mw-heading mw-heading2"><h2 id="Materials">Materials</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Piezoelectricity&action=edit&section=8" title="Edit section: Materials"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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">See also: <a href="/wiki/List_of_piezoelectric_materials" title="List of piezoelectric materials">List of piezoelectric materials</a></div> <p>Many materials exhibit piezoelectricity. </p> <div class="mw-heading mw-heading3"><h3 id="Crystalline_materials">Crystalline materials</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Piezoelectricity&action=edit&section=9" title="Edit section: Crystalline materials"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Langasite" class="mw-redirect" title="Langasite">Langasite</a> (La<sub>3</sub>Ga<sub>5</sub>SiO<sub>14</sub>) – a quartz-analogous crystal</li> <li><a href="/wiki/Gallium_orthophosphate" class="mw-redirect" title="Gallium orthophosphate">Gallium orthophosphate</a> (GaPO<sub>4</sub>) – a quartz-analogous crystal</li> <li><a href="/wiki/Lithium_niobate" title="Lithium niobate">Lithium niobate</a> (LiNbO<sub>3</sub>)</li> <li><a href="/wiki/Lithium_tantalate" title="Lithium tantalate">Lithium tantalate</a> (LiTaO<sub>3</sub>)</li> <li><a href="/wiki/Quartz" title="Quartz">Quartz</a></li> <li><a href="/wiki/Berlinite" title="Berlinite">Berlinite</a> (AlPO<sub>4</sub>) – a rare <a href="/wiki/Phosphate" title="Phosphate">phosphate</a> <a href="/wiki/Mineral" title="Mineral">mineral</a> that is structurally identical to quartz</li> <li><a href="/wiki/Potassium_sodium_tartrate" title="Potassium sodium tartrate">Rochelle salt</a></li> <li><a href="/wiki/Topaz" title="Topaz">Topaz</a> – piezoelectricity in topaz can probably be attributed to ordering of the (F,OH) in its lattice, which is otherwise centrosymmetric: orthorhombic bipyramidal (mmm). Topaz has anomalous optical properties, which are attributed to such ordering.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Tourmaline" title="Tourmaline">Tourmaline</a>-group minerals</li> <li><a href="/wiki/Lead_titanate" title="Lead titanate">Lead titanate</a> (PbTiO<sub>3</sub>) – although it occurs in nature as mineral macedonite,<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> it is synthesized for research and applications.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Ceramics">Ceramics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Piezoelectricity&action=edit&section=10" title="Edit section: Ceramics"><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:Perovskite.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Perovskite.svg/220px-Perovskite.svg.png" decoding="async" width="220" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Perovskite.svg/330px-Perovskite.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Perovskite.svg/440px-Perovskite.svg.png 2x" data-file-width="1615" data-file-height="883" /></a><figcaption>Tetragonal unit cell of lead titanate</figcaption></figure> <p>Ceramics with randomly oriented grains must be ferroelectric to exhibit piezoelectricity.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> The occurrence of <a href="/wiki/Abnormal_grain_growth" title="Abnormal grain growth">abnormal grain growth</a> (AGG) in sintered polycrystalline piezoelectric ceramics has detrimental effects on the piezoelectric performance in such systems and should be avoided, as the microstructure in piezoceramics exhibiting AGG tends to consist of few abnormally large elongated grains in a matrix of randomly oriented finer grains. Macroscopic piezoelectricity is possible in textured polycrystalline non-ferroelectric piezoelectric materials, such as AlN and ZnO. The families of ceramics with <a href="/wiki/Perovskite_(structure)" title="Perovskite (structure)">perovskite</a>, <a href="/wiki/Tungsten" title="Tungsten">tungsten</a>-<a href="/wiki/Bronze" title="Bronze">bronze</a>, and related structures exhibit piezoelectricity: </p> <ul><li><a href="/wiki/Lead_zirconate_titanate" title="Lead zirconate titanate">Lead zirconate titanate</a> (<style data-mw-deduplicate="TemplateStyles:r1123817410">.mw-parser-output .template-chem2-su{display:inline-block;font-size:80%;line-height:1;vertical-align:-0.35em}.mw-parser-output .template-chem2-su>span{display:block;text-align:left}.mw-parser-output sub.template-chem2-sub{font-size:80%;vertical-align:-0.35em}.mw-parser-output sup.template-chem2-sup{font-size:80%;vertical-align:0.65em}</style><span class="chemf nowrap"><a href="/wiki/Lead" title="Lead">Pb</a>[<a href="/wiki/Zirconium" title="Zirconium">Zr</a><sub class="template-chem2-sub"><i>x</i></sub><a href="/wiki/Titanium" title="Titanium">Ti</a><sub class="template-chem2-sub">1−<i>x</i></sub>]<a href="/wiki/Oxygen" title="Oxygen">O</a><sub class="template-chem2-sub">3</sub></span> with 0 ≤ <i>x</i> ≤ 1) – more commonly known as PZT, the most common piezoelectric ceramic in use today.</li> <li><a href="/wiki/Potassium_niobate" title="Potassium niobate">Potassium niobate</a> (KNbO<sub>3</sub>)<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Sodium_tungstate" title="Sodium tungstate">Sodium tungstate</a> (Na<sub>2</sub>WO<sub>3</sub>)</li> <li>Ba<sub>2</sub>NaNb<sub>5</sub>O<sub>5</sub></li> <li>Pb<sub>2</sub>KNb<sub>5</sub>O<sub>15</sub></li> <li><a href="/wiki/Zinc_oxide" title="Zinc oxide">Zinc oxide</a> (ZnO) – <a href="/wiki/Wurtzite_crystal_structure" class="mw-redirect" title="Wurtzite crystal structure">Wurtzite structure</a>. While single crystals of ZnO are piezoelectric and pyroelectric, polycrystalline (ceramic) ZnO with randomly oriented grains exhibits neither piezoelectric nor pyroelectric effect. Not being ferroelectric, polycrystalline ZnO cannot be poled like barium titanate or PZT. Ceramics and polycrystalline thin films of ZnO may exhibit macroscopic piezoelectricity and pyroelectricity only if they are <a href="/wiki/Texture_(crystalline)" class="mw-redirect" title="Texture (crystalline)">textured</a> (grains are preferentially oriented), such that the piezoelectric and pyroelectric responses of all individual grains do not cancel. This is readily accomplished in polycrystalline thin films.<sup id="cite_ref-DD1998_24-1" class="reference"><a href="#cite_note-DD1998-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Lead-free_piezoceramics">Lead-free piezoceramics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Piezoelectricity&action=edit&section=11" title="Edit section: Lead-free piezoceramics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=Sodium_potassium_niobate&action=edit&redlink=1" class="new" title="Sodium potassium niobate (page does not exist)">Sodium potassium niobate</a> ((K,Na)NbO<sub>3</sub>). This material is also known as NKN or KNN. In 2004, a group of Japanese researchers led by Yasuyoshi Saito discovered a sodium potassium niobate composition with properties close to those of PZT, including a high <i>T</i><sub>C</sub>.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> Certain compositions of this material have been shown to retain a high mechanical quality factor (<i>Q</i><sub>m</sub> ≈ 900) with increasing vibration levels, whereas the mechanical quality factor of hard PZT degrades in such conditions. This fact makes NKN a promising replacement for high power resonance applications, such as piezoelectric transformers.<sup id="cite_ref-GurdalUral2011_34-0" class="reference"><a href="#cite_note-GurdalUral2011-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Bismuth_ferrite" title="Bismuth ferrite">Bismuth ferrite</a> (BiFeO<sub>3</sub>)  – a promising candidate for the replacement of lead-based ceramics.</li> <li>Sodium niobate (NaNbO<sub>3</sub>)</li> <li><a href="/wiki/Barium_titanate" title="Barium titanate">Barium titanate</a> (BaTiO<sub>3</sub>) – Barium titanate was the first piezoelectric ceramic discovered.</li> <li><a href="/wiki/Bismuth_titanate" title="Bismuth titanate">Bismuth titanate</a> (Bi<sub>4</sub>Ti<sub>3</sub>O<sub>12</sub>)</li> <li><a href="/wiki/Sodium_bismuth_titanate" title="Sodium bismuth titanate">Sodium bismuth titanate</a> (NaBi(TiO<sub>3</sub>)<sub>2</sub>)</li></ul> <p>The fabrication of lead-free piezoceramics pose multiple challenges, from an environmental standpoint and their ability to replicate the properties of their lead-based counterparts. By removing the lead component of the piezoceramic, the risk of toxicity to humans decreases, but the mining and extraction of the materials can be harmful to the environment.