Month: February 2024

About Month: February 2024

Subscribe Feeds

The Quantum Pascal Pyramid

Posted on by Mohamed Sabba

In the previous blog post, I discussed a close relative of Pascal’s triangle – a Catalan triangle – and briefly alluded to z1 multiplets at the end.

In this post I will discuss another less well-known (but very useful!) combinatorial structure – the Pascal (difference) pyramid, which has a beautiful correspondence to the multispin (single-quantum) coherences observed in NMR experiments.

Suppose one has an INS (or AXN) spin system. Begin by considering the following cumulative tensor product:

The expansion may be organized by the spin product rank q, yielding a convenient operator basis of I-spin longitudinal (z-) spin-operators which Sorensen denoted ZNq. Some examples for the cases N=3 and N=4 are:


In my mind, I like to call these “the binomial Z-operators” since the number of operators forms binomial patterns much like NMR spectra (1:3:3:1, 1:4:6:4:1, etc.) for reasons I hope are obvious. (If it isn’t obvious, consider the basic combinatorial problem: How many unique words with length q can be formed by combining N different letters?)

Now, suppose you wanted to observe single-quantum coherences involving a product of any of these ZNq operators with, say, Sx. What would the S-spin spectrum look like? Some visual examples are given here:



Intuitively, one can reason that each ZNq.Sx operator has to be a sum of the “pure” single-transition operators corresponding to each m=J×{-N/2,…,+N/2} line component making up the S-spin spectrum. One might notice that each ZNq operator changes sign exactly q times. But it’s not so clear to see how.

It turns out there is an incredible direct map between the operators describing pure populations of states with azimuthal quantum number m (which I will denote PNm) and ZNq . The relationship is given by:

The map can be represented by an (N+1)×(N+1) matrix. Some matrices illustrating this relationship are shown below:

Naturally, this type of relationship has appeared in other fields of quantum mechanics (see equation 11). But to my knowledge, the only magnetic resonance paper which describes something resembling a “Pascal’s pyramid” is the excellent paper on relaxation in the AX4 spin system of 15NH4+ by Nicolas Werbeck and D. Flemming Hansen, where the combinatorial structure is referred to as a “modified Pascal triangle”.

EDIT (February 23, 2024, 15:56 UK time): after writing this article, I realized something neat. Recall the de Moivre-Laplace theorem, which is the famous mathematical result that the binomial distribution (corresponding to the multiplets associated with ZN0) converges to a normal (Gaussian) distribution as N→∞. I have observed a generalization of the de Moivre-Laplace theorem: that the columns of Pascal’s pyramid, i.e the multiplets associated with ZNq, converge to the q-th derivatives of a Gaussian distribution. The q-th derivative of a generic Gaussian function has a general form that can be written in terms of a regularized hypergeometric function:




That is to say, the ZNq operator may be approximated by a q-th order Gaussian derivative:

We know from basic mathematics that a derivative is a measure of a rate of change. We can appreciate the corresponding physical picture in more than one way:

1. each ZNq operator transforms under, say, an I-spin x-rotation of a flip angle β, with an increasing “responsiveness” to rotation that depends on the spin product rank q. The self-evolution of each ZNq operator would be given by:

2. each ZNq operator can be converted into I-spin multiple quantum coherence with a (maximum) coherence order q. This well-known property is commonly exploited in NMR experiments. Of course, by definition, an MQC of coherence order q (which I’ll dub qQC for q-quantum coherence) has the following property under an I-spin z-rotation with a flip-angle β:

EDIT (May 16, 2024, 14:20 UK time): I have written this blog post as a small paper on arXiV, which goes into a bit more detail.

Catalan Triangle and Clebsch-Gordan Multiplicities

Posted on by Mohamed Sabba

Most magnetic resonance spectroscopists will invariably have seen Pascal’s triangle (i.e. the triangle of binomial coefficients) in introductory undergraduate courses or school curricula:

Pascal’s triangle is widely used as a pedagogical tool to explain first-order multiplet patterns.

