Chemical Physics Letters 479 (2009) 165–170

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Chemical Physics Letters

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Harmonic peaks in 1D NMR spectra induced by radiation damping fields

Ling Peng a, Shuhui Cai a, Riqiang Fu b,*, Chaohui Ye c, Zhong Chen a,c,*

a Department of Physics and Fujian Key Laboratory of Plasma and Magnetic Resonance, State Key Laboratory of Physical Chemistry of Solid Surfaces, Xiamen University, Xiamen,Fujian 361005, Chinab Center for Interdisciplinary Magnetic Resonance, National High Magnetic Field Laboratory, 1800 E Paul Dirac Drive, Tallahassee, FL 32310, USAc State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, The Chinese Academy of Science, Wuhan 430071, China

a r t i c l e i n f o

Article history:Received 4 June 2009In final form 4 August 2009Available online 7 August 2009

0009-2614/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.cplett.2009.08.010

* Corresponding authors.E-mail addresses: rfu@magnet.fsu.edu (R. Fu), chen

a b s t r a c t

It is found experimentally that there exist some harmonic peaks in one-dimensional NMR spectra of solu-tion samples with two dominant solvents. Theoretical analysis indicates that such harmonic peaks resultfrom the interaction between two radiation damping fields induced by the dominant solvents. Using ahomo-nuclear decoupling pulse as a radiation damping field, a novel detection approach is proposedto investigate these harmonic peaks without interferences caused by the nonlinear amplification of thereceiver. The results presented herein give a new insight into the radiation damping effects in sampleswith solvent mixtures (e.g. natural products) at high fields.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

Since it was first forecasted by Suryan [1] and quantitativelystudied by Bloembergen and Pound [2], radiation damping effecthas been well documented in modern solution nuclear magneticresonance (NMR) [3–6]. Radiation damping is a dominant bulk sol-vent effect, which results from the large bulk magnetization of sol-vent. Radiation damping field is a macroscopic reaction field thatacts back on spins through the precessing magnetization inducedcurrent in the receiver coil, and is proportional to the transversemagnetization in intensity but lags 90� in phase. Stronger staticmagnetic fields and more sensitive probes (such as cryoprobes)help offset intrinsic limitations in signal sensitivity and spectralresolution, but at the same time intensify such concentrated sol-vent effects. In fact, a strong radiation damping field acts like aradio-frequency (RF) pulse [7], resulting in some unusual phenom-ena in 1D and 2D conventional NMR experiments [8,9]. Conse-quently, a simple experiment may become complex. To avoidthe interference from radiation damping fields, well-designedpulse sequences or improved experimental methods/equipmentsmust be utilized [10–13]. For example, radiation damping effectcan be counteracted in selective excitation experiments by cor-recting the offset based on the knowledge of radiation dampingtime constant and the ideal trajectory of magnetization in thecourse of selective excitation [14,15]. On the other hand, radiationdamping effect can be utilized to improve experimental results insome applications [16,17], such as spin excitation [18,19] andmagnetic resonance imaging (MRI) [20,21]. Detailed analysis with

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z@xmu.edu.cn (Z. Chen).

and without radiation damping effect has also been reported forintermolecular multiple-quantum coherences (iMQCs) [22,23].

So far, the radiation damping effect has been studied primarilyin samples with a single solvent. For samples having solvent mix-tures, harmonic peaks induced by the radiation damping effectwere observed in the indirect dimension of 2D COSY spectra [9],but they have not yet been reported in simple single-pulse 1Dexperiments. Since more and more solution NMR experimentsare dealing with solvent mixtures, such as LC–NMR in naturalproduct analysis, where binary-solvent systems (e.g. water–MeCNand water–methanol) are used [24], there is an imminent need toknow how the radiation damping effects from solvent mixturesaffect 1D NMR spectra in order to interpret the spectra accurately.In this Letter, we consider a sample consisting of two solvents, eachof which is an isolated spin system and generates a strong radia-tion damping field. Our experimental and simulated results indi-cate that the two isolated spin systems induce some harmonicpeaks in 1D single-pulse NMR spectra when the radiation dampingtime is short.