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> Analysis of the environmental profile of PZT versus sodium potassium niobate (NKN or KNN) shows that across the four indicators considered (primary energy consumption, toxicological footprint, eco-indicator 99, and input-output upstream greenhouse gas emissions), KNN is actually more harmful to the environment. Most of the concerns with KNN, specifically its Nb<sub>2</sub>O<sub>5</sub> component, are in the early phase of its life cycle before it reaches manufacturers. Since the harmful impacts are focused on these early phases, some actions can be taken to minimize the effects. Returning the land as close to its original form after Nb<sub>2</sub>O<sub>5</sub> mining via dam deconstruction or replacing a stockpile of utilizable soil are known aids for any extraction event. For minimizing air quality effects, modeling and simulation still needs to occur to fully understand what mitigation methods are required. The extraction of lead-free piezoceramic components has not grown to a significant scale at this time, but from early analysis, experts encourage caution when it comes to environmental effects. </p><p>Fabricating lead-free piezoceramics faces the challenge of maintaining the performance and stability of their lead-based counterparts. In general, the main fabrication challenge is creating the "morphotropic phase boundaries (MPBs)" that provide the materials with their stable piezoelectric properties without introducing the "polymorphic phase boundaries (PPBs)" that decrease the temperature stability of the material.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> New phase boundaries are created by varying additive concentrations so that the <a href="/wiki/Phase_transition" title="Phase transition">phase transition</a> temperatures converge at room temperature. The introduction of the MPB improves piezoelectric properties, but if a PPB is introduced, the material becomes negatively affected by temperature. Research is ongoing to control the type of phase boundaries that are introduced through phase engineering, diffusing phase transitions, domain engineering, and chemical modification. </p> <div class="mw-heading mw-heading3"><h3 id="III–V_and_II–VI_semiconductors"><span id="III.E2.80.93V_and_II.E2.80.93VI_semiconductors"></span>III–V and II–VI semiconductors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Piezoelectricity&action=edit&section=12" title="Edit section: III–V and II–VI semiconductors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A piezoelectric potential can be created in any bulk or nanostructured semiconductor crystal having non central symmetry, such as the <a href="/wiki/List_of_semiconductor_materials#Types_of_semiconductor_materials" title="List of semiconductor materials">Group</a> <a href="/wiki/Boron_group" title="Boron group">III</a>–<a href="/wiki/Nitrogen_group" class="mw-redirect" title="Nitrogen group">V</a> and <a href="/wiki/Group_12_element" title="Group 12 element">II</a>–<a href="/wiki/Chalcogen" title="Chalcogen">VI</a> materials, due to polarization of ions under applied stress and strain. This property is common to both the <a href="/wiki/Zincblende" class="mw-redirect" title="Zincblende">zincblende</a> and <a href="/wiki/Wurtzite" title="Wurtzite">wurtzite</a> crystal structures. To first order, there is only one independent piezoelectric coefficient in <a href="/wiki/Zincblende" class="mw-redirect" title="Zincblende">zincblende</a>, called e<sub>14</sub>, coupled to shear components of the strain. In <a href="/wiki/Wurtzite" title="Wurtzite">wurtzite</a>, there are instead three independent piezoelectric coefficients: <i>e</i><sub>31</sub>, <i>e</i><sub>33</sub> and <i>e</i><sub>15</sub>. The semiconductors where the strongest piezoelectricity is observed are those commonly found in the <a href="/wiki/Wurtzite" title="Wurtzite">wurtzite</a> structure, i.e. <a href="/wiki/GaN" class="mw-redirect" title="GaN">GaN</a>, <a href="/wiki/InN" class="mw-redirect" title="InN">InN</a>, <a href="/wiki/AlN" class="mw-redirect" title="AlN">AlN</a> and <a href="/wiki/ZnO" class="mw-redirect" title="ZnO">ZnO</a> (see <a href="/wiki/Piezotronics" title="Piezotronics">piezotronics</a>). </p><p>Since 2006, there have also been a number of reports of strong <a href="/wiki/Non_linear_piezoelectric_effects_in_polar_semiconductors" title="Non linear piezoelectric effects in polar semiconductors">non linear piezoelectric effects in polar semiconductors</a>.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> Such effects are generally recognized to be at least important if not of the same order of magnitude as the first order approximation. </p> <div class="mw-heading mw-heading3"><h3 id="Polymers">Polymers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Piezoelectricity&action=edit&section=13" title="Edit section: Polymers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The piezo-response of <a href="/wiki/Polymer" title="Polymer">polymers</a> is not as high as the response for ceramics; however, polymers hold properties that ceramics do not. Over the last few decades, non-toxic, piezoelectric polymers have been studied and applied due to their flexibility and smaller <a href="/wiki/Acoustic_impedance" title="Acoustic impedance">acoustical impedance</a>.<sup id="cite_ref-:0_38-0" class="reference"><a href="#cite_note-:0-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> Other properties that make these materials significant include their <a href="/wiki/Biocompatibility" title="Biocompatibility">biocompatibility</a>, <a href="/wiki/Biodegradation" title="Biodegradation">biodegradability</a>, low cost, and low power consumption compared to other piezo-materials (ceramics, etc.).<sup id="cite_ref-:1_39-0" class="reference"><a href="#cite_note-:1-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> Piezoelectric polymers and non-toxic polymer composites can be used given their different physical properties. </p><p>Piezoelectric polymers can be classified by bulk polymers, voided charged polymers ("piezoelectrets"), and polymer composites. A piezo-response observed by bulk polymers is mostly due to its molecular structure. There are two types of bulk polymers: <a href="/wiki/Amorphous_solid" title="Amorphous solid">amorphous</a> and <a href="/wiki/Semi-crystalline_polymer" class="mw-redirect" title="Semi-crystalline polymer">semi-crystalline</a>. Examples of semi-crystalline polymers are <a href="/wiki/Polyvinylidene_fluoride" title="Polyvinylidene fluoride">polyvinylidene fluoride</a> (PVDF) and its <a href="/wiki/Copolymer" title="Copolymer">copolymers</a>, <a href="/wiki/Polyamide" title="Polyamide">polyamides</a>, and <a href="/wiki/Parylene" title="Parylene">parylene-C</a>. Non-crystalline polymers, such as <a href="/wiki/Polyimide" title="Polyimide">polyimide</a> and <a href="/wiki/Polyvinylidene_chloride" title="Polyvinylidene chloride">polyvinylidene chloride</a> (PVDC), fall under amorphous bulk polymers. Voided charged polymers exhibit the piezoelectric effect due to charge induced by poling of a porous polymeric film. Under an electric field, charges form on the surface of the voids forming dipoles. Electric responses can be caused by any deformation of these voids. The piezoelectric effect can also be observed in polymer composites by integrating piezoelectric ceramic particles into a polymer film. A polymer does not have to be piezo-active to be an effective material for a polymer composite.<sup id="cite_ref-:1_39-1" class="reference"><a href="#cite_note-:1-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> In this case, a material could be made up of an inert matrix with a separate piezo-active component. </p><p>PVDF exhibits piezoelectricity several times greater than quartz. The piezo-response observed from PVDF is about 20–30 pC/N. That is an order of 5–50 times less than that of piezoelectric ceramic lead zirconate titanate (PZT).<sup id="cite_ref-:0_38-1" class="reference"><a href="#cite_note-:0-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_39-2" class="reference"><a href="#cite_note-:1-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> The thermal stability of the piezoelectric effect of polymers in the PVDF family (i.e. vinylidene fluoride co-poly trifluoroethylene) goes up to 125 °C. Some applications of PVDF are pressure sensors, hydrophones, and shock wave sensors.<sup id="cite_ref-:0_38-2" class="reference"><a href="#cite_note-:0-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> </p><p>Due to their flexibility, piezoelectric composites have been proposed as energy harvesters and nanogenerators. In 2018, it was reported by Zhu et al. that a piezoelectric response of about 17 pC/N could be obtained from PDMS/PZT nanocomposite at 60% porosity.<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> Another PDMS nanocomposite was reported in 2017, in which BaTiO<sub>3</sub> was integrated into PDMS to make a stretchable, transparent nanogenerator for self-powered physiological monitoring.<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> In 2016, polar molecules were introduced into a polyurethane foam in which high responses of up to 244 pC/N were reported.<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Other_materials">Other materials</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Piezoelectricity&action=edit&section=14" title="Edit section: Other materials"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Most materials exhibit at least weak piezoelectric responses. Trivial examples include <a href="/wiki/Sucrose" title="Sucrose">sucrose</a> (table sugar), <a href="/wiki/DNA" title="DNA">DNA</a>, viral proteins, including those from <a href="/wiki/Bacteriophage" title="Bacteriophage">bacteriophage</a>.<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> An actuator based on wood fibers, called <a href="/wiki/Cellulose_fiber" title="Cellulose fiber">cellulose fibers</a>, has been reported.