Chemical Physics 351 (2008) 33–36

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Chemical Physics

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Double quantum CRAZED NMR signal in inhomogeneous fields

Bin Jiang a,b, Huili Liu a,b, Maili Liu a,*, Chaohui Ye a, Xian Mao a,c,*

a Wuhan Center for Magnetic Resonance, State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics,Chinese Academy of Sciences, Wuhan 430071, Chinab Graduate School of Chinese Academy of Sciences, Beijing 100049, Chinac Department of Physiology and Biophysics, Case Western Reserve University School of Medicine, Cleveland, OH 44106, USA

a r t i c l e i n f o

Article history:Received 19 June 2007Accepted 28 March 2008Available online 3 April 2008

Keywords:NMRCRAZED experimentDouble quantumEchoInhomogeneous fields

0301-0104/$ - see front matter � 2008 Elsevier B.V. Adoi:10.1016/j.chemphys.2008.03.027

* Corresponding authors. Address: Department ofCase Western Reserve University School of MedicineFax: +1 216 368 3952 (X.A. Mao); fax: +86 27 8719 9

E-mail addresses: ml.liu@wipm.ac.cn (M.L. Liu), xia

a b s t r a c t

It has been well accepted that the double quantum (DQ) correlated-spectroscopy revamped by asymmet-ric z-gradient echo detection (CRAZED) signal is enveloped in the profile function t2 exp[�(t2 + 2t1)/T2],but this function is too simple to describe the spin echo characteristics of the CRAZED free inductiondecay signal. In this paper the DQ CRAZED experiment is analyzed by including the homogeneous andinhomogeneous broadening effects, and a formula for the time domain DQ CRAZED signal is obtained.This formula includes the chemical shift echo and the inhomogeneous echo, both appearing at t2 = 2t1.Experiments have confirmed the theory.

� 2008 Elsevier B.V. All rights reserved.

1. Introduction

In recent years the correlated-spectroscopy revamped by asym-metric z-gradient echo detection (CRAZED) experiment [1,2] hasattracted much attention. CRAZED is the modification of the pulsedfield gradient enhanced correlation spectroscopy (PFG-COSY)experiment whose symmetric 1:1 gradient pulse pair has been re-placed by the asymmetric 1:n (n = 0,2) gradient pulse pair. Withthe CRAZED experiment, correlation peaks between chemicallyuncorrelated species (like chemicals in separate glass tubes) andhigh-resolution spectra from inhomogeneous fields can be de-tected [1–4]. Applications of CRAZED are being expanded in vari-ous fields of nuclear magnetic resonance (NMR). Because theCRAZED signal intensity has a unique M0 (the magnetization atthermal equilibrium) square dependence, CRAZED magnetic reso-nance imaging (MRI) has shown a different contrast from normalMRI [5–7] and would be very important in MRI studies. In themeantime, relaxation and diffusion measurements based on theCRAZED experiment have also been proposed [8–11].

The double quantum (DQ) CRAZED signal is known to have thesimple form in Eq. (1), when the longitudinal relaxation and themolecular diffusion are neglected and when the condition of aweak demagnetization field (cl0M0t2� 1) is satisfied [9,12]

sðt1; t2Þ ¼ K expð�i2x0t1 � 2t1=T2Þ½t2 expðix0t2 � t2=T2Þ� ð1Þ

ll rights reserved.

Physiology and Biophysics,, Cleveland, OH 44106, USA.101 (M.L. Liu).n.mao@case.edu (X.A. Mao).

