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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">Larmor precession</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 26 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-26" 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">26 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A8%D8%AF%D8%A7%D8%B1%D9%8A%D8%A9_%D9%84%D8%A7%D8%B1%D9%85%D9%88%D8%B1" title="بدارية لارمور – Arabic" lang="ar" hreflang="ar" data-title="بدارية لارمور" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D1%80%D0%B5%D1%86%D0%B5%D1%81%D0%B8%D1%8F_%D0%B8_%D1%87%D0%B5%D1%81%D1%82%D0%BE%D1%82%D0%B0_%D0%BD%D0%B0_%D0%9B%D0%B0%D1%80%D0%BC%D0%BE%D1%80" title="Прецесия и честота на Лармор – Bulgarian" lang="bg" hreflang="bg" data-title="Прецесия и честота на Лармор" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Precessi%C3%B3_de_Larmor" title="Precessió de Larmor – Catalan" lang="ca" hreflang="ca" data-title="Precessió de Larmor" 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/Larmorova_precese" title="Larmorova precese – Czech" lang="cs" hreflang="cs" data-title="Larmorova precese" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Larmorpr%C3%A4zession" title="Larmorpräzession – German" lang="de" hreflang="de" data-title="Larmorpräzession" 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/Larmori_pretsessioon" title="Larmori pretsessioon – Estonian" lang="et" hreflang="et" data-title="Larmori pretsessioon" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Precesi%C3%B3n_de_Larmor" title="Precesión de Larmor – Spanish" lang="es" hreflang="es" data-title="Precesión de Larmor" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D8%B1%DA%A9%D8%A7%D9%86%D8%B3_%D9%84%D8%A7%D8%B1%D9%85%D9%88%D8%B1" title="فرکانس لارمور – Persian" lang="fa" hreflang="fa" data-title="فرکانس لارمور" 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/Pr%C3%A9cession_de_Larmor" title="Précession de Larmor – French" lang="fr" hreflang="fr" data-title="Précession de Larmor" 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/Luain%C3%ADocht_Larmor" title="Luainíocht Larmor – Irish" lang="ga" hreflang="ga" data-title="Luainíocht Larmor" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Larmorova_precesija" title="Larmorova precesija – Croatian" lang="hr" hreflang="hr" data-title="Larmorova precesija" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Precessione_di_Larmor" title="Precessione di Larmor – Italian" lang="it" hreflang="it" data-title="Precessione di Larmor" 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%A0%D7%A7%D7%99%D7%A4%D7%AA_%D7%9C%D7%A8%D7%9E%D7%95%D7%A8" title="נקיפת לרמור – Hebrew" lang="he" hreflang="he" data-title="נקיפת לרמור" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Larmoro_precesija" title="Larmoro precesija – Lithuanian" lang="lt" hreflang="lt" data-title="Larmoro precesija" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Larmor-precesszi%C3%B3" title="Larmor-precesszió – Hungarian" lang="hu" hreflang="hu" data-title="Larmor-precesszió" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%A9%E3%83%BC%E3%83%A2%E3%82%A2%E6%AD%B3%E5%B7%AE%E9%81%8B%E5%8B%95" title="ラーモア歳差運動 – Japanese" lang="ja" hreflang="ja" data-title="ラーモア歳差運動" 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-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Larmorfrekwens" title="Larmorfrekwens – Northern Frisian" lang="frr" hreflang="frr" data-title="Larmorfrekwens" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Larmorpresesjon" title="Larmorpresesjon – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Larmorpresesjon" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Precesja_Larmora" title="Precesja Larmora – Polish" lang="pl" hreflang="pl" data-title="Precesja Larmora" 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/Frequ%C3%AAncia_de_Larmor" title="Frequência de Larmor – Portuguese" lang="pt" hreflang="pt" data-title="Frequência de Larmor" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9B%D0%B0%D1%80%D0%BC%D0%BE%D1%80%D0%BE%D0%B2%D1%81%D0%BA%D0%B0%D1%8F_%D0%BF%D1%80%D0%B5%D1%86%D0%B5%D1%81%D1%81%D0%B8%D1%8F" title="Ларморовская прецессия – Russian" lang="ru" hreflang="ru" data-title="Ларморовская прецессия" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Larmorjeva_frekvenca" title="Larmorjeva frekvenca – Slovenian" lang="sl" hreflang="sl" data-title="Larmorjeva frekvenca" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Larmorin_prekessio" title="Larmorin prekessio – Finnish" lang="fi" hreflang="fi" data-title="Larmorin prekessio" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%AB%E0%B8%A1%E0%B8%B8%E0%B8%99%E0%B8%84%E0%B8%A7%E0%B8%87%E0%B8%A5%E0%B8%B2%E0%B8%A3%E0%B9%8C%E0%B9%80%E0%B8%A1%E0%B8%AD%E0%B8%A3%E0%B9%8C" title="การหมุนควงลาร์เมอร์ – Thai" lang="th" hreflang="th" data-title="การหมุนควงลาร์เมอร์" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9B%D0%B0%D1%80%D0%BC%D0%BE%D1%80%D0%BE%D0%B2%D0%B0_%D1%87%D0%B0%D1%81%D1%82%D0%BE%D1%82%D0%B0" title="Ларморова частота – Ukrainian" lang="uk" hreflang="uk" data-title="Ларморова частота" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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<div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Movement of an object's magnetic moment axis about a magnetic field</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Precession_in_magnetic_field.