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id="toc-General_applications" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>General applications</span> </div> </a> <ul id="toc-General_applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Biomedical_applications" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Biomedical_applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Biomedical applications</span> </div> </a> <ul id="toc-Biomedical_applications-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <button aria-controls="toc-References-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 References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</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" 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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%A1%D1%83%D0%BF%D0%B5%D1%80%D0%BF%D0%B0%D1%80%D0%B0%D0%BC%D0%B0%D0%B3%D0%BD%D0%B5%D1%82%D0%B8%D0%B7%D1%8A%D0%BC" title="Суперпарамагнетизъм – Bulgarian" lang="bg" hreflang="bg" data-title="Суперпарамагнетизъм" 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/Superparamagnetisme" title="Superparamagnetisme – Catalan" lang="ca" hreflang="ca" data-title="Superparamagnetisme" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Superparamagnetismus" title="Superparamagnetismus – German" lang="de" hreflang="de" data-title="Superparamagnetismus" 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/Superparamagnetism" title="Superparamagnetism – Estonian" lang="et" hreflang="et" data-title="Superparamagnetism" 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/Superparamagnetismo" title="Superparamagnetismo – Spanish" lang="es" hreflang="es" data-title="Superparamagnetismo" 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/%D8%B3%D9%88%D9%BE%D8%B1%D9%BE%D8%A7%D8%B1%D8%A7%D9%85%D8%BA%D9%86%D8%A7%D8%B7%DB%8C%D8%B3" title="سوپرپارامغناطیس – Persian" lang="fa" hreflang="fa" data-title="سوپرپارامغناطیس" 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/Superparamagn%C3%A9tisme" title="Superparamagnétisme – French" lang="fr" hreflang="fr" data-title="Superparamagnétisme" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Superparamagnetismo" title="Superparamagnetismo – Galician" lang="gl" hreflang="gl" data-title="Superparamagnetismo" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Superparamagnetismo" title="Superparamagnetismo – Italian" lang="it" hreflang="it" data-title="Superparamagnetismo" 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%90%D7%A8%D7%90%D7%9E%D7%92%D7%A0%D7%98%D7%99%D7%95%D7%AA_%D7%A2%D7%9C" title="פאראמגנטיות על – Hebrew" lang="he" hreflang="he" data-title="פאראמגנטיות על" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A1%D1%83%D0%BF%D0%B5%D1%80%D0%BC%D0%B0%D0%B3%D0%BD%D0%B5%D1%82%D0%B8%D0%B7%D0%BC" title="Супермагнетизм – Kazakh" lang="kk" hreflang="kk" data-title="Супермагнетизм" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%A1%D1%83%D0%BF%D0%B5%D1%80%D0%BF%D0%B0%D1%80%D0%B0%D0%BC%D0%B0%D0%B3%D0%BD%D0%B5%D1%82%D0%B8%D0%B7%D0%BC" title="Суперпарамагнетизм – Kyrgyz" lang="ky" hreflang="ky" data-title="Суперпарамагнетизм" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%B6%85%E5%B8%B8%E7%A3%81%E6%80%A7" title="超常磁性 – Japanese" lang="ja" hreflang="ja" data-title="超常磁性" 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-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Superparamagnetizm" title="Superparamagnetizm – Uzbek" lang="uz" hreflang="uz" data-title="Superparamagnetizm" 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/Superparamagnetyzm" title="Superparamagnetyzm – Polish" lang="pl" hreflang="pl" data-title="Superparamagnetyzm" 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/Superparamagnetismo" title="Superparamagnetismo – Portuguese" lang="pt" hreflang="pt" data-title="Superparamagnetismo" 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%A1%D1%83%D0%BF%D0%B5%D1%80%D0%BF%D0%B0%D1%80%D0%B0%D0%BC%D0%B0%D0%B3%D0%BD%D0%B5%D1%82%D0%B8%D0%B7%D0%BC" title="Суперпарамагнетизм – Russian" lang="ru" hreflang="ru" data-title="Суперпарамагнетизм" 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/Superparamagnetism" title="Superparamagnetism – Simple English" lang="en-simple" hreflang="en-simple" data-title="Superparamagnetism" 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/Superparamagnetizmus" title="Superparamagnetizmus – Slovak" lang="sk" hreflang="sk" data-title="Superparamagnetizmus" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A1%D1%83%D0%BF%D0%B5%D1%80%D0%BF%D0%B0%D1%80%D0%B0%D0%BC%D0%B0%D0%B3%D0%BD%D0%B5%D1%82%D0%B8%D0%B7%D0%B0%D0%BC" title="Суперпарамагнетизам – Serbian" lang="sr" hreflang="sr" data-title="Суперпарамагнетизам" 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-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/S%C3%BCperparamanyetizma" title="Süperparamanyetizma – Turkish" lang="tr" hreflang="tr" data-title="Süperparamanyetizma" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link 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href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks hlist"><tbody><tr><th class="sidebar-title"><a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">Condensed matter physics</a></th></tr><tr><td class="sidebar-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:QuantumPhaseTransition.