{"id":215,"date":"2024-06-13T19:19:36","date_gmt":"2024-06-13T19:19:36","guid":{"rendered":"https:\/\/sabba.me\/nmr\/?p=215"},"modified":"2024-06-17T18:44:34","modified_gmt":"2024-06-17T18:44:34","slug":"hartmann-hahn-evolution-from-solution-to-solids-part-ii-quasiperiodicity-and-nuclear-spin-plasma-dispersion-as-consequences-of-the-high-temperature-approximation","status":"publish","type":"post","link":"https:\/\/sabba.me\/nmr\/2024\/06\/13\/hartmann-hahn-evolution-from-solution-to-solids-part-ii-quasiperiodicity-and-nuclear-spin-plasma-dispersion-as-consequences-of-the-high-temperature-approximation\/","title":{"rendered":"Hartmann-Hahn Evolution from Solution to Solids, Part II: Quasiperiodicity and Nuclear Spin Plasma Dispersion as Consequences of the High-Temperature Approximation"},"content":{"rendered":"\n

In the previous blog post, we derived the effective Hamiltonian of (anisotropic) Hartmann-Hahn evolution:<\/p>\n\n\n\n

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

In this blog post, we will derive and explore the general form of the (Iz<\/sub><\/em> \u2192 Sz<\/sub><\/em>) polarization transfer trajectory under Hartmann-Hahn conditions in IN<\/sub>S spin systems.<\/p>\n\n\n\n

First, let’s recall some basic quantum mechanical properties. The operator amplitude of B <\/em>in A<\/em> (i.e. how much of operator B is in operator A) is given by:<\/p>\n\n\n\n

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

In which we’ve used the concept of a Frobenius norm<\/a> and a funky operation, common within the field of NMR at least, called the “Liouville bracket” (see the paper by Jean Jeener<\/a>):<\/p>\n\n\n\n

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

Part A: Hartmann-Hahn transfer from a virtual spin-\u2113 particle to a single spin-1\/2 particle<\/strong><\/p>\n\n\n\n

We’ll begin by considering the basic problem of spin order transfer from a spin-\u2113 particle I<\/em> to a spin-1\/2 particle S<\/em>. We have (at least as a start):<\/p>\n\n\n\n

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

In which it is crucial to note a consequence of the high-temperature approximation: the initial density operator is \u2113z<\/sub>. This is the fundamental root cause of a host of complications we will see later.<\/p>\n\n\n\n

If we define the following effective angle (just so the notation doesn’t get too horrible):<\/p>\n\n\n\n

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

We calculate the following polarization trajectories for the first values of \u2113, which can be expressed as a sum of oscillators:<\/p>\n\n\n\n

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

I’ve written the equations in this weird way to make the otherwise nonobvious pattern clear. Using mathematical induction, we have the general result:<\/p>\n\n\n\n

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

In which the apparently mysterious coefficient is related to a basic identity of angular momentum raising\/lowering\/ladder operators<\/a>:<\/p>\n\n\n\n

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

The argument inside the sine functions represents eigenvalues that are essentially (discretely) distributed along a Wigner semicircle<\/a>.<\/p>\n\n\n\n

Now, let’s look at the actual functions representing the polarization transfer trajectories. The first two are simple periodic functions:<\/p>\n\n\n\n

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

But all hell breaks loose once we have \u2113 \u2265 3\/2:<\/p>\n\n\n\n

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

Why does this happen? Due to the fact that the functions are sums of oscillators with frequencies that are incommensurate<\/em> (irrational multiples with respect to each other), the trajectory is now not <\/strong>periodic but quasiperiodic.<\/em> This means that:<\/p>\n\n\n\n

    \n
  1. In general, the trajectory never repeats itself exactly in time.<\/li>\n\n\n\n
  2. In general, we cannot <\/em>find closed-form simple expressions for when a local maximum, minimum, or zero occurs.<\/li>\n<\/ol>\n\n\n\n

    Essentially, we are dealing with a quantum many-body problem. However, a form of order does <\/em>emerge for large values of \u2113:<\/p>\n\n\n\n

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

    Here, H<\/strong>k<\/sub>(z)<\/em> denotes the Struve function of the first kind of order k<\/em><\/a>. The Struve function is a cousin of Bessel functions, which are commonly encountered<\/a> in solid-state NMR. This equation implies that at high-temperature conditions, the maximum possible transfer from a high-spin particle I<\/em> via a Hartmann-Hahn sequence is:<\/p>\n\n\n\n

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

    Struve and Bessel functions naturally arise in integrals of trigonometric functions inside trigonometric functions. Here’s a hint:<\/p>\n\n\n\n

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

    The particular mathematical trick I used to obtain this problem was the following identity:<\/p>\n\n\n\n

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

    I have not been able to find this identity in the mathematical literature, so it’s worth revealing the works my formal proof:<\/p>\n\n\n\n

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

    All we have to do to get the final result is plug in z = [\u2113(\u2113+1)]1\/2<\/sup> \u00d7 <\/em>\u03b2J<\/sub><\/em>.<\/p>\n\n\n\n