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Precession_in_magnetic_field.svg/220px-Precession_in_magnetic_field.svg.png" decoding="async" width="220" height="551" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Precession_in_magnetic_field.svg/330px-Precession_in_magnetic_field.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Precession_in_magnetic_field.svg/440px-Precession_in_magnetic_field.svg.png 2x" data-file-width="304" data-file-height="761" /></a><figcaption>Direction of precession for a particle with positive gyromagnetic ratio. <style data-mw-deduplicate="TemplateStyles:r981673959">.mw-parser-output .legend{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .legend-color{display:inline-block;min-width:1.25em;height:1.25em;line-height:1.25;margin:1px 0;text-align:center;border:1px solid black;background-color:transparent;color:black}.mw-parser-output .legend-text{}</style><div class="legend"><span class="legend-line mw-no-invert" style="display: inline-block; vertical-align: middle; width: 1.67em; height: 0; border-style: none; border-top: 2px dotted black;border-top:solid green 4px;"> </span> External magnetic field</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959"><div class="legend"><span class="legend-line mw-no-invert" style="display: inline-block; vertical-align: middle; width: 1.67em; height: 0; border-style: none; border-top: 2px dotted black;border-top:#222222 solid 3px;"> </span> Magnetic dipole moment</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959"><div class="legend"><span class="legend-line mw-no-invert" style="display: inline-block; vertical-align: middle; width: 1.67em; height: 0; border-style: none; border-top: 2px dotted black;border-top:#222222 dashed 2px;"> </span> Precession of the dipole axis</div> </figcaption></figure> <p>In <a href="/wiki/Physics" title="Physics">physics</a>, <b>Larmor precession</b> (named after <a href="/wiki/Joseph_Larmor" title="Joseph Larmor">Joseph Larmor</a>) is the <a href="/wiki/Precession" title="Precession">precession</a> of the <a href="/wiki/Magnetic_moment" title="Magnetic moment">magnetic moment</a> of an object about an external <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a>. The phenomenon is conceptually similar to the precession of a tilted classical <a href="/wiki/Gyroscope" title="Gyroscope">gyroscope</a> in an external torque-exerting gravitational field. Objects with a magnetic moment also have <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> and effective internal <a href="/wiki/Electric_current" title="Electric current">electric current</a> proportional to their angular momentum; these include <a href="/wiki/Electron" title="Electron">electrons</a>, <a href="/wiki/Proton" title="Proton">protons</a>, other <a href="/wiki/Fermion" title="Fermion">fermions</a>, many <a href="/wiki/Atom" title="Atom">atomic</a> and <a href="/wiki/Atomic_nucleus" title="Atomic nucleus">nuclear</a> systems, as well as classical macroscopic systems. The external magnetic field exerts a <a href="/wiki/Torque" title="Torque">torque</a> on the magnetic moment, </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 {\vec {\tau }}={\vec {\mu }}\times {\vec {B}}=\gamma {\vec {J}}\times {\vec {B}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>J</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\tau }}={\vec {\mu }}\times {\vec {B}}=\gamma {\vec {J}}\times {\vec {B}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2bc572253b5ca57df2413932166ac5a9c49e900" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.838ex; height:3.343ex;" alt="{\displaystyle {\vec {\tau }}={\vec {\mu }}\times {\vec {B}}=\gamma {\vec {J}}\times {\vec {B}},}"></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 {\vec {\tau }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\tau }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9476c9325cf30de081340cd070b30c3bd93f311a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.333ex; height:2.343ex;" alt="{\displaystyle {\vec {\tau }}}"></span> is the torque, <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 {\vec {\mu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\mu }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddeda8d3995d1518e6f2d38e812641f579126be4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.843ex;" alt="{\displaystyle {\vec {\mu }}}"></span> is the magnetic dipole moment, <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 {\vec {J}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>J</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {J}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f64ab02528d2b80e1df79bc2a8762489a986afa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.788ex; height:2.843ex;" alt="{\displaystyle {\vec {J}}}"></span> is the <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> vector, <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 {\vec {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83ae7d80cab55b606de217162280b2279142bbb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.843ex;" alt="{\displaystyle {\vec {B}}}"></span> is the external magnetic field, <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 \times }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>×</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \times }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffafff1ad26cbe49045f19a67ce532116a32703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.019ex; margin-bottom: -0.19ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \times }"></span> symbolizes the <a href="/wiki/Cross_product" title="Cross product">cross product</a>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> is the <a href="/wiki/Gyromagnetic_ratio" title="Gyromagnetic ratio">gyromagnetic ratio</a> which gives the proportionality constant between the magnetic moment and the angular momentum. The angular momentum vector <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 {\vec {J}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>J</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {J}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f64ab02528d2b80e1df79bc2a8762489a986afa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.788ex; height:2.843ex;" alt="{\displaystyle {\vec {J}}}"></span> precesses about the external field axis with an <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a> known as the <b>Larmor frequency</b>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =\vert \gamma B\vert }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mo fence="false" stretchy="false">|</mo> <mi>γ</mi> <mi>B</mi> <mo fence="false" stretchy="false">|</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =\vert \gamma B\vert }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e0863f6e5954ccd16667c7eaaecb71266a010ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.864ex; height:2.843ex;" alt="{\displaystyle \omega =\vert \gamma B\vert }"></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 \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> is the <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> is the magnitude of the applied magnetic field, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> is the <a href="/wiki/Gyromagnetic_ratio" title="Gyromagnetic ratio">gyromagnetic ratio</a> for a <a href="/wiki/Elementary_charge" title="Elementary charge">particle of charge</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07ba5105b33e9ac88f605cc50663a9e2c16fcc63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.892ex; height:2.176ex;" alt="{\displaystyle -e}"></span>,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {eg}{2m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>e</mi> <mi>g</mi> </mrow> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {eg}{2m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4b43ff321ae8c8c355ae40b2042fa14baefe25f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.847ex; height:4.843ex;" alt="{\displaystyle -{\frac {eg}{2m}}}"></span>, 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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> is the mass of the precessing system, while <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 g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> is the <a href="/wiki/G-factor_(physics)" title="G-factor (physics)"><i>g</i>-factor</a> of the system. The <i>g</i>-factor is the unit-less proportionality factor relating the system's angular momentum to the intrinsic magnetic moment; in classical physics it is 1 for any rigid object in which the charge and mass density are identically distributed. The Larmor frequency is independent of the angle between <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 {\vec {J}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>J</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {J}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f64ab02528d2b80e1df79bc2a8762489a986afa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.788ex; height:2.843ex;" alt="{\displaystyle {\vec {J}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83ae7d80cab55b606de217162280b2279142bbb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.843ex;" alt="{\displaystyle {\vec {B}}}"></span>. </p><p>In <a href="/wiki/Nuclear_physics" title="Nuclear physics">nuclear physics</a> the <i>g</i>-factor of a given system includes the effect of the <a href="/wiki/Nucleon" title="Nucleon">nucleon</a> spins, their orbital angular momenta, and their <a href="/wiki/Angular_momentum_coupling" title="Angular momentum coupling">couplings</a>. Generally, the <i>g</i>-factors are very difficult to calculate for such many-body systems, but they have been measured to high precision for most nuclei. The Larmor frequency is important in <a href="/wiki/NMR_spectroscopy" class="mw-redirect" title="NMR spectroscopy">NMR spectroscopy</a>. The gyromagnetic ratios, which give the Larmor frequencies at a given magnetic field strength, have been measured and tabulated.<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> </p><p>Crucially, the Larmor frequency is independent of the polar angle between the applied magnetic field and the magnetic moment direction. This is what makes it a key concept in fields such as <a href="/wiki/Nuclear_magnetic_resonance" title="Nuclear magnetic resonance">nuclear magnetic resonance</a> (NMR) and <a href="/wiki/Electron_paramagnetic_resonance" title="Electron paramagnetic resonance">electron paramagnetic resonance</a> (EPR), since the precession rate does not depend on the spatial orientation of the spins. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Including_Thomas_precession">Including Thomas precession</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Larmor_precession&action=edit&section=1" title="Edit section: Including Thomas precession"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The above equation is the one that is used in most applications. However, a full treatment must include the effects of <a href="/wiki/Thomas_precession" title="Thomas precession">Thomas precession</a>, yielding the equation (in <a href="/wiki/CGS_units" class="mw-redirect" title="CGS units">CGS units</a>, which are used so that <i>E</i> has the same units as <i>B</i>): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{s}={\frac {geB}{2mc}}+(1-\gamma ){\frac {eB}{mc\gamma }}={\frac {eB}{2mc}}\left(g-2+{\frac {2}{\gamma }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>g</mi> <mi>e</mi> <mi>B</mi> </mrow> <mrow> <mn>2</mn> <mi>m</mi> <mi>c</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>e</mi> <mi>B</mi> </mrow> <mrow> <mi>m</mi> <mi>c</mi> <mi>γ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>e</mi> <mi>B</mi> </mrow> <mrow> <mn>2</mn> <mi>m</mi> <mi>c</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>g</mi> <mo>−</mo> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>γ</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{s}={\frac {geB}{2mc}}+(1-\gamma ){\frac {eB}{mc\gamma }}={\frac {eB}{2mc}}\left(g-2+{\frac {2}{\gamma }}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83f4658dc3e8dd0ed93a11a8911cb86e32374fe9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.664ex; height:6.176ex;" alt="{\displaystyle \omega _{s}={\frac {geB}{2mc}}+(1-\gamma ){\frac {eB}{mc\gamma }}={\frac {eB}{2mc}}\left(g-2+{\frac {2}{\gamma }}\right)}"></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 \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> is the relativistic <a href="/wiki/Lorentz_factor" title="Lorentz factor">Lorentz factor</a> (not to be confused with the gyromagnetic ratio above). Notably, for the electron <i>g</i> is very close to 2 (<span class="nowrap"><span data-sort-value="7000200199999999999♠"></span>2.002...