2. Numerical simulation

The radiation damping effect was simulated using the theoret-ical formalism called ‘modified Bloch–Redfield equations’ [25,26].A sample with two isolated spin systems was studied. The longitu-dinal relaxation, transverse relaxation, self-diffusion and distantdipolar field (DDF) effects were neglected. For such a sample, theevolutions of these two spin systems are nonlinear under strongradiation damping effect. The status of either system at any mo-ment depends on its previous status. Our simulations were donewith a fifth-order Cash–Karp Runge–Kutta formulism. The step size

166 L. Peng et al. / Chemical Physics Letters 479 (2009) 165–170

during the evolution was selected based on the balance betweencalculation speed and numerical accuracy. For a high numericalaccuracy and avoiding the emanation of magnetization trajectory,a short step size of 0.01 ms was used. The two precessing frequen-cies were set to 100 Hz and �100 Hz in the rotating frame. Thesmall precessing frequencies help to enhance numerical accuracyonce the step size is determined [27,28].

When the radiation damping time was set to a very small value,some harmonic peaks were found besides the main peaks from thetwo components in the 1D single-pulse NMR spectrum (c.f. Fig. 1a).The frequency offset of each harmonic peak from its adjacent mainpeak is equal to the frequency difference between the two mainpeaks. The intensities of the harmonic peaks are related to the radi-ation damping time. As shown in Fig. 1b, the harmonic peaks be-come stronger as the radiation damping time decreases. Thusthese harmonic peaks appear to be induced by the strong radiationdamping effect.

3. Theoretical formalism and magnetization vector model

In order to describe the NMR signals at time, tfinal, the Bloch–Redfield equations must be integrated from t ¼ 0 to tfinal. For ahighly concentrated sample, the radiation damping effect is a dom-inant bulk solvent effect and the radiation damping time can bevery short under certain experimental conditions. Thus for simplic-ity, after the hard pulse is applied, a very short tfinal is chosen sothat the longitudinal relaxation, transverse relaxation, and self-dif-fusion effects can be conveniently neglected. In addition, we as-sume there is no DDF effect. With these approximations, we onlyneed to consider the radiation damping effect and chemical shifts.

Fig. 1. (a) Simulated 1H NMR spectrum of a sample with two isolated spin systemshaving equal magnetizations after a hard pulse with 150� flip-angle. The radiationdamping time was set to 6 ms. Harmonic peaks appear at two positions. (b)intensity variation of the simulated harmonic peaks vs radiation damping time.

The motion equation of the magnetization vector MðiÞðr; tÞ can thusbe simplified to:

dMðiÞðr; tÞdt

¼ cMðiÞðr; tÞ � xi

czþ

XBðjÞr ðr; tÞ

� �; ð1Þ

where the superscript i represents the i th type of spins, c is themagnetogyric ratio, xi is the processing frequency of spin systemi, BðjÞr ðr; tÞ is the radiation damping field given by:

BðjÞr ðr; tÞ ¼ �hMðjÞ

y ðtÞicMðjÞ

0 sðjÞr

xþ hMðjÞx ðtÞi

cMðjÞ0 sðjÞr

y; ð2Þ

where sðjÞr ¼ 1=ð2pgMðjÞ0 QcÞ is the characteristic radiation damping

time, g is the filling factor, Q is the probe’s Q-factor, and hMðjÞx ðtÞi

and hMðjÞy ðtÞi are average magnetizations in the transverse plane.

