Pinch (plasma physics)

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class="vector-toc-numb">2</span> <span>Types</span> </div> </a> <ul id="toc-Types-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Common_behavior" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Common_behavior"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Common behavior</span> </div> </a> <ul id="toc-Common_behavior-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications_and_devices" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications_and_devices"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Applications and devices</span> </div> </a> <button aria-controls="toc-Applications_and_devices-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 Applications and devices subsection</span> </button> <ul id="toc-Applications_and_devices-sublist" class="vector-toc-list"> <li id="toc-Crushing_cans_with_the_pinch_effect" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Crushing_cans_with_the_pinch_effect"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Crushing cans with the pinch effect</span> </div> </a> <ul id="toc-Crushing_cans_with_the_pinch_effect-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equilibrium_analysis" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Equilibrium_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Equilibrium analysis</span> </div> </a> <button aria-controls="toc-Equilibrium_analysis-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 Equilibrium analysis subsection</span> </button> <ul id="toc-Equilibrium_analysis-sublist" class="vector-toc-list"> <li id="toc-One_dimension" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#One_dimension"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>One dimension</span> </div> </a> <ul id="toc-One_dimension-sublist" class="vector-toc-list"> <li id="toc-The_θ-pinch" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#The_θ-pinch"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1.1</span> <span>The θ-pinch</span> </div> </a> <ul id="toc-The_θ-pinch-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_Z-pinch" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#The_Z-pinch"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1.2</span> <span>The Z-pinch</span> </div> </a> <ul id="toc-The_Z-pinch-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_screw_pinch" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#The_screw_pinch"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1.3</span> <span>The screw pinch</span> </div> </a> <ul id="toc-The_screw_pinch-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_screw_pinch_via_colliding_optical_vortices" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#The_screw_pinch_via_colliding_optical_vortices"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1.4</span> <span>The screw pinch via colliding optical vortices</span> </div> </a> <ul id="toc-The_screw_pinch_via_colliding_optical_vortices-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Two_dimensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Two_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Two dimensions</span> </div> </a> <ul id="toc-Two_dimensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Three_dimensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Three_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Three dimensions</span> </div> </a> <ul id="toc-Three_dimensions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Formal_treatment" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Formal_treatment"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Formal treatment</span> </div> </a> <button aria-controls="toc-Formal_treatment-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 Formal treatment subsection</span> </button> <ul id="toc-Formal_treatment-sublist" class="vector-toc-list"> <li id="toc-The_Bennett_relation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Bennett_relation"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>The Bennett relation</span> </div> </a> <ul id="toc-The_Bennett_relation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_generalized_Bennett_relation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_generalized_Bennett_relation"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>The generalized Bennett relation</span> </div> </a> <ul id="toc-The_generalized_Bennett_relation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_Carlqvist_relation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Carlqvist_relation"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>The Carlqvist relation</span> </div> </a> <ul id="toc-The_Carlqvist_relation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-References_in_culture" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References_in_culture"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References in culture</span> </div> </a> <ul id="toc-References_in_culture-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div 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class="firstHeading mw-first-heading"><span class="mw-page-title-main">Pinch (plasma physics)</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. 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href="https://bn.wikipedia.org/wiki/%E0%A6%AA%E0%A6%BF%E0%A6%9E%E0%A7%8D%E0%A6%9A_%E0%A6%95%E0%A7%8D%E0%A6%B0%E0%A6%BF%E0%A6%AF%E0%A6%BC%E0%A6%BE" title="পিঞ্চ ক্রিয়া – Bangla" lang="bn" hreflang="bn" data-title="পিঞ্চ ক্রিয়া" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Confinament_per_pin%C3%A7ament" title="Confinament per pinçament – Catalan" lang="ca" hreflang="ca" data-title="Confinament per pinçament" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Pinch-effekt_(elektrodynamik)" title="Pinch-effekt (elektrodynamik) – Danish" lang="da" hreflang="da" data-title="Pinch-effekt (elektrodynamik)" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Pinch-Effekt_(Elektrodynamik)" title="Pinch-Effekt (Elektrodynamik) – German" lang="de" hreflang="de" data-title="Pinch-Effekt (Elektrodynamik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%95%80%EC%B9%98_(%EB%AC%BC%EB%A6%AC%ED%95%99)" title="핀치 (물리학) – Korean" lang="ko" hreflang="ko" data-title="핀치 (물리학)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8A%D5%AB%D5%B6%D5%B9_%D5%A7%D6%86%D5%A5%D5%AF%D5%BF" title="Պինչ էֆեկտ – Armenian" lang="hy" hreflang="hy" data-title="Պինչ էֆեկտ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Effetto_pinch" title="Effetto pinch – Italian" lang="it" hreflang="it" data-title="Effetto pinch" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Pincheffekt" title="Pincheffekt – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Pincheffekt" 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/Skurcz_plazmy" title="Skurcz plazmy – Polish" lang="pl" hreflang="pl" data-title="Skurcz plazmy" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%B8%D0%BD%D1%87-%D1%8D%D1%84%D1%84%D0%B5%D0%BA%D1%82" title="Пинч-эффект – Russian" lang="ru" hreflang="ru" data-title="Пинч-эффект" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D1%96%D0%BD%D1%87-%D0%B5%D1%84%D0%B5%D0%BA%D1%82" title="Пінч-ефект – Ukrainian" lang="uk" hreflang="uk" data-title="Пінч-ефект" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit 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<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">Compression of an electrically conducting filament by magnetic forces</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">For other uses, see <a href="/wiki/Pinch_(disambiguation)" class="mw-redirect mw-disambig" title="Pinch (disambiguation)">Pinch (disambiguation)</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:204px;max-width:204px"><div class="trow"><div class="theader">Pinch phenomena</div></div><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Lightning_over_Oradea_Romania_3.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Lightning_over_Oradea_Romania_3.jpg/200px-Lightning_over_Oradea_Romania_3.jpg" decoding="async" width="200" height="309" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Lightning_over_Oradea_Romania_3.jpg/300px-Lightning_over_Oradea_Romania_3.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/13/Lightning_over_Oradea_Romania_3.jpg/400px-Lightning_over_Oradea_Romania_3.jpg 2x" data-file-width="1935" data-file-height="2989" /></a></span></div><div class="thumbcaption"><a href="/wiki/Lightning_discharge" class="mw-redirect" title="Lightning discharge">Lightning discharge</a> bolts showing electromagnetically pinched plasma filaments</div></div></div><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Crushed_rod_pollock_barraclough.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Crushed_rod_pollock_barraclough.jpg/200px-Crushed_rod_pollock_barraclough.jpg" decoding="async" width="200" height="647" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Crushed_rod_pollock_barraclough.jpg/300px-Crushed_rod_pollock_barraclough.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Crushed_rod_pollock_barraclough.jpg/400px-Crushed_rod_pollock_barraclough.jpg 2x" data-file-width="480" data-file-height="1552" /></a></span></div><div class="thumbcaption">A 1905 study of pinches, where electric lightning was used to create a Z-pinch inside a metal tube.<sup id="cite_ref-pollock_1-0" class="reference"><a href="#cite_note-pollock-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></div></div></div><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:TWK_pinch.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/TWK_pinch.jpg/200px-TWK_pinch.jpg" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/TWK_pinch.jpg/300px-TWK_pinch.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bb/TWK_pinch.jpg/400px-TWK_pinch.jpg 2x" data-file-width="1875" data-file-height="1875" /></a></span></div><div class="thumbcaption">A current-driven toroidal Z-pinch in a krypton plasma</div></div></div></div></div> <p>A <b>pinch</b> (or: <b>Bennett pinch</b><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> (after <a href="/wiki/Willard_Harrison_Bennett" title="Willard Harrison Bennett">Willard Harrison Bennett</a>), <b>electromagnetic pinch</b>,<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> <b>magnetic pinch</b>,<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> <b>pinch effect</b>,<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> or <b>plasma pinch</b>.<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>) is the compression of an electrically conducting <a href="/wiki/Electrical_filament" class="mw-redirect" title="Electrical filament">filament</a> by <a href="/wiki/Magnetic" class="mw-redirect" title="Magnetic">magnetic</a> forces, or a device that does such. The conductor is usually a <a href="/wiki/Plasma_(physics)" title="Plasma (physics)">plasma</a>, but could also be a solid or liquid <a href="/wiki/Metal" title="Metal">metal</a>. Pinches were the first type of device used for experiments in controlled nuclear <a href="/wiki/Fusion_power" title="Fusion power">fusion power</a>.