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/QuantumPhaseTransition.svg/220px-QuantumPhaseTransition.svg.png" decoding="async" width="220" height="159" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/QuantumPhaseTransition.svg/330px-QuantumPhaseTransition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/QuantumPhaseTransition.svg/440px-QuantumPhaseTransition.svg.png 2x" data-file-width="512" data-file-height="369" /></a></span></td></tr><tr><td class="sidebar-content"> <div class="hlist"><ul><li><a href="/wiki/Phase_(matter)" title="Phase (matter)">Phases</a></li><li><a href="/wiki/Phase_transition" title="Phase transition">Phase transition</a></li><li><a href="/wiki/Quantum_critical_point" title="Quantum critical point">QCP</a></li></ul></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/State_of_matter" title="State of matter">States of matter</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Solid" title="Solid">Solid</a></li><li><a href="/wiki/Liquid" title="Liquid">Liquid</a></li><li><a href="/wiki/Gas" title="Gas">Gas</a></li><li><a href="/wiki/Plasma_(physics)" title="Plasma (physics)">Plasma</a></li><li><a href="/wiki/Bose%E2%80%93Einstein_condensate" title="Bose–Einstein condensate">Bose–Einstein condensate</a></li><li><a href="/wiki/Bose_gas" title="Bose gas">Bose gas</a></li><li><a href="/wiki/Fermionic_condensate" title="Fermionic condensate">Fermionic condensate</a></li><li><a href="/wiki/Fermi_gas" title="Fermi gas">Fermi gas</a></li><li><a href="/wiki/Fermi_liquid_theory" title="Fermi liquid theory">Fermi liquid</a></li><li><a href="/wiki/Supersolid" title="Supersolid">Supersolid</a></li><li><a href="/wiki/Superfluidity" title="Superfluidity">Superfluidity</a></li><li><a href="/wiki/Luttinger_liquid" title="Luttinger liquid">Luttinger liquid</a></li><li><a href="/wiki/Time_crystal" title="Time crystal">Time crystal</a></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Phase phenomena</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Order_parameter" class="mw-redirect" title="Order parameter">Order parameter</a></li><li><a href="/wiki/Phase_transition" title="Phase transition">Phase transition</a></li><li><a href="/wiki/Quantum_critical_point" title="Quantum critical point">QCP</a></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Electronic phases</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Electronic_band_structure" title="Electronic band structure">Electronic band structure</a></li><li><a href="/wiki/Plasma_(physics)" title="Plasma (physics)">Plasma</a></li><li><a href="/wiki/Insulator_(electricity)" title="Insulator (electricity)">Insulator</a></li><li><a href="/wiki/Mott_insulator" title="Mott insulator">Mott insulator</a></li><li><a href="/wiki/Semiconductor" title="Semiconductor">Semiconductor</a></li><li><a href="/wiki/Semimetal" title="Semimetal">Semimetal</a></li><li><a href="/wiki/Electrical_conductor" title="Electrical conductor">Conductor</a></li><li><a href="/wiki/Superconductivity" title="Superconductivity">Superconductor</a></li><li><a href="/wiki/Thermoelectric_effect" title="Thermoelectric effect">Thermoelectric</a></li><li><a href="/wiki/Piezoelectricity" title="Piezoelectricity">Piezoelectric</a></li><li><a href="/wiki/Ferroelectricity" title="Ferroelectricity">Ferroelectric</a></li><li><a href="/wiki/Topological_insulator" title="Topological insulator">Topological insulator</a></li><li><a href="/wiki/Spin_gapless_semiconductor" title="Spin gapless semiconductor">Spin gapless semiconductor</a></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Electronic phenomena</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Quantum_Hall_effect" title="Quantum Hall effect">Quantum Hall effect</a></li><li><a href="/wiki/Spin_Hall_effect" title="Spin Hall effect">Spin Hall effect</a></li><li><a href="/wiki/Kondo_effect" title="Kondo effect">Kondo effect</a></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Magnetic phases</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Diamagnetism" title="Diamagnetism">Diamagnet</a></li><li><a href="/wiki/Superdiamagnetism" title="Superdiamagnetism">Superdiamagnet</a><br /> <a href="/wiki/Paramagnetism" title="Paramagnetism">Paramagnet</a></li><li><a class="mw-selflink selflink">Superparamagnet</a><br /><a href="/wiki/Ferromagnetism" title="Ferromagnetism">Ferromagnet</a></li><li><a href="/wiki/Antiferromagnetism" title="Antiferromagnetism">Antiferromagnet</a><br /><a href="/wiki/Metamagnetism" title="Metamagnetism">Metamagnet</a></li><li><a href="/wiki/Spin_glass" title="Spin glass">Spin glass</a></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Quasiparticle" title="Quasiparticle">Quasiparticles</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Phonon" title="Phonon">Phonon</a></li><li><a href="/wiki/Exciton" title="Exciton">Exciton</a></li><li><a href="/wiki/Plasmon" title="Plasmon">Plasmon</a><br /><a href="/wiki/Polariton" title="Polariton">Polariton</a></li><li><a href="/wiki/Polaron" title="Polaron">Polaron</a></li><li><a href="/wiki/Magnon" title="Magnon">Magnon</a></li><li><a href="/wiki/Roton" title="Roton">Roton</a></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Soft_matter" title="Soft matter">Soft matter</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Amorphous_solid" title="Amorphous solid">Amorphous solid</a></li><li><a href="/wiki/Colloid" title="Colloid">Colloid</a></li><li><a href="/wiki/Granular_material" title="Granular material">Granular material</a></li><li><a href="/wiki/Liquid_crystal" title="Liquid crystal">Liquid crystal</a></li><li><a href="/wiki/Polymer" title="Polymer">Polymer</a></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Johannes_Diderik_van_der_Waals" title="Johannes Diderik van der Waals">Van der Waals</a></li><li><a href="/wiki/Heike_Kamerlingh_Onnes" title="Heike Kamerlingh Onnes">Onnes</a></li><li><a href="/wiki/Max_von_Laue" title="Max von Laue">von Laue</a></li><li><a href="/wiki/William_Henry_Bragg" title="William Henry Bragg">Bragg</a></li><li><a href="/wiki/Peter_Debye" title="Peter Debye">Debye</a></li><li><a href="/wiki/Felix_Bloch" title="Felix Bloch">Bloch</a></li><li><a href="/wiki/Lars_Onsager" title="Lars Onsager">Onsager</a></li><li><a href="/wiki/Neville_Mott" class="mw-redirect" title="Neville Mott">Mott</a></li><li><a href="/wiki/Rudolf_Peierls" title="Rudolf Peierls">Peierls</a></li><li><a href="/wiki/Lev_Landau" title="Lev Landau">Landau</a></li><li><a