Category: Pulse Sequences

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J-Synchronized Echoes in the Pantheon of Symmetry-based Sequences?

Posted on by Mohamed Sabba

I often make the point that PulsePol is expected to replace the “default” option of M2S as the go-to windowed (meaning there are gaps between pulses) or hard-pulse sequence for exciting nuclear singlet order in strongly-coupled spin systems. (Shoutout to the archetypal soft-pulse/low-power sequence SLIC.)

That being said, I religiously avoid the misleading statement “symmetry-based sequences will replace M2S” and wince uncomfortably at the frequently-asked question “do you think symmetry-based sequences will replace M2S?”

This is because the building block of the M2S sequence, the J-synchronized echo (JSE; see refs. 1, 2, and 3), is technically a symmetry-based sequence.

Figure 1: The J-synchronized spin echo pulse sequence.

The fundamental difference between the JSE and PulsePol is that the JSE is designed to selectively swap the |S0〉and |T0〉states (via a zero-quantum effective Hamiltonian) while PulsePol is designed to selectively swap |S0〉and either of the |T±〉states (via a single-quantum effective Hamiltonian):

Figure 2: illustration of transitions excited by different symmetry-based sequences.

This is painfully obvious when you phrase it the right way: the R-element (i.e. basic inversion element) of the JSE is simply a windowed 1800 pulse, and the nominal duration of the R-element is half a “rotor” period 1/(2J). With regards to the first-order selection rules, no phase cycling is applied*. Hence, we may denote the JSE as R210 in the notation describing symmetry-based sequences. Or you could denote it C110 too (assuming you count a full cycle including two spin echoes).

As an aside, it is easy to see why the M2S sequence ended up appearing clunky with awkwardly placed pulses and delays interrupting the echoes: the selective transition being engineered by the JSE building block was the wrong one, assuming one was starting from thermal equilibrium magnetization. Which suggests that the JSE would’ve worked handsomely for nuclear singlet excitation if one was starting from dipolar order (an excess population of the central triplet state)…

*Much like PulsePol, “riffling” in the form of supercycles is necessary for the optimal performance of the JSE/M2S sequences vis-a-vis pulse strength and detuning errors.

Symmetry-based Sequences for Singlet-Triplet Excitation (PulsePol)

Posted on by Mohamed Sabba

In 2016 a Master’s student at Ulm’s Institute of Theoretical Physics, by the name of Benedikt Tratzmiller, described a simple yet powerful control sequence for optical DNP in diamond NV centres. The pulse sequence had the attractive name of PulsePol. The sequence made further appearances in an excellent paper and Tratzmiller’s PhD thesis.

Some time ago I listened to an inspiring online talk by Nino Wili (who quite convincingly spoke about cross-pollination between different fields of magnetic resonance) and we got to talking. After a few discussions/simulations eventually we realized that the PulsePol sequence could be applied to singlet NMR, and that application forms the basis of our paper, which chose to explain PulsePol using the language of symmetry-based sequence design borrowed from solid-state NMR.

PulsePol has some advantages over the M2S sequence, which was essentially a “default” option in NMR groups for generating nuclear singlet order (alongside variations of the arguably more elegant SLIC method invented by DeVience, Walsworth, and Rosen). These advantages include superior robustness (PulsePol is generally less sensitive to rf errors than M2S), simplicity, and even a small time advantage (PulsePol is ~1.21x faster than M2S). One should expect PulsePol to replace M2S in the future.

Here I provide some ready-to-use pulse programs, written for Bruker TopSpin, that the NMR community may use to actually implement the PulsePol sequence.

The pulse programs have a number of features:

  1. The so-called “riffling” 180 phase shift modification on the central 180° pulses, which improves robustness against pulse strength (Rabi frequency) and resonance offset (detuning) errors.
  2. A T00 gradient filter for selective filtration of singlet order, and a z-filter to select longitudinal magnetization.
  3. A singlet order destruction (SOD) element before the relaxation delay to purge residual singlet order, which may interfere with experiments.
  4. Wimperis’ BB1 composite pulse (in the symmetrized implementation) replacing the pulses in the filters, improving robustness against pulse strength errors.

