{"id":45,"date":"2024-02-19T05:52:36","date_gmt":"2024-02-19T05:52:36","guid":{"rendered":"https:\/\/sabba.me\/nmr\/?p=45"},"modified":"2024-02-19T20:12:01","modified_gmt":"2024-02-19T20:12:01","slug":"higher-order-terms-of-the-bloch-siegert-shift-hamiltonian","status":"publish","type":"post","link":"https:\/\/sabba.me\/nmr\/2024\/02\/19\/higher-order-terms-of-the-bloch-siegert-shift-hamiltonian\/","title":{"rendered":"Higher-Order Terms of the Bloch-Siegert Shift Hamiltonian"},"content":{"rendered":"\n<p>The <a href=\"https:\/\/en.wikipedia.org\/wiki\/Bloch%E2%80%93Siegert_shift\" data-type=\"link\" data-id=\"https:\/\/en.wikipedia.org\/wiki\/Bloch%E2%80%93Siegert_shift\">Bloch-Siegert shift<\/a> is a well-known perturbation to the <a href=\"https:\/\/www.scirp.org\/journal\/paperinformation?paperid=80654\" data-type=\"link\" data-id=\"https:\/\/www.scirp.org\/journal\/paperinformation?paperid=80654\">rotating-wave approximation<\/a> which becomes prominent when the nutation frequency of the driving rf field becomes comparable in magnitude with the Larmor frequency of the driven spins.<\/p>\n\n\n\n<p>Consider the following scenario: we are equipped with a set of B<sub>1<\/sub> coils at low-field (the longitudinal B<sub>0 <\/sub>field is~1.1 mT). Our untuned B<sub>1<\/sub> coils are capable of broadband irradiation of spins with a transverse B<sub>1<\/sub> field whose nutation frequency, \u03c9<sub>1<\/sub>\/(2<em>\u03c0<\/em>), can reach around ~2 kHz*.<\/p>\n\n\n\n<p>A 2 kHz nutation frequency presents little food-for-thought for <sup>1<\/sup>H spins (\u03c9<sub>0<\/sub>\/(2<em>\u03c0<\/em>) @ 1.1 mT <strong>\u2248<\/strong> 47 kHz) but really makes you reconsider your life choices if, <a href=\"https:\/\/pubs.aip.org\/aip\/jcp\/article\/160\/1\/014305\/2932365\" data-type=\"link\" data-id=\"https:\/\/pubs.aip.org\/aip\/jcp\/article\/160\/1\/014305\/2932365\">like us<\/a>, you have been working with <sup>103<\/sup>Rh (\u03c9<sub>0<\/sub>\/(2<em>\u03c0<\/em>) @ 1.1 mT <strong>\u2248<\/strong> 1.5 kHz).<\/p>\n\n\n\n<p>In this scenario it is necessary to calculate <a href=\"https:\/\/doi.org\/10.1016\/j.aop.2020.168327\" data-type=\"link\" data-id=\"https:\/\/doi.org\/10.1016\/j.aop.2020.168327\">higher-order terms<\/a> of the Bloch-Siegert shift Hamiltonian, which I have done to 12th (or 11th, depending on your convention) order. <\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"800\" height=\"800\" src=\"https:\/\/sabba.me\/nmr\/wp-content\/uploads\/2024\/02\/BlochSiegertTerms.png\" alt=\"\" class=\"wp-image-46\" srcset=\"https:\/\/sabba.me\/nmr\/wp-content\/uploads\/2024\/02\/BlochSiegertTerms.png 800w, https:\/\/sabba.me\/nmr\/wp-content\/uploads\/2024\/02\/BlochSiegertTerms-300x300.png 300w, https:\/\/sabba.me\/nmr\/wp-content\/uploads\/2024\/02\/BlochSiegertTerms-150x150.png 150w, https:\/\/sabba.me\/nmr\/wp-content\/uploads\/2024\/02\/BlochSiegertTerms-768x768.png 768w\" sizes=\"auto, (max-width: 800px) 100vw, 800px\" \/><\/figure>\n\n\n\n<p>*there is a subtle point here: with untuned coils operating in the high-inductance limit, the B<sub>1<\/sub> field strength is proportional to 1\/(\u03c9<sub>0<\/sub>), whereas the nutation frequency is of course proportional to \u03c9<sub>0<\/sub>. These factors cancel out as shown in pages 90-91 [Figs. 39-40 of <a href=\"https:\/\/eprints.soton.ac.uk\/474035\/\" data-type=\"link\" data-id=\"https:\/\/eprints.soton.ac.uk\/474035\/\">my thesis<\/a>], leading to a nutation frequency that is independent of frequency and hence \u03b3:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"830\" height=\"600\" src=\"https:\/\/sabba.me\/nmr\/wp-content\/uploads\/2024\/02\/nu1HighInductanceLimit.png\" alt=\"\" class=\"wp-image-49\" style=\"width:555px;height:auto\" srcset=\"https:\/\/sabba.me\/nmr\/wp-content\/uploads\/2024\/02\/nu1HighInductanceLimit.png 830w, https:\/\/sabba.me\/nmr\/wp-content\/uploads\/2024\/02\/nu1HighInductanceLimit-300x217.png 300w, https:\/\/sabba.me\/nmr\/wp-content\/uploads\/2024\/02\/nu1HighInductanceLimit-768x555.png 768w\" sizes=\"auto, (max-width: 830px) 100vw, 830px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>The Bloch-Siegert shift is a well-known perturbation to the rotating-wave approximation which becomes prominent when the nutation frequency of the driving rf field becomes comparable in magnitude with the Larmor frequency of the driven spins. Consider the following scenario: we are equipped with a set of B1 coils at low-field (the longitudinal B0 field is~1.1 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[7],"tags":[],"class_list":["post-45","post","type-post","status-publish","format-standard","hentry","category-theory"],"_links":{"self":[{"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts\/45","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/comments?post=45"}],"version-history":[{"count":2,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts\/45\/revisions"}],"predecessor-version":[{"id":50,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts\/45\/revisions\/50"}],"wp:attachment":[{"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/media?parent=45"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/categories?post=45"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/tags?post=45"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}