Radiation damping strongly perturbs remote resonances in presence of homo-nuclear mixing sequences

In this work, it is experimentally shown that the weak oscillating magnetic field (known as the “radiation damping” field) caused by the inductive coupling between the transverse magnetization of nuclei and the radio frequency circuit perturbs remote resonances when homo-nuclear total correlation mixing sequences are applied. Numerical simulations are used to rationalize this effect.

1 Introduction 5 Figure 1. The RD field lies in the xy-plane. It has an amplitude that is proportional to the projection of the water magnetization onto the same plane and a phase of ψ − π/2 with respect to this projection.
The coupling between precessing magnetization and a radio-frequency (RF) circuit induces an RF field, which in turn affects the evolution of the magnetization and hence the appearance of NMR spectra. The existence of this phenomenon was first hypothesized by Suryan (Suryan (1949)), while a more rigorous theoretical description was later provided by Bloembergen and Pound (Bloembergen and Pound (1954)). The latter introduced the term radiation damping (RD), an expression which, as several authors have stated before (Abragam (1961); Vlassenbroek et al. (1995); Hoult and Bhakar (1997); Krishnan and Murali (1997)) with the nuclei that induce RD. Subtle effects on remote resonances (Sobol et al. (1998)) can affect sensitive difference experiments. Homo-nuclear isotropic mixing sequences which have been designed for total correlation spectroscopy (TOCSY), however, are very efficient at removing the chemical shift differences from the effective Hamiltonian (Braunschweiler and Ernst (1983); Bax and Davis (1985)). In this work, it will be shown that RD, in presence of suitable mixing sequences, can heavily 25 perturb spins over wide range of resonance frequencies.

Materials and Methods
All experiments have been performed on a Bruker NMR spectrometer in a field of 14.1 T (600 MHz proton frequency) equipped with a probe cooled by liquid nitrogen ("Prodigy") with coils to generate pulsed field gradients along the z-axis. This study has been done on a standard calibration sample that contained, among other substances, about 80% H 2 O and 20% HDO (i.e., close 30 to 100 M solvent protons) and 0.5 mM Sodium-trimethyl-silyl-propane-sulfonate (DSS). At the experimental temperature of 298 K, the chemical shift difference between the solvent and the methyl protons is ca. 4.78 ppm (2868 Hz at 14.1 T, the water resonance being "downfield", i.e., precessing at a higher negative frequency).
The selective TOCSY experiment ; Kessler et al. (1986)) used in this work, with an optional bipolar gradient pair for coherence pathway selection (Dalvit and Bovermann (1995)), is described in figure 2. A selective pulse 35 applied to the solvent A, followed by a pulsed field gradient, can be inserted before the sequence so that the magnitude of the longitudinal magnetization M A z can be controlled and hence the strength of the RD effect. If, instead of the transverse magnetization, one wishes to monitor the z-component of the magnetization that remains after the homo-nuclear mixing sequence, a gradient followed by a π/2 pulse can be inserted just before acquisition. For homonuclear transfer, an isotropic mixing pulse train, DIPSI-2 (Rucker and Shaka (1989)), has been chosen with an RF amplitude γB 1 /2π = 4.17 kHz (which 40 corresponds to a duration of 60 µs for a π/2 pulse). Selective excitation, either on the water or on the methyl protons, has been achieved with a Gaussian π/2 pulse of 5 ms. Figure 2. Selective TOCSY sequence. The magnetization of one nuclear spin species is rotated into the transverse plane by the selective π/2 pulse, followed by a DIPSI-2 pulse train which is repeated nM times. Neglecting relaxation and coherence transfer, the isotropic mixing DIPSI-2 sequence is designed to leave the magnetization unchanged (spin-locked) across a wide band of frequencies centered on the RF carrier frequency. The selective pulse is cycled through (y, −y, −y, y) with a concomitant alternation of the receiver phase. (a) A selective pulse of duration τA applied to the water resonance followed by a pulsed field gradient can be inserted at position 1 to tune the amplitude of the longitudinal components of the water magnetization between +M eq and −M eq . (c) At position 3, a pulsed field gradient followed by a π/2 pulse permits the detection of Mz. (b) An optional bipolar pulsed field gradient pair at positions 2 and 3 on either side of the mixing interval leads to a cleaner coherence pathway selection and a higher signal-to-noise ratio if the receiver gain can be increased, albeit at the cost of some signal decay due to translational diffusion. In this work the carrier frequency was set either on the three methyl group resonances of DSS (leading to the situation -immediately after the selective π/2 pulse-shown in d) or on the water resonance (e).
The programs for numerical simulations of the trajectories of the magnetization and to extract the experimental peak intensities were written in the Python language. In particular, the evolution of the magnetization under the TOCSY pulse train (governed by the set of non-linear coupled differential equations 1-3) was numerically evaluated with the SciPy integration 45 libraries (Virtanen et al. (2020)) using an explicit Runge-Kutta method of order 5 (RK5(4)) (Shampine (1986)).

Experimental Results
The selective TOCSY experiment of figure 2 was applied with the RF carrier frequency set on the protons of the three methyl groups of DSS. The isotropic mixing module, DIPSI-2, consists of 36 RF pulses of constant amplitude and varying duration, applied along +x or −x and is repeated n M times (Rucker and Shaka (1989)). Since the excited methyl spins S are not 50 coupled, the mixing sequence acts as a spin-lock and only a relaxation-induced decay should be observed as n M increases. Nevertheless, the spectra of figure 3 (left) show a clear phase-drift, making nearly a full turn at n M = 26. The change of phase depends strongly on the water M A z magnetization at the beginning of the experiment, as can be seen on the right of the figure: for n M = 26, immediately before the selective TOCSY sequence, an RF pulse applied to the H 2 O resonance of varying length τ A , followed by a gradient, has been inserted, so as to modify at will M A z before the isotropic mixing sequence.

