Breakdown of Carr-Purcell Meiboom-Gill spin echoes in inhomogeneous fields
BBreakdown of Carr-Purcell Meiboom-Gill spin echoes in inhomogeneous fields
Nanette N. Jarenwattananon and Louis-S. Bouchard ∗ Department of Chemistry and Biochemistry, University of California Los Angeles,607 Charles E Young Drive East, Los Angeles, CA 90095-1059 (Dated: November 9, 2018)The Carr-Purcell Meiboom-Gill (CPMG) experiment has been used for decades to measurenuclear-spin transverse ( T ) relaxation times. In the presence of magnetic-field inhomogeneities,the limit of short interpulse spacings yields the intrinsic T time. Here we show that the signaldecay in such experiments exhibits fundamentally different behaviors between liquids and gases. Ingases, CPMG unexpectedly fails to eliminate the inhomogeneous broadening due to the non-Fickiannature of the motional averaging. PACS numbers: 51.20.+d, 66.10.C-, 82.56.Lz, 82.56.-b, 33.25.+k
Previously, we found that the decay of nuclear induc-tion signal in a magnetic field gradient differs fundamen-tally in gases compared to liquids, as manifested in thetemperature dependence of the nuclear magnetic reso-nance (NMR) linewidth . In the conventional descrip-tion of NMR, spectral lines should broaden in a gradientof the magnetic field as temperature increases , which re-sults in a larger diffusion coefficient. However, we foundexperimentally that in gases, the NMR linewidth instead decreases with temperature, which is consistent with amotional narrowing effect. The importance of this mo-tional narrowing effect was not predicted by the conven-tional theory. In this article, we demonstrate that thisdifference in motional averaging between gases and liq-uids also manifests itself in the signal decay of CPMGspin echoes. In liquids, a series of spin echoes in thelimit of short interpulse spacings minimizes signal decayeffects due to diffusion in a gradient (as expected fromthe conventional theory ). For gases, however, we findthat CPMG is unable to eliminate this signal decay inthe limit of short interpulse spacings. This result impliesthat any T − or diffusion-weighted NMR measurementsof gases made in the presence of magnetic susceptibilitygradients or applied field-gradients are potentially com-promised. r.f.g ACQUISITION180 o o τ τ τ loop n times FIG. 1. Measurement of T in an inhomogeneous field (gradi-ent, g ) using a CPMG sequence. A 90 ◦ radio-frequency (r.f.)pulse tips the magnetization, which is refocussed by a seriesof n ◦ pulses at intervals of 2 τ . In a 90 ◦ − τ − ◦ − τ spin-echo experiment the free in-duction signal in the presence of a magnetic field gradientis given by Hahn’s famous result : S ( τ ) = exp ( − τ /T ) exp (cid:18) − γ g Dτ (cid:19) , (1) in which 2 τ is the evolution time (total length of thespin-echo experiment), T is the intrinsic spin-spin re-laxation time, γ is the nuclear gyromagnetic ratio, D is the self-diffusion coefficient, and g is the magneticfield gradient ( g = ∂H z /∂r , where r is a spatial direc-tion). In the absence of a large magnetic field inhomo-geneity and large diffusivity the expression collapses toexp ( − τ /T ). The result (1) assumes that the Einstein-Fick limit holds, implying that we can make the approx-imation to the position autocorrelation function (PAF), (cid:104) x ( t ) x (0) (cid:105) ≈ (cid:104) x ( t ) x ( t ) (cid:105) = 2 Dt , i.e., the PAF is approxi-mated by the mean-square displacement.According to the same conventional theory, the signalin the CPMG experiment, which features a train of n echoes (Fig. 1), decays according to : S ( τ ) = exp ( − n τ /T ) exp (cid:18) − γ g Dτ ( n τ ) (cid:19) , (2)where n τ is the total duration of the sequence, which isheld fixed. In practice, an upper bound on 2 nτ is imposedby the relaxation time of the sample. In the limit of shortecho spacing ( τ →
0, holding n τ constant) CPMG mini-mizes the effect of molecular self-diffusion on nuclear spindecoherence in inhomogeneous magnetic fields. Underthese conditions, the contribution of the inhomogeneousterm becomes negligible, recovering exp ( − n τ /T ).However, the expression (2) only holds for substanceswhose diffusional properties obey the Einstein-Fick limit,which mainly applies to liquids. Gases typically lie out-side the Einstein-Fick limit. We have derived in priorwork an expression for signal decay in gases : S ( τ ) = exp ( − τ /T ) exp (cid:18) − γ g κτ (cid:19) , (3)where κ is a term that depends on the PAF of diffusingmolecules. It depends, among other things, on temper-ature and viscosity. We note that the time dependenceis τ , not τ . This difference in exponents