<sup id="cite_ref-Phillips,_James_2013_7-0" class="reference"><a href="#cite_note-Phillips,_James_2013-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>Pinches occur naturally in electrical discharges such as <a href="/wiki/Lightning" title="Lightning">lightning bolts</a>,<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> planetary <a href="/wiki/Aurora_(astronomy)" class="mw-redirect" title="Aurora (astronomy)">auroras</a>,<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> <a href="/wiki/Current_sheet" title="Current sheet">current sheets</a>,<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Solar_flare" title="Solar flare">solar flares</a>.<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> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Basic_mechanism">Basic mechanism</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pinch_(plasma_physics)&action=edit&section=1" title="Edit section: Basic mechanism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Plasma_Pinch_Mechanism.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Plasma_Pinch_Mechanism.png/600px-Plasma_Pinch_Mechanism.png" decoding="async" width="600" height="169" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Plasma_Pinch_Mechanism.png/900px-Plasma_Pinch_Mechanism.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Plasma_Pinch_Mechanism.png/1200px-Plasma_Pinch_Mechanism.png 2x" data-file-width="1659" data-file-height="466" /></a><figcaption>This is a basic explanation of how a pinch works. (<b>1</b>) Pinches apply a high voltage and current across a tube. This tube is filled with a gas, typically a fusion fuel such as deuterium. If the product of the voltage & the charge is higher than the ionization energy of the gas the gas ionizes. (<b>2</b>) Current jumps across this gap. (<b>3</b>) The current makes a magnetic field which is perpendicular to the current. This magnetic field pulls the material together. (<b>4</b>) These atoms can get close enough to fuse.</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="Types">Types</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pinch_(plasma_physics)&action=edit&section=2" title="Edit section: Types"><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:Plasma-filaments.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Plasma-filaments.jpg/300px-Plasma-filaments.jpg" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Plasma-filaments.jpg/450px-Plasma-filaments.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Plasma-filaments.jpg/600px-Plasma-filaments.jpg 2x" data-file-width="2592" data-file-height="1944" /></a><figcaption>An example of a man-made pinch. Here Z-pinches constrain a <a href="/wiki/Plasma_(physics)" title="Plasma (physics)">plasma</a> inside filaments of <a href="/wiki/Electrical_discharge" class="mw-redirect" title="Electrical discharge">electrical discharge</a> from a <a href="/wiki/Tesla_coil" title="Tesla coil">Tesla coil</a></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:MagLif_Concept.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/MagLif_Concept.png/220px-MagLif_Concept.png" decoding="async" width="220" height="190" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/MagLif_Concept.png/330px-MagLif_Concept.png 1.5x, //upload.wikimedia.org/wikipedia/commons/e/ec/MagLif_Concept.png 2x" data-file-width="360" data-file-height="311" /></a><figcaption>The MagLIF concept, a combination of a Z-pinch and a laser beam</figcaption></figure> <p>Pinches exist in nature and in laboratories. Pinches differ in their geometry and operating forces.<sup id="cite_ref-adsabs.harvard.edu_12-0" class="reference"><a href="#cite_note-adsabs.harvard.edu-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> These include: </p> <ul><li><i>Uncontrolled</i> – Any time an electric current moves in large amounts (e.g., lightning, arcs, sparks, discharges) a magnetic force can pull together plasma. This can be insufficient for fusion.</li> <li><i><a href="/wiki/Current_sheet" title="Current sheet">Sheet pinch</a></i> – An astrophysical effect, this arises from vast sheets of charged particles.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></li> <li><i><a href="/wiki/Z-pinch" title="Z-pinch">Z-pinch</a></i> – The current runs down the axis, or walls, of a cylinder while the <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a> is <a href="/wiki/Azimuth" title="Azimuth">azimuthal</a></li> <li><i><a href="/wiki/Theta_pinch" title="Theta pinch">Theta pinch</a></i> – The magnetic field runs down the axis of a cylinder, while the electric field is in the <a href="/wiki/Azimuth" title="Azimuth">azimuthal</a> direction (also called a thetatron<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup>)</li> <li><i>Screw pinch</i> – A combination of a Z-pinch and theta pinch<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> (also called a stabilized Z-pinch, or θ-Z pinch)<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup></li> <li><i><a href="/wiki/Reversed_field_pinch" title="Reversed field pinch">Reversed field pinch</a></i> or <i>toroidal pinch</i> – This is a Z-pinch arranged in the shape of a <a href="/wiki/Torus" title="Torus">torus</a>. The plasma has an internal magnetic field. As distance increases from the center of this ring, the magnetic field reverses direction.</li> <li><i>Inverse pinch</i> – An early fusion concept, this device consisted of a rod surrounded by plasma. Current traveled through the plasma and returned along the center rod.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> This geometry was slightly different than a z-pinch in that the conductor was in the center, not the sides.</li> <li><i>Cylindrical pinch</i></li> <li><i>Orthogonal pinch effect</i></li> <li><i>Ware pinch</i> – A pinch that occurs inside a Tokamak plasma, when particles inside the banana orbit condense together.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Magnetized_Liner_Inertial_Fusion" class="mw-redirect" title="Magnetized Liner Inertial Fusion"><i>Magnetized liner inertial fusion</i></a> (MagLIF) – A Z-pinch of preheated, premagnetized fuel inside a metal liner, which could lead to ignition and practical fusion energy with a larger pulsed-power driver.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Common_behavior">Common behavior</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pinch_(plasma_physics)&action=edit&section=3" title="Edit section: Common behavior"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pinches may become <a href="/wiki/Plasma_stability#Plasma_instabilities" title="Plasma stability">unstable</a>.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> They radiate energy across the whole <a href="/wiki/Electromagnetic_spectrum" title="Electromagnetic spectrum">electromagnetic spectrum</a> including <a href="/wiki/Radio_wave" title="Radio wave">radio waves</a>, <a href="/wiki/Microwave" title="Microwave">microwaves</a>, <a href="/wiki/Infrared" title="Infrared">infrared</a>, <a href="/wiki/X-ray" title="X-ray">x-rays</a>,<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Gamma_ray" title="Gamma ray">gamma rays</a>,<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Synchrotron_radiation" title="Synchrotron radiation">synchrotron radiation</a>,<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Visible_spectrum" title="Visible spectrum">visible light</a>. They also produce <a href="/wiki/Neutron" title="Neutron">neutrons</a>, as a product of fusion.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Kink_Modes_Model.png" class="mw-file-description"><img alt="Model of the kink modes that form inside a pinch" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Kink_Modes_Model.png/220px-Kink_Modes_Model.png" decoding="async" width="220" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Kink_Modes_Model.png/330px-Kink_Modes_Model.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Kink_Modes_Model.png/440px-Kink_Modes_Model.png 2x" data-file-width="572" data-file-height="260" /></a><figcaption>Model of the kink modes that form inside a pinch</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="Applications_and_devices">Applications and devices</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pinch_(plasma_physics)&action=edit&section=4" title="Edit section: Applications and devices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pinches are used to generate <a href="/wiki/X-rays" class="mw-redirect" title="X-rays">X-rays</a> and the intense magnetic fields generated are used in <a href="/wiki/Electromagnetic_forming" title="Electromagnetic forming">electromagnetic forming</a> of metals. They also have applications in <a href="/wiki/Particle_beam" title="Particle beam">particle beams</a><sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> including <a href="/wiki/Particle_beam_weapon" class="mw-redirect" title="Particle beam weapon">particle beam weapons</a>,<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> astrophysics studies<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> and it has been proposed to use them in space propulsion.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> A number of large pinch machines have been built to study <a href="/wiki/Fusion_power" title="Fusion power">fusion power</a>; here are several: </p> <ul><li><a href="/wiki/MAGPIE" title="MAGPIE">MAGPIE</a> A Z-pinch at Imperial College. This dumps a large amount of current across a wire. Under these conditions, the wire becomes plasma and compresses to produce fusion.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Z_Pulsed_Power_Facility" title="Z Pulsed Power Facility">Z Pulsed Power Facility</a> at Sandia National Laboratories.</li> <li><a href="/wiki/ZETA_(fusion_reactor)" title="ZETA (fusion reactor)">ZETA</a> device in Culham, England</li> <li><a href="/wiki/Madison_Symmetric_Torus" title="Madison Symmetric Torus">Madison Symmetric Torus</a> at the University of Wisconsin, Madison</li> <li><a href="/wiki/Reversed-Field_eXperiment" title="Reversed-Field eXperiment">Reversed-Field eXperiment</a> in Italy.</li> <li><a href="/wiki/Dense_plasma_focus" title="Dense plasma focus">Dense plasma focus</a> in New Jersey</li> <li><a href="/wiki/University_of_Nevada,_Reno" title="University of Nevada, Reno">University of Nevada, Reno</a> (USA)</li> <li><a href="/wiki/Cornell_University" title="Cornell University">Cornell University</a> (USA)</li> <li><a href="/wiki/University_of_Michigan" title="University of Michigan">University of Michigan</a> (USA)</li> <li><a href="/wiki/University_of_California,_San_Diego" title="University of California, San Diego">University of California, San Diego</a> (USA)</li> <li><a href="/wiki/University_of_Washington" title="University of Washington">University of Washington</a> (USA)</li> <li><a href="/wiki/Ruhr_University" class="mw-redirect" title="Ruhr University">Ruhr University</a> (Germany)</li> <li><a href="/wiki/%C3%89cole_Polytechnique" class="mw-redirect" title="École Polytechnique">École Polytechnique</a> (France)</li> <li><a href="/wiki/Weizmann_Institute_of_Science" title="Weizmann Institute of Science">Weizmann Institute of Science</a> (Israel)</li> <li><a href="/wiki/Universidad_Aut%C3%B3noma_Metropolitana" title="Universidad Autónoma Metropolitana">Universidad Autónoma Metropolitana</a> (Mexico).