The Hartmann-Hahn (HH) experiment [1. the original 1962 paper] is one of the cornerstones of modern magnetic resonance, with a history so deeply tied to “cross-polarization” that the two are often considered synonymous. Several different ways of engineering an HH Hamiltonian exist [2], and despite the stereotypical association of HH evolution with solid-state experiments (due to the stringent matching condition associated with the conventional experiment, and the comparatively miniscule J-couplings encountered in solution), it has enjoyed a rich history in the solution-state [3. the beautiful paper by Chingas et al.; 4. the informative chapter by Glaser and Quant, 5, 6, 7], and repeatedly stimulated the minds of the greatest theorists [8, 9, 10, 11]. In practice, for reasons of robustness, the most common variation on a “Hartmann-Hahn” experiment in solution will typically exploit sequences originally designed for heteronuclear decoupling [12. Levitt’s “MOIST” experiment, which is just MLEV-4, 13. M. Ernst et al.’s study of cross-polarization].
I thought I would write a series of blog posts to talk about some neat and underappreciated features of Hartmann-Hahn evolution, whose solution-state theory (which I have spent a good deal of time thinking about) appears to have been totally understudied since the paper by Chingas and coworkers at the US Naval Research Laboratory [3]. But the deeper theory will first require a solid introduction as foundation, which by itself is likely to be more useful to a general audience. So here is Part I.
1. Classic Hartmann-Hahn evolution
Consider a heteronuclear spin-1/2 system consisting of two spin species I and S. In general, each spin species may have an arbitrary number of spins. The spins precess at Larmor frequencies ω0I and ω0S respectively, and are connected via a mutual J-coupling JIS.
Under the secular assumption (the difference in Larmor frequencies of the two spins greatly exceeds the J-coupling):
The J-coupling Hamiltonian is given by:
In the doubly-rotating frame, this can be treated as the total Hamiltonian.
Under the tacit assumption of the rotating-wave approximation:
Consider the general propagator of a matched (that is to say, ω1I = ω1S = ω1) rf-field along the x-axis, synchronously applied on the I and S channels, on resonance, following a simultaneous 90y rotation:
The Hamiltonian may be rewritten in the interaction frame of the rf-field to give the interaction frame Hamiltonian:
And the effective propagator Ueff(t) can be taken as the complex exponential of the average Hamiltonian, which may be expressed as a sum of Magnus expansion terms:
Note that we have used the Nielsen convention [14] for indexing of the Magnus expansion terms that differs from the older literature [15] by one. The first term is given by:
Whose solution in this case is given by:
Containing undesirable double-quantum “error” terms:
In the limit of a nutation frequency greatly exceeding the heteronuclear J-coupling (ω1 >> JIS), the first-order average Hamiltonian (Hav) simply reduces to:
In which the scaling factor κ is simply 1/2 in ideal circumstances (the absence of pulse imperfections, errors, etc.). The corresponding propagator, Uav(t), is:
Irrespective of the number of spins I and S, Equation (13) holds under the previous assumptions of:
- The Hartmann-Hahn condition; the rf field applied to both spin species synchronously, and along the same axis, must be matched.
- The nutation frequency overwhelmingly exceeds the heteronuclear J-coupling.
- The rotating-wave approximation.
- The absence of pulse imperfections in the form of either resonance offset and/or pulse strength errors on either the I or S channels.
Equation (13) above corresponds to the well-known effective Hamiltonian describing evolution under anisotropic Hartmann-Hahn transfer conditions [3, 4].
2. Hartmann-Hahn transfer via DualPol
Hartmann-Hahn transfer need not be achieved via matched rf fields applied on both spin channels synchronously – it is possible to engineer an alternative route of polarization transfer using pulse-interrupted free precession.
Consider the following multiple-pulse sequence applied to the secular J-coupling Hamiltonian HJ, under the assumption that the pulses (applied synchronously on the I and S channels) represent perfect rotations of negligible duration:
The evolution has a piecewise form. If the repetition rate greatly exceeds the coupling constant (τ-1 >> JIS), the average Hamiltonian may be taken as the first-order term of a Baker-Campbell-Hausdorff expansion:
Which, in this case, is simply:
It is trivial to engineer the previous sequence, but we have only considered the bilinear term(s) IzSz present within HJ. It is also obvious that – due to the lack of a refocusing element – the sequence is particularly vulnerable to inhomogeneous broadening i.e. resonance offset effects.
More formally, the presence of a single resonance offset term, such as Iz, leads to a first-order average error Hamiltonian of the form:
Clearly, unless the offset terms are much smaller than the J-coupling, polarization transfer is compromised. In standard solution-state NMR, where heteronuclear J-couplings are on the order of 1-100 Hz, this means that the sequences are strictly restricted to on-resonance applications. The stringent requirement on the matching condition is analogous to the sensitivity of the conventional Hartmann-Hahn experiment to the condition ω1I = ω1S.
The problem is easy to remedy by simply introducing a refocusing element in the pulse sequence, as follows:
This leads to a pulse sequence we have called DualPol [see the trilogy of rhodium papers I coauthored with Harry Harbor-Collins 16, 17, 18]. DualPol is a dual-channel variant of the PulsePol sequence [19] invented by Tratzmiller [20] of the Ulm group, originally in the context of optical DNP in NV centres, which we had shown could be applied for excitation of nuclear singlet order [21]. This diversity of applications is of course no coincidence and directly follows from the premise of archetypal pulse sequences.
DualPol engineers a Hartmann-Hahn Hamiltonian Hav that is highly robust against resonance offset and pulse strength errors. Unusually for solution-state cross-polarization sequences… DualPol is windowed.
3. Some Notes on Windowed and Windowless Sequences
During the development of heteronuclear spin decoupling, there were competing arguments as to the final form of the decoupling sequence. Should the decoupling sequence be windowed (“hard” strong pulses separated by comparatively long delays) or windowless (“soft” pulses, with the rf field applied continuously without any gaps)?
As the legend goes, the late Ray Freeman had insisted that windowed decoupling was superior. His PhD student at the time, Malcolm Levitt, had proven his advisor wrong by showing [Levitt’s PhD thesis 22] that windowless decoupling was more efficient, in the sense of less power deposition and less sample-heating (the most serious concerns for solution-state applications at the time). This became common wisdom among solution-state NMR spectroscopists, and indeed, windowed sequences for decoupling/cross-polarization became rather unusual.
I can relay a vaguely similar personal anecdote: during our testing of the DualPol sequence (note that the problem now is cross-polarization, not decoupling), my advisor had pointed out the previous point to us, and argued against us in favor of windowless sequences, which conventional wisdom as well as literature evidence had conclusively shown were more efficient. We exhaustively ran comparisons with every windowless sequence there was – from MOIST/MLEV-4 to DIPSI and FLOPSY – and found that in all cases DualPol had superior behavior with regards to compensation against resonance offset errors (our main concern at the time due to the massive bandwidth expected for heavy spins such as 103Rh). So I won this argument and a grand cash prize of £0.
How could this be? The bandwidth of a pulse sequence is determined by the peak nutation frequency. It depends on how much beating your probe can take, but a windowed pulse sequence (such as DualPol), in general, allows one to use much more powerful pulses – with a larger nutation frequency – than windowless sequences, which have 100% duty cycles by definition. For the same average rf power deposition, windowed pulse sequences are associated with a wider effective bandwidth! Indeed, the basic logic of “windowed = wider” is generally a feature of ultra-broadband sequences in the solution-state [23].