has impor-tant consequences. Namely, the application of a CPMGsequence with n echoes, S ( τ ) = exp ( − nτ /T ) exp (cid:18) − γ g κ (2 nτ ) (cid:19) . (4) a r X i v : . [ c ond - m a t . o t h e r] M a r no longer eliminates the second term describinginhomogeneous-field decay in the limit of short interpulsespacing τ → nτ is fixed). Thus, any measurement of T in a gas using a CPMG sequence will yield an apparent T value that is affected by diffusivity effects in the inho-mogeneous magnetic field. This could include, for exam-ple, unwanted weightings due to temperature, viscosity,external hardware, pulse sequence design, magnetic sus-ceptibility and pore geometry. This is in contrast to thecase of liquids, where diffusion effects can be mitigatedby extrapolation to extract the true (intrinsic) T time.Consider the CPMG experiment (Fig. 1) with a 90 ◦ broadband pulse and a series of 180 ◦ pulses to refo-cus the magnetization at intervals of 2 τ . The follow-ing phases were applied in the CPMG sequence: 90 ◦ x − τ − (180 ◦ y − τ − τ ) n , where the 180 ◦ pulse is repeated n times. For each τ value, the echo envelope was ac-quired in a single-shot experiment. An external mag-netic field gradient during the course of the experimentcreates an inhomogeneous magnetic field. For gas-phaseexperiments, a sealable 5-mm diameter J. Young NMRtube was filled with liquid and freeze-pump-thawed toevacuate excess air. The NMR tube was heated by theNMR spectrometer’s variable temperature unit until thetube was vaporized. For liquid-phase experiments, a so-lution of 0.5% weight/volume tetramethylsilane (TMS)in acetone- d was degassed and flame-sealed in an NMRtube. Measurements were performed on a 14.1 T verticalbore Bruker AV 600 MHz NMR spectrometer equippedwith a 5 mm broadband probe with a z -gradient. The re-ceiver was operated in qsim mode (forward Fourier trans-form, quadrature detection). The pulses were hard pulseswhose lengths were 16 µ s and 32 µ s for the 90 ◦ and 180 ◦ pulses, respectively. The size of the sensitive RF regionis less than 1 cm. All NMR signals were analyzed inmagnitude mode and decay functions included baselinesubtraction.We take an explicit look at the time-decay of theCPMG echo train, which the theory (c.f. Eq. 1 and3) predicts should exhibit fundamentally different be-havior ( t vs t , respectively). A direct verification isobtained by plotting the NMR signal in the CPMG ex-periment versus time along the echo train (see Fig. 2).The normalized NMR signal decay of the CPMG spinecho (with an interpulse spacing of 5 ms) for liquid-phaseTMS is plotted in Fig. 2a. The normalized NMR signaldecay of the CPMG spin echo (with an interpulse spac-ing of 5 ms) is plotted in Fig. 2b for gas-phase TMS.For gas, the normalized NMR signal decays exponen-tially, according to exp( − t/T ) exp( − t/b ), in agreementwith our revised theory of the NMR linewidth. We notethat neither alteration of the phase cycling scheme to90 ◦ x − (180 ◦ y − ◦− y ) n nor replacement of the 180 ◦ pulse with a 90 ◦ x / ◦ y / ◦ x composite pulse affected theresults.Measurements of the CPMG echo train signal decay asa function of interpulse spacing τ for TMS in the liquidphase are shown in Fig. 3a. Figure 3b shows the corre- S ( t ) no r m a li z ed N M R s i gna l time (s) no applied gradient0.01 G/cm0.05 G/cm S ( t ) no r m a li z ed N M R s i gna l time (s) no applied gradient0.01 G/cm0.05 G/cm (a) (b) gasliquid FIG. 2. Direct verification that signal decay in thepresence of a magnetic-field gradient follows a time depen-dence of the form exp( − t/T ) exp( − t/b ) for the gas andexp( − t/T ) exp( − ( t/b ) ) for the liquid (solid lines, fit; dots,data). Here we show sample CPMG decay curves for τ =5 msand g =0, 0.01, 0.05 and 0.5 G/cm. (a) Liquid-phase TMS( t ). Number of scans (ns) = 1. (b) Gas-phase TMS ( t ). ns= 8. In both liquid and gas cases, we scanned multiple acqui-sitions (ns = 1 and 8 for liquid and gas, respectively) and a T value with experimental error bars was derived for Figure3. The fits to the respective models are excellent. Fits to theconverse equation ( t ↔ t ) do not yield acceptable fits (notshown here). sponding experiment in the gas phase. TMS is a liquidat room temperature but a gas at 26 ◦ C; thus, a mod-est temperature increase enables comparison of the samesubstance in two different phases. TMS was also chosendue to its long relaxation times in both liquid and gasphases, enabling us to apply a large number of refocusingpulses even at long τ values. For liquids, as the interpulsespacing decreases the measured T value approaches asingle value