</li> <li><a href="/wiki/Zap_Energy" title="Zap Energy">Zap Energy</a> Inc. (USA)</li></ul> <div class="mw-heading mw-heading3"><h3 id="Crushing_cans_with_the_pinch_effect">Crushing cans with the pinch effect</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pinch_(plasma_physics)&action=edit&section=5" title="Edit section: Crushing cans with the pinch effect"><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:Aluminium-can-white.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Aluminium-can-white.jpg/150px-Aluminium-can-white.jpg" decoding="async" width="150" height="270" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Aluminium-can-white.jpg/225px-Aluminium-can-white.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Aluminium-can-white.jpg/300px-Aluminium-can-white.jpg 2x" data-file-width="591" data-file-height="1063" /></a><figcaption>Pinched aluminium can, produced via a <a href="/wiki/Pulsed_power" title="Pulsed power">pulsed</a> magnetic field created by rapidly discharging 2 kilojoules from a high voltage <a href="/wiki/Capacitor" title="Capacitor">capacitor</a> bank into a 3-turn coil of heavy gauge wire.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Electromagnetic_forming" title="Electromagnetic forming">Electromagnetic forming</a></div><p>Many high-voltage electronics enthusiasts make their own crude electromagnetic forming devices.<sup id="cite_ref-LaPointe_32-0" class="reference"><a href="#cite_note-LaPointe-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Tristan_33-0" class="reference"><a href="#cite_note-Tristan-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Borros_34-0" class="reference"><a href="#cite_note-Borros-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> They use <a href="/wiki/Pulsed_power" title="Pulsed power">pulsed power</a> techniques to produce a theta pinch able to crush an aluminium soft drink can using the <a href="/wiki/Lorentz_force" title="Lorentz force">Lorentz forces</a> created when large currents are induced in the can by the strong magnetic field of the primary coil.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p><p>An electromagnetic aluminium can crusher consists of four main components: a <a href="/wiki/High_voltage" title="High voltage">high-voltage</a> <a href="/wiki/Direct_current" title="Direct current">DC</a> <a href="/wiki/Power_supply" title="Power supply">power supply</a>, which provides a source of <a href="/wiki/Electrical_energy" title="Electrical energy">electrical energy</a>, a large <i>energy discharge</i> <a href="/wiki/Capacitor" title="Capacitor">capacitor</a> to accumulate the electrical energy, a high voltage switch or <a href="/wiki/Spark_gap" title="Spark gap">spark gap</a>, and a robust coil (capable of surviving high magnetic pressure) through which the stored electrical energy can be quickly discharged in order to generate a correspondingly strong pinching magnetic field (see diagram below). </p> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Can-pincher.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Can-pincher.png/300px-Can-pincher.png" decoding="async" width="300" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Can-pincher.png/450px-Can-pincher.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Can-pincher.png/600px-Can-pincher.png 2x" data-file-width="1119" data-file-height="670" /></a><figcaption>Electromagnetic pinch "can crusher": schematic diagram</figcaption></figure> <p>In practice, such a device is somewhat more sophisticated than the schematic diagram suggests, including electrical components that control the current in order to maximize the resulting pinch, and to ensure that the device works safely. For more details, see the notes.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pinch_(plasma_physics)&action=edit&section=6" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Ieee-emblem.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/e/e9/Ieee-emblem.jpg/128px-Ieee-emblem.jpg" decoding="async" width="128" height="158" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/e/e9/Ieee-emblem.jpg/192px-Ieee-emblem.jpg 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/e/e9/Ieee-emblem.jpg/256px-Ieee-emblem.jpg 2x" data-file-width="283" data-file-height="350" /></a><figcaption>The <a href="/wiki/Institute_of_Electrical_and_Electronics_Engineers" title="Institute of Electrical and Electronics Engineers">Institute of Electrical and Electronics Engineers</a> emblem shows the basic features of an azimuthal magnetic pinch.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>The first creation of a Z-pinch in the laboratory may have occurred in 1790 in Holland when <a href="/wiki/Martin_van_Marum" title="Martin van Marum">Martinus van Marum</a> created an explosion by discharging 100 <a href="/wiki/Leyden_jar" title="Leyden jar">Leyden jars</a> into a wire.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> The phenomenon was not understood until 1905, when Pollock and Barraclough<sup id="cite_ref-pollock_1-1" class="reference"><a href="#cite_note-pollock-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> investigated a compressed and distorted length of copper tube from a <a href="/wiki/Lightning_rod" title="Lightning rod">lightning rod</a> after it had been struck by lightning. Their analysis showed that the forces due to the interaction of the large current flow with its own magnetic field could have caused the compression and distortion.<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> A similar, and apparently independent, theoretical analysis of the pinch effect in liquid metals was published by Northrup in 1907.<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> The next major development was the publication in 1934 of an analysis of the radial pressure balance in a static Z-pinch by <a href="/wiki/Willard_Harrison_Bennett" title="Willard Harrison Bennett">Bennett</a><sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> (see the following section for details). </p><p>Thereafter, the experimental and theoretical progress on pinches was driven by <a href="/wiki/Fusion_power" title="Fusion power">fusion power</a> research. In their article on the "Wire-array Z-pinch: a powerful x-ray source for <a href="/wiki/Inertial_confinement_fusion" title="Inertial confinement fusion">ICF</a>", M G Haines <i>et al.</i>, wrote on the "Early history of Z-pinches".<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> </p> <dl><dd>In 1946 Thompson and Blackman submitted a patent for a <a href="/wiki/Fusion_reactor" class="mw-redirect" title="Fusion reactor">fusion reactor</a> based on a toroidal Z-pinch<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> with an additional vertical magnetic field. But in 1954 Kruskal and Schwarzschild<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> published their theory of MHD instabilities in a Z-pinch. In 1956, Kurchatov gave his famous Harwell lecture showing nonthermal neutrons and the presence of <i>m</i> = 0 and <i>m</i> = 1 instabilities in a deuterium pinch.<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> In 1957 Pease<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> and Braginskii<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> independently predicted radiative collapse in a Z-pinch under pressure balance when in hydrogen the current exceeds 1.4 MA. (The viscous rather than resistive dissipation of magnetic energy discussed above and in<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> would however prevent radiative collapse).</dd></dl> <p>In 1958, the world's first controlled thermonuclear fusion experiment was accomplished using a theta-pinch machine named Scylla I at the <a href="/wiki/Los_Alamos_National_Laboratory" title="Los Alamos National Laboratory">Los Alamos National Laboratory</a>. A cylinder full of deuterium was converted into a plasma and compressed to 15 million degrees Celsius under a theta-pinch effect.<sup id="cite_ref-Phillips,_James_2013_7-1" class="reference"><a href="#cite_note-Phillips,_James_2013-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Lastly, at Imperial College in 1960, led by R Latham, the <a href="/wiki/Plateau%E2%80%93Rayleigh_instability" title="Plateau–Rayleigh instability">Plateau–Rayleigh instability</a> was shown, and its growth rate measured in a dynamic Z-pinch.<sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Equilibrium_analysis">Equilibrium analysis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pinch_(plasma_physics)&action=edit&section=7" title="Edit section: Equilibrium analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="One_dimension">One dimension</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pinch_(plasma_physics)&action=edit&section=8" title="Edit section: One dimension"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Plasma_physics" class="mw-redirect" title="Plasma physics">plasma physics</a> three pinch geometries are commonly studied: the θ-pinch, the <a href="/wiki/Z-pinch" title="Z-pinch">Z-pinch</a>, and the screw pinch. These are cylindrically shaped. The cylinder is symmetric in the axial (<i>z</i>) direction and the azimuthal (θ) directions. The one-dimensional pinches are named for the direction the current travels. </p> <div class="mw-heading mw-heading4"><h4 id="The_θ-pinch"><span id="The_.CE.B8-pinch"></span>The θ-pinch</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pinch_(plasma_physics)&action=edit&section=9" title="Edit section: The θ-pinch"><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:Thet_pinch.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Thet_pinch.png/200px-Thet_pinch.png" decoding="async" width="200" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Thet_pinch.png/300px-Thet_pinch.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Thet_pinch.png/400px-Thet_pinch.png 2x" data-file-width="1024" data-file-height="768" /></a><figcaption>A sketch of the θ-pinch equilibrium. The <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><span class="legend-color mw-no-invert" style="background-color:purple; color:white;"> </span> z-directed magnetic field corresponds to a <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959"><span class="legend-color mw-no-invert" style="background-color:yellow; color:black;"> </span> θ-directed plasma current.