</span>), so if one sets <i>g</i> = 2, one arrives at </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{s(g=2)}={\frac {eB}{mc\gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo>=</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>e</mi> <mi>B</mi> </mrow> <mrow> <mi>m</mi> <mi>c</mi> <mi>γ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{s(g=2)}={\frac {eB}{mc\gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a917ba0f00298c86d6378b2f3dd265e37ba6a93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.862ex; height:5.676ex;" alt="{\displaystyle \omega _{s(g=2)}={\frac {eB}{mc\gamma }}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Bargmann–Michel–Telegdi_equation"><span id="Bargmann.E2.80.93Michel.E2.80.93Telegdi_equation"></span>Bargmann–Michel–Telegdi equation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Larmor_precession&action=edit&section=2" title="Edit section: Bargmann–Michel–Telegdi equation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The spin precession of an electron in an external electromagnetic field is described by the Bargmann–Michel–Telegdi (BMT) equation (named after <a href="/wiki/Valentine_Bargmann" title="Valentine Bargmann">Valentine Bargmann</a>, <a href="/wiki/Louis_Michel_(physicist)" title="Louis Michel (physicist)">Louis Michel</a> and <a href="/wiki/Valentine_Telegdi" title="Valentine Telegdi">Valentine Telegdi</a>)<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {da^{\tau }}{ds}}={\frac {e}{m}}u^{\tau }u_{\sigma }F^{\sigma \lambda }a_{\lambda }+2\mu (F^{\tau \lambda }-u^{\tau }u_{\sigma }F^{\sigma \lambda })a_{\lambda },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <mi>s</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>e</mi> <mi>m</mi> </mfrac> </mrow> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msup> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msub> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mi>λ</mi> </mrow> </msup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <mi>μ</mi> <mo stretchy="false">(</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> <mi>λ</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msup> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msub> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mi>λ</mi> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {da^{\tau }}{ds}}={\frac {e}{m}}u^{\tau }u_{\sigma }F^{\sigma \lambda }a_{\lambda }+2\mu (F^{\tau \lambda }-u^{\tau }u_{\sigma }F^{\sigma \lambda })a_{\lambda },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/586dd21770c2b937f6b18200914df88d75f78127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:47.455ex; height:5.509ex;" alt="{\displaystyle {\frac {da^{\tau }}{ds}}={\frac {e}{m}}u^{\tau }u_{\sigma }F^{\sigma \lambda }a_{\lambda }+2\mu (F^{\tau \lambda }-u^{\tau }u_{\sigma }F^{\sigma \lambda })a_{\lambda },}"></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 a^{\tau }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{\tau }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae50d66cbdf50a7e020a2e011e7ae81bb17154d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.312ex; height:2.343ex;" alt="{\displaystyle a^{\tau }}"></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 e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> are polarization four-vector, charge, mass, and magnetic moment, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u^{\tau }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u^{\tau }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a99b7cbe3b66baffbe2de444ba498bf5578ef4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.412ex; height:2.343ex;" alt="{\displaystyle u^{\tau }}"></span> is four-velocity of electron (in a system of units in which <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 c=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e3467f9e219a5ea38a30da5c3a02c2c23f61a79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c=1}"></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 a^{\tau }a_{\tau }=-u^{\tau }u_{\tau }=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msup> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{\tau }a_{\tau }=-u^{\tau }u_{\tau }=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f84e349a5501940f9dbc699f6ce643cd20f01553" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.423ex; height:2.676ex;" alt="{\displaystyle a^{\tau }a_{\tau }=-u^{\tau }u_{\tau }=-1}"></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 u^{\tau }a_{\tau }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u^{\tau }a_{\tau }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaa7367b2792e04887b1719b923ac31d0af27a52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.985ex; height:2.676ex;" alt="{\displaystyle u^{\tau }a_{\tau }=0}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{\tau \sigma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> <mi>σ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F^{\tau \sigma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bddf99cb8056100bc533783261121aabb792ea8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.837ex; height:2.343ex;" alt="{\displaystyle F^{\tau \sigma }}"></span> is electromagnetic field-strength tensor. Using equations of motion, </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 m{\frac {du^{\tau }}{ds}}=eF^{\tau \sigma }u_{\sigma },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <mi>s</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>e</mi> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> <mi>σ</mi> </mrow> </msup> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m{\frac {du^{\tau }}{ds}}=eF^{\tau \sigma }u_{\sigma },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f578c0657af77a284fb7a6ab7a394b21bfb94922" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.672ex; height:5.509ex;" alt="{\displaystyle m{\frac {du^{\tau }}{ds}}=eF^{\tau \sigma }u_{\sigma },}"></span></dd></dl> <p>one can rewrite the first term on the right side of the BMT equation as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-u^{\tau }w^{\lambda }+u^{\lambda }w^{\tau })a_{\lambda }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msup> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msup> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-u^{\tau }w^{\lambda }+u^{\lambda }w^{\tau })a_{\lambda }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb0f66e138c1ba67202782239b9ac3e2cbf6c6c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.411ex; height:3.176ex;" alt="{\displaystyle (-u^{\tau }w^{\lambda }+u^{\lambda }w^{\tau })a_{\lambda }}"></span>, 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 w^{\tau }=du^{\tau }/ds}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msup> <mo>=</mo> <mi>d</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w^{\tau }=du^{\tau }/ds}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e71b4ea6c8e750e61ee06a32b294bb8798ac8814" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.941ex; height:2.843ex;" alt="{\displaystyle w^{\tau }=du^{\tau }/ds}"></span> is four-acceleration. This term describes <a href="/wiki/Fermi%E2%80%93Walker_transport" title="Fermi–Walker transport">Fermi–Walker transport</a> and leads to <a href="/wiki/Thomas_precession" title="Thomas precession">Thomas precession</a>. The second term is associated with Larmor precession. </p><p>When electromagnetic fields are uniform in space or when gradient forces like <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 ({\boldsymbol {\mu }}\cdot {\boldsymbol {B}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">μ</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">B</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla ({\boldsymbol {\mu }}\cdot {\boldsymbol {B}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e91514308895af3c0c656bd8243cd74224707221" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.082ex; height:2.843ex;" alt="{\displaystyle \nabla ({\boldsymbol {\mu }}\cdot {\boldsymbol {B}})}"></span> can be neglected, the particle's translational motion is described by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {du^{\alpha }}{d\tau }}={\frac {e}{m}}F^{\alpha \beta }u_{\beta }\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>e</mi> <mi>m</mi> </mfrac> </mrow> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {du^{\alpha }}{d\tau }}={\frac {e}{m}}F^{\alpha \beta }u_{\beta }\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a81c0e1411708ab0b5508d0fa27e1c047163a12c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.477ex; height:5.509ex;" alt="{\displaystyle {\frac {du^{\alpha }}{d\tau }}={\frac {e}{m}}F^{\alpha \beta }u_{\beta }\;.}"></span></dd></dl> <p>The BMT equation is then written as<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> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dS^{\alpha }}{d\tau }}={\frac {e}{m}}{\bigg [}{g \over 2}F^{\alpha \beta }S_{\beta }+\left({g \over 2}-1\right)u^{\alpha }\left(S_{\lambda }F^{\lambda \mu }u_{\mu }\right){\bigg ]}\;,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>e</mi> <mi>m</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>g</mi> <mn>2</mn> </mfrac> </mrow> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>g</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mi>μ</mi> </mrow> </msup> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dS^{\alpha }}{d\tau }}={\frac {e}{m}}{\bigg [}{g \over 2}F^{\alpha \beta }S_{\beta }+\left({g \over 2}-1\right)u^{\alpha }\left(S_{\lambda }F^{\lambda \mu }u_{\mu }\right){\bigg ]}\;,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/288f28feae2533d89e29de501c17fa7b8ad53aa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:49.905ex; height:6.176ex;" alt="{\displaystyle {\frac {dS^{\alpha }}{d\tau }}={\frac {e}{m}}{\bigg [}{g \over 2}F^{\alpha \beta }S_{\beta }+\left({g \over 2}-1\right)u^{\alpha }\left(S_{\lambda }F^{\lambda \mu }u_{\mu }\right){\bigg ]}\;,}"></span></dd></dl> <p>The Beam-Optical version of the Thomas-BMT, from the <i>Quantum Theory of Charged-Particle Beam Optics</i>, applicable in accelerator optics.