href="/wiki/Luttinger" class="mw-redirect" title="Luttinger">Luttinger</a></li><li><a href="/wiki/Philip_Warren_Anderson" class="mw-redirect" title="Philip Warren Anderson">Anderson</a></li><li><a href="/wiki/John_Hasbrouck_Van_Vleck" title="John Hasbrouck Van Vleck">Van Vleck</a></li><li><a href="/wiki/John_Hubbard_(physicist)" title="John Hubbard (physicist)">Hubbard</a></li><li><a href="/wiki/William_Bradford_Shockley" class="mw-redirect" title="William Bradford Shockley">Shockley</a></li><li><a href="/wiki/John_Bardeen" title="John Bardeen">Bardeen</a></li><li><a href="/wiki/Leon_Cooper" title="Leon Cooper">Cooper</a></li><li><a href="/wiki/John_Robert_Schrieffer" title="John Robert Schrieffer">Schrieffer</a></li><li><a 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In sufficiently small nanoparticles, magnetization can randomly flip direction under the influence of temperature. The typical time between two flips is called the <a href="/wiki/N%C3%A9el_relaxation_theory" title="Néel relaxation theory">Néel relaxation time</a>. In the absence of an external magnetic field, when the time used to measure the magnetization of the nanoparticles is much longer than the Néel relaxation time, their magnetization appears to be on average zero; they are said to be in the superparamagnetic state. In this state, an external magnetic field is able to magnetize the nanoparticles, similarly to a <a href="/wiki/Paramagnet" class="mw-redirect" title="Paramagnet">paramagnet</a>. However, their <a href="/wiki/Magnetic_susceptibility" title="Magnetic susceptibility">magnetic susceptibility</a> is much larger than that of paramagnets. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="The_Néel_relaxation_in_the_absence_of_magnetic_field"><span id="The_N.C3.A9el_relaxation_in_the_absence_of_magnetic_field"></span>The Néel relaxation in the absence of magnetic field</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superparamagnetism&action=edit&section=1" title="Edit section: The Néel relaxation in the absence of magnetic field"><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">Main article: <a href="/wiki/N%C3%A9el_relaxation_theory" title="Néel relaxation theory">Néel relaxation theory</a></div> <p>Normally, any ferromagnetic or ferrimagnetic material undergoes a transition to a paramagnetic state above its <a href="/wiki/Curie_temperature" title="Curie temperature">Curie temperature</a>. Superparamagnetism is different from this standard transition since it occurs below the Curie temperature of the material. </p><p>Superparamagnetism occurs in nanoparticles which are <a href="/wiki/Single_domain_(magnetic)" title="Single domain (magnetic)">single-domain</a>, i.e. composed of a single <a href="/wiki/Magnetic_domain" title="Magnetic domain">magnetic domain</a>. This is possible when their diameter is below 3–50 nm, depending on the materials. In this condition, it is considered that the magnetization of the nanoparticles is a single giant magnetic moment, sum of all the individual magnetic moments carried by the atoms of the nanoparticle. Those in the field of superparamagnetism call this "macro-spin approximation". </p><p>Because of the nanoparticle’s <a href="/wiki/Magnetic_anisotropy" title="Magnetic anisotropy">magnetic anisotropy</a>, the magnetic moment has usually only two stable orientations antiparallel to each other, separated by an <a href="/wiki/Energy_barrier" class="mw-redirect" title="Energy barrier">energy barrier</a>. The stable orientations define the nanoparticle’s so called “easy axis”. At finite temperature, there is a finite probability for the magnetization to flip and reverse its direction. The mean time between two flips is called the Néel relaxation time <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 \tau _{\text{N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>N</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{\text{N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eae89f9963ded0fe3729f5c9ed89eacf8ac339ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.481ex; height:2.009ex;" alt="{\displaystyle \tau _{\text{N}}}"></span> and is given by the following Néel–Arrhenius equation:<sup id="cite_ref-Neel49_1-0" class="reference"><a href="#cite_note-Neel49-1"><span class="cite-bracket">[</span>1<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 \tau _{\text{N}}=\tau _{0}\exp \left({\frac {KV}{k_{\text{B}}T}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>N</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>K</mi> <mi>V</mi> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{\text{N}}=\tau _{0}\exp \left({\frac {KV}{k_{\text{B}}T}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a705b5474069b77b10c092b3b74a584d4f2c45b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.09ex; height:6.176ex;" alt="{\displaystyle \tau _{\text{N}}=\tau _{0}\exp \left({\frac {KV}{k_{\text{B}}T}}\right)}"></span>,</dd></dl> <p>where: </p> <ul><li><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 \tau _{\text{N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>N</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{\text{N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eae89f9963ded0fe3729f5c9ed89eacf8ac339ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.481ex; height:2.009ex;" alt="{\displaystyle \tau _{\text{N}}}"></span> is thus the average length of time that it takes for the nanoparticle’s magnetization to randomly flip as a result of <a href="/wiki/Thermal_fluctuations" title="Thermal fluctuations">thermal fluctuations</a>.</li> <li><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 \tau _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56dad457e274b970f5d98b9dc40bef7f895c7f6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.07ex; height:2.009ex;" alt="{\displaystyle \tau _{0}}"></span> is a length of time, characteristic of the material, called the <i>attempt time</i> or <i>attempt period</i> (its reciprocal is called the <i>attempt frequency</i>); its typical value is between 10<sup>−9</sup> and 10<sup>−10</sup> second.