</figcaption></figure> <p>The θ-pinch has a magnetic field directed in the z direction and a large diamagnetic current directed in the θ direction. Using <a href="/wiki/Amp%C3%A8re%27s_circuital_law" title="Ampère's circuital law">Ampère's circuital law</a> (discarding the displacement term) </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\vec {B}}&=B_{z}(r){\hat {z}}\\\mu _{0}{\vec {J}}&=\nabla \times {\vec {B}}\\&={\frac {1}{r}}{\frac {d}{d\theta }}B_{z}(r){\hat {r}}-{\frac {d}{dr}}B_{z}(r){\hat {\theta }}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>J</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>θ</mi> </mrow> </mfrac> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\vec {B}}&=B_{z}(r){\hat {z}}\\\mu _{0}{\vec {J}}&=\nabla \times {\vec {B}}\\&={\frac {1}{r}}{\frac {d}{d\theta }}B_{z}(r){\hat {r}}-{\frac {d}{dr}}B_{z}(r){\hat {\theta }}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c53055aa8eb9f86db67bc19faa1d6a189f3af3e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.411ex; margin-bottom: -0.261ex; width:33.071ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}{\vec {B}}&=B_{z}(r){\hat {z}}\\\mu _{0}{\vec {J}}&=\nabla \times {\vec {B}}\\&={\frac {1}{r}}{\frac {d}{d\theta }}B_{z}(r){\hat {r}}-{\frac {d}{dr}}B_{z}(r){\hat {\theta }}\end{aligned}}}"></span></dd></dl> <p>Since <i>B</i> is only a function of <i>r</i> we can simplify this to </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 \mu _{0}{\vec {J}}=-{\frac {d}{dr}}B_{z}(r){\hat {\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>J</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{0}{\vec {J}}=-{\frac {d}{dr}}B_{z}(r){\hat {\theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50541e981d9d2fcb1ec885e367a6acb6b6e06d5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.231ex; height:5.509ex;" alt="{\displaystyle \mu _{0}{\vec {J}}=-{\frac {d}{dr}}B_{z}(r){\hat {\theta }}}"></span></dd></dl> <p>So <i>J</i> points in the θ direction. </p><p>Thus, the equilibrium condition (<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 p=\mathbf {j} \times \mathbf {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla p=\mathbf {j} \times \mathbf {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce0645250103410d5eec031d4e3d8c86dc75687c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.762ex; height:2.509ex;" alt="{\displaystyle \nabla p=\mathbf {j} \times \mathbf {B} }"></span>) for the θ-pinch reads: </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 {d}{dr}}\left(p+{\frac {B_{z}^{2}}{2\mu _{0}}}\right)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <mn>2</mn> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dr}}\left(p+{\frac {B_{z}^{2}}{2\mu _{0}}}\right)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b05b9be1f1a566a2146f880936152ca186ed4c7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.634ex; height:6.176ex;" alt="{\displaystyle {\frac {d}{dr}}\left(p+{\frac {B_{z}^{2}}{2\mu _{0}}}\right)=0}"></span></dd></dl> <p>θ-pinches tend to be resistant to plasma instabilities; This is due in part to <a href="/wiki/Alfv%C3%A9n%27s_theorem" title="Alfvén's theorem">Alfvén's theorem</a> (also known as the frozen-in flux theorem). </p> <div class="mw-heading mw-heading4"><h4 id="The_Z-pinch">The Z-pinch</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pinch_(plasma_physics)&action=edit&section=10" title="Edit section: The Z-pinch"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Z-pinch" title="Z-pinch">Z-pinch</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Z_pinch.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Z_pinch.png/200px-Z_pinch.png" decoding="async" width="200" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Z_pinch.png/300px-Z_pinch.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Z_pinch.png/400px-Z_pinch.png 2x" data-file-width="1024" data-file-height="768" /></a><figcaption>A sketch of the Z-pinch equilibrium. A <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959"><span class="legend-color mw-no-invert" style="background-color:purple; color:white;"> </span> θ-directed magnetic field corresponds to a <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959"><span class="legend-color mw-no-invert" style="background-color:yellow; color:black;"> </span> z-directed plasma current.</figcaption></figure> <p>The Z-pinch has a magnetic field in the θ direction and a current <i>J</i> flowing in the <i>z</i> direction. Again, by electrostatic Ampère's law, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\vec {B}}&=B_{\theta }(r){\hat {\theta }}\\\mu _{0}{\vec {J}}&=\nabla \times {\vec {B}}\\&={\frac {1}{r}}{\frac {d}{dr}}\left(rB_{\theta }(r)\right){\hat {z}}-{\frac {d}{dz}}B_{\theta }(r){\hat {r}}\\&={\frac {1}{r}}{\frac {d}{dr}}\left(rB_{\theta }(r)\right){\hat {z}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>J</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>r</mi> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>r</mi> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\vec {B}}&=B_{\theta }(r){\hat {\theta }}\\\mu _{0}{\vec {J}}&=\nabla \times {\vec {B}}\\&={\frac {1}{r}}{\frac {d}{dr}}\left(rB_{\theta }(r)\right){\hat {z}}-{\frac {d}{dz}}B_{\theta }(r){\hat {r}}\\&={\frac {1}{r}}{\frac {d}{dr}}\left(rB_{\theta }(r)\right){\hat {z}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/563881e21033e39b40300eb67457c57edc1489be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.505ex; width:36.644ex; height:18.176ex;" alt="{\displaystyle {\begin{aligned}{\vec {B}}&=B_{\theta }(r){\hat {\theta }}\\\mu _{0}{\vec {J}}&=\nabla \times {\vec {B}}\\&={\frac {1}{r}}{\frac {d}{dr}}\left(rB_{\theta }(r)\right){\hat {z}}-{\frac {d}{dz}}B_{\theta }(r){\hat {r}}\\&={\frac {1}{r}}{\frac {d}{dr}}\left(rB_{\theta }(r)\right){\hat {z}}\end{aligned}}}"></span></dd></dl> <p>Thus, the equilibrium condition, <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 p=\mathbf {j} \times \mathbf {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla p=\mathbf {j} \times \mathbf {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce0645250103410d5eec031d4e3d8c86dc75687c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.762ex; height:2.509ex;" alt="{\displaystyle \nabla p=\mathbf {j} \times \mathbf {B} }"></span>, for the Z-pinch reads: </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 {d}{dr}}\left(p+{\frac {B_{\theta }^{2}}{2\mu _{0}}}\right)+{\frac {B_{\theta }^{2}}{\mu _{0}r}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <mn>2</mn> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dr}}\left(p+{\frac {B_{\theta }^{2}}{2\mu _{0}}}\right)+{\frac {B_{\theta }^{2}}{\mu _{0}r}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a7f134afffc50db08a7c3f65c0c85ba3d26af2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.075ex; height:7.509ex;" alt="{\displaystyle {\frac {d}{dr}}\left(p+{\frac {B_{\theta }^{2}}{2\mu _{0}}}\right)+{\frac {B_{\theta }^{2}}{\mu _{0}r}}=0}"></span></dd></dl> <p>Although Z-pinches satisfy the MHD equilibrium condition, it is important to note that this is an unstable equilibrium, resulting in various instabilies such as the m = 0 instability ('sausage'), m = 1 instability ('kink'), and various other higher order instabilities.<sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="The_screw_pinch">The screw pinch</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pinch_(plasma_physics)&action=edit&section=11" title="Edit section: The screw pinch"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The screw pinch is an effort to combine the stability aspects of the θ-pinch and the confinement aspects of the Z-pinch. Referring once again to Ampère's law, </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 \nabla \times {\vec {B}}=\mu _{0}{\vec {J}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <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> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <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 \nabla \times {\vec {B}}=\mu _{0}{\vec {J}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fa5014e194a995f85161212fe2d70ec347f22ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.883ex; height:3.343ex;" alt="{\displaystyle \nabla \times {\vec {B}}=\mu _{0}{\vec {J}}}"></span></dd></dl> <p>But this time, the <i>B</i> field has a θ component <i>and</i> a <i>z</i> component </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\vec {B}}&=B_{\theta }{\hat {\theta }}+B_{z}{\hat {z}}\\\mu _{0}{\vec {J}}&={\frac {1}{r}}{\frac {d}{dr}}\left(rB_{\theta }\right){\hat {z}}-{\frac {d}{dr}}B_{z}{\hat {\theta }}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>J</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>r</mi> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\vec {B}}&=B_{\theta }{\hat {\theta }}+B_{z}{\hat {z}}\\\mu _{0}{\vec {J}}&={\frac {1}{r}}{\frac {d}{dr}}\left(rB_{\theta }\right){\hat {z}}-{\frac {d}{dr}}B_{z}{\hat {\theta }}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3074df4150553d76b3a536bcc24693157fc1e19c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.588ex; margin-bottom: -0.25ex; width:30.952ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}{\vec {B}}&=B_{\theta }{\hat {\theta }}+B_{z}{\hat {z}}\\\mu _{0}{\vec {J}}&={\frac {1}{r}}{\frac {d}{dr}}\left(rB_{\theta }\right){\hat {z}}-{\frac {d}{dr}}B_{z}{\hat {\theta }}\end{aligned}}}"></span></dd></dl> <p>So this time <i>J</i> has a component in the <i>z</i> direction and a component in the θ direction. </p><p>Finally, the equilibrium condition (<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 p=\mathbf {j} \times \mathbf {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla p=\mathbf {j} \times \mathbf {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce0645250103410d5eec031d4e3d8c86dc75687c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.762ex; height:2.509ex;" alt="{\displaystyle \nabla p=\mathbf {j} \times \mathbf {B} }"></span>) for the screw pinch reads: </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 {d}{dr}}\left(p+{\frac {B_{z}^{2}+B_{\theta }^{2}}{2\mu _{0}}}\right)+{\frac {B_{\theta }^{2}}{\mu _{0}r}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mn>2</mn> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dr}}\left(p+{\frac {B_{z}^{2}+B_{\theta }^{2}}{2\mu _{0}}}\right)+{\frac {B_{\theta }^{2}}{\mu _{0}r}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12246f9f9206dd4f690d928f83bb894a0389bafc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.934ex; height:7.509ex;" alt="{\displaystyle {\frac {d}{dr}}\left(p+{\frac {B_{z}^{2}+B_{\theta }^{2}}{2\mu _{0}}}\right)+{\frac {B_{\theta }^{2}}{\mu _{0}r}}=0}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="The_screw_pinch_via_colliding_optical_vortices">The screw pinch via colliding optical vortices</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pinch_(plasma_physics)&action=edit&section=12" title="Edit section: The screw pinch via colliding optical vortices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <i> screw pinch </i> might be produced in laser plasma by colliding optical vortices of ultrashort duration.<sup id="cite_ref-Okulov2010plasma_53-0" class="reference"><a href="#cite_note-Okulov2010plasma-53"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> For this purpose optical vortices should be phase-conjugated.<sup id="cite_ref-Okulov2008_phase_conjugation_54-0" class="reference"><a href="#cite_note-Okulov2008_phase_conjugation-54"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> The magnetic field distribution is given here again via Ampère's law: </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 \nabla \times {\vec {B}}=\mu _{0}{\vec {J}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <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> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <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 \nabla \times {\vec {B}}=\mu _{0}{\vec {J}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fa5014e194a995f85161212fe2d70ec347f22ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.883ex; height:3.343ex;" alt="{\displaystyle \nabla \times {\vec {B}}=\mu _{0}{\vec {J}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Two_dimensions">Two dimensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pinch_(plasma_physics)&action=edit&section=13" title="Edit section: Two dimensions"><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:Toroidal_coord.