<sup id="cite_ref-:1_39-3" class="reference"><a href="#cite_note-:1-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> D33 responses for cellular polypropylene are around 200 pC/N. Some applications of cellular polypropylene are musical key pads, microphones, and ultrasound-based echolocation systems.<sup id="cite_ref-:0_38-3" class="reference"><a href="#cite_note-:0-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> Recently, single amino acid such as β-glycine also displayed high piezoelectric (178 pmV<sup>−1</sup>) as compared to other biological materials.<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Ionic_liquid" title="Ionic liquid">Ionic liquids</a> were recently identified as the first piezoelectric liquid.<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Application">Application</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Piezoelectricity&action=edit&section=15" title="Edit section: Application"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="High_voltage_and_power_sources">High voltage and power sources</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Piezoelectricity&action=edit&section=16" title="Edit section: High voltage and power sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Direct piezoelectricity of some substances, like quartz, can generate <a href="/wiki/Potential_difference" class="mw-redirect" title="Potential difference">potential differences</a> of thousands of volts. </p> <ul><li>The best-known application is the electric <a href="/wiki/Lighter" title="Lighter">cigarette lighter</a>: pressing the button causes a spring-loaded hammer to hit a piezoelectric crystal, producing a sufficiently high-voltage <a href="/wiki/Electric_current" title="Electric current">electric current</a> that flows across a small <a href="/wiki/Spark_gap" title="Spark gap">spark gap</a>, thus heating and igniting the gas. The portable sparkers used to ignite <a href="/wiki/Gas_stove" title="Gas stove">gas stoves</a> work the same way, and many types of gas burners now have built-in piezo-based ignition systems.</li> <li>A similar idea is being researched by <a href="/wiki/DARPA" title="DARPA">DARPA</a> in the United States in a project called <a href="/wiki/Energy_harvesting" title="Energy harvesting">energy harvesting</a>, which includes an attempt to power battlefield equipment by piezoelectric generators embedded in <a href="/wiki/Soldier" title="Soldier">soldiers</a>' boots. However, these energy harvesting sources by association affect the body. DARPA's effort to harness 1–2 watts from continuous shoe impact while walking were abandoned due to the impracticality and the discomfort from the additional energy expended by a person wearing the shoes. Other energy harvesting ideas include <a href="/wiki/Crowd_Farm" title="Crowd Farm">Crowd Farm</a>, harvesting the energy from human movements in train stations or other public places<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> and converting a dance floor to generate electricity.<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> Vibrations from industrial machinery can also be harvested by piezoelectric materials to charge batteries for backup supplies or to power low-power microprocessors and wireless radios.<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup></li> <li>A piezoelectric <a href="/wiki/Transformer" title="Transformer">transformer</a> is a type of AC voltage multiplier. Unlike a conventional transformer, which uses magnetic coupling between input and output, the piezoelectric transformer uses <a href="/wiki/Acoustic_coupling" class="mw-redirect" title="Acoustic coupling">acoustic coupling</a>. An input voltage is applied across a short length of a bar of piezoceramic material such as <a href="/wiki/Lead_zirconate_titanate" title="Lead zirconate titanate">PZT</a>, creating an alternating stress in the bar by the inverse piezoelectric effect and causing the whole bar to vibrate. The vibration frequency is chosen to be the <a href="/wiki/Resonance" title="Resonance">resonant</a> frequency of the block, typically in the 100 <a href="/wiki/Kilohertz" class="mw-redirect" title="Kilohertz">kilohertz</a> to 1 megahertz range. A higher output voltage is then generated across another section of the bar by the piezoelectric effect. Step-up ratios of more than 1,000:1 have been demonstrated.<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. (May 2013)">citation needed</span></a></i>]</sup> An extra feature of this transformer is that, by operating it above its resonant frequency, it can be made to appear as an <a href="/wiki/Inductor" title="Inductor">inductive</a> load, which is useful in circuits that require a controlled soft start.<sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> These devices can be used in DC–AC inverters to drive <a href="/wiki/Cold_cathode_fluorescent_lamp" class="mw-redirect" title="Cold cathode fluorescent lamp">cold cathode fluorescent lamps</a>. Piezo transformers are some of the most compact high voltage sources.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Sensors">Sensors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Piezoelectricity&action=edit&section=17" title="Edit section: Sensors"><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:Piezoelectric_pickup1.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Piezoelectric_pickup1.jpg/220px-Piezoelectric_pickup1.jpg" decoding="async" width="220" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Piezoelectric_pickup1.jpg/330px-Piezoelectric_pickup1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/Piezoelectric_pickup1.jpg/440px-Piezoelectric_pickup1.jpg 2x" data-file-width="545" data-file-height="446" /></a><figcaption>Piezoelectric disk used as a <a href="/wiki/Guitar_pickup" class="mw-redirect" title="Guitar pickup">guitar pickup</a></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:RPG-7_detached.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/RPG-7_detached.jpg/220px-RPG-7_detached.jpg" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/RPG-7_detached.jpg/330px-RPG-7_detached.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/RPG-7_detached.jpg/440px-RPG-7_detached.jpg 2x" data-file-width="2304" data-file-height="1536" /></a><figcaption>Many rocket-propelled grenades used a piezoelectric <a href="https://en.wiktionary.org/wiki/fuse" class="extiw" title="wikt:fuse">fuse</a>. Pictured, a Russian <a href="/wiki/RPG-7" title="RPG-7">RPG-7</a><sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup></figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Piezoelectric_sensor" title="Piezoelectric sensor">Piezoelectric sensor</a></div> <p>The principle of operation of a piezoelectric <a href="/wiki/Sensor" title="Sensor">sensor</a> is that a physical dimension, transformed into a force, acts on two opposing faces of the sensing element. Depending on the design of a sensor, different "modes" to load the piezoelectric element can be used: longitudinal, transversal and shear. </p><p>Detection of pressure variations in the form of sound is the most common sensor application, e.g. piezoelectric <a href="/wiki/Microphone" title="Microphone">microphones</a> (sound waves bend the piezoelectric material, creating a changing voltage) and piezoelectric <a href="/wiki/Pick_up_(music_technology)" class="mw-redirect" title="Pick up (music technology)">pickups</a> for <a href="/wiki/Acoustic-electric_guitar" title="Acoustic-electric guitar">acoustic-electric guitars</a>. A piezo sensor attached to the body of an instrument is known as a <a href="/wiki/Contact_microphone" title="Contact microphone">contact microphone</a>. </p><p>Piezoelectric sensors especially are used with high frequency sound in ultrasonic transducers for medical imaging and also industrial <a href="/wiki/Nondestructive_testing" title="Nondestructive testing">nondestructive testing</a> (NDT). </p><p>For many sensing techniques, the sensor can act as both a sensor and an actuator—often the term <i>transducer</i> is preferred when the device acts in this dual capacity, but most piezo devices have this property of reversibility whether it is used or not. Ultrasonic transducers, for example, can inject ultrasound waves into the body, receive the returned wave, and convert it to an electrical signal (a voltage). Most medical ultrasound transducers are piezoelectric. </p><p>In addition to those mentioned above, various sensor and transducer applications include: </p> <ul><li>Piezoelectric elements are also used in the detection and generation of sonar waves.</li> <li>Piezoelectric materials are used in single-axis and dual-axis tilt sensing.<sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup></li> <li>Power monitoring in high power applications (e.g. medical treatment, <a href="/wiki/Sonochemistry" title="Sonochemistry">sonochemistry</a> and industrial processing).</li> <li><a href="/wiki/Piezoelectric_microbalance" class="mw-redirect" title="Piezoelectric microbalance">Piezoelectric microbalances</a> are used as very sensitive chemical and biological sensors.</li> <li>Piezoelectrics are sometimes used in <a href="/wiki/Strain_gauge" title="Strain gauge">strain gauges</a>. More commonly however, a <a href="/wiki/Piezoresistive_effect" title="Piezoresistive effect">Piezoresistive effect</a> element is used.</li> <li>A piezoelectric transducer was used in the penetrometer instrument on the <a href="/wiki/Huygens_Probe" class="mw-redirect" title="Huygens Probe">Huygens Probe</a>.