According to Eq. (2), the radiation damping field after a hardpulse is given by:

Brðr; tÞ ¼ �hMð1Þ

y ðtÞicMð1Þ

0 sð1Þr

þhMð2Þ

y ðtÞicMð2Þ

0 sð2Þr

" #x

þ hMð1Þx ðtÞi

cMð1Þ0 sð1Þr

þ hMð2Þx ðtÞi

cMð2Þ0 sð2Þr

" #y; ð3Þ

where the superscripts ð1Þ and ð2Þ represent the two distinct iso-lated spin systems. Each spin system experiences two radiationdamping fields stemming from the two isolated spin systems.

After a hard pulse with a flip-angle h0 along the x direction(Fig. 2a), the two spin systems begin to interact with each otherthrough the radiation damping fields. Their evolution model isillustrated in Fig. 2c.

In a homogeneous magnetic field, the magnetizations are posi-tion-independent. Within a short time Dt immediately after thehard pulse, the magnetizations can be described as:

MðiÞx ðtÞ ¼ MðiÞ

0 sin h0 sinðxitÞMðiÞ

y ðtÞ ¼ MðiÞ0 sin h0 cosðxitÞ

MðiÞz ðtÞ ¼ MðiÞ

0 cos h0

8>><>>: ; i ¼ 1 or 2: ð4Þ

Since the two spin magnetizations from the two isolated spinsystems can be written separately and they do not rotate back totheir equilibrium states in such a short period, the radiation damp-ing field induced by each spin system can be approximately takenas only a function of its spin magnetization. Thus, with theassumption of sð1Þr ¼ sð2Þr ¼ sr , the radiation damping field of Eq.(3) can be separated into two independent parts:

BðiÞr ðtÞ ¼ �MðiÞ

y ðtÞcMðiÞ

0 sr

xþ MðiÞx ðtÞ

cMðiÞ0 sr

y; i ¼ 1 or 2: ð5Þ

As shown in Fig. 1a, the relative position of the harmonic peaksis only related to the frequency difference of the two spin systems.From Eq. (1), the evolution of the magnetization of the spin system(1) under the influence of the radiation damping field introducedby the spin system (2) is given by:

dMð1ÞðtÞdt

¼ cMð1ÞðtÞ � �Mð2Þ

y ðtÞcMð2Þ

0 sr

xþ Mð2Þx ðtÞ

cMð2Þ0 sr

y

( ): ð6Þ

Introducing Eq. (4) to Eq. (6), we have:

Mð1Þx ðtÞ ¼

Mð1Þ0

2x2srsin 2h0 cosðx2tÞ

Mð1Þy ðtÞ ¼ �

Mð1Þ0

2x2srsin 2h0 sinðx2tÞ

Mð1Þz ðtÞ ¼

Mð1Þ0 sin2 h0

srðx2 �x1Þsinðx2 �x1Þt:

8>>>>>>>><>>>>>>>>:

ð7Þ

Fig. 3. Vector model illustrating different effects of the radiation damping fields.The transverse magnetization of the spin system (1) keeps static in the rotatingframe. (a) The angle between the transverse magnetizations of the two spin systemsis acute. The radiation damping field from the spin system (2) is in the grey area,thus the magnetization of the spin system (1) will be rotated back to its equilibriumdirection; and (b) the angle is obtuse. The magnetization of the spin system (1) willbe rotated to the opposite direction.

Fig. 4. Visual model illustrating the effect of radiation damping field on different-frequency signal. Assume the spin system (1) has zero precessing frequency in therotating frame. The radiation damping field from the spin system (1) is along the �xdirection after a hard pulse and rotates the different-frequency signal (Sdf ) to the xyplane.

Fig. 2. (a) Pulse sequence used to directly detect the harmonic peaks in simulationsand experiments; (b) pulse sequence for detecting the harmonic peaks using ahomo-nuclear decoupling field during acquisition as one of the radiation dampingfields. A hard pulse with 150� flip-angle was used in simulations and experiments;and (c) visual model illustrating the magnetization precession after a hard pulse.The precessing frequencies of the blue and red magnetizations are x1 and x2,respectively, and x1 > x2 in the rotating frame.