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/Toroidal_coord.png/250px-Toroidal_coord.png" decoding="async" width="250" height="188" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/Toroidal_coord.png/375px-Toroidal_coord.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/db/Toroidal_coord.png/500px-Toroidal_coord.png 2x" data-file-width="1024" data-file-height="768" /></a><figcaption>A <i>toroidal coordinate system</i> in common use in plasma physics. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959"><div class="legend"><span class="legend-color mw-no-invert" style="background-color:red; color:black;"> </span> The red arrow denotes the <i>poloidal</i> direction (θ)</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959"><div class="legend"><span class="legend-color mw-no-invert" style="background-color:blue; color:white;"> </span> The blue arrow denotes the <i>toroidal</i> direction (φ)</div> </figcaption></figure> <p>A common problem with one-dimensional pinches is the end losses. Most of the motion of particles is along the magnetic field. With the θ-pinch and the screw-pinch, this leads particles out of the end of the machine very quickly, leading to a loss of mass and energy. Along with this problem, the Z-pinch has major stability problems. Though particles can be reflected to some extent with <a href="/wiki/Magnetic_mirror" title="Magnetic mirror">magnetic mirrors</a>, even these allow many particles to pass. A common method of beating these end losses, is to bend the cylinder around into a torus. Unfortunately this breaks θ symmetry, as paths on the inner portion (inboard side) of the torus are shorter than similar paths on the outer portion (outboard side). Thus, a new theory is needed. This gives rise to the famous <a href="/wiki/Grad%E2%80%93Shafranov_equation" title="Grad–Shafranov equation">Grad–Shafranov equation</a>. Numerical solutions to the Grad–Shafranov equation have also yielded some equilibria, most notably that of the <a href="/wiki/Reversed_field_pinch" title="Reversed field pinch">reversed field pinch</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Three_dimensions">Three dimensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pinch_(plasma_physics)&action=edit&section=14" title="Edit section: Three dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As of 2015<sup class="plainlinks noexcerpt noprint asof-tag update" style="display:none;"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Pinch_(plasma_physics)&action=edit">[update]</a></sup>, there is no coherent analytical theory for three-dimensional equilibria. The general approach to finding such equilibria is to solve the vacuum ideal MHD equations. Numerical solutions have yielded designs for <a href="/wiki/Stellarator" title="Stellarator">stellarators</a>. Some machines take advantage of simplification techniques such as helical symmetry (for example University of Wisconsin's Helically Symmetric eXperiment). However, for an arbitrary three-dimensional configuration, an equilibrium relation, similar to that of the 1-D configurations exists:<sup id="cite_ref-55" class="reference"><a href="#cite_note-55"><span class="cite-bracket">[</span>55<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 \nabla _{\perp }\left(p+{\frac {B^{2}}{2\mu _{0}}}\right)-{\frac {B^{2}}{\mu _{0}}}{\vec {\kappa }}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>κ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{\perp }\left(p+{\frac {B^{2}}{2\mu _{0}}}\right)-{\frac {B^{2}}{\mu _{0}}}{\vec {\kappa }}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4208826362cf4d5a647fc6a47bbe97f9ea6daa83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.814ex; height:6.343ex;" alt="{\displaystyle \nabla _{\perp }\left(p+{\frac {B^{2}}{2\mu _{0}}}\right)-{\frac {B^{2}}{\mu _{0}}}{\vec {\kappa }}=0}"></span></dd></dl> <p>Where κ is the curvature vector defined as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\kappa }}=\left({\vec {b}}\cdot \nabla \right){\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>κ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> </mrow> <mo>)</mo> </mrow> <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 {\kappa }}=\left({\vec {b}}\cdot \nabla \right){\vec {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d02013013d49d2086658fa2d2581d5a2f1d3e298" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.403ex; height:4.843ex;" alt="{\displaystyle {\vec {\kappa }}=\left({\vec {b}}\cdot \nabla \right){\vec {b}}}"></span></dd></dl> <p>with <i>b</i> the unit vector tangent to <i>B</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Formal_treatment">Formal treatment</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pinch_(plasma_physics)&action=edit&section=15" title="Edit section: Formal treatment"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Water-pinching.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/7/7f/Water-pinching.jpg/256px-Water-pinching.jpg" decoding="async" width="256" height="315" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/7/7f/Water-pinching.jpg/384px-Water-pinching.jpg 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/7/7f/Water-pinching.jpg/512px-Water-pinching.jpg 2x" data-file-width="862" data-file-height="1062" /></a><figcaption><b>A stream of water pinching</b> into droplets has been suggested as an analogy to the electromagnetic pinch.<sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> Gravity accelerates free-falling water which causes the water column to constrict. <a href="/wiki/Surface_tension" title="Surface tension">Surface tension</a> breaks the narrowing water column into droplets (not shown, see <a href="/wiki/Plateau%E2%80%93Rayleigh_instability" title="Plateau–Rayleigh instability">Plateau–Rayleigh instability</a>). This is analogous to the <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a> suggested as the cause of pinching in bead lightning.<sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> The morphology (shape) is similar to the so-called sausage <a href="/wiki/Plasma_stability#Plasma_instabilities" title="Plasma stability">instability</a> in plasma.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="The_Bennett_relation">The Bennett relation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pinch_(plasma_physics)&action=edit&section=16" title="Edit section: The Bennett relation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider a cylindrical column of fully ionized quasineutral plasma, with an axial electric field, producing an axial current density, <b>j</b>, and associated azimuthal magnetic field, <b>B</b>. As the current flows through its own magnetic field, a pinch is generated with an inward radial force density of <b>j x B</b>. In a steady state with forces balancing: </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 \nabla p=\nabla (p_{e}+p_{i})=\mathbf {j} \times \mathbf {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mi>p</mi> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla p=\nabla (p_{e}+p_{i})=\mathbf {j} \times \mathbf {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd49fdddf104b9c063e5ba23f910a6f759b3dc90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.583ex; height:2.843ex;" alt="{\displaystyle \nabla p=\nabla (p_{e}+p_{i})=\mathbf {j} \times \mathbf {B} }"></span></dd></dl> <p>where ∇<i>p</i> is the magnetic pressure gradient, and <i>p</i><sub>e</sub> and <i>p</i><sub>i</sub> are the electron and ion pressures, respectively. Then using <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equation</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 \nabla \times \mathbf {B} =\mu _{0}\mathbf {j} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {j} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8efee9146b42e6e4cff08829d9692e0ca11c758e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.048ex; height:2.676ex;" alt="{\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {j} }"></span> and the <a href="/wiki/Ideal_gas_law" title="Ideal gas law">ideal gas law</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 p=NkT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>N</mi> <mi>k</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=NkT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b113b351ba46e84f9d6cb797bef15fd9965a1ef4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.268ex; height:2.509ex;" alt="{\displaystyle p=NkT}"></span>, we derive: </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 2Nk(T_{e}+T_{i})={\frac {\mu _{0}}{4\pi }}I^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>N</mi> <mi>k</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2Nk(T_{e}+T_{i})={\frac {\mu _{0}}{4\pi }}I^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aad30d52b5b7926ba39824b3f026347bc04a15fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.3ex; height:4.843ex;" alt="{\displaystyle 2Nk(T_{e}+T_{i})={\frac {\mu _{0}}{4\pi }}I^{2}}"></span> (the Bennett relation)</dd></dl> <p>where <i>N</i> is the number of electrons per unit length along the axis, <i>T<sub>e</sub></i> and <i>T<sub>i</sub></i> are the electron and ion temperatures, <i>I</i> is the total beam current, and <i>k</i> is the <a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a>. </p> <div class="mw-heading mw-heading3"><h3 id="The_generalized_Bennett_relation">The generalized Bennett relation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pinch_(plasma_physics)&action=edit&section=17" title="Edit section: The generalized Bennett relation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Generalized_Bennett_Relation_diagram.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/3/3f/Generalized_Bennett_Relation_diagram.png/200px-Generalized_Bennett_Relation_diagram.png" decoding="async" width="200" height="321" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/3/3f/Generalized_Bennett_Relation_diagram.png/300px-Generalized_Bennett_Relation_diagram.