</li> <li>Piezoelectric <a href="/wiki/Transducer" title="Transducer">transducers</a> are used in <a href="/wiki/Electronic_drum" title="Electronic drum">electronic drum pads</a> to detect the impact of the drummer's sticks, and to detect muscle movements in medical <a href="/wiki/Acceleromyography" class="mw-redirect" title="Acceleromyography">acceleromyography</a>.</li> <li>Automotive <a href="/wiki/Engine_Control_Unit" class="mw-redirect" title="Engine Control Unit">engine management systems</a> use piezoelectric transducers to detect Engine knock (Knock Sensor, KS), also known as detonation, at certain hertz frequencies. A piezoelectric transducer is also used in fuel injection systems to measure manifold absolute pressure (MAP sensor) to determine engine load, and ultimately the fuel injectors milliseconds of on time.</li> <li>Ultrasonic piezo sensors are used in the detection of acoustic emissions in <a href="/wiki/Acoustic_emission_testing" class="mw-redirect" title="Acoustic emission testing">acoustic emission testing</a>.</li> <li>Piezoelectric transducers can be used in transit-time <a href="/wiki/Ultrasonic_flow_meters" class="mw-redirect" title="Ultrasonic flow meters">ultrasonic flow meters</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Actuators">Actuators</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Piezoelectricity&action=edit&section=18" title="Edit section: Actuators"><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:2007-07-24_Piezoelectric_buzzer.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/2007-07-24_Piezoelectric_buzzer.jpg/220px-2007-07-24_Piezoelectric_buzzer.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/2007-07-24_Piezoelectric_buzzer.jpg/330px-2007-07-24_Piezoelectric_buzzer.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d3/2007-07-24_Piezoelectric_buzzer.jpg/440px-2007-07-24_Piezoelectric_buzzer.jpg 2x" data-file-width="1400" data-file-height="1050" /></a><figcaption>Metal disk with piezoelectric disk attached, used in a <a href="/wiki/Buzzer" title="Buzzer">buzzer</a></figcaption></figure> <p>As very high electric fields correspond to only tiny changes in the width of the crystal, this width can be changed with better-than-<a href="/wiki/Micrometre" title="Micrometre">μm</a> precision, making piezo crystals the most important tool for positioning objects with extreme accuracy—thus their use in <a href="/wiki/Actuators" class="mw-redirect" title="Actuators">actuators</a>.<sup id="cite_ref-shabestari_55-0" class="reference"><a href="#cite_note-shabestari-55"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> Multilayer ceramics, using layers thinner than <span class="nowrap">100 μm</span>, allow reaching high electric fields with voltage lower than <span class="nowrap">150 V</span>. These ceramics are used within two kinds of actuators: direct piezo actuators and <a href="/wiki/Amplified_piezoelectric_actuator" title="Amplified piezoelectric actuator">amplified piezoelectric actuators</a>. While direct actuator's stroke is generally lower than <span class="nowrap">100 μm</span>, amplified piezo actuators can reach millimeter strokes. </p> <ul><li><a href="/wiki/Piezoelectric_loudspeaker" class="mw-redirect" title="Piezoelectric loudspeaker">Loudspeakers</a>: Voltage is converted to mechanical movement of a metallic diaphragm.</li> <li><a href="/wiki/Ultrasonic_cleaning" title="Ultrasonic cleaning">Ultrasonic cleaning</a> usually uses piezoelectric elements to produce intense sound waves in liquid.</li> <li><a href="/wiki/Piezoelectric_motor" title="Piezoelectric motor">Piezoelectric motors</a>: Piezoelectric elements apply a directional force to an <a href="/wiki/Axle" title="Axle">axle</a>, causing it to rotate. Due to the extremely small distances involved, the piezo motor is viewed as a high-precision replacement for the <a href="/wiki/Stepper_motor" title="Stepper motor">stepper motor</a>.</li> <li>Piezoelectric elements can be used in <a href="/wiki/Laser" title="Laser">laser</a> mirror alignment, where their ability to move a large mass (the mirror mount) over microscopic distances is exploited to electronically align some laser mirrors. By precisely controlling the distance between mirrors, the laser electronics can accurately maintain optical conditions inside the laser cavity to optimize the beam output.</li> <li>A related application is the <a href="/wiki/Acousto-optic_modulator" title="Acousto-optic modulator">acousto-optic modulator</a>, a device that scatters light off soundwaves in a crystal, generated by piezoelectric elements. This is useful for fine-tuning a laser's frequency.</li> <li><a href="/wiki/Atomic_force_microscope" class="mw-redirect" title="Atomic force microscope">Atomic force microscopes</a> and <a href="/wiki/Scanning_tunneling_microscope" title="Scanning tunneling microscope">scanning tunneling microscopes</a> employ converse piezoelectricity to keep the sensing needle close to the specimen.<sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Inkjet_printer" class="mw-redirect" title="Inkjet printer">Inkjet printers</a>: On many inkjet printers, piezoelectric crystals are used to drive the ejection of ink from the inkjet print head towards the paper.</li> <li><a href="/wiki/Diesel_engine" title="Diesel engine">Diesel engines</a>: High-performance <a href="/wiki/Common_rail" title="Common rail">common rail</a> diesel engines use piezoelectric <a href="/wiki/Fuel_injector" class="mw-redirect" title="Fuel injector">fuel injectors</a>, first developed by <a href="/wiki/Robert_Bosch_GmbH" class="mw-redirect" title="Robert Bosch GmbH">Robert Bosch GmbH</a>, instead of the more common <a href="/wiki/Solenoid_valve" title="Solenoid valve">solenoid valve</a> devices.</li> <li>Active vibration control using amplified actuators.</li> <li><a href="/wiki/X-ray" title="X-ray">X-ray</a> shutters.</li> <li>XY stages for micro scanning used in infrared cameras.</li> <li>Moving the patient precisely inside active <a href="/wiki/X-ray_computed_tomography" class="mw-redirect" title="X-ray computed tomography">CT</a> and <a href="/wiki/Magnetic_resonance_imaging" title="Magnetic resonance imaging">MRI</a> scanners where the strong radiation or magnetism precludes electric motors.<sup id="cite_ref-piezact_57-0" class="reference"><a href="#cite_note-piezact-57"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Crystal_earpiece" title="Crystal earpiece">Crystal earpieces</a> are sometimes used in old or low power radios.</li> <li><a href="/wiki/High-intensity_focused_ultrasound" class="mw-redirect" title="High-intensity focused ultrasound">High-intensity focused ultrasound</a> for localized heating or creating a localized <a href="/wiki/Cavitation" title="Cavitation">cavitation</a> can be achieved, for example, in patient's body or in an industrial chemical process.</li> <li><a href="/wiki/Refreshable_braille_display" title="Refreshable braille display">Refreshable braille display</a>. A small crystal is expanded by applying a current that moves a lever to raise individual braille cells.</li> <li>Piezoelectric actuator. A single crystal or a number of crystals are expanded by applying a voltage for moving and controlling a mechanism or system.<sup id="cite_ref-shabestari_55-1" class="reference"><a href="#cite_note-shabestari-55"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup></li> <li>Piezoelectric actuators are used for fine servo positioning in hard disc drives.<sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Frequency_standard">Frequency standard</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Piezoelectricity&action=edit&section=19" title="Edit section: Frequency standard"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The piezoelectrical properties of quartz are useful as a <a href="/wiki/Frequency_standard" title="Frequency standard">standard of frequency</a>. </p> <ul><li><a href="/wiki/Quartz_clock" title="Quartz clock">Quartz clocks</a> employ a <a href="/wiki/Crystal_oscillator" title="Crystal oscillator">crystal oscillator</a> made from a quartz crystal that uses a combination of both direct and converse piezoelectricity to generate a regularly timed series of electrical pulses that is used to mark time. The quartz crystal (like any <a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">elastic</a> material) has a precisely defined <a href="/wiki/Resonance" title="Resonance">natural frequency</a> (caused by its shape and size) at which it prefers to <a href="/wiki/Oscillator" class="mw-redirect" title="Oscillator">oscillate</a>, and this is used to stabilize the frequency of a periodic voltage applied to the crystal.</li> <li>The same principle is used in some <a href="/wiki/Radio" title="Radio">radio</a> <a href="/wiki/Transmitter" title="Transmitter">transmitters</a> and <a href="/wiki/Receiver_(radio)" class="mw-redirect" title="Receiver (radio)">receivers</a>, and in <a href="/wiki/Computer" title="Computer">computers</a> where it creates a <a href="/wiki/Clock_pulse" class="mw-redirect" title="Clock pulse">clock pulse</a>. Both of these usually use a <a href="/wiki/Frequency_multiplier" title="Frequency multiplier">frequency multiplier</a> to reach gigahertz ranges.