Evolution-Constrained Bounds on Polarization Transfer in I_{N}S Spin Systems Part I: Refocused INEPT

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 – may be associated with a distinct upper bound on the maximum achievable spin order transfer, an upper value that is in general lower than the absolute upper bound, irrespective of whether the absolute upper bound happens to be symmetry-constrained [1] or not [2].

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):

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] as a sequel to the famous INEPT sequence devised by Morris and Freeman [4].

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 in the 1980s [5, 6, 7, 8, 9, 10]. To save the reader time ruffling through references, these expressions can be found in a very brief JACS paper [11].

Let’s start with the INEPT sequence, which has a very simple theory in I_NS spin systems. In a nutshell, INEPT accomplishes the conversion I_z → [I_x → 2I_yS_z → 2I_zS_z →] 2I_zS_y (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:

1. The I-spins are tipped to the transverse plane by a 90_y pulse i.e. something like I_z → I_x. [Alternatively, from another frame of reference, the Hamiltonian is tipped from 2πJ×I_zS_z → 2πJ×I_xS_z].
2. Evolution for a total time period of τ = 1/(2J) leads to a total rotation τ×2πJ = π about the I_xS_z axis (more properly π/2 about the 2I_xS_z axis). This basically accomplishes the transformation I_z → 2I_yS_z.
3. The I spins are tipped to the longitudinal plane by a 90_x pulse; i.e. 2I_yS_z → 2I_zS_z
4. The S spins are tipped to the transverse plane for detection by 90_-x pulse; i.e. 2I_zS_z → 2I_zS_y, leading to an antiphase multiplet most people associate with INEPT, that you can read more about in my paper [12].

[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.]

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 – there is only one in I_NS spin systems. Hence, the optimal duration of the INEPT sequence is always going to be 1/(2J). It’s a no-brainer.

Now, let’s consider the eponymous refocusing block present in refocused INEPT. In a nutshell, the refocusing block tries to accomplish the transformation 2I_zS_z [→ 2I_zS_y → S_x] → S_z. 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:

Where:

All we have to do is evaluate the first few solutions. We get:

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 2I_zS_z to S_z:

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

Leading to:

In the thermodynamic limit, we encounter yet another high-school calculus problem:

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].