Introduction (skip if you know what singlet NMR is)
Suppose you had a coupled 2-spin-1/2 system where the two nuclei can be said to be identical i.e. in the sense of, they experience the exact same spin interactions. As you may know from introductory courses (or expect from the good ol’ Catalan triangle) the spin system would have a set of eigenstates described by a spin-0 (singlet) and a spin-1 (triplet) manifold, where the eigenkets can be expressed in a number of languages:
The Hamiltonian of this spin system would look something like this in the product operator formalism, in which “J” denotes the isotropic part of the J-coupling tensor:
And something like this in the matrix representation:
Now, introduce a small chemical shift difference between the spins, which we will call Δ, with the corresponding Hamiltonian:
It is pretty useful to express the previous Hamiltonians in terms of the single-transition operators for the zero-quantum subspace (the subspace of |T0> and |S0>):
I’ll also include the cheeky unity operator of the double quantum subspace:
To make it clear that I1zI2z consists of unity operators:
We have:
And we really only need to consider the truncated version of HJ (denoted HJθ) when it comes to the overall dynamics.
In the strong-coupling limit, we may express the perturbation HΔ in the interaction frame of HJ, and it shouldn’t be rocket science to see that the time-dependent interaction-frame Hamiltonian is:
That is to say, the net effect of the J-coupling Hamiltonian is to modulate the chemical shift difference term in the xy-plane of the ZQ subspace.
Zero-quantum effective Hamiltonians
Consider the traditional J-CPMG building block of the old-school M2S experiment, which involves a basic spin echo sequence [1/(4J) – 180 – 1/(4J)]N. Its average Hamiltonian is trivial to calculate using a first-order (numbered per the modern Nielsen-Levitt convention; this would be zeroth-order in the traditional convention favored by the prominent American groups i.e. Warren) Magnus expansion:
Which works out like this:
In a nutshell, we’ve generated an effective Hamiltonian that generates a pure x-rotation in the ZQ subspace! What if we explored the alternative sequence {180-[1/(4J) – 180 – 1/(4J)]N-180} ? We would instead have a rotation about the (-x) axis:
Now, a little thought reveals you can generate a zero-quantum effective Hamiltonian of any phase. For a phase ϕ in the interval [-π/2,+π/2]:
And for a phase ϕ in the interval [+π/2,+3π/2]:
Realized via an actual pulse sequence, we have the following pair of sequences which correspond to a rotation of phase ϕ and flip angle β in the zero-quantum subspace:
You can use this generalized control protocol to generate many fancy trajectories in the ZQ subspace, but I originally invented this for another reason: composite pulses. You can see what I’m talking about below:
This is a simulation for a hypothetical spin system with J = 100 and Δ = 1, using a naive train of J-CPMG echoes (the ideal number Nideal is 156) vs. a composite rotation employing 5 ZQ-phase-shifted J-CPMG trains (the composite pulse itself is nothing a special and part of a family of composite pulses I came up with a while ago while generalizing the work of Tycko and Pines). You can see that you can achieve excellent compensation against flip-angle errors just like you would do with a normal pulse, and you could try off-resonance errors too (see Pages 82-85 of Tayler’s excellent thesis). But for now, I leave it at this, and leave you with the suggestion that there are many more tricks to be played in the zero-quantum world…
Departing from our usual programming, this will not be a blog post about anything related (at least directly) to magnetic resonance or quantum dynamics. It will be a post about mathematics, particularly continued fractions.
How I met continued fractions
After graduating high school at 16, I discovered continued fractions, having been intrigued by much of the mythology surrounding the Indian mathematician Srinavasa Ramanujan. Continued fractions were one (of several) areas of mathematics Ramanujan made incredible contributions to.
[It goes without saying that Ramanujan was one of the most naturally gifted mathematicians in history, despite never having received any formal training, with a signature habit of providing extraordinary results without proof.]
So I spent a lot of my time on random number theory problems, and published a couple of my continued fractions in the Online Encyclopedia of Integer Sequences (OEIS) you can find here.
Continued fractions can be more neatly expressed in the “K notation” commonly ascribed to Carl Friedrich Gauss (from Kettenbruch, the German name for a continued fraction):
One of the first cool simple patterns I stumbled upon in mathematics was the following relationship relating to powers of the golden ratio:
The simple continued fraction for π may appear to show no consistent pattern:
But there are incredible non-simple continued fractions for Pi. The most famous is Lord Brouncker‘s continued fraction:
Lord Brouncker had a fascinating biography which included being the first president of the Royal Society and having a prolonged affair with the wife of the first cousin of Oliver Cromwell.