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/3/3f/Generalized_Bennett_Relation_diagram.png/400px-Generalized_Bennett_Relation_diagram.png 2x" data-file-width="594" data-file-height="954" /></a><figcaption>The generalized Bennett relation considers a current-carrying magnetic-field-aligned cylindrical plasma pinch undergoing rotation at angular frequency ω</figcaption></figure><p>The <i>generalized Bennett relation</i> considers a current-carrying magnetic-field-aligned cylindrical plasma pinch undergoing rotation at angular frequency ω. Along the axis of the plasma cylinder flows a current density j<sub>z</sub>, resulting in an azimuthal magnetic field Β<sub>φ</sub>. Originally derived by Witalis,<sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> the generalized Bennett relation results in:<sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup> </p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {1}{4}}{\frac {\partial ^{2}J_{0}}{\partial t^{2}}}={}&W_{\perp {\text{kin}}}+\Delta W_{E_{z}}+\Delta W_{B_{z}}+\Delta W_{k}-{\frac {\mu _{0}}{8\pi }}I^{2}(a)\\[8pt]&{}-{\frac {1}{2}}G{\overline {m}}^{2}N^{2}(a)+{\frac {1}{2}}\pi a^{2}\epsilon _{0}\left(E_{r}^{2}(a)-E_{\phi }^{2}(a)\right)\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.1em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>kin</mtext> </mrow> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mn>8</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>G</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>π</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {1}{4}}{\frac {\partial ^{2}J_{0}}{\partial t^{2}}}={}&W_{\perp {\text{kin}}}+\Delta W_{E_{z}}+\Delta W_{B_{z}}+\Delta W_{k}-{\frac {\mu _{0}}{8\pi }}I^{2}(a)\\[8pt]&{}-{\frac {1}{2}}G{\overline {m}}^{2}N^{2}(a)+{\frac {1}{2}}\pi a^{2}\epsilon _{0}\left(E_{r}^{2}(a)-E_{\phi }^{2}(a)\right)\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71e972f92a0d232a7244df3fa2ed04d57711196f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:55.828ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {1}{4}}{\frac {\partial ^{2}J_{0}}{\partial t^{2}}}={}&W_{\perp {\text{kin}}}+\Delta W_{E_{z}}+\Delta W_{B_{z}}+\Delta W_{k}-{\frac {\mu _{0}}{8\pi }}I^{2}(a)\\[8pt]&{}-{\frac {1}{2}}G{\overline {m}}^{2}N^{2}(a)+{\frac {1}{2}}\pi a^{2}\epsilon _{0}\left(E_{r}^{2}(a)-E_{\phi }^{2}(a)\right)\\\end{aligned}}}"></span></dd></dl> <ul><li>where a current-carrying, magnetic-field-aligned cylindrical plasma has a radius <i>a</i>,</li> <li><i>J</i><sub>0</sub> is the total moment of inertia with respect to the z axis,</li> <li><i>W</i><sub>⊥kin</sub> is the <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a> per unit length due to beam motion transverse to the beam axis</li> <li><i>W</i><sub>B<sub>z</sub></sub> is the self-consistent B<sub>z</sub> energy per unit length</li> <li><i>W</i><sub>E<sub>z</sub></sub> is the self-consistent E<sub>z</sub> energy per unit length</li> <li><i>W</i><sub>k</sub> is thermokinetic energy per unit length</li> <li><i>I</i>(<i>a</i>) is the axial current inside the radius <i>a</i> (<i>r</i> in diagram)</li> <li><i>N</i>(<i>a</i>) is the total number of particles per unit length</li> <li><i>E</i><sub>r</sub> is the radial electric field</li> <li><i>E</i><sub>φ</sub> is the rotational electric field</li></ul> <p>The positive terms in the equation are expansional forces while the negative terms represent beam compressional forces. </p> <div class="mw-heading mw-heading3"><h3 id="The_Carlqvist_relation">The Carlqvist relation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pinch_(plasma_physics)&action=edit&section=18" title="Edit section: The Carlqvist relation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Carlqvist relation, published by <a href="/wiki/Per_Carlqvist" title="Per Carlqvist">Per Carlqvist</a> in 1988,<sup id="cite_ref-adsabs.harvard.edu_12-1" class="reference"><a href="#cite_note-adsabs.harvard.edu-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> is a specialization of the generalized Bennett relation (above), for the case that the kinetic pressure is much smaller at the border of the pinch than in the inner parts. It takes the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mu _{0}}{8\pi }}I^{2}(a)+{\frac {1}{2}}G{\overline {m}}^{2}N^{2}(a)=\Delta W_{B_{z}}+\Delta W_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mn>8</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>G</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mu _{0}}{8\pi }}I^{2}(a)+{\frac {1}{2}}G{\overline {m}}^{2}N^{2}(a)=\Delta W_{B_{z}}+\Delta W_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/473e676551b78a26fd5ce2a3cd0083f385cdedbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:42.287ex; height:5.176ex;" alt="{\displaystyle {\frac {\mu _{0}}{8\pi }}I^{2}(a)+{\frac {1}{2}}G{\overline {m}}^{2}N^{2}(a)=\Delta W_{B_{z}}+\Delta W_{k}}"></span></dd></dl> <p>and is applicable to many space plasmas. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Bennett_Pinch_graph.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/b/b9/Bennett_Pinch_graph.png/400px-Bennett_Pinch_graph.png" decoding="async" width="400" height="334" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/b/b9/Bennett_Pinch_graph.png/600px-Bennett_Pinch_graph.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/b/b9/Bennett_Pinch_graph.png/800px-Bennett_Pinch_graph.png 2x" data-file-width="1053" data-file-height="878" /></a><figcaption>The Bennett pinch showing the total current (I) versus the number of particles per unit length (N). The chart illustrates four physically distinct regions. The plasma temperature is 20 K, the mean particle mass 3×10<sup>−27</sup> kg, and ΔW<sub>Bz</sub> is the excess magnetic energy per unit length due to the axial magnetic field B<sub>z</sub>. The plasma is assumed to be non-rotational, and the kinetic pressure at the edges is much smaller than inside.</figcaption></figure> <p>The Carlqvist relation can be illustrated (see right), showing the total current (<i>I</i>) versus the number of particles per unit length (<i>N</i>) in a Bennett pinch. The chart illustrates four physically distinct regions. The plasma temperature is quite cold (<i>T</i><sub>i</sub> = <i>T</i><sub>e</sub> = <i>T</i><sub>n</sub> = 20 K), containing mainly hydrogen with a mean particle mass 3×10<sup>−27</sup> kg. The thermokinetic energy <i>W</i><sub><i>k</i></sub> >> <i>πa</i><sup>2</sup> <i>p</i><sub><i>k</i></sub>(a). The curves, ΔW<sub>Bz</sub> show different amounts of excess magnetic energy per unit length due to the axial magnetic field B<sub>z</sub>. The plasma is assumed to be non-rotational, and the kinetic pressure at the edges is much smaller than inside. </p><p><b>Chart regions:</b> (a) In the top-left region, the pinching force dominates. (b) Towards the bottom, outward kinetic pressures balance inwards magnetic pressure, and the total pressure is constant. (c) To the right of the vertical line Δ<i>W</i><sub><i>B</i>z</sub> = 0, the magnetic pressures balances the gravitational pressure, and the pinching force is negligible. (d) To the left of the sloping curve Δ<i>W</i><sub><i>B</i>z</sub> = 0, the gravitational force is negligible. Note that the chart shows a special case of the Carlqvist relation, and if it is replaced by the more general Bennett relation, then the designated regions of the chart are not valid. </p><p>Carlqvist further notes that by using the relations above, and a derivative, it is possible to describe the Bennett pinch, the <a href="/wiki/Jeans_instability" title="Jeans instability">Jeans criterion</a> (for gravitational instability,<sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup> in one and two dimensions), <a href="/wiki/Birkeland_current" title="Birkeland current">force-free magnetic fields</a>, gravitationally balanced magnetic pressures, and continuous transitions between these states. </p> <div class="mw-heading mw-heading2"><h2 id="References_in_culture">References in culture</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pinch_(plasma_physics)&action=edit&section=19" title="Edit section: References in culture"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A fictionalized <a href="/wiki/Explosively_pumped_flux_compression_generator" title="Explosively pumped flux compression generator">pinch-generating device</a> was used in <i><a href="/wiki/Ocean%27s_Eleven_(2001_film)" class="mw-redirect" title="Ocean's Eleven (2001 film)">Ocean's Eleven</a></i>, where it was used to disrupt Las Vegas's power grid just long enough for the characters to begin their heist.<sup id="cite_ref-61" class="reference"><a href="#cite_note-61"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pinch_(plasma_physics)&action=edit&section=20" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Electromagnetic_forming" title="Electromagnetic forming">Electromagnetic forming</a></li> <li><a href="/wiki/Explosively_pumped_flux_compression_generator" title="Explosively pumped flux compression generator">Explosively pumped flux compression generator</a></li> <li><a href="/wiki/Fusion_power" title="Fusion power">Fusion power</a></li> <li><a href="/wiki/Madison_Symmetric_Torus" title="Madison Symmetric Torus">Madison Symmetric Torus</a> (reversed field pinch)</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=Pinch_(plasma_physics)&action=edit&section=21" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-pollock-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-pollock_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-pollock_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Pollock J A and Barraclough S (1905) <i>Proc. R. Soc. New South Wales</i> 39 131</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="CITEREFBuneman1961" class="citation book cs1">Buneman, O. 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New York: McGraw-Hill. p. 202. <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/1961plph.conf..202B">1961plph.conf..202B</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Bennett+Pinch&rft.btitle=Plasma+Physics&rft.place=New+York&rft.pages=202&rft.pub=McGraw-Hill&rft.date=1961&rft_id=info%3Abibcode%2F1961plph.conf..202B&rft.aulast=Buneman&rft.aufirst=O.&rft_id=https%3A%2F%2Fui.adsabs.harvard.edu%2Fabs%2F1961plph.conf..202B%2Fabstract&rfr_id=info%3Asid%2Fen.wikipedia.org%3APinch+%28plasma+physics%29" class="Z3988"></span> LOC 60-12766.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLee1983" class="citation journal cs1">Lee, S. 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S.; Vikhrev, V. V.; Ivanov, V. V.; Rozanova, G. A. (1990). "Pinch Mechanism of Energy Release of Stellar Flares". <i>Astrophysics</i>. <b>32</b> (3): 230–235. <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/1990Ap.....32..230A">1990Ap.....32..230A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf01005504">10.1007/bf01005504</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:120883355">120883355</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Astrophysics&rft.atitle=Pinch+Mechanism+of+Energy+Release+of+Stellar+Flares&rft.volume=32&rft.issue=3&rft.pages=230-235&rft.date=1990&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120883355%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fbf01005504&rft_id=info%3Abibcode%2F1990Ap.....32..230A&rft.aulast=Airapetyan&rft.aufirst=V.+S.&rft.au=Vikhrev%2C+V.+V.&rft.au=Ivanov%2C+V.+V.&rft.au=Rozanova%2C+G.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APinch+%28plasma+physics%29" class="Z3988"></span></span> </li> <li id="cite_note-adsabs.harvard.edu-12"><span class="mw-cite-backlink">^ <a href="#cite_ref-adsabs.harvard.edu_12-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-adsabs.harvard.edu_12-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarlqvist1988" class="citation journal cs1">Carlqvist, Per (May 1988). <a rel="nofollow" class="external text" href="https://link.springer.com/article/10.1007/BF00793173">"Cosmic electric currents and the generalized Bennett relation"</a>. <i>Astrophysics and Space Science</i>. <b>144</b> (1–2): 73–84. <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/1988Ap&SS.144...73C">1988Ap&SS.144...73C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00793173">10.1007/BF00793173</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:119719745">119719745</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Astrophysics+and+Space+Science&rft.atitle=Cosmic+electric+currents+and+the+generalized+Bennett+relation&rft.volume=144&rft.issue=1%E2%80%932&rft.pages=73-84&rft.date=1988-05&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119719745%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF00793173&rft_id=info%3Abibcode%2F1988Ap%26SS.144...73C&rft.aulast=Carlqvist&rft.aufirst=Per&rft_id=https%3A%2F%2Flink.springer.com%2Farticle%2F10.1007%2FBF00793173&rfr_id=info%3Asid%2Fen.wikipedia.org%3APinch+%28plasma+physics%29" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBiskamp1997" class="citation book cs1">Biskamp, Dieter (1997). <i>Nonlinear Magnetohydrodynamics</i>. <a href="/wiki/Cambridge" title="Cambridge">Cambridge</a>, England: Cambridge University Press. p. 130. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-59918-0" title="Special:BookSources/0-521-59918-0"><bdi>0-521-59918-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Nonlinear+Magnetohydrodynamics&rft.place=Cambridge%2C+England&rft.pages=130&rft.pub=Cambridge+University+Press&rft.date=1997&rft.isbn=0-521-59918-0&rft.aulast=Biskamp&rft.aufirst=Dieter&rfr_id=info%3Asid%2Fen.wikipedia.org%3APinch+%28plasma+physics%29" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBasu2018" class="citation book cs1">Basu, Dipak K. (8 October 2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vvJazZZpB1QC&dq=thetatron+pinch&pg=PA315"><i>Dictionary of Material Science and High Energy Physics</i></a>. CRC Press. p. 315. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8493-2889-3" title="Special:BookSources/978-0-8493-2889-3"><bdi>978-0-8493-2889-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dictionary+of+Material+Science+and+High+Energy+Physics&rft.pages=315&rft.pub=CRC+Press&rft.date=2018-10-08&rft.isbn=978-0-8493-2889-3&rft.aulast=Basu&rft.aufirst=Dipak+K.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DvvJazZZpB1QC%26dq%3Dthetatron%2Bpinch%26pg%3DPA315&rfr_id=info%3Asid%2Fen.wikipedia.org%3APinch+%28plasma+physics%29" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSrivastavaVyas1982" class="citation journal cs1">Srivastava, K. M.; Vyas, D. N. (August 1982). <a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1982Ap%26SS..86...71S/abstract">"Non-linear analysis of the stability of the screw pinch"</a>. <i>Astrophysics and Space Science</i>. <b>86</b> (1): 71–89. <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/1982Ap&SS..86...71S">1982Ap&SS..86...71S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00651831">10.1007/BF00651831</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:121575638">121575638</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Astrophysics+and+Space+Science&rft.atitle=Non-linear+analysis+of+the+stability+of+the+screw+pinch&rft.volume=86&rft.issue=1&rft.pages=71-89&rft.date=1982-08&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121575638%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF00651831&rft_id=info%3Abibcode%2F1982Ap%26SS..86...71S&rft.aulast=Srivastava&rft.aufirst=K.+M.&rft.au=Vyas%2C+D.+N.&rft_id=https%3A%2F%2Fui.adsabs.harvard.edu%2Fabs%2F1982Ap%2526SS..86...71S%2Fabstract&rfr_id=info%3Asid%2Fen.wikipedia.org%3APinch+%28plasma+physics%29" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">See "<a rel="nofollow" class="external text" href="http://silas.psfc.mit.edu/introplasma/chap4.html#tth_sEc4.7">MHD Equilibria" in Introduction to Plasma Physics by I.H.Hutchinson (2001)</a></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSrivastavaWaelbroeck1976" class="citation journal cs1">Srivastava, K. M.; Waelbroeck, F. (1976). "On the stability of the screw pinch in the CGL model". <i>Journal of Plasma Physics</i>. <b>16</b> (3): 261. <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/1976JPlPh..16..261S">1976JPlPh..16..261S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2Fs0022377800020201">10.1017/s0022377800020201</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:123689314">123689314</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Plasma+Physics&rft.atitle=On+the+stability+of+the+screw+pinch+in+the+CGL+model&rft.volume=16&rft.issue=3&rft.pages=261&rft.date=1976&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A123689314%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2Fs0022377800020201&rft_id=info%3Abibcode%2F1976JPlPh..16..261S&rft.aulast=Srivastava&rft.aufirst=K.+M.&rft.au=Waelbroeck%2C+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APinch+%28plasma+physics%29" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAndersonFurthStoneWright1958" class="citation journal cs1">Anderson, O. A.; Furth, H. P.; Stone, J. M.; Wright, R. E. (November 1958). "Inverse Pinch Effect". <i>Physics of Fluids</i>. <b>1</b> (6): 489–494. <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/1958PhFl....1..489A">1958PhFl....1..489A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1724372">10.1063/1.1724372</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+of+Fluids&rft.atitle=Inverse+Pinch+Effect&rft.volume=1&rft.issue=6&rft.pages=489-494&rft.date=1958-11&rft_id=info%3Adoi%2F10.1063%2F1.1724372&rft_id=info%3Abibcode%2F1958PhFl....1..489A&rft.aulast=Anderson&rft.aufirst=O.+A.&rft.au=Furth%2C+H.+P.&rft.au=Stone%2C+J.+M.&rft.au=Wright%2C+R.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APinch+%28plasma+physics%29" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHelanderAkersValovič2005" class="citation journal cs1">Helander, P.; Akers, R. J.; Valovič, M. (3 November 2005). "The effect of non-inductive current drive on tokamak transport". <i>Plasma Physics and Controlled Fusion</i>. <b>47</b> (12B): B151–B163. <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/2005PPCF...47B.151H">2005PPCF...47B.151H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0741-3335%2F47%2F12b%2Fs12">10.1088/0741-3335/47/12b/s12</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:121961613">121961613</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Plasma+Physics+and+Controlled+Fusion&rft.atitle=The+effect+of+non-inductive+current+drive+on+tokamak+transport&rft.volume=47&rft.issue=12B&rft.pages=B151-B163&rft.date=2005-11-03&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121961613%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F0741-3335%2F47%2F12b%2Fs12&rft_id=info%3Abibcode%2F2005PPCF...47B.151H&rft.aulast=Helander&rft.aufirst=P.&rft.au=Akers%2C+R.+J.&rft.au=Valovi%C4%8D%2C+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APinch+%28plasma+physics%29" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNishikawaWakatani2000" class="citation book cs1">Nishikawa, K.; Wakatani, M. (2000-01-24). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4cHkd77TSHcC&dq=Ware+pinch&pg=PA266"><i>Plasma Physics: Third Edition</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-65285-4" title="Special:BookSources/978-3-540-65285-4"><bdi>978-3-540-65285-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Plasma+Physics%3A+Third+Edition&rft.pub=Springer+Science+%26+Business+Media&rft.date=2000-01-24&rft.isbn=978-3-540-65285-4&rft.aulast=Nishikawa&rft.aufirst=K.&rft.au=Wakatani%2C+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D4cHkd77TSHcC%26dq%3DWare%2Bpinch%26pg%3DPA266&rfr_id=info%3Asid%2Fen.wikipedia.org%3APinch+%28plasma+physics%29" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSlutzVesey2012" class="citation journal cs1">Slutz, Stephen; Vesey, Roger A. (2012). <a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.108.025003">"High-Gain Magnetized Inertial Fusion"</a>. <i>Physical Review Letters</i>. <b>108</b> (2): 025003. <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/2012PhRvL.108b5003S">2012PhRvL.108b5003S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.108.025003">10.1103/PhysRevLett.108.025003</a></span>. <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/22324693">22324693</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=High-Gain+Magnetized+Inertial+Fusion&rft.volume=108&rft.issue=2&rft.pages=025003&rft.date=2012&rft_id=info%3Apmid%2F22324693&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.108.025003&rft_id=info%3Abibcode%2F2012PhRvL.108b5003S&rft.aulast=Slutz&rft.aufirst=Stephen&rft.au=Vesey%2C+Roger+A.&rft_id=https%3A%2F%2Fdoi.org%2F10.1103%252FPhysRevLett.108.025003&rfr_id=info%3Asid%2Fen.wikipedia.org%3APinch+%28plasma+physics%29" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHardee1982" class="citation journal cs1">Hardee, P. E. (1982). <a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F160008">"Helical and pinching instability of supersonic expanding jets in extragalactic radio sources"</a>. <i>Astrophysical Journal</i>. <b>257</b>: 509–526. <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/1982ApJ...257..509H">1982ApJ...257..509H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F160008">10.1086/160008</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Astrophysical+Journal&rft.atitle=Helical+and+pinching+instability+of+supersonic+expanding+jets+in+extragalactic+radio+sources&rft.volume=257&rft.pages=509-526&rft.date=1982&rft_id=info%3Adoi%2F10.1086%2F160008&rft_id=info%3Abibcode%2F1982ApJ...257..509H&rft.aulast=Hardee&rft.aufirst=P.+E.&rft_id=https%3A%2F%2Fdoi.org%2F10.1086%252F160008&rfr_id=info%3Asid%2Fen.wikipedia.org%3APinch+%28plasma+physics%29" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPereira1988" class="citation journal cs1">Pereira, N. R.; et al. (1988). "X-rays from Z-pinches on relativistic electron-beam generators". <i>Journal of Applied Physics</i>. <b>64</b> (3): R1–R27. <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/1988JAP....64....1P">1988JAP....64....1P</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.341808">10.1063/1.341808</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Applied+Physics&rft.atitle=X-rays+from+Z-pinches+on+relativistic+electron-beam+generators&rft.volume=64&rft.issue=3&rft.pages=R1-R27&rft.date=1988&rft_id=info%3Adoi%2F10.1063%2F1.341808&rft_id=info%3Abibcode%2F1988JAP....64....1P&rft.aulast=Pereira&rft.aufirst=N.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APinch+%28plasma+physics%29" class="Z3988"></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWuChenLi2005" class="citation journal cs1">Wu, Mei; Chen, Li; Li, Ti-Pei (2005). "Polarization in Gamma-Ray Bursts Produced by Pinch Discharge". <i>Chinese Journal of Astronomy & Astrophysics</i>. <b>5</b> (1): 57–64. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/astro-ph/0501334">astro-ph/0501334</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005ChJAA...5...57W">2005ChJAA...5...57W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1009-9271%2F5%2F1%2F007">10.1088/1009-9271/5/1/007</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:121943">121943</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Chinese+Journal+of+Astronomy+%26+Astrophysics&rft.atitle=Polarization+in+Gamma-Ray+Bursts+Produced+by+Pinch+Discharge&rft.volume=5&rft.issue=1&rft.pages=57-64&rft.date=2005&rft_id=info%3Aarxiv%2Fastro-ph%2F0501334&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121943%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F1009-9271%2F5%2F1%2F007&rft_id=info%3Abibcode%2F2005ChJAA...5...57W&rft.aulast=Wu&rft.aufirst=Mei&rft.au=Chen%2C+Li&rft.au=Li%2C+Ti-Pei&rfr_id=info%3Asid%2Fen.wikipedia.org%3APinch+%28plasma+physics%29" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text">Peratt, A.L., "<a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZSlJRAeL95sC&dq=synchrotron+pinch&pg=PA62">Synchrotron radiation from pinched particle beams</a>", (1998) Plasma Physics: VII Lawpp 97: Proceedings of the 1997 Latin American Workshop on Plasma Physics, Edited by Pablo Martin, Julio Puerta, Pablo Martmn, with reference to Meierovich, B. E., "<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1984PhR...104..259M/abstract">Electromagnetic collapse. Problems of stability, emission of radiation and evolution of a dense pinch</a>" (1984) <i>Physics Reports</i>, Volume 104, Issue 5, p. 259-346.