L. Peng et al. / Chemical Physics Letters 479 (2009) 165–170 167

Clearly, Mð1Þ� ðtÞ has the precessing frequency x2 of the spin sys-

tem (2), while Mð1Þz ðtÞ contains the difference of the precessing fre-

quencies of the spin systems (1) and (2). This can be understood asfollows. After a hard pulse, the magnetization vectors of the twospin systems precess with their own frequencies. The spin system(1) experiences two radiation damping fields, one from itself withthe tendency of rotating its magnetization vector back to the equi-librium position, and the other from the spin system (2) whichgoes around the magnetization vector of the spin system (1) andis not always perpendicular to its transverse magnetization. Sincethe radiation damping field lags 90� in phase to the transversemagnetization it comes from, the magnetization of the spin system(1) will be rotated to its equilibrium position by the radiationdamping field from the spin system (2) when the angle betweenthe transverse magnetizations of these two spin systems is smallerthan 90� (Fig. 3a). On the other hand, when the angle is larger than90� (Fig. 3b), the magnetization will be rotated to the directionopposite to equilibrium position. As a result, a signal emerges inthe z direction, and its frequency is equal to the precessing fre-quency difference of these two spin systems. Apparently, the effectof the radiation damping field induced by the spin system (2) onthe spin system (1) results in a different-frequency signal position-ing at the frequency difference between the two spin systems. Suchan effect gets larger when the frequency difference between the

two spin systems is smaller. Similarly, the effect of the radiationdamping field induced by the spin system (1) on the spin system(2) also results in a different-frequency signal positioning at thefrequency difference between the two spin systems.

Now, if we suppose the precessing frequency of the spin system(1) is 0 in the rotating frame, under the influence of the radiationdamping field from the spin system (1), the radiation damping fieldinduced signal will be partly rotated into the xy plane, as illustratedin Fig. 4.

The related magnetization M(h) (i.e. harmonic signals) in the ydirection can be expressed as:

MðhÞy ðtÞ ¼ �a

Mð1Þ0 sin2 h0

srx2sinx2t; ð8Þ

where a represents the fraction of the magnetization projected ontothe xy plane and is proportional to the amplitude of the radiationdamping field from the spin system (1). Eq. (8) can be rewrittenas follows:

MðhÞy ðtÞ ¼ i � a Mð1Þ

0 sin2 h0

2srx2ðeix2t � e�ix2tÞ: ð9Þ

168 L. Peng et al. / Chemical Physics Letters 479 (2009) 165–170

Obviously, MðhÞy ðtÞ describes a pair of harmonic peaks on the left

and right sides of the main peak of the spin system (1), as shown inFig. 1a. (In practice, only one harmonic peak is observed, the otherone is submerged in the main peak of the spin system (2).) Whenthe precessing goes on, the magnetization rotates back to its equi-librium position gradually, but the expression of harmonic signals(Eq. (9)) still holds. The evolution of the spin system (2) can bededuced accordingly.

4. Experimental observation

Based on the simulated results and theoretical analysis, we triedto detect the harmonic signals from some highly concentratedsamples mixed with two solvents such as water–acetone or ace-tone–DMSO solution on a Varian NMR System 500 MHz spectrom-eter with an indirect detection (ID) probe. Fig. 5a shows the 1D 1HNMR spectrum of the water–acetone solution. Obviously, besidesthe two strong resonances at ±612 Hz from water and acetone,there exist two small additional peaks at ±1836 Hz. Although thesesmall peaks appear at the same positions as expected for the har-monic peaks induced by the radiation damping fields, they do notpossess the same phases as the simulated harmonic signals (c.f.Fig. 1a). It is worth noting that the radiation damping fields underour experimental conditions are much weaker than the ones usedin the simulations, as indicated by the linewidth of the main peaksand the rapid decay of the FID. When the probe was optimallytuned, the linewidth of water or acetone peak was about 15 Hz(Fig. 5a), much smaller than 40 Hz used in the simulations whenthe harmonic peaks could be observed (c.f. Fig. 1a). The intensityof the harmonic peaks would thus be very weak, as illustrated inFig. 1b.