</li> <li><i>K</i> is the nanoparticle’s magnetic anisotropy energy density and <i>V</i> its volume. <i>KV</i> is therefore the <a href="/wiki/Energy_barrier" class="mw-redirect" title="Energy barrier">energy barrier</a> associated with the magnetization moving from its initial easy axis direction, through a “hard plane”, to the other easy axis direction.</li> <li><i>k</i><sub>B</sub> is the <a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a>.</li> <li><i>T</i> is the temperature.</li></ul> <p>This length of time can be anywhere from a few nanoseconds to years or much longer. In particular, it can be seen that the Néel relaxation time is an exponential function of the grain volume, which explains why the flipping probability becomes rapidly negligible for bulk materials or large nanoparticles. </p> <div class="mw-heading mw-heading3"><h3 id="Blocking_temperature">Blocking temperature</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superparamagnetism&action=edit&section=2" title="Edit section: Blocking temperature"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let us imagine that the magnetization of a single superparamagnetic nanoparticle is measured and let us define <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 \tau _{\text{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{\text{m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93f2b63d7ea5521a2e5898e9e53b6c45b09a898e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.617ex; height:2.009ex;" alt="{\displaystyle \tau _{\text{m}}}"></span> as the measurement time. If <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 \tau _{\text{m}}\gg \tau _{\text{N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </msub> <mo>≫</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>N</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{\text{m}}\gg \tau _{\text{N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0c330120ececb60098fb12badaf8b1a0949134b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.712ex; height:2.176ex;" alt="{\displaystyle \tau _{\text{m}}\gg \tau _{\text{N}}}"></span>, the nanoparticle magnetization will flip several times during the measurement, then the measured magnetization will average to zero. If <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 \tau _{\text{m}}\ll \tau _{\text{N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </msub> <mo>≪</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>N</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{\text{m}}\ll \tau _{\text{N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e6a1c6f3b519d6ea99fbfd0587562afdb5b49b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.712ex; height:2.176ex;" alt="{\displaystyle \tau _{\text{m}}\ll \tau _{\text{N}}}"></span>, the magnetization will not flip during the measurement, so the measured magnetization will be what the instantaneous magnetization was at the beginning of the measurement. In the former case, the nanoparticle will appear to be in the superparamagnetic state whereas in the latter case it will appear to be “blocked” in its initial state. </p><p>The state of the nanoparticle (superparamagnetic or blocked) depends on the measurement time. A transition between superparamagnetism and blocked state occurs when <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 \tau _{\text{m}}=\tau _{\text{N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>N</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{\text{m}}=\tau _{\text{N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92600bb878c2836b6666e159c92fbfe6ee711c5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.197ex; height:2.009ex;" alt="{\displaystyle \tau _{\text{m}}=\tau _{\text{N}}}"></span>. In several experiments, the measurement time is kept constant but the temperature is varied, so the transition between superparamagnetism and blocked state is seen as a function of the temperature. The temperature for 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 \tau _{\text{m}}=\tau _{\text{N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>N</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{\text{m}}=\tau _{\text{N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92600bb878c2836b6666e159c92fbfe6ee711c5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.197ex; height:2.009ex;" alt="{\displaystyle \tau _{\text{m}}=\tau _{\text{N}}}"></span> is called the <i>blocking temperature</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 T_{\text{B}}={\frac {KV}{k_{\text{B}}\ln \left({\frac {\tau _{\text{m}}}{\tau _{0}}}\right)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>K</mi> <mi>V</mi> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </msub> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\text{B}}={\frac {KV}{k_{\text{B}}\ln \left({\frac {\tau _{\text{m}}}{\tau _{0}}}\right)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adf5660da0a3a043be9649ad82a94a0d59a08316" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:17.227ex; height:7.843ex;" alt="{\displaystyle T_{\text{B}}={\frac {KV}{k_{\text{B}}\ln \left({\frac {\tau _{\text{m}}}{\tau _{0}}}\right)}}}"></span></dd></dl> <p>For typical laboratory measurements, the value of the logarithm in the previous equation is in the order of 20–25. </p><p>Equivalently, blocking temperature is the temperature below which a material shows slow relaxation of magnetization.<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> </p> <div class="mw-heading mw-heading2"><h2 id="Effect_of_a_magnetic_field">Effect of a magnetic field</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superparamagnetism&action=edit&section=3" title="Edit section: Effect of a magnetic field"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Langevin_function.