irrespective of the applied gradient strength g , as if there were no external gradient (Fig. 3a,c). Thiscorresponds to the limit τ → T to the limit τ → T times in the presenceof magnetic-field inhomogeneities (from external or in-ternal fields). For gases, however, the T values in thelimit τ → g . This fundamentally differ-ent behavior implies that the inhomogeneous-field decayterm is still present, as predicted by Eq. 4.In this study we have confirmed the t dependence inthe NMR signal decay function of gases in the presenceof an external gradient (Eq. 3 and 4). In our priorwork we had verified the temperature dependence of thelinewidth . The verification of the t time dependencecan be considered the missing part of the puzzle whichnow unambiguously confirms the validity of the revisedlinewidth theory presented in Ref. . The g dependencehas already been verified in our previous paper .The fundamentally different motional averaging behav-ior of the NMR experiment in gases has important impli-cations for several experiments. This behavior has pre-viously led to the development of a novel non-invasivemethod for mapping temperatures of gases . Gas-phaseMRI experiments that utilize frequency-encoding gradi- ents could be affected; gradients during readout affectmeasurements of T or diffusion, introducing an appar-ent coupling between them. This means that a quanti-tative interpretation of these parameters would need toaccount for the non-Fickian nature of the diffusion. Dy-namic decoupling schemes such as the Uhrig sequence ,which aim at generating the longest possible coherencetimes, are also expected to break down in the case of gasesbecause short τ values no longer guarantee the elimina-tion of environmental factors. Finally, the interpretationof restricted diffusion results in porous media andother confined geometries may require new theoreticaldevelopments that model the signal decay in restrictedenvironments in light of the new theory of signal decay. ACKNOWLEDGMENTS
This work was partially funded by a Beckman YoungInvestigator Award and the National Science Foundationthrough grants CHE-1153159 and CHE-1508707. ∗ [email protected] N. N. Jarenwattananon and L.-S. Bouchard, Phys. Rev.Lett. , 197601 (2015). N. N. Jarenwattananon and L.-S. Bouchard, Phys. Rev.Lett. , 249702 (2016). C. P. Slichter,
Principles of Magnetic Resonance (Sl, 1990). E. L. Hahn, Phys. Rev. , 580 (1950). H. Y. Carr and E. M. Purcell, Phys. Rev. , 630 (1954). S. Meiboom and D. Gill, Rev. Sci. Instrum. , 688 (1958). N. N. Jarenwattananon, S. Gl¨oggler, T. Otto, A. Melko-nian, W. Morris, S. R. Burt, O. M. Yaghi, and L.-S.Bouchard, Nature , 537 (2013). G. Uhrig, Phys. Rev. Lett. , 100504 (2007). P. T. Callaghan,
Translational dynamics and magnetic res-onance: principles of pulsed gradient spin echo NMR (Ox-ford University Press, 2011). D. S. Grebenkov, Reviews of Modern Physics , 1077(2007). G. Q. Zhang and G. J. Hirasaki, Journal of Magnetic Res-onance , 81 (2003). L. J. Zielinski and M. D. H¨urlimann, Journal of MagneticResonance , 161 (2005). R. W. Mair, G. Wong, D. Hoffmann, M. D. H¨urlimann,S. Patz, L. M. Schwartz, and R. L. Walsworth, Physicalreview letters , 3324 (1999). M. Hurlimann, K. Helmer, T. Deswiet, and P. Sen, Journalof Magnetic Resonance, Series A , 260 (1995). T. M. de Swiet and P. N. Sen, The Journal of chemicalphysics , 5597 (1994). P. N. Sen, A. Andr´e, and S. Axelrod, The Journal ofchemical physics , 6548 (1999). P. Le Doussal and P. N. Sen, Phys. Rev. B , 3465 (1992). -2-10
23 -7 -6 -5 -4 -3 -2 l n T applied gradient strength in G/cm ln τ τ → τ ln τ l n T -6 -5 -4-3-2-10 -7 -6 -5 -4 -3 -2 l n T applied gradient strength in G/cm ln τ -3.3-2.3-1.3-0.3 -7 -6 ln τ l n T → τ -10123 0 1 2 3 4 l n T → τ liquid τ, echo spacing (ms) -3.75 -2.75-1.75 -0.75 l n T → τ gas (a) (b)(c) (d) τ, echo spacing (ms) FIG. 3. T relaxation time of tetramethylsilane (TMS) vs interpulse spacing τ under conditions of magnetic field inhomogeneity(field gradient g , in G/cm). (a) Liquid-phase T values approach the limit of no applied gradient as τ → T values shownrange from 50 ms to 20 s. τ values shown range from 1 ms to 100 ms. Inset: Expansion of red boxed region. (b) Gas-phase T values do not converge to a single value as τ → T values shown range from 18 ms to 1 s. τ values shown range from 1 msto 100 ms. Inset: Expansion of red boxed region in A. (c) Data from (a) plotted on a linear scale. The straight lines are linearextrapolations as τ →
0. The g values are the same as in (a). (d) Data from (b) plotted on a linear scale. The straight linesare linear extrapolations as τ →
0. The gg