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Piezoelectric_motors">Piezoelectric motors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Piezoelectricity&action=edit&section=20" title="Edit section: Piezoelectric motors"><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:Slip-stick_actuator_operation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Slip-stick_actuator_operation.svg/220px-Slip-stick_actuator_operation.svg.png" decoding="async" width="220" height="238" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Slip-stick_actuator_operation.svg/330px-Slip-stick_actuator_operation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Slip-stick_actuator_operation.svg/440px-Slip-stick_actuator_operation.svg.png 2x" data-file-width="300" data-file-height="325" /></a><figcaption>A stick-slip actuator</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Piezoelectric_motor" title="Piezoelectric motor">Piezoelectric motor</a></div> <p>Types of piezoelectric motor include: </p> <ul><li>The <a href="/wiki/Ultrasonic_motor" title="Ultrasonic motor">ultrasonic motor</a> used for <a href="/wiki/Auto-focus" class="mw-redirect" title="Auto-focus">auto-focus</a> in <a href="/wiki/Single_lens_reflex" class="mw-redirect" title="Single lens reflex">reflex cameras</a></li> <li><a href="/wiki/Inchworm_motor" title="Inchworm motor">Inchworm motors</a> for linear motion</li> <li>Rectangular four-quadrant motors with high power density (2.5 <a href="/wiki/Watt" title="Watt">W</a>/cm<sup>3</sup>) and speed ranging from 10 nm/s to 800 mm/s.</li> <li>Stepping piezo motor, using <a href="/wiki/Stick-slip" class="mw-redirect" title="Stick-slip">stick-slip</a> effect.</li></ul> <p>Aside from the stepping stick-slip motor, all these motors work on the same principle. Driven by dual orthogonal vibration modes with a <a href="/wiki/Phase_(waves)" title="Phase (waves)">phase</a> difference of 90°, the contact point between two surfaces vibrates in an <a href="/wiki/Ellipse" title="Ellipse">elliptical</a> path, producing a <a href="/wiki/Friction" title="Friction">frictional</a> force between the surfaces. Usually, one surface is fixed, causing the other to move. In most piezoelectric motors, the piezoelectric crystal is excited by a <a href="/wiki/Sine_wave" title="Sine wave">sine wave</a> signal at the resonant frequency of the motor. Using the resonance effect, a much lower voltage can be used to produce a high vibration amplitude. </p><p>A stick-slip motor works using the inertia of a mass and the friction of a clamp. Such motors can be very small. Some are used for camera sensor displacement, thus allowing an anti-shake function. </p> <div class="mw-heading mw-heading3"><h3 id="Reduction_of_vibrations_and_noise">Reduction of vibrations and noise</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Piezoelectricity&action=edit&section=21" title="Edit section: Reduction of vibrations and noise"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Different teams of researchers have been investigating ways to reduce vibrations in materials by attaching piezo elements to the material. When the material is bent by a vibration in one direction, the vibration-reduction system responds to the bend and sends electric power to the piezo element to bend in the other direction. Future applications of this technology are expected in cars and houses to reduce noise. Further applications to flexible structures, such as shells and plates, have also been studied for nearly three decades. </p><p>In a demonstration at the Material Vision Fair in <a href="/wiki/Frankfurt" title="Frankfurt">Frankfurt</a> in November 2005, a team from <a href="/wiki/Darmstadt_University_of_Technology" class="mw-redirect" title="Darmstadt University of Technology">TU Darmstadt</a> in <a href="/wiki/Germany" title="Germany">Germany</a> showed several panels that were hit with a rubber mallet, and the panel with the piezo element immediately stopped swinging. </p><p>Piezoelectric ceramic fiber technology is being used as an electronic damping system on some <a href="/wiki/Head_(company)" title="Head (company)">HEAD</a> <a href="/wiki/Tennis_racket" class="mw-redirect" title="Tennis racket">tennis rackets</a>.<sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup> </p><p>All piezo transducers have a fundamental resonant frequency and many harmonic frequencies. Piezo driven Drop-On-Demand fluid systems are sensitive to extra vibrations in the piezo structure that must be reduced or eliminated. One inkjet company, Howtek, Inc solved this problem by replacing glass(rigid) inkjet nozzles with Tefzel (soft) inkjet nozzles. This novel idea popularized single nozzle inkjets and they are now used in 3D Inkjet printers that run for years if kept clean inside and not overheated (Tefzel creeps under pressure at very high temperatures) </p> <div class="mw-heading mw-heading3"><h3 id="Infertility_treatment">Infertility treatment</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Piezoelectricity&action=edit&section=22" title="Edit section: Infertility treatment"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In people with previous <a href="/wiki/Total_fertilization_failure" class="mw-redirect" title="Total fertilization failure">total fertilization failure</a>, piezoelectric activation of <a href="/wiki/Oocyte" title="Oocyte">oocytes</a> together with <a href="/wiki/Intracytoplasmic_sperm_injection" title="Intracytoplasmic sperm injection">intracytoplasmic sperm injection</a> (ICSI) seems to improve fertilization outcomes.<sup id="cite_ref-61" class="reference"><a href="#cite_note-61"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Surgery">Surgery</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Piezoelectricity&action=edit&section=23" title="Edit section: Surgery"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Piezosurgery" class="mw-redirect" title="Piezosurgery">Piezosurgery</a><sup id="cite_ref-Manbachi,_A._and_Cobbold_R.S.C._2011_187–196_62-0" class="reference"><a href="#cite_note-Manbachi,_A._and_Cobbold_R.S.C._2011_187–196-62"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup> is a minimally invasive technique that aims to cut a target tissue with little damage to neighboring tissues. For example, Hoigne <i>et al.</i><sup id="cite_ref-63" class="reference"><a href="#cite_note-63"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup> uses frequencies in the range 25–29 kHz, causing microvibrations of 60–210 μm. It has the ability to cut mineralized tissue without cutting neurovascular tissue and other soft tissue, thereby maintaining a blood-free operating area, better visibility and greater precision.<sup id="cite_ref-64" class="reference"><a href="#cite_note-64"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Potential_applications">Potential applications</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Piezoelectricity&action=edit&section=24" title="Edit section: Potential applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 2015, Cambridge University researchers working in conjunction with researchers from the National Physical Laboratory and Cambridge-based dielectric antenna company Antenova Ltd, using thin films of piezoelectric materials found that at a certain frequency, these materials become not only efficient resonators, but efficient radiators as well, meaning that they can potentially be used as antennas. The researchers found that by subjecting the piezoelectric thin films to an asymmetric excitation, the symmetry of the system is similarly broken, resulting in a corresponding symmetry breaking of the electric field, and the generation of electromagnetic radiation.<sup id="cite_ref-65" class="reference"><a href="#cite_note-65"><span class="cite-bracket">[</span>65<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-66" class="reference"><a href="#cite_note-66"><span class="cite-bracket">[</span>66<span class="cite-bracket">]</span></a></sup> </p><p>Several attempts at the macro-scale application of the piezoelectric technology have emerged<sup id="cite_ref-67" class="reference"><a href="#cite_note-67"><span class="cite-bracket">[</span>67<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-68" class="reference"><a href="#cite_note-68"><span class="cite-bracket">[</span>68<span class="cite-bracket">]</span></a></sup> to harvest kinetic energy from walking pedestrians. </p><p>In this case, locating high traffic areas is critical for optimization of the energy harvesting efficiency, as well as the orientation of the tile pavement significantly affects the total amount of the harvested energy.<sup id="cite_ref-69" class="reference"><a href="#cite_note-69"><span class="cite-bracket">[</span>69<span class="cite-bracket">]</span></a></sup> A density flow evaluation is recommended to qualitatively evaluate the piezoelectric power harvesting potential of the considered area based on the number of pedestrian crossings per unit time.