<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><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Larmor_precession&action=edit&section=3" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A 1935 paper published by <a href="/wiki/Lev_Landau" title="Lev Landau">Lev Landau</a> and <a href="/wiki/Evgeny_Lifshitz" title="Evgeny Lifshitz">Evgeny Lifshitz</a> predicted the existence of <a href="/wiki/Ferromagnetic_resonance" title="Ferromagnetic resonance">ferromagnetic resonance</a> of the Larmor precession, which was independently verified in experiments by J. H. E. Griffiths (UK)<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Yevgeny_Zavoisky" title="Yevgeny Zavoisky">E. K. Zavoiskij</a> (USSR) in 1946.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><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> </p><p>Larmor precession is important in <a href="/wiki/Nuclear_magnetic_resonance" title="Nuclear magnetic resonance">nuclear magnetic resonance</a>, <a href="/wiki/Magnetic_resonance_imaging" title="Magnetic resonance imaging">magnetic resonance imaging</a>, <a href="/wiki/Electron_paramagnetic_resonance" title="Electron paramagnetic resonance">electron paramagnetic resonance</a>, <a href="/wiki/Muon_spin_spectroscopy" title="Muon spin spectroscopy">muon spin resonance</a>, and <a href="/wiki/Neutron_spin_echo" title="Neutron spin echo">neutron spin echo</a>. It is also important for the alignment of <a href="/wiki/Cosmic_dust" title="Cosmic dust">cosmic dust</a> grains, which is a cause of the <a href="/wiki/Polarization_of_starlight" class="mw-redirect" title="Polarization of starlight">polarization of starlight</a>. </p><p>To calculate the spin of a particle in a magnetic field, one must in general also take into account <a href="/wiki/Thomas_precession" title="Thomas precession">Thomas precession</a> if the particle is moving. </p> <div class="mw-heading mw-heading2"><h2 id="Precession_direction">Precession direction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Larmor_precession&action=edit&section=4" title="Edit section: Precession direction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The spin angular momentum of an electron precesses counter-clockwise about the direction of the magnetic field. An electron has a negative charge, so the direction of its magnetic moment is opposite to that of its spin. </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=Larmor_precession&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/LARMOR_neutron_microscope" title="LARMOR neutron microscope">LARMOR neutron microscope</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Larmor_precession&action=edit&section=6" title="Edit section: Notes"><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"><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">Spin Dynamics, Malcolm H. Levitt, Wiley, 2001,</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</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="CITEREFLouis_N._Hand_and_Janet_D._Finch.1998" class="citation book cs1">Louis N. Hand and Janet D. Finch. (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1J2hzvX2Xh8C&q=Larmor's+Theorem&pg=PA192"><i>Analytical Mechanics</i></a>. Cambridge, England: <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. p. 192. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-57572-0" title="Special:BookSources/978-0-521-57572-0"><bdi>978-0-521-57572-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analytical+Mechanics&rft.place=Cambridge%2C+England&rft.pages=192&rft.pub=Cambridge+University+Press&rft.date=1998&rft.isbn=978-0-521-57572-0&rft.au=Louis+N.+Hand+and+Janet+D.+Finch.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D1J2hzvX2Xh8C%26q%3DLarmor%27s%2BTheorem%26pg%3DPA192&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALarmor+precession" 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"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20180201154413/http://www-lcs.ensicaen.fr/pyPulsar/index.php/List_of_NMR_isotopes">List of NMR isotopes</a></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 href="/wiki/Valentine_Bargmann" title="Valentine Bargmann">V. Bargmann</a>, <a href="/wiki/Louis_Michel_(physicist)" title="Louis Michel (physicist)">L. Michel</a>, and <a href="/wiki/Valentine_Telegdi" title="Valentine Telegdi">V. L. Telegdi</a>, <i>Precession of the Polarization of Particles Moving in a Homogeneous Electromagnetic Field</i>, Phys. Rev. Lett. 2, 435 (1959).</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">Jackson, J. D., <i>Classical Electrodynamics</i>, 3rd edition, Wiley, 1999, p. 563.