    There have been at least two reports of the Struve function popping up in magnetic resonance problems [1<\/a>, 2<\/a>]. For those who don’t believe me, here is a comparison of plots of the polarization transfer trajectory for a hypothetical spin-100 particle vs. the Struve function approximation:<\/p>\n\n\n\n

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

    Part B: Hartmann-Hahn transfer in IN<\/sub>S spin systems (from N<\/em> spin-1\/2 particles to a single spin-1\/2 particle<\/strong>)<\/p>\n\n\n\n

    The previous results can be used to great effect to automatically obtain the form of the polarization transfer trajectory in IN<\/sub>S spin systems. We merely have to do some sneaky normalization (the old I-spin basis had 2\u2113+1 states, the new I-spin basis has 2N<\/sup> states) and employ our friend the Catalan triangle<\/a> to account for multiplicities:<\/p>\n\n\n\n

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

    Here’s what the first few functions look like. Just like before, we have a quantum many-body problem where quasiperiodicity kicks in when N > 2<\/em>.<\/p>\n\n\n\n

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

    You will notice something peculiar for larger values of N – neglecting the transient “revivals”\/Loschmidt echoes, the time period between t = 0<\/em> and t = 1\/J<\/em> begins to vaguely resemble a cross-polarization curve in solids. There is a gradual loss of coherent behavior and a prominent “surge” in the beginning. <\/p>\n\n\n\n

    In fact, there is an exact <\/em>solution for the Hartmann-Hahn polarization transfer trajectory in the thermodynamic limit of large N<\/em>:<\/p>\n\n\n\n

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

    In which “F<\/em>” denotes the Dawson integral<\/a> (also called Dawson’s integral or the Dawson function). <\/p>\n\n\n\n

    Deriving this result was not completely straightforward, requiring both the Struve function result shown earlier and the (1st derivative of the!) Gaussian limit for the Catalan triangle coefficients obtained via what I’ve called the “extended de Moivre-Laplace theorem” (see my paper<\/a>):<\/p>\n\n\n\n

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

    … as well as a couple of tricks. Essentially, the limiting form of the Catalan triangle coefficients gives you a Laplace transform<\/a>-looking cursed object that spits out the required result:<\/p>\n\n\n\n

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

    You can compare the Dawson function approximation (blue) to the analytical expression (black) for a hypothetical N = 1000<\/em> spin system:<\/p>\n\n\n\n

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

    In the high-N thermodynamic limit, there are no oscillations or coherent behavior whatsoever<\/em>. We have a surge followed by a plateau to 1. This resembles the sort of cross-polarization kinetics<\/a> you would see in solid-state magnetic resonance, usually with some kind of relaxation\/damping:<\/p>\n\n\n\n

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

    But in the absence <\/em>of relaxation, we have the following results for Hartmann-Hahn evolution in the thermodynamic limit:<\/p>\n\n\n\n

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

    Nuclear spin plasmas?<\/strong><\/p>\n\n\n\n

    The Dawson integral is also known as the Fried-Conte plasma dispersion function<\/a> or simply “the plasma dispersion function”. What the hell is a function associated with plasma physics doing in magnetic resonance? If you’re feeling crazy enough, there is a simple physical analogy you could make:<\/p>\n\n\n\n

    The criteria for plasma formation are typically a combination of (very) high temperature and high density – which obviously combine to to produce high plasma pressure. The “plasma-like” behavior we have just characterized emerges in nuclear spin systems that display a combination of high spin temperature (a very easy condition to satisfy unless you are doing some form of hyperpolarization) and high spin density (a large sqrt(N)<\/em>), leading to a virtual quantity one could perhaps call… a high nuclear spin pressure? But only if you really wanted to.<\/p>\n\n\n\n

    Speaking of hyperpolarization, you can appreciate that there are vivid consequences for polarization transfer in highly-polarized spin systems – for now I can say that the dynamics become simpler (think one populated ground state, one oscillator) and the enhancement is no longer plagued by destructive interference. But that aspect might be better suited for another blog post\/paper.<\/p>\n\n\n\n

    EDIT<\/strong> (17\/06\/2024, 19:44 UK time): I fixed a couple of typos and wasn’t too happy with the lack of mathematical exposition so I expanded the mathematics a bit. <\/p>\n","protected":false},"excerpt":{"rendered":"

    In the previous blog post, we derived the effective Hamiltonian of (anisotropic) Hartmann-Hahn evolution: In this blog post, we will derive and explore the general form of the (Iz \u2192 Sz) polarization transfer trajectory under Hartmann-Hahn conditions in INS spin systems. First, let’s recall some basic quantum mechanical properties. The operator amplitude of B in […]<\/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-215","post","type-post","status-publish","format-standard","hentry","category-theory"],"_links":{"self":[{"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts\/215","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=215"}],"version-history":[{"count":5,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts\/215\/revisions"}],"predecessor-version":[{"id":279,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts\/215\/revisions\/279"}],"wp:attachment":[{"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/media?parent=215"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/categories?post=215"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/tags?post=215"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}