{"id":161,"date":"2024-06-10T13:54:20","date_gmt":"2024-06-10T13:54:20","guid":{"rendered":"https:\/\/sabba.me\/nmr\/?p=161"},"modified":"2024-06-10T13:58:41","modified_gmt":"2024-06-10T13:58:41","slug":"evolution-constrained-bounds-on-polarization-transfer-in-i_ns-spin-systems-part-i-refocused-inept","status":"publish","type":"post","link":"https:\/\/sabba.me\/nmr\/2024\/06\/10\/evolution-constrained-bounds-on-polarization-transfer-in-i_ns-spin-systems-part-i-refocused-inept\/","title":{"rendered":"Evolution-Constrained Bounds on Polarization Transfer in I_{N}S Spin Systems Part I: Refocused INEPT"},"content":{"rendered":"\n

It is seldom appreciated that the bounds on spin-order transfer are evolution-constrained<\/em>. That is to say, a particular type<\/em> of evolution (corresponding to some pulse sequence, effective Hamiltonian, or equivalent definition) – designed to convert some well-defined initial spin order configuration (be it a population, coherence, or arbitrary mixture thereof) into another well-defined target – may be associated with a distinct upper bound on the maximum achievable spin order transfer, an upper value that is in general lower <\/em>than the absolute <\/em>upper bound, irrespective of whether the absolute upper bound happens to be symmetry-constrained [1<\/a>] or not [2<\/a>].<\/p>\n\n\n\n

Oof, that was a mouthful. Here’s what I’m talking about (a table from a paper I’ve been writing for a few forevers):<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

For now, let’s talk about just one of the columns from my table. The refocused INEPT pulse sequence is perhaps the most commonly used polarization transfer experiment in solution-state NMR spectroscopy. The experiment was described by Burum and Ernst [3]<\/a> as a sequel to the famous INEPT sequence devised by Morris and Freeman [4<\/a>].<\/p>\n\n\n\n

To my surprise, most NMR spectroscopists are unaware of the simple general expressions for both the optimal refocusing duration and the ensuing maximum enhancement factor, despite the fact that they were derived by David Doddrell and coworkers in the land Down Under<\/a> in the 1980s [5<\/a>, 6<\/a>, 7<\/a>, 8<\/a>, 9<\/a>, 10<\/a>]. To save the reader time ruffling through references, these expressions can be found in a very brief JACS paper [11<\/a>]. <\/p>\n\n\n\n

Let’s start with the INEPT sequence, which has a very simple theory in IN<\/sub>S spin systems. In a nutshell, INEPT accomplishes the conversion Iz<\/sub> \u2192 [Ix<\/sub> \u2192 2Iy<\/sub>Sz<\/sub> \u2192 2Iz<\/sub>Sz<\/sub> \u2192] 2Iz<\/sub>Sy<\/sub> (from pure I-spin longitudinal order to maximally I-correlated S-spin order, via a zz- or two-spin longitudinal- order intermediate), via the following steps:<\/p>\n\n\n\n

    \n
  1. The I-spins are tipped to the transverse plane by a 90y<\/sub> pulse i.e. something like Iz<\/sub> \u2192 Ix<\/sub>. [Alternatively, from another frame of reference, the Hamiltonian <\/em>is tipped from 2\u03c0J\u00d7Iz<\/sub>Sz<\/sub> \u2192 2\u03c0J\u00d7Ix<\/sub>Sz<\/sub>].<\/li>\n\n\n\n
  2. Evolution for a total time period of \u03c4 = 1\/(2J) leads to a total rotation \u03c4\u00d72\u03c0J = \u03c0 about the Ix<\/sub>Sz<\/sub> axis (more properly \u03c0\/2 about the 2Ix<\/sub>Sz<\/sub> axis). This basically accomplishes the transformation Iz<\/sub> \u2192 2Iy<\/sub>Sz<\/sub>.<\/li>\n\n\n\n
  3. The I spins are tipped to the longitudinal plane by a 90x<\/sub> pulse; i.e. 2Iy<\/sub>Sz<\/sub> \u2192 2Iz<\/sub>Sz<\/sub><\/li>\n\n\n\n
  4. The S spins are tipped to the transverse plane for detection by 90-x<\/sub> pulse; i.e. 2Iz<\/sub>Sz<\/sub> \u2192 2Iz<\/sub>Sy<\/sub>, leading to an antiphase multiplet most people associate with INEPT, that you can read more about in my paper [12<\/a>]. <\/li>\n<\/ol>\n\n\n\n

    [Note: there are some sign changes involved depending on the sign of the J-coupling, whether a 180 degree pulse is used, its phase, etc. which I will not waste our time worrying about.]<\/p>\n\n\n\n

    The key thing to note about the INEPT sequence is that the I-spins are evolving under the action of the S-spins, of which – by definition<\/em> – there is only one in IN<\/sub>S spin systems. Hence, the optimal duration of the INEPT sequence is always going to be 1\/(2J). It’s a no-brainer. <\/p>\n\n\n\n

    Now, let’s consider the eponymous refocusing block present in refocused INEPT. In a nutshell, the refocusing block tries to accomplish the transformation 2Iz<\/sub>Sz<\/sub> [\u2192 2Iz<\/sub>Sy<\/sub> \u2192 Sx<\/sub>] \u2192 Sz<\/sub>. Note the symmetry with the INEPT sequence; it’s mirrored. Due to the fact that the S-spin is evolving under the action of (in general) multiple I-spins, things get a bit tricky, but it’s nothing we can’t handle. The problem essentially amounts to solving:<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    Where:<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n
    \"\"<\/figure>\n\n\n\n

    All we have to do is evaluate the first few solutions. We get:<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    Now, the solution is staring us right in the face. It is easy to see by mathematical induction that we have a general solution for the transformation amplitude of longitudinal 2Iz<\/sub>Sz<\/sub> to Sz<\/sub>:<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    Obtaining the optimal duration of the refocusing tau (aka the time at which the maximum occurs) is a simple matter of high-school calculus:<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    Leading to:<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    In the thermodynamic limit, we encounter yet another high-school calculus problem:<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    The reader is encouraged to compare these values with the so-called “entropy bound” (a term I despise with a passion) and the thermodynamic limit of Sorensen’s bound [13<\/a>].<\/p>\n","protected":false},"excerpt":{"rendered":"

    It is seldom appreciated that the bounds on spin-order transfer are evolution-constrained. That is to say, a particular type of evolution (corresponding to some pulse sequence, effective Hamiltonian, or equivalent definition) – designed to convert some well-defined initial spin order configuration (be it a population, coherence, or arbitrary mixture thereof) into another well-defined target – […]<\/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-161","post","type-post","status-publish","format-standard","hentry","category-theory"],"_links":{"self":[{"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts\/161","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=161"}],"version-history":[{"count":4,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts\/161\/revisions"}],"predecessor-version":[{"id":180,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts\/161\/revisions\/180"}],"wp:attachment":[{"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/media?parent=161"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/categories?post=161"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/tags?post=161"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}