<sup id="cite_ref-ReferenceA_70-0" class="reference"><a href="#cite_note-ReferenceA-70"><span class="cite-bracket">[</span>70<span class="cite-bracket">]</span></a></sup> In X. Li's study, the potential application of a commercial piezoelectric energy harvester in a central hub building at Macquarie University in Sydney, Australia is examined and discussed. Optimization of the piezoelectric tile deployment is presented according to the frequency of pedestrian mobility and a model is developed where 3.1% of the total floor area with the highest pedestrian mobility is paved with piezoelectric tiles. The modelling results indicate that the total annual energy harvesting potential for the proposed optimized tile pavement model is estimated at 1.1 MWh/year, which would be sufficient to meet close to 0.5% of the annual energy needs of the building.<sup id="cite_ref-ReferenceA_70-1" class="reference"><a href="#cite_note-ReferenceA-70"><span class="cite-bracket">[</span>70<span class="cite-bracket">]</span></a></sup> In Israel, there is a company which has installed piezoelectric materials under a busy highway. The energy generated is enough to power street lights, billboards, and signs.<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. (June 2016)">citation needed</span></a></i>]</sup> </p><p>Tire company <a href="/wiki/Goodyear_Tire_and_Rubber_Company" title="Goodyear Tire and Rubber Company">Goodyear</a> has plans to develop an electricity generating tire which has piezoelectric material lined inside it. As the tire moves, it deforms and thus electricity is generated.<sup id="cite_ref-71" class="reference"><a href="#cite_note-71"><span class="cite-bracket">[</span>71<span class="cite-bracket">]</span></a></sup> </p><p>The efficiency of a hybrid <a href="/wiki/Photovoltaic_cell" class="mw-redirect" title="Photovoltaic cell">photovoltaic cell</a> that contains piezoelectric materials can be increased simply by placing it near a source of ambient noise or vibration. The effect was demonstrated with organic cells using <a href="/wiki/Zinc_oxide" title="Zinc oxide">zinc oxide</a> nanotubes. The electricity generated by the piezoelectric effect itself is a negligible percentage of the overall output. Sound levels as low as 75 decibels improved efficiency by up to 50%. Efficiency peaked at 10 kHz, the resonant frequency of the nanotubes. The electrical field set up by the vibrating nanotubes interacts with electrons migrating from the organic polymer layer. This process decreases the likelihood of recombination, in which electrons are energized but settle back into a hole instead of migrating to the electron-accepting ZnO layer.<sup id="cite_ref-72" class="reference"><a href="#cite_note-72"><span class="cite-bracket">[</span>72<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-73" class="reference"><a href="#cite_note-73"><span class="cite-bracket">[</span>73<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=Piezoelectricity&action=edit&section=25" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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: 20em;"> <ul><li><a href="/wiki/Charge_amplifier" title="Charge amplifier">Charge amplifier</a></li> <li><a href="/wiki/Electret" title="Electret">Electret</a></li> <li><a href="/wiki/Electronic_component" title="Electronic component">Electronic component</a></li> <li><a href="/wiki/Electrostriction" title="Electrostriction">Electrostriction</a></li> <li><a href="/wiki/Flexoelectricity" title="Flexoelectricity">Flexoelectricity</a></li> <li><a href="/wiki/Magnetostriction" title="Magnetostriction">Magnetostriction</a></li> <li><a href="/wiki/Photoelectric_effect" title="Photoelectric effect">Photoelectric effect</a></li> <li><a href="/wiki/Piezoelectric_speaker" title="Piezoelectric speaker">Piezoelectric speaker</a></li> <li><a href="/wiki/Piezoluminescence" title="Piezoluminescence">Piezoluminescence</a></li> <li><a href="/wiki/Piezomagnetism" title="Piezomagnetism">Piezomagnetism</a></li> <li><a href="/wiki/Piezoresistive_effect" title="Piezoresistive effect">Piezoresistive effect</a></li> <li><a href="/wiki/Piezosurgical" class="mw-redirect" title="Piezosurgical">Piezosurgical</a></li> <li><a href="/wiki/Quartz_crystal_microbalance" title="Quartz crystal microbalance">Quartz crystal microbalance</a></li> <li><a href="/wiki/Sonomicrometry" title="Sonomicrometry">Sonomicrometry</a></li> <li><a href="/wiki/Surface_acoustic_wave" title="Surface acoustic wave">Surface acoustic wave</a></li> <li><a href="/wiki/Thermoelectric_generator" title="Thermoelectric generator">Thermoelectric generator</a></li> <li><a href="/wiki/Triboluminescence" title="Triboluminescence">Triboluminescence</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=Piezoelectricity&action=edit&section=26" 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="CITEREFWells2008" class="citation book cs1"><a href="/wiki/John_C._Wells" title="John C. Wells">Wells, John C.</a> (2008). <i>Longman Pronunciation Dictionary</i> (3rd ed.). Longman. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4058-8118-0" title="Special:BookSources/978-1-4058-8118-0"><bdi>978-1-4058-8118-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Longman+Pronunciation+Dictionary&rft.edition=3rd&rft.pub=Longman&rft.date=2008&rft.isbn=978-1-4058-8118-0&rft.aulast=Wells&rft.aufirst=John+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APiezoelectricity" class="Z3988"></span></span> </li> <li id="cite_note-InstrumentAnalysis-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-InstrumentAnalysis_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHollerSkoogCrouch2007" class="citation book cs1">Holler, F. James; Skoog, Douglas A. & Crouch, Stanley R. (2007). <i>Principles of Instrumental Analysis</i> (6th ed.). <a href="/wiki/Cengage" class="mw-redirect" title="Cengage">Cengage</a> Learning. p. 9. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-495-01201-6" title="Special:BookSources/978-0-495-01201-6"><bdi>978-0-495-01201-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Instrumental+Analysis&rft.pages=9&rft.edition=6th&rft.pub=Cengage+Learning&rft.date=2007&rft.isbn=978-0-495-01201-6&rft.aulast=Holler&rft.aufirst=F.+James&rft.au=Skoog%2C+Douglas+A.&rft.au=Crouch%2C+Stanley+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APiezoelectricity" 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="CITEREFHarper" class="citation web cs1">Harper, Douglas. <a rel="nofollow" class="external text" href="https://www.etymonline.com/?term=piezoelectric">"piezoelectric"</a>. <i><a href="/wiki/Online_Etymology_Dictionary" title="Online Etymology Dictionary">Online Etymology Dictionary</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Online+Etymology+Dictionary&rft.atitle=piezoelectric&rft.aulast=Harper&rft.aufirst=Douglas&rft_id=https%3A%2F%2Fwww.etymonline.com%2F%3Fterm%3Dpiezoelectric&rfr_id=info%3Asid%2Fen.wikipedia.org%3APiezoelectricity" class="Z3988"></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"><a rel="nofollow" class="external text" href="https://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0057:entry=pie/zw"><span title="Ancient Greek (to 1453)-language text"><span lang="grc">πιέζειν</span></span></a>, <a rel="nofollow" class="external text" href="https://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0057:entry=h)/lektron"><span title="Ancient Greek (to 1453)-language text"><span lang="grc">ἤλεκτρον</span></span></a>. <a href="/wiki/Henry_Liddell" title="Henry Liddell">Liddell, Henry George</a>; <a href="/wiki/Robert_Scott_(philologist)" title="Robert Scott (philologist)">Scott, Robert</a>; <i><a href="/wiki/A_Greek%E2%80%93English_Lexicon" title="A Greek–English Lexicon">A Greek–English Lexicon</a></i> at the <a href="/wiki/Perseus_Project" class="mw-redirect" title="Perseus Project">Perseus Project</a>.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarper" class="citation web cs1">Harper, Douglas. <a rel="nofollow" class="external text" href="https://www.etymonline.com/?term=piezoelectric">"piezoelectric"</a>. <i><a href="/wiki/Online_Etymology_Dictionary" title="Online Etymology Dictionary">Online Etymology Dictionary</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Online+Etymology+Dictionary&rft.atitle=piezoelectric&rft.aulast=Harper&rft.aufirst=Douglas&rft_id=https%3A%2F%2Fwww.etymonline.com%2F%3Fterm%3Dpiezoelectric&rfr_id=info%3Asid%2Fen.wikipedia.org%3APiezoelectricity" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHankel1881" class="citation journal cs1 cs1-prop-foreign-lang-source">Hankel, W. G. (1881). "Elektrische Untersuchungen. Fünfzehnte Abhandlung. Über die aktino- und piezoelektrischen Eigenschaften des Bergkrystalles und ihre Beziehung zu den thermoelektrischen" [Electrical researches. Fifteenth treatise. On the radiative- and piezoelectric properties of rock crystal [i.e., quartz] and their relation to the thermoelectric [ones].]. <i>Abhandlungen der Mathematisch-Physischen Klasse der Königlichen-Säschsischen Gesellschaft der Wissenschaften</i> (in German). <b>12</b>: 459–547.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Abhandlungen+der+Mathematisch-Physischen+Klasse+der+K%C3%B6niglichen-S%C3%A4schsischen+Gesellschaft+der+Wissenschaften&rft.atitle=Elektrische+Untersuchungen.+F%C3%BCnfzehnte+Abhandlung.+%C3%9Cber+die+aktino-+und+piezoelektrischen+Eigenschaften+des+Bergkrystalles+und+ihre+Beziehung+zu+den+thermoelektrischen.&rft.volume=12&rft.pages=459-547&rft.date=1881&rft.aulast=Hankel&rft.aufirst=W.+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APiezoelectricity" class="Z3988"></span> From p. 462: <i>"Da die durch Druck erzeugte Elektricität sonach auch besonderen Gesetzen unterliegt, so wird es angemessen sein, derselben gleichfalls einen besonderen Namen beizulegen, und es dürfte sich dazu die Bezeichnung Piezoelektricität eignen."