{"id":22,"date":"2024-02-18T21:02:15","date_gmt":"2024-02-18T21:02:15","guid":{"rendered":"https:\/\/sabba.me\/nmr\/?p=22"},"modified":"2024-02-18T21:02:16","modified_gmt":"2024-02-18T21:02:16","slug":"quadrupolar-linewidth-patterns-in-the-solution-state","status":"publish","type":"post","link":"https:\/\/sabba.me\/nmr\/2024\/02\/18\/quadrupolar-linewidth-patterns-in-the-solution-state\/","title":{"rendered":"Quadrupolar Linewidth Patterns in the Solution-State"},"content":{"rendered":"\n

Most nuclei in the periodic table that possess spin are quadrupolar (i.e. spin \u2113<\/strong> > 1\/2). It is well-known that, in the solution-state, a J-coupling between the I<\/em> (\u2113<\/strong> = 1\/2) and S<\/em> (\u2113<\/strong> >1\/2) spins “vanishes” (or more precisely induces a scalar relaxation effect) in the spectrum of the I<\/em> spin(s) when the following condition is satisfied:<\/p>\n\n\n\n

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

That is to say, J-couplings to quadrupolar spins are not directly observable when the rate of relaxation (1\/T1<\/sub>) of the S<\/em> spins significantly exceeds the J-coupling. Sometimes this is referred to as “self-decoupling”, a term coined by Spiess, Haeberlen, and Zimmermann<\/a>.<\/p>\n\n\n\n

But there are exceptions. The relaxation of the S spins is usually overwhelmingly dominated by the quadrupolar mechanism, whose contribution (in the extreme narrowing limit) is given by:<\/p>\n\n\n\n

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

Here, the norm of the quadrupolar coupling tensor is related to the quadrupolar coupling constant (QCC):<\/p>\n\n\n\n

\"\"<\/p>\n\n\n\n

And the quadrupolar coupling constant itself is:<\/p>\n\n\n\n

\"\"
Here, it is worth explaining why I have written the equation in an unconventional way. The nuclear quadrupole moment Q<\/em> (an intrinsic nuclear property which is non-zero for \u2113<\/strong> > 1\/2) is proportional to the eccentricity of the spheroid charge distribution unique to each nuclear species, and has units of area. On the other hand, Vzz<\/sub><\/em> (in units voltage\/area) represents the principal component of the electric field gradient (EFG) tensor, and EFGs may technically be present even at spin-0 or spin-1\/2 nuclei. It is clear that Q<\/em> and Vzz<\/sub><\/em> couple to produce a voltage<\/em> at the nucleus. Finally, half the Josephson constant (\u00bdKJ<\/sub><\/em>, in units of frequency\/voltage) is the fundamental quantum of voltage-driven oscillation. This basic physical picture is rarely appreciated; a run-of-the-mill QCC of ~150 kHz corresponds to a ~100 picovolt potential difference at the nucleus. <\/p>\n\n\n\n

Now it is simple to figure out the exceptions when quadrupolar relaxation may be slow enough to observe J-couplings:<\/p>\n\n\n\n

    \n
  1. Quadrupolar spins with an intrinsically small nuclear quadrupole moment Q<\/em>. Examples include 2<\/sup>H and 17<\/sup>O.<\/li>\n\n\n\n
  2. Molecules where the quadrupolar spin experiences a small electric field gradient V<\/em>. Examples are molecules with intrinsically high symmetry, such as 14<\/sup>NH4<\/sub>+ , but it is worth noting that this criterion may not be sufficient for spins where Q<\/em> is massive. <\/li>\n\n\n\n
  3. A larger heteronuclear J-coupling, which is increasingly possible when the I<\/em> and\/or S<\/em> spin(s) has a larger atomic number and gyromagnetic ratio. But again, the J-coupling must be “large” relative to the quadrupolar relaxation rate.<\/li>\n<\/ol>\n\n\n\n

    Now, suppose you actually observe, in the I<\/em> spectrum, a well-resolved coupling to a single quadrupolar spin S<\/em>. It is well-known that one would observe 2S<\/em>+1 Lorentzian peaks with equal areas\/integrals. It is less well-known that the Lorentzian linewidths <\/em>of these peaks would not <\/strong><\/em>be equal:<\/p>\n\n\n\n

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

    This remarkable effect was described by the Nobel laureate John Pople in 1958<\/a>, as well as Masuo Suzuki & Ryogo Kubo in 1963<\/a>, but it appears that the name “Pople-Suzuki-Kubo effect” never really caught on.<\/p>\n\n\n\n

    Prominent examples where the “PSK effect” have been observed include the hexafluorides, e.g. of niobium<\/a> (93<\/sup>Nb, I=9\/2), antimony<\/a> (121\/123<\/sup>Sb with I=5\/2 and 7\/2 respectively), or bismuth<\/a> (209<\/sup>Bi, I=9\/2). Insanely enough, there are even examples<\/a> involving the unpleasant nucleus 235<\/sup>U (I=7\/2).

    However, my favorite example (and I must admit some personal bias) is certainly     the fine paper<\/a> by Stuart J. Elliott et al. from the Levitt group. The paper not only shows a beautiful spectrum:<\/p>\n\n\n\n

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

    But impressively, the paper also provides a hidden gem in the supporting information<\/a>: a general expression for the linewidths that was derived with the help of the commutator relations of spherical tensor operators. The expression may be written:<\/p>\n\n\n\n

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

    And I have provided a triangle of the linewidth coefficients k(m)<\/em> [note that the intensities <\/em>would be provided by 1\/k(m)<\/em>] here:<\/p>\n\n\n\n

    \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

    Most nuclei in the periodic table that possess spin are quadrupolar (i.e. spin \u2113 > 1\/2). It is well-known that, in the solution-state, a J-coupling between the I (\u2113 = 1\/2) and S (\u2113 >1\/2) spins “vanishes” (or more precisely induces a scalar relaxation effect) in the spectrum of the I spin(s) when the following […]<\/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-22","post","type-post","status-publish","format-standard","hentry","category-theory"],"_links":{"self":[{"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts\/22","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=22"}],"version-history":[{"count":2,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts\/22\/revisions"}],"predecessor-version":[{"id":35,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts\/22\/revisions\/35"}],"wp:attachment":[{"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/media?parent=22"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/categories?post=22"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/tags?post=22"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}