where K is a constant when the pulse flip angles are fixed. The re-corded free induction decay (FID) of the DQ CRAZED signal is envel-oped in the profile function t2 exp(�t2/T2), which is known tobehave like an echo [1]: when t2 is short, the FID grows up becausethe function is proportional to t2. When t2 is long, the FID decaysdue to the transverse relaxation. However, this description is toosimple to depict the spin echo characteristics of the DQ CRAZED sig-nal in fairly inhomogeneous fields, which was documented as earlyas 1996 [13], but so far no equation has been available for the echo.As most CRAZED experiments are performed in inhomogeneousfields, to quantitatively describe the echo is of great importance.In this study, we have found that the DQ CRAZED signal is best de-scribed by

sðt2; t1Þ ¼ Kt2 exp½ix0ðt2 � 2t1Þ� exp½�ðt2 þ 2t1Þ=T2�

� 1p

Z 1

�1exp½ix0ðt2 � 2t1Þ�gðx0Þdx0 ð2Þ

where g(x0) is the inhomogeneous distribution function. Eq. (2) con-tains not only the chemical shift echo exp[ix0(t2 � 2t1)] and thehomogeneous relaxation exp[�(t2 + 2t1)/T2], which are seen in Eq.(1), but also the inhomogeneity echo which is expressed as the inte-gration in the frequency space. The inhomogeneous echo can beeasily observed in experiments.

The CRAZED experiment in inhomogeneous fields has becomethe subject of a number of papers [3,4], but the inhomogeneous ef-fect has not been quantitatively described. Therefore, Eq. (2) is use-ful in the CRAZED studies, particularly when the inhomogeneouseffect is important.

34 B. Jiang et al. / Chemical Physics 351 (2008) 33–36

2. Methods and materials

The sample used in this study was the doped H2O (1% v/v) inD2O with the presence of 0.1 g/L GdCl3 which serves as relaxationreagent. The double quantum CRAZED experiment shown in Fig. 1was performed at 25 �C on a Bruker AVANCE 600 spectrometerwith the cryoprobe with z-gradient under homogeneous and inho-mogeneous conditions. The intrinsic relaxation time of the dopedwater proton was 0.166 s, as measured by the saturation-recoverymethod. The gradient pulse duration d was 1 ms, and the gradientstrength G was 6 Gauss/cm.

Fig. 2. Double quantum CRAZED echoes from a sample of doped H2O in D2O withvaried t1: (A) 10 ms, (B) 40 ms and (C) 80 ms. All echoes appeared precisely att2 = 2t1. The experiments were conducted in an inhomogeneous field with an eff-ective T�2 ¼ 9 ms (line with 35 Hz), while the homogeneous T2 was 166 ms (contr-ibuting 1.9 Hz to the total line width). The water resonance was purposely moved to�200 Hz off the carrier so that the echo shape was better seen.

Fig. 3. The same experiments as Fig. 2 (t1 = 10, 40 and 80 ms for A, B and C, resp-

3. Results and discussion

3.1. Double quantum CRAZED echo in inhomogeneous fields

The DQ CRAZED signal in the time domain is in fact a DQ echo,as has been directly compared to simple spin echoes [14]. The sec-ond gradient pulse, whose area is twice of the first one, refocusesthe dephasing caused by the first one. This effect has been wellknown since the CRAZED experiments was proposed [1,2] and evensince the demagnetization field effect in NMR was first discovered[15]. However, just like the conventional echoes, the DQ CRAZEDecho can also refocus the inhomogeneous effect at the same timethe chemical shift effect is refocused. According to the nonlinearityof the echo phenomena [16], all two-pulse experiments are inher-ently associated with echoes that are capable of refocusing theinhomogeneous effect. The ability of the CRAZED pulse sequencerefocusing the chemical shifts has been shown in Eq. (1), wherethe chemical shift echo occurs at t2 = 2t1. The refocusing of theinhomogeneous effects, however, is not included in this wellknown equation.

Before the theoretical analysis of the DQ inhomogeneous echo isgiven in the following section, here a series of DQ CRAZED echoeswith different t1 recorded on a Bruker Avance 600 spectrometerwith a cryoprobe are presented in Fig. 2. The inhomogeneous effecthad been deliberately enhanced by deshimming the magnet so thatthe line width of the doped water signal in the conventional single-pulse experiment was 35 Hz. Compared with the contribution ofthe natural (homogeneous) relaxation (1.9 Hz as determined inrelaxation measurements) to the line width, the inhomogeneouscontribution was overwhelming. In this case, the inhomogeneouseffect on the DQ CRAZED echoes was particularly obvious. All ech-oes precisely appeared at t2 = 2t1, although the echo looked asym-metric when the evolution time was short (see the top echo in thefigure).