The pulse sequences can be downloaded here:

  1. R431
  2. R873
  3. R411
  4. R612
  5. R813
  6. R1014

I also attach a quick reference table which provides the 2 experimental parameters that are actually relevant for optimal excitation of singlet order, because it has always annoyed me that the first one was not explicitly stated in our somewhat cryptic original paper:



Note that I have given the total duration of the PulsePol in terms of the SLIC duration 1/(√2 Δ) – which is the fastest currently known way to fully excite singlet order from longitudinal magnetization. The minimum of the total duration is around n = 3, N = 4, where the total evolution is ~1.38x longer than the SLIC sequence. The (fixed) total duration of the M2S sequence (~1.67x longer than SLIC) is given for comparison.

The “default” sequence for most people working with nearly equivalent spin-1/2 pairs would be R431 – or possibly R873, which is only ~3.5% slower than R431 but provides improved robustness against resonance offset/detuning errors. However, in the common case where a spin system is at intermediate inequivalence (the chemical shift has a comparable magnitude with respect to the J coupling), one can benefit from the sequences in the series R411, R612, R813, R1014… which perform over an increasingly wider range of inequivalence angles at the expense of being increasingly slower, as I discussed in the appendix to this paper.

Archetypal Pulse Sequences

Posted on by Mohamed Sabba

Archetype (noun); the original pattern or model of which all things of the same type are representations or copies.

The Merriam-Webster Dictionary

In many spin dynamical problems, I like to say there is an “archetypal” pulse sequence. The archetypal pulse sequence may be defined as the simplest possible way of achieving a unitary transformation from Operator A to Operator B. Typically the relevant operators are spin-state populations, so the behaviour expected under these sequences would correspond to the classic scenario of Rabi oscillations (i.e. motion from Population A to Population B, and back again). A recurrent motif in pulse sequence development is that the archetypal pulse sequence consists of nothing more than simple cw irradiation (perhaps preceded by a phase-shifted 90 pulse), whose amplitude is adjusted to satisfy a particular matching condition.

Examples of archetypal pulse sequences include:

1. The classic Hartmann-Hahn sequence for polarization transfer, which exchanges populations between spin(s) I and spin(s) S. The matching condition of the Hartmann-Hahn sequence is that the nutation frequency of the simultaneous driving field on the I and S spins coincides.
2. The NOVEL (nuclear orientation via electron spin-locking) sequence for DNP, which exchanges populations between electron (e) and nuclear (n) spins. The matching condition of NOVEL is that the nutation frequency of the driving field on the electron spins coincides with the Larmor frequency of the nuclear spins.
3. The SLIC (spin-lock induced crossing) sequence from the field of singlet NMR, which exchanges populations between (either of) the outer triplet states and the nuclear singlet state. The SLIC matching condition is that the nutation frequency of the driving field applied to a spin pair coincides with their homonuclear J-coupling.

All these pulse sequences are not that different:





This interpretation is quite useful. The archetypal pulse sequences are a direct map between fields of quantum dynamics that appear totally different at first sight. It immediately allows us to translate pulse sequences that were developed in one context, into another. Irrespective of the minutiae of the actual target effective Hamiltonian, and whatever populations are actually being exchanged, we can see that pulse sequences developed for, say, singlet NMR, may be adapted for DNP purposes, and vice versa.

In addition, archetypal pulse sequences possess interesting properties. For example, the SLIC sequence sets the bound on the fastest possible transfer from longitudinal magnetization to nuclear singlet order, which is just:


This basic but powerful correspondence was used by Nino Wili and myself as inspiration to adapt the PulsePol sequence (originally designed by Benedikt Tratzmiller of the Ulm group for optical DNP in NV centres) to the context of singlet NMR, and has also been noticed by other authors such as Pang et al. and Korzeczek et al..

As a pulse sequence designer, I (and the previous authors, evidently) think this is how we should be teaching these sequences to the coming generations of magnetic resonance. And I think that people in different fields (DNP, ssNMR, NV centres, solution-state singlet NMR, etc.) should talk to each other.