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Langevin_function.png/300px-Langevin_function.png" decoding="async" width="300" height="235" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Langevin_function.png/450px-Langevin_function.png 1.5x, //upload.wikimedia.org/wikipedia/commons/b/b6/Langevin_function.png 2x" data-file-width="564" data-file-height="442" /></a><figcaption>Langevin function (red line), compared with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \tanh \left({\frac {1}{3}}x\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>tanh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \tanh \left({\frac {1}{3}}x\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd1b8f847c1c8133d79d33ba14a2c4115a004bb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.415ex; height:4.843ex;" alt="{\textstyle \tanh \left({\frac {1}{3}}x\right)}"></span> (blue line).</figcaption></figure> <p>When an external magnetic field <i>H</i> is applied to an assembly of superparamagnetic nanoparticles, their magnetic moments tend to align along the applied field, leading to a net magnetization. The magnetization curve of the assembly, i.e. the magnetization as a function of the applied field, is a reversible S-shaped <a href="/wiki/Increasing_function" class="mw-redirect" title="Increasing function">increasing function</a>. This function is quite complicated but for some simple cases: </p> <ol><li>If all the particles are identical (same energy barrier and same magnetic moment), their easy axes are all oriented parallel to the applied field and the temperature is low enough (<i>T</i><sub>B</sub> < <i>T</i> ≲ <i>KV</i>/(10 <i>k</i><sub>B</sub>)), then the magnetization of the assembly is <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(H)\approx n\mu \tanh \left({\frac {\mu _{0}H\mu }{k_{\text{B}}T}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mi>n</mi> <mi>μ</mi> <mi>tanh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>H</mi> <mi>μ</mi> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(H)\approx n\mu \tanh \left({\frac {\mu _{0}H\mu }{k_{\text{B}}T}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/753eeb4a0015dc18cbd9f08380124340dd4daf50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.428ex; height:6.176ex;" alt="{\displaystyle M(H)\approx n\mu \tanh \left({\frac {\mu _{0}H\mu }{k_{\text{B}}T}}\right)}"></span>.</dd></dl></li> <li>If all the particles are identical and the temperature is high enough (<i>T</i> ≳ <i>KV</i>/<i>k</i><sub>B</sub>), then, irrespective of the orientations of the easy axes: <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(H)\approx n\mu L\left({\frac {\mu _{0}H\mu }{k_{\text{B}}T}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mi>n</mi> <mi>μ</mi> <mi>L</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>H</mi> <mi>μ</mi> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(H)\approx n\mu L\left({\frac {\mu _{0}H\mu }{k_{\text{B}}T}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71c3267130030c408746e8dea72e70fade35ed2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.358ex; height:6.176ex;" alt="{\displaystyle M(H)\approx n\mu L\left({\frac {\mu _{0}H\mu }{k_{\text{B}}T}}\right)}"></span></dd></dl></li></ol> <p>In the above equations: </p> <ul><li><i>n</i> is the density of nanoparticles in the sample</li> <li><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="{\textstyle \mu _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mu _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa21a17517ea72a8644be5988fb438abf91d6ca3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.176ex;" alt="{\textstyle \mu _{0}}"></span> is the <a href="/wiki/Magnetic_permeability" class="mw-redirect" title="Magnetic permeability">magnetic permeability</a> of vacuum</li> <li><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="{\textstyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/259577540a13444806174d5a1ae7662974f58085" 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="{\textstyle \mu }"></span> is the magnetic moment of a nanoparticle</li> <li><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="{\textstyle L(x)={\frac {1}{\tanh(x)}}-{\frac {1}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>tanh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle L(x)={\frac {1}{\tanh(x)}}-{\frac {1}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5a48c965a693e4727ab11f2e6ce6d6bf7cb04a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.782ex; height:4.176ex;" alt="{\textstyle L(x)={\frac {1}{\tanh(x)}}-{\frac {1}{x}}}"></span> is the <a href="/wiki/Langevin_function" class="mw-redirect" title="Langevin function">Langevin function</a></li></ul> <p>The initial slope of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(H)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(H)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f141d4fc1fd6fe07bb052d2f0c72ea996ee4ac22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.315ex; height:2.843ex;" alt="{\displaystyle M(H)}"></span> function is the magnetic susceptibility of the sample <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 \chi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/656111758322ace96d80a9371771aa6d3de25437" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }"></span>: </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 \chi ={\begin{cases}\displaystyle {\frac {n\mu _{0}\mu ^{2}}{k_{\text{B}}T}}&{\text{for the 1st case}}\\\displaystyle {\frac {n\mu _{0}\mu ^{2}}{3k_{\text{B}}T}}&{\text{for the 2nd case}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> </mstyle> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for the 1st case</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>3</mn> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> </mstyle> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for the 2nd case</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi ={\begin{cases}\displaystyle {\frac {n\mu _{0}\mu ^{2}}{k_{\text{B}}T}}&{\text{for the 1st case}}\\\displaystyle {\frac {n\mu _{0}\mu ^{2}}{3k_{\text{B}}T}}&{\text{for the 2nd case}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7f3c5e46e1b50c8f10039f417da14001fea5498" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:32.485ex; height:12.176ex;" alt="{\displaystyle \chi ={\begin{cases}\displaystyle {\frac {n\mu _{0}\mu ^{2}}{k_{\text{B}}T}}&{\text{for the 1st case}}\\\displaystyle {\frac {n\mu _{0}\mu ^{2}}{3k_{\text{B}}T}}&{\text{for the 2nd case}}\end{cases}}}"></span></dd></dl> <p>The latter susceptibility is also valid for all temperatures <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 T>T_{\text{B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>></mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T>T_{\text{B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4afd284169fa4b1e75757e320d814b4d342e9b68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.488ex; height:2.509ex;" alt="{\displaystyle T>T_{\text{B}}}"></span> if the easy axes of the nanoparticles are randomly oriented. </p><p>It can be seen from these equations that large nanoparticles have a larger <i>μ</i> and so a larger susceptibility. This explains why superparamagnetic nanoparticles have a much larger susceptibility than standard paramagnets: they behave exactly as a paramagnet with a huge magnetic moment. </p> <div class="mw-heading mw-heading3"><h3 id="Time_dependence_of_the_magnetization">Time dependence of the magnetization</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superparamagnetism&action=edit&section=4" title="Edit section: Time dependence of the magnetization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There is no time-dependence of the magnetization when the nanoparticles are either completely blocked (<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 T\ll T_{\text{B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>≪</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\ll T_{\text{B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2002775fffba43809001421ee1fcb5d8a3efeabe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.004ex; height:2.509ex;" alt="{\displaystyle T\ll T_{\text{B}}}"></span>) or completely superparamagnetic (<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 T\gg T_{\text{B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>≫</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\gg T_{\text{B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8022ebfc5999e1b9c6178d3789908225a6638582" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.004ex; height:2.509ex;" alt="{\displaystyle T\gg T_{\text{B}}}"></span>). There is, however, a narrow window around <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 T_{\text{B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\text{B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0bdc35cd82271fd68f4db02e94ec7fd04cc0d2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.753ex; height:2.509ex;" alt="{\displaystyle T_{\text{B}}}"></span> where the measurement time and the relaxation time have comparable magnitude. In this case, a frequency-dependence of the susceptibility can be observed. For a randomly oriented sample, the complex susceptibility<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> is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi (\omega )={\frac {\chi _{\text{sp}}+i\omega \tau \chi _{\text{b}}}{1+i\omega \tau }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>sp</mtext> </mrow> </msub> <mo>+</mo> <mi>i</mi> <mi>ω</mi> <mi>τ</mi> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>b</mtext> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mi>ω</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi (\omega )={\frac {\chi _{\text{sp}}+i\omega \tau \chi _{\text{b}}}{1+i\omega \tau }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a2dd26713b0fbbbb04bd639951fda64f1c2f74d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.786ex; height:5.676ex;" alt="{\displaystyle \chi (\omega )={\frac {\chi _{\text{sp}}+i\omega \tau \chi _{\text{b}}}{1+i\omega \tau }}}"></span></dd></dl> <p>where </p> <ul><li><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="{\textstyle {\frac {\omega }{2\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\omega }{2\pi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add39c366c8c0389400441855b174d77beb8d3a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.6ex; height:3.176ex;" alt="{\textstyle {\frac {\omega }{2\pi }}}"></span> is the frequency of the applied field</li> <li><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="{\textstyle \chi _{\text{sp}}={\frac {n\mu _{0}\mu ^{2}}{3k_{\text{B}}T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>sp</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>3</mn> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \chi _{\text{sp}}={\frac {n\mu _{0}\mu ^{2}}{3k_{\text{B}}T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d31477d9a716faf3adec8b571f7d65adbc3ce82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:11.815ex; height:4.843ex;" alt="{\textstyle \chi _{\text{sp}}={\frac {n\mu _{0}\mu ^{2}}{3k_{\text{B}}T}}}"></span> is the susceptibility in the superparamagnetic state</li> <li><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="{\textstyle \chi _{\text{b}}={\frac {n\mu _{0}\mu ^{2}}{3KV}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>b</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>3</mn> <mi>K</mi> <mi>V</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \chi _{\text{b}}={\frac {n\mu _{0}\mu ^{2}}{3KV}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3490315777c167caeaef171eaf214e69c4fa21d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.168ex; height:4.509ex;" alt="{\textstyle \chi _{\text{b}}={\frac {n\mu _{0}\mu ^{2}}{3KV}}}"></span> is the susceptibility in the blocked