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 → 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 A (i.e. how much of operator B is in operator A) is given by:
In which we’ve used the concept of a Frobenius norm and a funky operation, common within the field of NMR at least, called the “Liouville bracket” (see the paper by Jean Jeener):
Part A: Hartmann-Hahn transfer from a virtual spin-ℓ particle to a single spin-1/2 particle
We’ll begin by considering the basic problem of spin order transfer from a spin-ℓ particle I to a spin-1/2 particle S. We have (at least as a start):
In which it is crucial to note a consequence of the high-temperature approximation: the initial density operator is ℓz. This is the fundamental root cause of a host of complications we will see later.
If we define the following effective angle (just so the notation doesn’t get too horrible):
We calculate the following polarization trajectories for the first values of ℓ, which can be expressed as a sum of oscillators:
I’ve written the equations in this weird way to make the otherwise nonobvious pattern clear. Using mathematical induction, we have the general result:
The argument inside the sine functions represents eigenvalues that are essentially (discretely) distributed along a Wigner semicircle.
Now, let’s look at the actual functions representing the polarization transfer trajectories. The first two are simple periodic functions:
But all hell breaks loose once we have ℓ ≥ 3/2:
Why does this happen? Due to the fact that the functions are sums of oscillators with frequencies that are incommensurate (irrational multiples with respect to each other), the trajectory is now not periodic but quasiperiodic. This means that:
In general, the trajectory never repeats itself exactly in time.
In general, we cannot find closed-form simple expressions for when a local maximum, minimum, or zero occurs.
Essentially, we are dealing with a quantum many-body problem. However, a form of order does emerge for large values of ℓ:
Here, Hk(z) denotes the Struve function of the first kind of order k. The Struve function is a cousin of Bessel functions, which are commonly encountered in solid-state NMR. This equation implies that at high-temperature conditions, the maximum possible transfer from a high-spin particle I via a Hartmann-Hahn sequence is:
Struve and Bessel functions naturally arise in integrals of trigonometric functions inside trigonometric functions. Here’s a hint:
The particular mathematical trick I used to obtain this problem was the following identity:
I have not been able to find this identity in the mathematical literature, so it’s worth revealing the works my formal proof:
All we have to do to get the final result is plug in z = [ℓ(ℓ+1)]1/2 × βJ.
There have been at least two reports of the Struve function popping up in magnetic resonance problems [1, 2]. 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:
Part B: Hartmann-Hahn transfer in INS spin systems (from N spin-1/2 particles to a single spin-1/2 particle)
The previous results can be used to great effect to automatically obtain the form of the polarization transfer trajectory in INS spin systems. We merely have to do some sneaky normalization (the old I-spin basis had 2ℓ+1 states, the new I-spin basis has 2N states) and employ our friend the Catalan triangle to account for multiplicities:
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.
You will notice something peculiar for larger values of N – neglecting the transient “revivals”/Loschmidt echoes, the time period between t = 0 and t = 1/J 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.
In fact, there is an exact solution for the Hartmann-Hahn polarization transfer trajectory in the thermodynamic limit of large N:
In which “F” denotes the Dawson integral (also called Dawson’s integral or the Dawson function).
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):
… as well as a couple of tricks. Essentially, the limiting form of the Catalan triangle coefficients gives you a Laplace transform-looking cursed object that spits out the required result:
You can compare the Dawson function approximation (blue) to the analytical expression (black) for a hypothetical N = 1000 spin system:
In the high-N thermodynamic limit, there are no oscillations or coherent behavior whatsoever. We have a surge followed by a plateau to 1. This resembles the sort of cross-polarization kinetics you would see in solid-state magnetic resonance, usually with some kind of relaxation/damping:
But in the absence of relaxation, we have the following results for Hartmann-Hahn evolution in the thermodynamic limit:
Nuclear spin plasmas?
The Dawson integral is also known as the Fried-Conte plasma dispersion function 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:
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)), leading to a virtual quantity one could perhaps call… a high nuclear spin pressure? But only if you really wanted to.
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.
EDIT (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.
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.