</span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnderson1958" class="citation journal cs1">Anderson, Oscar A.; et al. (1958). <a rel="nofollow" class="external text" href="https://escholarship.org/uc/item/0gj9x38q">"Neutron Production in Linear Deuterium Pinches"</a>. <i>Physical Review</i>. <b>110</b> (6): 1375–1387. <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/1958PhRv..110.1375A">1958PhRv..110.1375A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysrev.110.1375">10.1103/physrev.110.1375</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review&rft.atitle=Neutron+Production+in+Linear+Deuterium+Pinches&rft.volume=110&rft.issue=6&rft.pages=1375-1387&rft.date=1958&rft_id=info%3Adoi%2F10.1103%2Fphysrev.110.1375&rft_id=info%3Abibcode%2F1958PhRv..110.1375A&rft.aulast=Anderson&rft.aufirst=Oscar+A.&rft_id=https%3A%2F%2Fescholarship.org%2Fuc%2Fitem%2F0gj9x38q&rfr_id=info%3Asid%2Fen.wikipedia.org%3APinch+%28plasma+physics%29" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRyutovDerzonMatzen2000" class="citation journal cs1">Ryutov, D. D.; Derzon, M. S.; Matzen, M. K (2000). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1233969">"The physics of fast Z pinches"</a>. <i>Reviews of Modern Physics</i>. <b>72</b> (1): 167–223. <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/2000RvMP...72..167R">2000RvMP...72..167R</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Frevmodphys.72.167">10.1103/revmodphys.72.167</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reviews+of+Modern+Physics&rft.atitle=The+physics+of+fast+Z+pinches&rft.volume=72&rft.issue=1&rft.pages=167-223&rft.date=2000&rft_id=info%3Adoi%2F10.1103%2Frevmodphys.72.167&rft_id=info%3Abibcode%2F2000RvMP...72..167R&rft.aulast=Ryutov&rft.aufirst=D.+D.&rft.au=Derzon%2C+M.+S.&rft.au=Matzen%2C+M.+K&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1233969&rfr_id=info%3Asid%2Fen.wikipedia.org%3APinch+%28plasma+physics%29" class="Z3988"></span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text">Andre Gsponer, "<a rel="nofollow" class="external text" href="https://arxiv.org/abs/physics/0409157">Physics of high-intensity high-energy particle beam propagation in open air and outer-space plasmas</a>" (2004) <a rel="nofollow" class="external free" href="https://arxiv.org/abs/physics/0409157">https://arxiv.org/abs/physics/0409157</a></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text">Peratt, Anthony L., "<a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/abs/1988LaPaB...6..471P">The role of particle beams and electrical currents in the plasma universe</a>" (1988) <i>Laser and Particle Beams</i> (ISSN 0263-0346), vol. 6, Aug. 1988, p. 471-491.</span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text">"Z-Pinch Pulsed Plasma Propulsion Technology Development" Final Report Advanced Concepts Office (ED04) Marshall Space Flight Center October 8, 2010, Tara Polsgrove, Et Al.</span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external free" href="http://dorland.pp.ph.ic.ac.uk/magpie/?page_id=239">http://dorland.pp.ph.ic.ac.uk/magpie/?page_id=239</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20141105113922/http://dorland.pp.ph.ic.ac.uk/magpie/?page_id=239">Archived</a> 2014-11-05 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> "Wire Arrays Z-Pinch" accessed: 3-27-2015</span> </li> <li id="cite_note-LaPointe-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-LaPointe_32-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLaPointe" class="citation web cs1">LaPointe, Robert. <a rel="nofollow" class="external text" href="https://members.tm.net/lapointe/Main.html">"High Voltage Devices and Experiments"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">February 21,</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=High+Voltage+Devices+and+Experiments&rft.aulast=LaPointe&rft.aufirst=Robert&rft_id=https%3A%2F%2Fmembers.tm.net%2Flapointe%2FMain.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APinch+%28plasma+physics%29" class="Z3988"></span></span> </li> <li id="cite_note-Tristan-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-Tristan_33-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTristan" class="citation web cs1">Tristan. <a rel="nofollow" class="external text" href="https://members.tripod.com/extreme_skier/cancrusher/">"Electromagnetic Can Crusher"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">February 21,</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Electromagnetic+Can+Crusher&rft.au=Tristan&rft_id=https%3A%2F%2Fmembers.tripod.com%2Fextreme_skier%2Fcancrusher%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3APinch+%28plasma+physics%29" class="Z3988"></span></span> </li> <li id="cite_note-Borros-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-Borros_34-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBorros" class="citation web cs1">Borros, Sam. <a rel="nofollow" class="external text" href="http://www.powerlabs.org/pssecc.htm">"Solid State Can Crusher"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">February 21,</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Solid+State+Can+Crusher&rft.aulast=Borros&rft.aufirst=Sam&rft_id=http%3A%2F%2Fwww.powerlabs.org%2Fpssecc.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3APinch+%28plasma+physics%29" class="Z3988"></span></span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20030522114102/http://magnet-physik.de/st_magnetopuls.html#method">"MagnetoPulS"</a>. <i>Magnet-Physik, Dr. Steingroever GmbH</i>. 2002. Archived from <a rel="nofollow" class="external text" href="http://magnet-physik.de/st_magnetopuls.html#method">the original</a> on 2003-05-22<span class="reference-accessdate">. Retrieved <span class="nowrap">February 21,</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Magnet-Physik%2C+Dr.+Steingroever+GmbH&rft.atitle=MagnetoPulS&rft.date=2002&rft_id=http%3A%2F%2Fmagnet-physik.de%2Fst_magnetopuls.html%23method&rfr_id=info%3Asid%2Fen.wikipedia.org%3APinch+%28plasma+physics%29" class="Z3988"></span></span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110715125631/http://www.english.pstproducts.com/index_htm_files/English%20White%20Paper%20by%20PSTproducts.pdf">"Industrial Application of the Electromagnetic Pulse Technology"</a> <span class="cs1-format">(PDF)</span>. <i>white paper</i>. PSTproducts GmbH. June 2009. Archived from <a rel="nofollow" class="external text" href="http://www.english.pstproducts.com/index_htm_files/English%20White%20Paper%20by%20PSTproducts.pdf">the original</a> <span class="cs1-format">(PDF)</span> on July 15, 2011<span class="reference-accessdate">. Retrieved <span class="nowrap">February 21,</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=white+paper&rft.atitle=Industrial+Application+of+the+Electromagnetic+Pulse+Technology&rft.date=2009-06&rft_id=http%3A%2F%2Fwww.english.pstproducts.com%2Findex_htm_files%2FEnglish%2520White%2520Paper%2520by%2520PSTproducts.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APinch+%28plasma+physics%29" class="Z3988"></span></span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text">Examples of electromagnetic pinch can crushers can be found at (a) Bob LaPointe's site on <a rel="nofollow" class="external text" href="http://members.tm.net/lapointe/Main.html">High Voltage Devices and Experiments</a> (b) Tristran's <a rel="nofollow" class="external text" href="https://members.tripod.com/extreme_skier/cancrusher/">Electromagnetic Can Crusher</a> (including schematic) (c) Sam Borros's <a rel="nofollow" class="external text" href="http://www.powerlabs.org/pssecc.htm">Solid State Can Crusher</a></span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text">See also the IEEE History Center, "<a rel="nofollow" class="external text" href="https://web.archive.org/web/20020414172744/http://www.ieee.org/organizations/history_center/ieee_emblem.html">Evolution of the IEEE Logo</a>" March 1963; see also the comments in "<a rel="nofollow" class="external text" href="http://public.lanl.gov/alp/plasma/lab_astro.html">Laboratory Astrophysics</a>"</span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text">van Marum M 1790 <i>Proc. 4th Int. Conf. on Dense Z-Pinches</i> (Vancouver 1997) (Am. Inst. Phys. Woodbury, New York, 1997) Frontispiece and p ii</span> </li> <li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text"><a href="/wiki/Bas_Pease" title="Bas Pease">R. S. Pease</a>, "The Electromagnetic Pinch: From <a href="/wiki/James_Arthur_Pollock" title="James Arthur Pollock">Pollock</a> to the <a href="/wiki/Joint_European_Torus" title="Joint European Torus">Joint European Torus</a>", "<a rel="nofollow" class="external text" href="http://nsw.royalsoc.org.au/journal/118_12.html#pease">Pollock Memorial Lecture for 1984 delivered at the University of Sydney, 28 November, 1984"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060529053852/http://nsw.royalsoc.org.au/journal/118_12.html#pease">Archived</a> 2006-05-29 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNorthrup1907" class="citation journal cs1">Northrup, Edwin F. (1907). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1649961">"Some Newly Observed Manifestations of Forces in the Interior of an Electric Conductor"</a>. <i>Physical Review</i>. Series I. <b>24</b> (6). 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March 2002.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The+Con-Artist+Physics+of+%27Ocean%27s+Eleven%27&rft.date=2002-03&rft_id=https%3A%2F%2Fwww.aps.org%2Fpublications%2Fapsnews%2F200203%2Foceans-eleven.cfm&rfr_id=info%3Asid%2Fen.wikipedia.org%3APinch+%28plasma+physics%29" class="Z3988"></span></span> </li> </ol></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=Pinch_(plasma_physics)&action=edit&section=22" 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://www.capturedlightning.com/frames/shrinkergallery.html">Examples of electromagnetically shrunken coins and crushed cans</a></li> <li><a rel="nofollow" class="external text" href="http://www.capturedlightning.com/frames/shrinker.html">Theory of electromagnetic coin shrinking</a></li> <li><a rel="nofollow" class="external text" href="http://www.capturedlightning.com/frames/Shrinking_History.htm">The Known History of "Quarter Shrinking"</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20060910014555/http://tesladownunder.iinet.net.au/CanCrushing.htm">Can crushing info using electromagnetism among other things</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20100406085526/http://dorland.pp.ph.ic.ac.uk/magpie/">The MAGPIE project at Imperial College London</a> is used to study wire array Z-pinch implosions.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist 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Pinch (plasma physics) - Wikipedia