Since the two main signals were so strong, we had to use theminimal receiver gain to detect the signals. However, such strong

Fig. 5. (a) 1H NMR experimental spectrum of a sample of 40% H2O, 40% acetone, and 20%as (a) but a 10 dB attenuator was utilized before signal was amplified; (c) and (d) areexperimental results shown in (a) and (b), respectively. The radiation damping time wa

signals could have reached beyond the linear amplification regionand gone into the nonlinear (or saturation) region of the receiveramplifiers, although there was no indication in our experimentsthat these signals overflowed the receiver. Fig. 5c shows the simu-lated spectrum with the FID nonlinearly amplified. Surprisingly,the phases of the small additional peaks were the same as theexperimentally observed ones (c.f. Fig. 5a), implying that the non-linear amplification of the FID could bring unwanted peaks into thespectra. On the other hand, when the FID is linearly amplified inthe simulation, there indeed exist the harmonic peaks induced bythe radiation damping fields, as shown in Fig. 5d. Since the radia-tion damping fields are not strong enough under our experimentalconditions, the harmonic peaks are much smaller in intensity thanthe nonlinear amplification induced artifacts. Unfortunately, thetwo unwanted peaks induced by the nonlinear amplification ap-pear at the same positions as the harmonic peaks, thus obscuringthe observation of the harmonic peaks.

To experimentally verify such weak harmonic signals withoutthe influence of nonlinear amplification, we need to attenuate theintensities of the strong signals so that the signals remain in thelinear amplification region of the receiver amplifiers. Fig. 5b showsthe 1D NMR spectrum of the water–acetone solution obtained witha 10 dB attenuator between the preamplifier and the receiver to re-duce the FID signals going into the receiver amplifiers. Clearly, thevery weak harmonic peaks can now be observed at the expectedpositions and are consistent with the simulated one (Fig. 5d).When the 10 dB attenuator was added before the preamplifier,the same harmonic peaks were observed, indicating that the pre-amplifier has a good linearity, while the amplifiers in the receiverare the main cause for the nonlinear amplification.

Since the harmonic peaks are induced by the radiation dampingfields, it is expected that at higher magnetic fields and with the useof more sensitive probes such as cryoprobes, the radiation

D2O after a 150� flip-angle pulse when the probe was optimally tuned; (b) the samesimulated spectra with nonlinear and linearly amplification, corresponding to thes set to 20 ms.

L. Peng et al. / Chemical Physics Letters 479 (2009) 165–170 169

damping fields would be much stronger, leading to stronger har-monic peaks. However, direct experimental observation of the har-monic peaks appears to be difficult because of nonlinearamplification. Such nonlinear amplification was found in all NMRspectrometers available to us. So we designed a novel method todetect the weak harmonic peaks induced by the radiation dampingfields.

As mentioned above, the harmonic peaks are brought by thetwo radiation damping fields from two distinct isolated spinsystems. These radiation damping fields can hardly be further en-hanced for any given sample once the probe is optimally tuned.However, since the radiation damping field plays a role as a RFpulse [6], we could use a continuous-wave pulse with a fixed fre-quency and a changeable intensity, e.g. a homo-nuclear decouplingpulse, to act as one of the radiation damping fields after the hardpulse is applied [29,30], as shown in Fig. 2b. This allow us to adjustits intensity at will, as if changing the amplitude of the radiationdamping field, and to use a sample containing only one spin sys-tem with a medium radiation damping effect. In order to avoidthe nonlinear amplification of the FID signal, the sample was di-luted by corresponding deuterated reagent. In this case, the realradiation damping field induced by the sample magnetization lags90� in phase to the transverse magnetization and its intensity isproportional to the transverse magnetization, while the artificialradiation damping field (i.e. the decoupling field) goes aroundthe real one at their different-frequency with a changeableintensity.