</i> (Since the electricity [which is] generated by pressure is therefore also subject to special laws, then it will likewise be appropriate to give it a special name, and for this, the designation "piezoelectricity" might be suitable.) <ul><li>The above article was also published separately as a pamphlet: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHankel1881" class="citation book cs1 cs1-prop-foreign-lang-source">Hankel, W. G. (1881). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kXXWkyYs8cEC&pg=PA455"><i>Elektrische Untersuchungen. Fünfzehnte Abhandlung. Über die aktino- und piezoelektrischen Eigenschaften des Bergkrystalles und ihre Beziehung zu den thermoelektrischen</i></a> (in German). Leipzig, Germany: S. Hirzel.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elektrische+Untersuchungen.+F%C3%BCnfzehnte+Abhandlung.+%C3%9Cber+die+aktino-+und+piezoelektrischen+Eigenschaften+des+Bergkrystalles+und+ihre+Beziehung+zu+den+thermoelektrischen.&rft.place=Leipzig%2C+Germany&rft.pub=S.+Hirzel&rft.date=1881&rft.aulast=Hankel&rft.aufirst=W.+G.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DkXXWkyYs8cEC%26pg%3DPA455&rfr_id=info%3Asid%2Fen.wikipedia.org%3APiezoelectricity" class="Z3988"></span> See p. 462.</li></ul> </span></li> <li id="cite_note-Piezoelectric_Sensorics-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Piezoelectric_Sensorics_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGautschi2002" class="citation book cs1">Gautschi, G. (2002). <i>Piezoelectric Sensorics: Force, Strain, Pressure, Acceleration and Acoustic Emission Sensors, Materials and Amplifiers</i>. <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</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%2F978-3-662-04732-3">10.1007/978-3-662-04732-3</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-662-04732-3" title="Special:BookSources/978-3-662-04732-3"><bdi>978-3-662-04732-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Piezoelectric+Sensorics%3A+Force%2C+Strain%2C+Pressure%2C+Acceleration+and+Acoustic+Emission+Sensors%2C+Materials+and+Amplifiers&rft.pub=Springer&rft.date=2002&rft_id=info%3Adoi%2F10.1007%2F978-3-662-04732-3&rft.isbn=978-3-662-04732-3&rft.aulast=Gautschi&rft.aufirst=G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APiezoelectricity" class="Z3988"></span></span> </li> <li id="cite_note-UT_of_Materials-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-UT_of_Materials_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKrautkrämerKrautkrämer1990" class="citation book cs1">Krautkrämer, J. & Krautkrämer, H. 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Retrieved <span class="nowrap">July 27,</span> 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=SparkFun+Electronics&rft.atitle=Piezo+Drum+Kit+Quickstart+Guide&rft.date=2011-10-31&rft.aulast=Taylor&rft.aufirst=C.&rft_id=https%3A%2F%2Fwww.sparkfun.com%2Ftutorials%2F330&rfr_id=info%3Asid%2Fen.wikipedia.org%3APiezoelectricity" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFErhart" class="citation web cs1">Erhart, Jiří. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140508030117/https://moodle.fp.tul.cz/nano/pluginfile.php/2476/mod_resource/content/3/FPM_Piezo_lecture1.pdf">"Piezoelectricity and ferroelectricity: Phenomena and properties"</a> <span class="cs1-format">(PDF)</span>. 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Archived from the original on May 8, 2014.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Piezoelectricity+and+ferroelectricity%3A+Phenomena+and+properties&rft.pub=Department+of+Physics%2C+Technical+University+of+Liberec&rft.aulast=Erhart&rft.aufirst=Ji%C5%99%C3%AD&rft_id=https%3A%2F%2Fmoodle.fp.tul.cz%2Fnano%2Fpluginfile.php%2F2476%2Fmod_resource%2Fcontent%2F3%2FFPM_Piezo_lecture1.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APiezoelectricity" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_web" title="Template:Cite web">cite web</a>}}</code>: CS1 maint: unfit URL (<a href="/wiki/Category:CS1_maint:_unfit_URL" title="Category:CS1 maint: unfit URL">link</a>)</span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCurieCurie1880" class="citation journal cs1"><a href="/wiki/Jacques_Curie" class="mw-redirect" title="Jacques Curie">Curie, Jacques</a>; <a href="/wiki/Pierre_Curie" title="Pierre Curie">Curie, Pierre</a> (1880). "Développement par compression de l'électricité polaire dans les cristaux hémièdres à faces inclinées" [Development, via compression, of electric polarization in hemihedral crystals with inclined faces]. <i>Bulletin de la Société Minérologique de France</i>. <b>3</b> (4): 90–93. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3406%2Fbulmi.1880.1564">10.3406/bulmi.1880.1564</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+de+la+Soci%C3%A9t%C3%A9+Min%C3%A9rologique+de+France&rft.atitle=D%C3%A9veloppement+par+compression+de+l%27%C3%A9lectricit%C3%A9+polaire+dans+les+cristaux+h%C3%A9mi%C3%A8dres+%C3%A0+faces+inclin%C3%A9es&rft.volume=3&rft.issue=4&rft.pages=90-93&rft.date=1880&rft_id=info%3Adoi%2F10.3406%2Fbulmi.1880.1564&rft.aulast=Curie&rft.aufirst=Jacques&rft.au=Curie%2C+Pierre&rfr_id=info%3Asid%2Fen.wikipedia.org%3APiezoelectricity" class="Z3988"></span><br /> Reprinted in: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCurieCurie1880" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Jacques_Curie" class="mw-redirect" title="Jacques Curie">Curie, Jacques</a>; <a href="/wiki/Pierre_Curie" title="Pierre Curie">Curie, Pierre</a> (1880). <a rel="nofollow" class="external text" href="http://gallica.bnf.fr/ark:/12148/bpt6k30485/f296.image">"Développement, par pression, de l'électricité polaire dans les cristaux hémièdres à faces inclinées"</a>. <i>Comptes Rendus</i> (in French). <b>91</b>: 294–295. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20121205083302/http://gallica.bnf.fr/ark:/12148/bpt6k30485/f296.image">Archived</a> from the original on 2012-12-05.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Comptes+Rendus&rft.atitle=D%C3%A9veloppement%2C+par+pression%2C+de+l%27%C3%A9lectricit%C3%A9+polaire+dans+les+cristaux+h%C3%A9mi%C3%A8dres+%C3%A0+faces+inclin%C3%A9es&rft.volume=91&rft.pages=294-295&rft.date=1880&rft.aulast=Curie&rft.aufirst=Jacques&rft.au=Curie%2C+Pierre&rft_id=http%3A%2F%2Fgallica.bnf.fr%2Fark%3A%2F12148%2Fbpt6k30485%2Ff296.image&rfr_id=info%3Asid%2Fen.wikipedia.org%3APiezoelectricity" class="Z3988"></span><br /> See also: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCurieCurie1880" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Jacques_Curie" class="mw-redirect" title="Jacques Curie">Curie, Jacques</a>; <a href="/wiki/Pierre_Curie" title="Pierre Curie">Curie, Pierre</a> (1880). <a rel="nofollow" class="external text" href="http://gallica.bnf.fr/ark:/12148/bpt6k30485/f385.image">"Sur l'électricité polaire dans les cristaux hémièdres à faces inclinées"</a> [On electric polarization in hemihedral crystals with inclined faces]. <i>Comptes Rendus</i> (in French). <b>91</b>: 383–386. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20121205090430/http://gallica.bnf.fr/ark:/12148/bpt6k30485/f385.image">Archived</a> from the original on 2012-12-05.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Comptes+Rendus&rft.atitle=Sur+l%27%C3%A9lectricit%C3%A9+polaire+dans+les+cristaux+h%C3%A9mi%C3%A8dres+%C3%A0+faces+inclin%C3%A9es&rft.volume=91&rft.pages=383-386&rft.date=1880&rft.aulast=Curie&rft.aufirst=Jacques&rft.au=Curie%2C+Pierre&rft_id=http%3A%2F%2Fgallica.bnf.fr%2Fark%3A%2F12148%2Fbpt6k30485%2Ff385.image&rfr_id=info%3Asid%2Fen.wikipedia.org%3APiezoelectricity" 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="CITEREFLippmann1881" class="citation journal cs1 cs1-prop-foreign-lang-source">Lippmann, G. 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Retrieved <span class="nowrap">2012-05-04</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Machinedesign.com&rft.atitle=Sensor+Sense%3A+Piezoelectric+Force+Sensors&rft.date=2008-02-07&rft.au=Robert+Repas&rft_id=http%3A%2F%2Fmachinedesign.com%2Farticle%2Fsensor-sense-piezoelectric-force-sensors-0207&rfr_id=info%3Asid%2Fen.wikipedia.org%3APiezoelectricity" 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">IEC 80000-6, item 6-12</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=121-11-40">"IEC 60050 – International Electrotechnical Vocabulary – Details for IEV number 121-11-40: "electric flux density"<span class="cs1-kern-right"></span>"</a>. <i>www.electropedia.org</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.electropedia.org&rft.atitle=IEC+60050+%E2%80%93+International+Electrotechnical+Vocabulary+%E2%80%93+Details+for+IEV+number+121-11-40%3A+%22electric+flux+density%22&rft_id=http%3A%2F%2Fwww.electropedia.org%2Fiev%2Fiev.nsf%2Fdisplay%3Fopenform%26ievref%3D121-11-40&rfr_id=info%3Asid%2Fen.wikipedia.org%3APiezoelectricity" class="Z3988"></span></span> </li> <li id="cite_note-ikeda-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-ikeda_23-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIkeda1996" class="citation book cs1">Ikeda, T. 