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:
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].
Everybody loves adiabatic sweeps. They are a popular way of achieving Iz → Sz polarization transfer because you can kill two birds with one stone: you usually get an impressive enhancement factor as well as automatic robustness against errors in whatever parameter you felt like sweeping, whether it was pulse strength, resonance offsets/detuning, or something else.
But what goes up must come down. Here’s the unfortunate catch: an adiabatic sweep can never give you the maximum possible enhancement factor. See my OEIS sequence A362534, and consult the Sacred Table:
In this blog post we’re going to try and deal with the adiabatic column of this table, but I should first provide a disclaimer: note that the basic observation that adiabatic sequences do not lead to the maximum possible transfer of polarization dates back (at least) to Hodgkinson and Pines’ paper from 1997 [1]. The adiabatic-associated bounds themselves were derived by Chingas, Garroway, and friends at the US Naval Research Laboratory [2, 3] even earlier.
The bounds on an adiabatic Iz → Sz transfer, in INS spin systems, are simply:
Where we have used our good friend the Catalan triangle to avoid doing more group theory than we really need to. (I shall not bear gruppenpest in this house.)
But hold on, where does the above result come from? The bound is very simply obtained by considering the much easier problem of an adiabatic Iz → Sz transfer in a two-spin subsystem (I = ℓ, S = 1/2):
All we’ve really done here is some sneaky normalization i.e. multiply by the (2ℓ+1) factor associated with the old 1-spin-ℓ basis, divide by the 2N factor associated with the new N-spin-1/2 basis, and you’re golden. Then, just consult the aforementioned Catalan triangle to account for the multiplicity of the spin-ℓ manifolds [As emphasized in that blog post, a common theme in tackling multispin problems where permutation symmetry is strictly satisfied, is solving for the irreducible subspaces and simply adding things up.]
Conveniently, it turns out that there is a closed form for the adiabatic-associated bounds:
Which I think is somewhat less intimidating than the equivalent form given in the aforementioned paper by Chingas et al.:
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 INS spin systems. In a nutshell, INEPT accomplishes the conversion Iz → [Ix → 2IySz → 2IzSz →] 2IzSy (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:
The I-spins are tipped to the transverse plane by a 90y pulse i.e. something like Iz → Ix. [Alternatively, from another frame of reference, the Hamiltonian is tipped from 2πJ×IzSz → 2πJ×IxSz].
Evolution for a total time period of τ = 1/(2J) leads to a total rotation τ×2πJ = π about the IxSz axis (more properly π/2 about the 2IxSz axis). This basically accomplishes the transformation Iz → 2IySz.
The I spins are tipped to the longitudinal plane by a 90x pulse; i.e. 2IySz → 2IzSz
The S spins are tipped to the transverse plane for detection by 90-x pulse; i.e. 2IzSz → 2IzSy, 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 INS 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 2IzSz [→ 2IzSy → Sx] → Sz. 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 2IzSz to Sz:
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].
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 archetypalsoft-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.
“Funny story: our neighbouring Chemistry institute can’t measure good quality HSQCs. I told them once they are missing the filter. Tried ours on theirs and showed it worked, but the responsible person called Bruker and they said they don’t need them.
Well, they still don’t measure HSQCs.“
– an anecdote from an anonymous NMR spectroscopist, shared with permission.
A fundamental basic principle in all technologies that involve radiofrequency circuits (NMR is no exception) is the importance of isolating the signals you want and removing interference from the signals you Do Not Want via the use of radiofrequency filters. Entire books have been written on this topic alone, but the gist of it is this summary:
“The primary objective of RF filters is to block undesired signals effectively. As passive devices, they operate seamlessly by selectively permitting or hindering the transmission of signals based on their frequencies. The critical nature of this function becomes apparent when considering scenarios where unwanted signals could interfere with the normal operation of electronic devices or communication systems. In essence, RF filters act as essential guardians, ensuring the integrity and efficiency of signal transmission by preventing interference from signals that fall outside the desired frequency range.“
Figure 1: a set of RF filters in my office, with a 50 MHz low-pass filter being tested on my NanoVNA-H4 network analyzer.