</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">M. Conte, <a rel="nofollow" class="external text" href="https://scholar.google.com/citations?user=mp7XSDAAAAAJ&hl=en">R. Jagannathan</a>, <a rel="nofollow" class="external text" href="https://scholar.google.com/citations?user=hZvL5eYAAAAJ&hl">S. A. Khan</a> and M. Pusterla, Beam optics of the Dirac particle with anomalous magnetic moment, Particle Accelerators, 56, 99–126 (1996); (Preprint: IMSc/96/03/07, INFN/AE-96/08).</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://inspirehep.net/author/S.A.Khan.5/">Khan, S. A.</a> (1997). <a rel="nofollow" class="external text" href="https://www.imsc.res.in/xmlui/handle/123456789/75?show=full">Quantum Theory of Charged-Particle Beam Optics</a>, <i>Ph.D Thesis</i>, <a href="/wiki/University_of_Madras" title="University of Madras">University of Madras</a>, <a href="/wiki/Chennai" title="Chennai">Chennai</a>, <a href="/wiki/India" title="India">India</a>. (complete thesis available from <a rel="nofollow" class="external text" href="https://www.imsc.res.in/xmlui/">Dspace of IMSc Library</a>, <a href="/wiki/Institute_of_Mathematical_Sciences,_Chennai" title="Institute of Mathematical Sciences, Chennai">The Institute of Mathematical Sciences</a>, where the doctoral research was done).</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJ._H._E._Griffiths1946" class="citation journal cs1">J. H. E. Griffiths (1946). "Anomalous high-frequency resistance of ferromagnetic metals". <i><a href="/wiki/Nature_(journal)" title="Nature (journal)">Nature</a></i>. <b>158</b> (4019): <span class="nowrap">670–</span>671. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1946Natur.158..670G">1946Natur.158..670G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F158670a0">10.1038/158670a0</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:4143499">4143499</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Anomalous+high-frequency+resistance+of+ferromagnetic+metals&rft.volume=158&rft.issue=4019&rft.pages=%3Cspan+class%3D%22nowrap%22%3E670-%3C%2Fspan%3E671&rft.date=1946&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A4143499%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1038%2F158670a0&rft_id=info%3Abibcode%2F1946Natur.158..670G&rft.au=J.+H.+E.+Griffiths&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALarmor+precession" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZavoisky1946" class="citation journal cs1"><a href="/wiki/Yevgeny_Zavoisky" title="Yevgeny Zavoisky">Zavoisky, E.</a> (1946). "Spin magnetic resonance in the decimeter-wave region". <i>Fizicheskiĭ Zhurnal</i>. <b>10</b>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Fizicheski%C4%AD+Zhurnal&rft.atitle=Spin+magnetic+resonance+in+the+decimeter-wave+region&rft.volume=10&rft.date=1946&rft.aulast=Zavoisky&rft.aufirst=E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALarmor+precession" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZavoisky1946" class="citation journal cs1"><a href="/wiki/Yevgeny_Zavoisky" title="Yevgeny Zavoisky">Zavoisky, E.</a> (1946). "Paramagnetic absorption in some salts in perpendicular magnetic fields". <i>Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki</i>. <b>16</b> (7): <span class="nowrap">603–</span>606.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Zhurnal+Eksperimental%27noi+i+Teoreticheskoi+Fiziki&rft.atitle=Paramagnetic+absorption+in+some+salts+in+perpendicular+magnetic+fields&rft.volume=16&rft.issue=7&rft.pages=%3Cspan+class%3D%22nowrap%22%3E603-%3C%2Fspan%3E606&rft.date=1946&rft.aulast=Zavoisky&rft.aufirst=E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALarmor+precession" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Larmor_precession&action=edit&section=7" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/larmor.html">Georgia State University HyperPhysics page on Larmor Frequency</a></li> <li><a rel="nofollow" class="external text" href="http://bio.groups.et.byu.net/LarmourFreqCal.phtml">Larmor Frequency Calculator</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Larmor_precession&oldid=1273047166">https://en.wikipedia.org/w/index.php?title=Larmor_precession&oldid=1273047166</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Atomic_physics" title="Category:Atomic physics">Atomic physics</a></li><li><a href="/wiki/Category:Electromagnetism" title="Category:Electromagnetism">Electromagnetism</a></li><li><a href="/wiki/Category:Nuclear_magnetic_resonance" title="Category:Nuclear magnetic resonance">Nuclear magnetic resonance</a></li><li><a href="/wiki/Category:Precession" title="Category:Precession">Precession</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 31 January 2025, at 12:17<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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