Spin noise gradient echoes

Abstract Nuclear spin noise spectroscopy in the absence of radio frequency pulses was studied under the influence of pulsed field gradients (PFGs) on pure and mixed liquids. Under conditions where the radiation-damping-induced line broadening is smaller than the gradient-dependent inhomogeneous broadening, echo responses can be observed in difference spectra between experiments employing pulsed field gradient pairs of the same and opposite signs. These observed spin noise gradient echoes (SNGEs) were analyzed through a simple model to describe the effects of transient phenomena. Experiments performed on high-resolution nuclear magnetic resonance (NMR) probes demonstrate how refocused spin noise behaves and how it can be exploited to determine sample properties. In bulk liquids and their mixtures, transverse relaxation times and translational diffusion constants can be determined from SNGE spectra recorded following tailored sequences of magnetic field gradient pulses.


Simulations of spin-noise gradient echo
To assess the effect of SNGE, we perform simulations using Mathematica (Wolfram Research Inc., 2012) based on the Bloch equations (Bloch, 1946), and neglecting any effect of radiation damping.This last assumption corresponds to the used experimental conditions where the inhomogeneous broadening by the gradients exceeds the radiation-damping rate.In these simulations, spin noise originates from a series of random excitation events of random timing, random phase and a random small flip angle supposed to be below 0.01°.For a single noise event, the evolution of magnetization in the rotating reference frame is simulated by assuming a weak randomized initial transverse magnetization  0 starting at a random time  within the first gradient  1 .Using  as the position coordinate along the gradient axis, we get for a single isochromat with offset 0 at the end of  1 : In the case of a distribution of components along , it is necessary to integrate: For the sample with length , it results in: Then, the evolution during the second gradient leads further to: Here we assume the delays ∆ and  (Fig. 1) to be negligibly short.After integration over the sample, we get the expression: By numeric summation over 100 uniformly random distributed (with respect to starting times, phases, and amplitudes) initial magnetizations  0 , we obtain the curves shown in Fig. S1.The case including  2 relaxation is described by Eq. (S6): These simulations assume that for small radiation damping contribution (gradient broadening of the linewidth larger than the initial linewidth due to transverse relaxation and radiation damping), spin noise can be envisaged as a series of spontaneous excitation events of random phase.Because of creating a series of random coherences, they cannot be applied to the cases with weaker gradients or for strong polarization situations, as it was the case in (Desvaux, 2013;Pöschko et al., 2017).After these excitations, the individual coherences can be defocused and refocused by field-gradient pulses, as usual.As a consequence of these simulations, the noise power amplitude of the (+) experiments (Fig. 1) is due to transverse coherences excited during the second gradient only.However, the additional contributions observed in the (-) experiments originate from events during first gradient with length of δ 1 refocused by the second gradient during  As in Fig. 5, Fig. S2 reveals a correct linear behaviour for not too small  1 2 values.This is particularly true for acetone, while for benzene which molecular diffusion coefficient is smaller, more scattered measurements are observed.From this SNGE attenuation vs  1 2 experiment (Fig. S2), an estimation of ratio of slope for acetone component in this mixture to that for benzene component results can be determined.This ratio is comparable to the independent determination of diffusion coefficients in pure solvents (acetone and benzene) measured with a SNGE NMR-diffusion experiment (Fig. 3) with a small RF pulse of 0.2µ added (2.4 ⋅ 10 −9 m 2 s for benzene and 4.58 ⋅ 10 −9 m 2 s for acetone).These self-diffusion coefficients agree well with literature values of benzene and acetone as obtained by canonical RF NMR spin-echo experiments.(Ertl, and Dullien, 1973) This experiment (Fig. S2) illustrates that in two-component mixture with different chemical shifts it is possible to acquire SNGE attenuation data (Fig. 4) and to extract SNGE attenuation curves separately for each diffusion component.(Fig. 5 and Fig. S2).This is still not enough to get data set for calculation of diffusion coefficient according to Eq. ( 3).We could not increase measurable gradient values to carry out reliable SNGE experiment at gradients higher than 75 mT m . So, implementation of diffusion experiment limited by the fast gradient duty cycle -this is too fast for our high-resolution probe.Therefore, a search of additional experimental facilities can result in new advantages for realization of spin-noise gradient echo measurements.

Molecular diffusion and relaxation measurements
Figure S3 illustrates how 3 gradient pulses scheme (Fig. 3) produces SNGE data sets in 1:1 acetone-benzene mixture vs ∆.The difference in the relaxation times ( 2 * ) determined from the spin noise line shape and the one from spin noise amplitude 85 decay in the inter-gradient delay is ascribed to different radiation damping times which are subject to influence of preamplifier feedback (see main text) (Pöschko et al., 2017)

Figure S1 .
Figure S1.Numerical simulation of the time course of refocused spin noise during the second gradient of a gradient echo sequence, using Eq.S6; (a) neglecting transverse relaxation and diffusion, (b) including transverse relaxation   =  .Experimental parameters corresponding to Fig 1(b) are used in the simulation:   = −.   ;   = .   ;  = . ×     ; M0=1;  =  .No additional non-spin random noise was added.It would exceed the echo amplitude without extensive averaging.
Fig.S1only shows the contribution from the refocused noise initiated during δ 1 .The signal envelope indicates an echo center at the point where the presence of transverse relaxation (Fig.S1(b)).The location of the maximum gradient echo is typically that observed with coherent excitation ( Figure S2 presents the results of SNGE attenuation vs gradient values for a mixture composed of acetone and benzene in 1:1 concentration.The associated spectra are reported in Fig. 4.

Figure S2 .
Figure S2.SNGE attenuations vs gradient increase for separate signals of acetone (top) and benzene (bottom) in a 1:1 mixture of

Figure S3 .,,,
Figure S3.SNGE experiments as a function of ∆ between   and   with a triple gradient sequence (Fig. 3) on a 1:1 mixture of acetone and benzene (with 10% of acetone-d6 for locking).The respective lower traces correspond to the (+) sub-experiment, the higher ones are to the (−) measurements.SNGE spectra are presented for 2 values of delay increasing from top to bottom:  ms, 70

Figure
Figure S5.(blue) Spin noise spectrum of a 1:1 mixture of benzene and acetone (as in Figure 6), but recorded without any gradients or pulses.(red) Lorentzian best fit of experimental peaks amounting to FWHM = 37.6Hz / 39.8Hz and hence   * = /( • ) = 8.5ms / 8.0ms for benzene / acetone respectively.(black) Simulated Lorentzian resonance peaks assuming   * 's as determined from three-gradient experiments (c.f.Fig. 3) (92ms / 62.5ms for benzene/acetone respectively).The asterisks (*) indicate signals of 13 C satellites as described in (Pöschko et al., 2017) So, qualitative comparison of how SNGE signal is attenuated in diffusion experiment in mixtures with differently diffusing (and chemically shifted) molecules.The issue resides in obtaining enough measurements in the range of gradient values (i.e., in such linear