state</li> <li><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="{\textstyle \tau ={\frac {\tau _{\text{N}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>N</mtext> </mrow> </msub> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \tau ={\frac {\tau _{\text{N}}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50f54ec82878ac9a3fc551af85aad2c7efee1ecc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.02ex; height:3.509ex;" alt="{\textstyle \tau ={\frac {\tau _{\text{N}}}{2}}}"></span> is the relaxation time of the assembly</li></ul> <p>From this frequency-dependent susceptibility, the time-dependence of the magnetization for low-fields can be derived: </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 \tau {\frac {\mathrm {d} M}{\mathrm {d} t}}+M=\tau \chi _{\text{b}}{\frac {\mathrm {d} H}{\mathrm {d} t}}+\chi _{\text{sp}}H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>M</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>M</mi> <mo>=</mo> <mi>τ</mi> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>b</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>H</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>sp</mtext> </mrow> </msub> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau {\frac {\mathrm {d} M}{\mathrm {d} t}}+M=\tau \chi _{\text{b}}{\frac {\mathrm {d} H}{\mathrm {d} t}}+\chi _{\text{sp}}H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7873e4d9591274309334629b82085e0499ad1dd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:30.303ex; height:5.509ex;" alt="{\displaystyle \tau {\frac {\mathrm {d} M}{\mathrm {d} t}}+M=\tau \chi _{\text{b}}{\frac {\mathrm {d} H}{\mathrm {d} t}}+\chi _{\text{sp}}H}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Measurements">Measurements</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superparamagnetism&action=edit&section=5" title="Edit section: Measurements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A superparamagnetic system can be measured with <a href="/wiki/Magnetic_susceptibility" title="Magnetic susceptibility">AC susceptibility</a> measurements, where an applied magnetic field varies in time, and the magnetic response of the system is measured. A superparamagnetic system will show a characteristic frequency dependence: When the frequency is much higher than 1/τ<sub>N</sub>, there will be a different magnetic response than when the frequency is much lower than 1/τ<sub>N</sub>, since in the latter case, but not the former, the ferromagnetic clusters will have time to respond to the field by flipping their magnetization.<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 precise dependence can be calculated from the Néel–Arrhenius equation, assuming that the neighboring clusters behave independently of one another (if clusters interact, their behavior becomes more complicated). It is also possible to perform magneto-optical AC susceptibility measurements with magneto-optically active superparamagnetic materials such as iron oxide nanoparticles in the visible wavelength range.<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> <div class="mw-heading mw-heading2"><h2 id="Effect_on_hard_drives">Effect on hard drives</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superparamagnetism&action=edit&section=6" title="Edit section: Effect on hard drives"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Superparamagnetism sets a limit on the storage density of <a href="/wiki/Hard_disk_drive" title="Hard disk drive">hard disk drives</a> due to the minimum size of particles that can be used. This limit on <a href="/wiki/Areal_density_(computer_storage)" class="mw-redirect" title="Areal density (computer storage)">areal-density</a> is known as the <b>superparamagnetic limit</b>. </p> <ul><li>Older hard disk technology uses <a href="/wiki/Longitudinal_recording" class="mw-redirect" title="Longitudinal recording">longitudinal recording</a>. It has an estimated limit of 100 to 200 Gbit/in<sup>2</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></li> <li>Current hard disk technology uses <a href="/wiki/Perpendicular_recording" title="Perpendicular recording">perpendicular recording</a>. As of July 2020<sup class="plainlinks noexcerpt noprint asof-tag update" style="display:none;"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Superparamagnetism&action=edit">[update]</a></sup> drives with densities of approximately 1 Tbit/in<sup>2</sup> are available commercially.<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> This is at the limit for conventional magnetic recording that was predicted in 1999.<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><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></li> <li>Future hard disk technologies currently in development include: <a href="/wiki/Heat-assisted_magnetic_recording" title="Heat-assisted magnetic recording">heat-assisted magnetic recording</a> (HAMR) and <a href="/w/index.php?title=Microwave-assisted_magnetic_recording&action=edit&redlink=1" class="new" title="Microwave-assisted magnetic recording (page does not exist)">microwave-assisted magnetic recording</a> (MAMR), which use materials that are stable at much smaller sizes.<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> They require localized heating or microwave excitation before the magnetic orientation of a bit can be changed. <a href="/wiki/Bit-patterned_recording" class="mw-redirect" title="Bit-patterned recording">Bit-patterned recording</a> (BPR) avoids the use of fine-grained media and is another possibility.<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> In addition, magnetic recording technologies based on topological distortions of the magnetization, known as <a href="/wiki/Skyrmion" title="Skyrmion">skyrmions</a>, have been proposed.