The experimental result from a water sample shows that be-sides the water signal at 550 Hz, there are three peaks in the spec-trum (c.f. Fig. 6a). One is at the position of homo-nucleardecoupling frequency (300 Hz). Its intensity can be controlled bythe RF amplitude of the homo-nuclear decoupling pulse. Theappearance of this signal is because the magnetization is partlymodulated by the strong homo-nuclear decoupling pulse when itreturns to its equilibrium position. The other two small signals,as indicated by arrows, are the harmonic peaks brought by thejoint action of the radiation damping field induced by water mag-netization and the applied homo-nuclear decoupling field. Becausethe artificial radiation damping field is much stronger than the realone, the harmonic peak in the right side of the decoupling carrierfrequency is stronger than the one in the left side of the water sig-

Fig. 6. (a) 1H NMR spectrum recorded using the pulse sequence in Fig. 2b. A sampleof 30% H2O and 70% D2O was used and the probe was optimally tuned; (b) thesimulated spectrum. The harmonic peaks are indicated by the arrows.

nal. The simulated result is consistent with the experimental one(c.f. Fig. 6b). As pointed out by Warren and co-workers [31], theirradiations above resonance and below resonance gave very dif-ferent and complex dynamics, resulting in spin magnetizationspiraled to a fixed point based on the collective effects of the radi-ation damping field, continuous irradiation field, and the chemicalshift. Furthermore, the harmonic peaks become stronger eitherwith the increase of the RF amplitude of the homo-nuclear decou-pling field or with the decrease of the frequency difference be-tween the water signal and the decoupling carrier, as predictedby Eqs. (7) and (9). Thus, by using the homo-nuclear decouplingfield, we can observe the harmonic peaks directly. More impor-tantly, this new detection method permits us to investigate howa radiation damping field interacts with another one at differentamplitudes and offsets without changing sample conditions.

5. Conclusions

We have demonstrated for the first time that harmonic peaksexist in single-pulse 1D solution NMR spectra of samples withtwo dominant solvents. Theoretical formula and magnetizationvector model were derived. The results indicate that these har-monic peaks come from the interaction between two radiationdamping fields induced by the large bulk magnetizations of the sol-vents. The harmonic peaks appear in the positions away from bothsides of the two solvent resonances by the frequency difference ofthe two main resonances, while their intensities strongly dependon the radiation damping fields. Due to the nonlinear amplifica-tions commonly existing in NMR consoles, the harmonic peakstend to be difficult to observe in experiments performed at rela-tively low magnetic field strength. A novel detection method wasproposed to convincingly observe the harmonic peaks. Since thestatic magnetic field strength and probe sensitivity have dramati-cally increased with the technical development of modern NMRspectrometers [32], radiation damping phenomena emerge inmore and more solution NMR experiments, bringing some newfeatures into NMR spectra, especially for samples having solventmixtures such as natural products and complex materials. Theexperimental observations and theoretical analyses presentedherein give a new insight into the radiation damping effect in sam-ples with solvent mixtures and will provide a foundation to under-stand the radiation damping effect in multi-dimensional NMRexperiments as well as to design new experiments utilizing theradiation damping effect.

Acknowledgments

This work was partially supported by the NNSF of China underGrants 10774125 and 10875101, the National Key Technology R&DProgram of China under Grant 2006BAK03A22, the Foundation ofState Key Laboratory of Magnetic Resonance and Atomic andMolecular Physics, and the Research Fund for the Doctoral Programof Higher Education of China under Grant 200803840019. Espe-cially thanks for Prof. J. Jeener to provide an electronic copy ofhis Encyclopedia paper.