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title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Advanced+Materials&rft.atitle=Acoustic+Enhancement+of+Polymer%2FZnO+Nanorod+Photovoltaic+Device+Performance&rft.volume=26&rft.issue=2&rft.pages=263-268&rft.date=2013&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A40624518%23id-name%3DS2CID&rft_id=info%3Apmid%2F24194369&rft_id=info%3Adoi%2F10.1002%2Fadma.201303304&rft.aulast=Shoaee&rft.aufirst=S.&rft.au=Briscoe%2C+J.&rft.au=Durrant%2C+J.+R.&rft.au=Dunn%2C+S.&rft_id=http%3A%2F%2Fqmro.qmul.ac.uk%2Fxmlui%2Fhandle%2F123456789%2F12456&rfr_id=info%3Asid%2Fen.wikipedia.org%3APiezoelectricity" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Piezoelectricity&action=edit&section=27" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>EN 50324 (2002) Piezoelectric properties of ceramic materials and components (3 parts)</li> <li>ANSI-IEEE 176 (1987) Standard on Piezoelectricity</li> <li>IEEE 177 (1976) Standard Definitions & Methods of Measurement for Piezoelectric Vibrators</li> <li>IEC 444 (1973) Basic method for the measurement of resonance freq & equiv series resistance of quartz crystal units by zero-phase technique in a pi-network</li> <li>IEC 302 (1969) Standard Definitions & Methods of Measurement for Piezoelectric Vibrators Operating over the Freq Range up to 30 MHz</li></ul> <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=Piezoelectricity&action=edit&section=28" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style 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(2002). <i>Piezoelectric Sensorics</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-42259-4" title="Special:BookSources/978-3-540-42259-4"><bdi>978-3-540-42259-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Piezoelectric+Sensorics&rft.pub=Springer&rft.date=2002&rft.isbn=978-3-540-42259-4&rft.aulast=Gautschi&rft.aufirst=Gustav+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APiezoelectricity" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20180905175615/http://www.aimspress.com/fileOther/PDF/Materials/matersci-05-05-845.pdf">Piezoelectric cellular polymer films: Fabrication, properties and applications</a></li> <li><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pubmed/19163430">Piezo motor based microdrive for neural signal recording</a></li> <li><a rel="nofollow" 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href="/wiki/Template:Marie_and_Pierre_Curie" title="Template:Marie and Pierre Curie"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Marie_and_Pierre_Curie" title="Template talk:Marie and Pierre Curie"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Marie_and_Pierre_Curie" title="Special:EditPage/Template:Marie and Pierre Curie"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Marie_and_Pierre_Curie" style="font-size:114%;margin:0 4em"><a href="/wiki/Marie_Curie" title="Marie Curie"> Marie</a> and <a href="/wiki/Pierre_Curie" title="Pierre Curie"> Pierre</a> Curie</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Discoveries</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/Curie%27s_law" title="Curie's law">Curie's law</a></li> <li><a href="/wiki/Curie%E2%80%93Weiss_law" title="Curie–Weiss law">Curie–Weiss law</a></li> <li><a href="/wiki/Curie_temperature" title="Curie temperature">Curie temperature</a></li> <li><a href="/wiki/Mean-field_theory" title="Mean-field theory">Mean-field theory</a></li> <li><a class="mw-selflink selflink">Piezoelectricity</a></li> <li><a href="/wiki/Polonium" title="Polonium">Polonium</a></li> <li><a href="/wiki/Radioactive_decay" title="Radioactive decay">Radioactivity</a></li> <li><a href="/wiki/Radium" title="Radium">Radium</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Publications</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/Curie%27s_principle" title="Curie's principle">Curie's principle</a></li> <li><i><a href="/wiki/Treatise_on_Radioactivity" title="Treatise on Radioactivity">Treatise on Radioactivity</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Museums</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/Maria_Sk%C5%82odowska-Curie_Museum" title="Maria Skłodowska-Curie Museum">Maria Skłodowska-Curie Museum</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Curie_family" title="Curie family">Family</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/Ir%C3%A8ne_Joliot-Curie" title="Irène Joliot-Curie">Irène Joliot-Curie</a> (daughter)</li> <li><a href="/wiki/%C3%88ve_Curie" title="Ève Curie">Ève Curie</a> (daughter)</li> <li><a href="/wiki/H%C3%A9l%C3%A8ne_Langevin-Joliot" title="Hélène Langevin-Joliot">Hélène Langevin-Joliot</a> (granddaughter)</li> <li><a href="/wiki/Pierre_Joliot" title="Pierre Joliot">Pierre Joliot</a> (grandson)</li> <li><a href="/wiki/Paul-Jacques_Curie" title="Paul-Jacques Curie">Paul-Jacques Curie</a> (Pierre's brother)</li> <li><a href="/wiki/Fr%C3%A9d%C3%A9ric_Joliot-Curie" title="Frédéric Joliot-Curie">Frédéric Joliot-Curie</a> (son-in-law)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Namesakes</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/Curie_Institute_(Paris)" title="Curie Institute (Paris)">Curie Institute</a></li> <li><a href="/wiki/Curie_(unit)" title="Curie (unit)">Curie</a></li> <li><a href="/wiki/Curium" title="Curium">Curium</a></li> <li><a href="/wiki/IEEE_Marie_Sklodowska-Curie_Award" title="IEEE Marie Sklodowska-Curie Award">IEEE Marie Sklodowska-Curie Award</a></li> <li><a href="/wiki/Marie_Curie_Medal" title="Marie Curie Medal">Marie Curie Medal</a></li> <li><a href="/wiki/Maria_Sk%C5%82odowska-Curie_Bridge,_Warsaw" title="Maria Skłodowska-Curie Bridge, Warsaw">Maria Skłodowska-Curie Bridge</a></li> <li><a href="/wiki/Maria_Sk%C5%82odowska-Curie_Park" title="Maria Skłodowska-Curie Park">Maria Skłodowska-Curie Park</a></li> <li><a href="/wiki/Maria_Curie-Sk%C5%82odowska_University" title="Maria Curie-Skłodowska University">Maria Curie-Skłodowska University</a></li> <li><a href="/wiki/Pierre_and_Marie_Curie_University" title="Pierre and Marie Curie University">Pierre and Marie Curie University</a></li> <li><a href="/wiki/Maria_Sk%C5%82odowska-Curie_National_Research_Institute_of_Oncology" title="Maria Skłodowska-Curie National Research Institute of Oncology">Maria Skłodowska-Curie National Research Institute of Oncology</a></li> <li><a href="/wiki/Curie_Island" title="Curie Island">Curie Island</a></li> <li><a href="/wiki/7000_Curie" class="mw-redirect" title="7000 Curie">7000 Curie</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Depictions</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/Maria_Sk%C5%82odowska-Curie_Monument_(Lublin)" title="Maria Skłodowska-Curie Monument (Lublin)">Maria Skłodowska-Curie Monument in Lublin</a></li> <li><a href="/wiki/Maria_Sk%C5%82odowska-Curie_Monument_(Downtown,_Warsaw)" title="Maria Skłodowska-Curie Monument (Downtown, Warsaw)">Maria Skłodowska-Curie Monument in Warsaw (Downtown)</a></li> <li><a href="/wiki/Maria_Sk%C5%82odowska-Curie_Monument_(Ochota)" title="Maria Skłodowska-Curie Monument (Ochota)">Maria Skłodowska-Curie Monument in Warsaw (Ochota)</a></li> <li><i><a href="/wiki/Marie_Curie_Gargoyle" title="Marie Curie Gargoyle">Marie Curie Gargoyle</a></i></li> <li><i><a href="/wiki/Maria_Sk%C5%82odowska-Curie_Medallion" title="Maria Skłodowska-Curie Medallion">Maria Skłodowska-Curie Medallion</a></i></li> <li><i><a href="/wiki/Madame_Curie_(film)" title="Madame Curie (film)">Madame Curie</a></i> (1943 film)</li> <li><i><a href="/wiki/Les_Palmes_de_M._Schutz" title="Les Palmes de M. Schutz">Les Palmes de M. Schutz</a></i> (1997 film)</li> <li><i><a href="/wiki/Marie_Curie,_une_femme_sur_le_front" title="Marie Curie, une femme sur le front">Marie Curie, une femme sur le front</a></i> (2014 film)</li> <li><i><a href="/wiki/Marie_Curie:_The_Courage_of_Knowledge" title="Marie Curie: The Courage of Knowledge">Marie Curie: The Courage of Knowledge</a></i> (2016 film)</li> <li><i><a href="/wiki/Radioactive_(film)" title="Radioactive (film)">Radioactive</a></i> (2019 film)</li></ul> </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><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"><link rel="mw-deduplicated-inline-style" 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class="external text" href="https://id.loc.gov/authorities/sh85102071">United States</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Piézoélectricité"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb12254070z">France</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Piézoélectricité"><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb12254070z">BnF data</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="圧電気"><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00560405">Japan</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="piezoelektrický jev"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph124074&CON_LNG=ENG">Czech Republic</a></span></span><ul><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="piezoelektřina"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph392423&CON_LNG=ENG">2</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="piezoelektrika"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph624207&CON_LNG=ENG">3</a></span></span></li></ul></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=987007548576305171">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" 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Piezoelectricity - Wikipedia