<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></li></ul> <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=Superparamagnetism&action=edit&section=7" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="General_applications">General applications</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superparamagnetism&action=edit&section=8" title="Edit section: General applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Ferrofluid" title="Ferrofluid">Ferrofluid</a>: tunable viscosity</li></ul> <div class="mw-heading mw-heading3"><h3 id="Biomedical_applications">Biomedical applications</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superparamagnetism&action=edit&section=9" title="Edit section: Biomedical applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Imaging: <a href="/wiki/Contrast_agents" class="mw-redirect" title="Contrast agents">contrast agents</a> in <a href="/wiki/Magnetic_resonance_imaging" title="Magnetic resonance imaging">magnetic resonance imaging</a> (MRI)</li> <li>Magnetic separation: cell-, DNA-, protein- separation, RNA fishing</li> <li>Treatments: <a href="/wiki/Targeted_drug_delivery" title="Targeted drug delivery">targeted drug delivery</a>, <a href="/wiki/Magnetic_hyperthermia" class="mw-redirect" title="Magnetic hyperthermia">magnetic hyperthermia</a>, <a href="/wiki/Magnetofection" title="Magnetofection">magnetofection</a></li></ul> <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=Superparamagnetism&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Iron_oxide_nanoparticles" class="mw-redirect" title="Iron oxide nanoparticles">Iron oxide nanoparticles</a></li> <li><a href="/wiki/Single-molecule_magnet" title="Single-molecule magnet">Single-molecule magnet</a></li></ul> <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=Superparamagnetism&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superparamagnetism&action=edit&section=12" 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 mw-references-columns"><ol class="references"> <li id="cite_note-Neel49-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Neel49_1-0">^</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="CITEREFNéel1949" class="citation journal cs1">Néel, L. 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"Skyrmions on the track". <i>Nature Nanotechnology</i>. <b>8</b> (3): 152–156. <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/2013NatNa...8..152F">2013NatNa...8..152F</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%2Fnnano.2013.29">10.1038/nnano.2013.29</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1748-3387">1748-3387</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/23459548">23459548</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature+Nanotechnology&rft.atitle=Skyrmions+on+the+track&rft.volume=8&rft.issue=3&rft.pages=152-156&rft.date=2013-03-01&rft_id=info%3Adoi%2F10.1038%2Fnnano.2013.29&rft.issn=1748-3387&rft_id=info%3Apmid%2F23459548&rft_id=info%3Abibcode%2F2013NatNa...8..152F&rft.aulast=Fert&rft.aufirst=Albert&rft.au=Cros%2C+Vincent&rft.au=Sampaio%2C+Jo%C3%A3o&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASuperparamagnetism" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Sources">Sources</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superparamagnetism&action=edit&section=13" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNéel1949" class="citation journal cs1 cs1-prop-foreign-lang-source">Néel, L. (1949). <a rel="nofollow" class="external text" href="http://hal.archives-ouvertes.fr/docs/00/23/42/17/PDF/ajp-jphysrad_1950_11_2_49_0.pdf">"Théorie du traînage magnétique des ferromagnétiques en grains fins avec applications aux terres cuites"</a> <span class="cs1-format">(PDF)</span>. <i>Ann. Géophys.</i> (in French). <b>5</b>: 99–136.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ann.+G%C3%A9ophys.&rft.atitle=Th%C3%A9orie+du+tra%C3%AEnage+magn%C3%A9tique+des+ferromagn%C3%A9tiques+en+grains+fins+avec+applications+aux+terres+cuites&rft.volume=5&rft.pages=99-136&rft.date=1949&rft.aulast=N%C3%A9el&rft.aufirst=L.&rft_id=http%3A%2F%2Fhal.archives-ouvertes.fr%2Fdocs%2F00%2F23%2F42%2F17%2FPDF%2Fajp-jphysrad_1950_11_2_49_0.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASuperparamagnetism" class="Z3988"></span> An English translation is available in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKurti1988" class="citation book cs1">Kurti, N., ed. (1988). <i>Selected Works of Louis Néel</i>. New York: Gordon and Breach. pp. 407–427. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-2-88124-300-4" title="Special:BookSources/978-2-88124-300-4"><bdi>978-2-88124-300-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Selected+Works+of+Louis+N%C3%A9el&rft.place=New+York&rft.pages=407-427&rft.pub=Gordon+and+Breach&rft.date=1988&rft.isbn=978-2-88124-300-4&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASuperparamagnetism" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWellerMoser1999" class="citation journal cs1">Weller, D.; Moser, A. (1999). "Thermal Effect Limits in Ultrahigh Density Magnetic Recording". <i><a href="/wiki/IEEE_Transactions_on_Magnetics" title="IEEE Transactions on Magnetics">IEEE Transactions on Magnetics</a></i>. <b>35</b> (6): 4423–4439. <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/1999ITM....35.4423W">1999ITM....35.4423W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F20.809134">10.1109/20.809134</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Magnetics&rft.atitle=Thermal+Effect+Limits+in+Ultrahigh+Density+Magnetic+Recording&rft.volume=35&rft.issue=6&rft.pages=4423-4439&rft.date=1999&rft_id=info%3Adoi%2F10.1109%2F20.809134&rft_id=info%3Abibcode%2F1999ITM....35.4423W&rft.aulast=Weller&rft.aufirst=D.&rft.au=Moser%2C+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASuperparamagnetism" class="Z3988"></span></li></ul> </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=Superparamagnetism&action=edit&section=14" 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" 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