References

[1] G. Suryan, Curr. Sci. 6 (1949) 203.[2] N. Bloembergen, R.V. Pound, Phys. Rev. 95 (1954) 8.[3] M.P. Augustine, E.L. Hahn, Concepts Magn. Reson. 13 (2001) 1.[4] X.A. Mao, J.X. Guo, C.H. Ye, Chem. Phys. Lett. 218 (1994) 249.[5] V.V. Krishnan, Curr. Sci. 74 (1998) 1049.[6] J. Jeener, in: D.M. Grant, R.K. Harris (Eds.), Encyclopedia of Nuclear Magnetic

Resonance, vol. 9, Wiley, New York, 2002, p. 642.[7] J.H. Chen, A. Jerschow, G. Bodenhausen, Chem. Phys. Lett. 308 (1999) 397.[8] J.H. Chen, X.A. Mao, C.H. Ye, J. Magn. Reson. 124 (1997) 490.[9] S. Ahn, S. Lee, W.S. Warren, Mol. Phys. 95 (1998) 769.

170 L. Peng et al. / Chemical Physics Letters 479 (2009) 165–170

[10] W.S. Price, M. Walchli, Magn. Reson. Chem. 40 (2002) S128.[11] P.S.C. Wu, G. Otting, J. Biomol. NMR 32 (2005) 243.[12] M.A. Connell, A.L. Davis, A.M. Kenwright, G.A. Morris, Anal. Bioanal. Chem. 378

(2004) 1568.[13] T.R. Eykyn, G.S. Payne, M.O. Leach, Phys. Med. Biol. 50 (2005) N371.[14] B. Cutting, J.H. Chen, D. Moskau, G. Bodenhausen, J. Biomol. NMR 17 (2000)

323.[15] J.H. Chen, B. Cutting, G. Bodenhausen, J. Chem. Phys. 112 (2000) 6511.[16] V.V. Krishnan, K.H. Thornton, M. Cosman, Chem. Phys. Lett. 302 (1999) 317.[17] L.J. Alain, D. Abergel, I. Lebars, J.Y. Lallemand, Chem. Phys. Lett. 337 (2001) 92.[18] J.D. Walls, S.Y. Huang, Y.Y. Lin, J. Phys. Chem. B 110 (2006) 19985.[19] J.D. Walls, S.Y. Huang, Y.Y. Lin, J. Chem. Phys. 127 (2007) 054507.[20] S. Datta, S.Y. Huang, Y.Y. Lin, J. Phys. Chem. B 110 (2006) 22071.[21] S.Y. Huang et al., Magn. Reson. Med. 56 (2006) 776.

[22] S. Kirsch, W.E. Hull, J. Chem. Phys. 129 (2008) 044505.[23] S. Kirsch, P. Bachert, J. Magn. Reson. 185 (2007) 183.[24] V. Exarchou, M. Krucker, T.A. van Beek, J. Vervoort, I.P. Gerothanassis, K. Albert,

Magn. Reson. Chem. 43 (2005) 681.[25] B.W. Zheng, Z. Chen, S.H. Cai, J.H. Zhong, C.H. Ye, J. Chem. Phys. 123 (2005)

074317.[26] L. Peng, S.H. Cai, Z. Chen, Physica B 396 (2007) 57.[27] C.B. Cai, Z. Chen, S.H. Cai, J.H. Zhong, J. Magn. Reson. 172 (2005) 242.[28] C.B. Cai, Z. Chen, S.H. Cai, L.P. Hwang, J.H. Zhong, Chem. Phys. Lett. 407 (2005)

438.[29] R. Freeman, E. Kupce, NMR Biomed, 10 (1997) 372.[30] D.M. Freeman, R. Hurd, NMR Biomed. 10 (1997) 381.[31] W.S. Warren, S.L. Hammes, J.L. Bates, J. Chem. Phys. 91 (1989) 5895.[32] D.J.Y. Marion, H. Desvaux, J. Magn. Reson. 193 (2008) 153.