Unless you are doing something like ZULF NMR, it is usually the case that conventional high-field multinuclear NMR experiments will involve more than one nuclear spin species – each precessing at a unique Larmor frequency – with the relevant frequencies typically (but not always) being well-separated by tens or hundreds of megahertz. Disregarding “true” spin-dynamical effects such as heteronuclear Bloch-Siegert shifts one would generally like – at least on the RF level – that a pulse applied on Channel 1 has no effect on Channel 2; i.e. we’d say there is no RF cross-talk, or the RF channels are isolated.
A common but erroneous assumption (sometimes encouraged by non-experimentalists, or PIs that struggled with far less forgiving hardware in previous eras of NMR) is that a modern spectrometer is now “perfect” or “it does it all for you“. Many students acquire the misleading expectation that a spectrometer will output exactly what is programmed into it and nothing else. This is not the case, and has never been the case for any real-life/non-fictional RF circuit.
Figure 2: 1H[13C] spectra of 13C-labelled sodium formate, with and without a bandpass filter installed on the 13C channel where decoupling is being applied (nutation frequency = 1 kHz). Broadband noise from the 13C channel bleeds into the 1H and 2Hchannel, severely compromising the S/N. It is worth noting that both cases have a bandpass filter on the 1H channel. The style of the graphic is an intentional homage to Glenn Facey’s blog.
The experimentalist’s reality is that broadband amplifier noise on one channel will bleed into other channels unless otherwise prevented from doing so. For example, when observing, say, the I-spin spectrum under S-spin decoupling, it is imperative to minimize cross-talk by inserting an RF bandpass (or lowpass) filter at the output of the preamplifier corresponding to the S-spin channel. This is a precaution in experiments that involve heteronuclear decoupling that is always mentioned in well-written experimental sections. If the S/N on the receiving channel is good enough, the signal attenuation by the filter is not a serious problem – so if there is no harm, bandpass filter everything.
Another less-appreciated source of annoying experimental artifacts is broadband noise leaking into the lock channel. On some of our spectrometers, even 13C pulses on the BB (broadband) channel are easily seen as prominent “spikes” in the 2H lock display, and during 13C decoupling or spin-locking the lock becomes extremely noisy. A noisy lock channel is bad news because it translates to noisy and/or artifact-ridden spectra. In this case, the problem can be entirely prevented by connecting a 13C bandpass filter (or alternatively 2H bandstop filter) to the output of the BB channel. Some spectrometers come pre-equipped with 2H bandstop filters, which is not an ideal solution since spectroscopists may prefer to retune the BB channel to perform 2H experiments (the lock channel has limitations better left for a future blog post).
Adding RF filters is not merely a fine detail that marginally improves the quality of spectra. The problem can already screw with mainstream 13C/15N experiments and tends to be far more severe for exotic lower-gamma (Cinderella) nuclei. Here it is worth noting that we simply never succeeded in measuring any 13C or 1H signals under 103Rh decoupling on some of our spectrometers without the appropriate RF filters (see our first and second103Rh papers where I tried to make this explicit in the experimental section).
Yes, it’s that bad. If you’re struggling with S/N in multinuclear NMR experiments this is one possible source of trouble you can look at. It is always better to listen to WHAT YOUR SPECTRA SAY than the guys who confidently swear “the filters are pointless, you don’t need them, take them off“, unhelpful mis-advice that has wasted days of innocent PhD students’ magnet time.
If you haven’t lost them, Bruker bandpass/bandstop filters do a good job. We buy our other filters mainly from K&L Microwave, and occasionally use archaeological remnants such as the now-extinct Chemagnetics/Varian filters, or an old (and magnetic) TV filter from a random company in Ohio that I used to great success in e.g. potassium-39 and krypton-83 experiments:
Figure 3: a set of old-school RF filters.
If you’re a hands-on person you can also build your own rf filter at little cost using references such as this paper and one of the several online filter design tools. The filter does not need to be built with nonmagnetic components, but you’d ideally avoid questionable design choices such as a pure steel shell. The current trend in electronics is “miniature and cheap” and a tool I’ve found to be indispensable is the incredible NanoVNA network analyzer. The nifty little devil of a tool does all of the jobs that the £3,000-£30,000 network analyzers can do (filter characterization, probe tuning) without costing too much money and lab members fighting “who took our network analyzer” wars. Several variants of the NanoVNA can be bought online for £50-100. I bought my own but unless your finance department is a real headache, you should ask your PI to get it using the grant money, which I’m sure they will appreciate!
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.
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:
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.