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In figure 4 (left), the phase variations of the latter experiment are plotted as a function of τ A . The theoretical curve in orange predicts the phase evolution if the phase is proportional to the initial water longitudinal magnetization, M A z , and the RF pulse on the water is ideal (i.e., with a nutation angle equal to ω 1 τ A ). The deviations between the curve and the experimental points are due to RF inhomogeneities, RD during the pulse applied to water, slight miss-calibrations of the RF power and a possible small miss-estimation of the initial phase-shift. At positions a (τ A = 0 ms, when the water magnetization is unperturbed), b 60 (τ A = 1.1 ms, when the water magnetization approximately vanishes) and c (τ A = 2.2 ms, when the water magnetization is approximately inverted) the phase evolution has been recorded as a function of number n M of isotropic mixing cycles, as shown in figure 4 (right). The dashed curve corresponds to a linear regression of the first half of the points of a, showing that a larger n M the dephasing slows down slightly (due to relaxation of the water magnetization). When a bipolar gradient pair is inserted to bracket the DIPSI-2 mixing sequence (black crosses) the effect of RD on the DSS resonance is almost undistinguishable from the same experiment that does not use gradients for coherence pathway selection.   the carrier frequency has been moved to the solvent resonance, and that the amplitude of the selective Gaussian pulse has been increased in order to overcome RD effects during this pulse, so that the solvent magnetization is rotated in the xy-plane. Here, 70 the z-component of the magnetization must be detected without changing the phase of the receiver for the different scans.
Without RD, the magnetization of the methyl groups is expected to stay along the z-axis. Clearly, effects of the RD field are also observed in the latter experiment.  correspond to the same components of the magnetization. The RD parameters for the simulations on the right were the same as in figure 5.

Theory and Discussion
In order to explain the experimental results, the homo-nuclear case of abundant spins A (H 2 O), whose magnetization induces an RD field in the coil as shown in figure 1, and sparse spins S (the three methyl groups in DSS), whose RD interaction with the coil can be neglected, will be considered. In the rotating frame, the evolution of the two (uncoupled) types of spins can be described by the modified Bloch equations (Bloom (1957)): with i either spin A or S, ω i 0 the difference between the resonance frequency of spin i and the carrier frequency, ω 1x and ω 1y the x and y components of the RF field during the mixing sequence, while the remaining terms in the equations are due to the RD field:

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where the amplitude of the RD field is ω R (t) = α R M A x (t) 2 + M A y (t) 2 and its phase is given by the angle ψ as indicated in figure 1. The proportionality constant α R depends on the characteristics of the RF circuit: with µ 0 the vacuum permeability, γ the gyro-magnetic ratio of the protons, η the filling factor of the sample and Q the quality factor of the RF circuit. By a multiplication of α R with the equilibrium magnetization of the abundant spins A, the use of the 90 RD rate allows one to use normalized magnetization vectors (i.e., divide all components of spin i by M i eq ) in equations 1-4. The evolution of the magnetization of both A and S nuclei during the TOCSY pulse train has been numerically simulated using the above equations. First the evolution of M A (t) was determined. For spin S equations 1-3 reduce to the traditional 95 Bloch equations, with the magnetization of spin A as a source of a time-dependent RF field. The values of the rate R R and the angle ψ were estimated to give a qualitative agreement with the data, as shown in figure 5, rather than an exact fit. The combination of the two parameters is not unique, a smaller angle ψ can be compensated by a larger value of R R . The use of the RD parameters extracted from the signal of H 2 0 after a simple pulse-acquire experiment does not lead to a good agreement. This is likely due to the fact that the RF circuit is not the same during signal acquisition as during the application of RF pulses 100 (Marion and Desvaux (2008)). In figure 5, the agreement between simulations and experiments is quite satisfactory. The decay of the experimental curves is not only due to relaxation but also to RF inhomogeneities: the precession frequency of the DSS signal varies slightly with the the RF amplitude, while the evolution of the z component is even more sensitive (simulations not shown).
For the curves on the right-hand side of figure 6 the same RD parameters in figure 5 have been used. In the simulations, the 105 fact that, due to RD effects, the water magnetization is not aligned along the x-axis after the first π/2 pulse has been taken into account (a phase shift of -18°was determined experimentally). The agreement between experiments and simulations is adequate, considering the fact that neither RF inhomogeneity and calibration errors nor relaxation effects have been taken into account. Moreover, the evolution is very sensitive to the exact position of the water magnetization after the selective Gaussian pulse.

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The phenomenon shown in this work strongly depends on the characteristics of the probe. Similar results (not shown), albeit much smaller in magnitude, have been obtained at 18.8 T (800 MHz proton frequency) on a traditional "room temperature" probe.

Conclusions
It has been shown RD can strongly perturb the evolution of magnetization of spins that are neither directly coupled by scalar or 115 dipolar interactions to the source spins nor have a nearby resonance frequency. Counter-intuitively, the RD field can thus cause the magnetization of remote resonances to precess notwithstanding the presence of a much stronger RF spin-locking pulsetrain. This effect increases with increasing RF amplitudes (results not shown). It can be prevented by saturating or dephasing the magnetization of the spins that cause radiation damping.