However, magnetic resonance is full of other, less well-known combinatorial structures. One of the most useful is closely related: a Catalan triangle* (so named due to the leftmost columns giving the Catalan numbers):

In basic terms, this Catalan triangle (which adds up just like Pascal’s triangle) provides the distribution of eigenstates in a symmetric N-spin-1/2 system, immensely simplifying the treatment of multispin problems. For example, an A2 spin system such as H2 can be treated as a sum of 1 spin-0 (singlet) and 1 spin-1 (triplet) particle. An A4 spin system such as CH4 can be treated as a sum of 2 spin-0, 3 spin-1, and 1 spin-2 particles:

Deep in the annals of NMR theory, the coefficients of this Catalan triangle are also known as the “Clebsch-Gordan multiplicities“.

The old NMR texts referred to the above decomposition as the composite particle method showing that this was a much easier way to treat multispin systems than going Rambo and invoking symmetry groups, character tables, molecular rotation, and what have you. The simple triangle itself appears in some form in early references such as Grimley’s paper (1963) and Corio’s famous book (1967):

It is rather unfortunate that in today’s literature, there is an apparent insistence on obfuscating such basic combinatorial results by invoking unnecessarily complicated, apocryphal chains of group theoretical arguments – behavior that early quantum physicists would have called gruppenpest.

A beautiful recent example of the appearance of the coefficients of this Catalan triangle in an unexpected context is this excellent paper, which rigorously explores weighted multiplets; for example, the intensity ratios of the simple z1 multiplets (generated by the INEPT sequence) are given by a “mirrored” Catalan triangle:

Historically, this Catalan triangle can be traced back to Wigner’s timeless book (1959) (where it appears as an expression) and the classic combinatorics paper of Forder (1961), although there are surely older, less relevant appearances in the literature.

*Note: unfortunately, due to the ubiquity of Catalan numbers in mathematics, “Catalan triangle” and “Catalan’s triangle” have been used in the literature to refer to several different number triangles, generating immense confusion. Since there are many Catalan triangles, care should be taken with terms such as “Catalan’s triangle” or “the Catalan triangle” without an appropriate reference, and one should use terms such as “a Catalan triangle” or simply “triangle of Clebsch-Gordan multiplicites”. I personally like “Catalan-type triangle”.

†There are some pretty results, such as the number of spin-0 (singlet) states in a (2N)-spin-1/2 system being given by the N-th Catalan number. This is also true for the number of spin-1/2 states in a (2N-1)-spin-1/2 system.

Higher-Order Terms of the Bloch-Siegert Shift Hamiltonian

Posted on by Mohamed Sabba

The Bloch-Siegert shift is a well-known perturbation to the rotating-wave approximation which becomes prominent when the nutation frequency of the driving rf field becomes comparable in magnitude with the Larmor frequency of the driven spins.

Consider the following scenario: we are equipped with a set of B1 coils at low-field (the longitudinal B0 field is~1.1 mT). Our untuned B1 coils are capable of broadband irradiation of spins with a transverse B1 field whose nutation frequency, ω1/(2π), can reach around ~2 kHz*.

A 2 kHz nutation frequency presents little food-for-thought for 1H spins (ω0/(2π) @ 1.1 mT ≈ 47 kHz) but really makes you reconsider your life choices if, like us, you have been working with 103Rh (ω0/(2π) @ 1.1 mT ≈ 1.5 kHz).

In this scenario it is necessary to calculate higher-order terms of the Bloch-Siegert shift Hamiltonian, which I have done to 12th (or 11th, depending on your convention) order.

*there is a subtle point here: with untuned coils operating in the high-inductance limit, the B1 field strength is proportional to 1/(ω0), whereas the nutation frequency is of course proportional to ω0. These factors cancel out as shown in pages 90-91 [Figs. 39-40 of my thesis], leading to a nutation frequency that is independent of frequency and hence γ:

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.

Quadrupolar Linewidth Patterns in the Solution-State

Posted on by Mohamed Sabba

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

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

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:

Here, the norm of the quadrupolar coupling tensor is related to the quadrupolar coupling constant (QCC):

And the quadrupolar coupling constant itself is:


Here, it is worth explaining why I have written the equation in an unconventional way. The nuclear quadrupole moment Q (an intrinsic nuclear property which is non-zero for ℓ > 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 (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 and Vzz couple to produce a voltage at the nucleus. Finally, half the Josephson constant (½KJ, 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.

Now it is simple to figure out the exceptions when quadrupolar relaxation may be slow enough to observe J-couplings:

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

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

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

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

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

But impressively, the paper also provides a hidden gem in the supporting information: 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:

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

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.