In order to see how the inhomogeneity affects the shape and theposition of the CRAZED echo, the same experiments were per-formed under improved homogeneous conditions withT�2 ¼ 78 ms and the echoes are shown in Fig. 3. The echo shape isnot sharp at all and the exact echo position is hard to determine.

t1RF

PFG

y y

Fig. 1. Double quantum CRAZED pulse sequences used in this study, which consistsof two (90�)y pulses and a pair of gradient pulses with the area ratio = 1:2.

ectively) but with improved homogeneity (T�2 ¼ 78 ms, T2 = 166 ms).

In this case the function t2 exp(�t2/T2) apparently plays a moreimportant role than the inhomogeneous contribution.

3.2. Formalism of the CRAZED echo by including the inhomogeneouseffect

To understand the DQ CRAZED echo in inhomogeneous fields, itis necessary to analyze the DQ CRAZED experiment. It is knownthat in the CRAZED experiment the demagnetization effect occursonly after the second pulse and that the demagnetization fieldforces the precessing transverse magnetization to be multipliedby the longitudinal magnetization [1,2,17,18]. Although the

B. Jiang et al. / Chemical Physics 351 (2008) 33–36 35

CRAZED experiment has been formally interpreted by the intermo-lecular multiple quantum coherence (iMQC) theory [1–3], it canalso be successfully interpreted by the classical demagnetizationfield effect theory as demonstrated early by Deville et al. [15],and later by Bowtell et al. [19], Augustine and Zilm [17], Levitt[18], and Kimmich and Ardelean [20]. Now it is known that theCRAZED signals result from the interactions between the trans-verse and longitudinal magnetizations. Here we follow the trans-verse–longitudinal magnetization interaction model, which iswithin the framework of the mature NMR theory [21], to analyzethe original DQ CRAZED experiment with the pulse sequence inFig. 1 in inhomogeneous field on the simple doped (1%) water sys-tem, where the radiation damping effects can be completelyignored. In the analysis we assume that the gradient effect is sub-stantially stronger than the inhomogeneous effect so that the co-sine spatially modulation on the magnetization is not seriouslyaffected [15]. Since in the experiment only 90� pulses are used,after the second 90� pulse and before the demagnetization fieldis effective, Mxðt1Þ ¼ 0 and the water magnetization consists ofonly two components, i.e., the y- and z-components (denoted inthe following by Mð1Þ

y , and Mð1Þz , respectively), all having been de-

phased by the gradient pulse G and the background inhomogeneityB0 (for the moment the homogeneous relaxation is not considered)

Mð1Þy ðt1Þ ¼M0 sinðx0t1 þ cGzdþ cB0t1Þ ð3aÞ

Mð1Þz ðt1Þ ¼ �M0 cosðx0t1 þ cGzdþ cB0t1Þ ð3bÞ

where d is the duration of the gradient pulse (see Section 2). InEq. (3), Mð1Þ

y is a typical single quantum coherence during t1 andMð1Þ

z is the source of the demagnetization field. In the detectionperiod the demagnetization field becomes effective, which is

Bdðt1Þ ¼ ð1=3Þl0Mð1Þz ðt1Þ

¼ �ð1=3Þl0M0 cosðx0t1 þ cGzdþ cB0t1Þ ð4Þ

where the factor 1/3 accounts for the Lorentzian cavity effect [22].Thus, the detected DQ CRAZED signal is modulated in amplitudeby the evolution during t1 and modulated in frequency by thedemagnetization field Bd that is also a function of t1

sðt2; t1Þ ¼ Mð1Þy ðt1Þ expðix0t2 þ ic2Gzdþ icB0t2Þ expðicBdt2Þ ð5Þ

Since the area of the second gradient pulse is twice as much as thefirst one, a factor 2 appears in Eq. (5). For convenience, we define

q ¼� ð1=3Þcl0M0t2; ð6Þh ¼x0t1 þ cGzdþ cB0t1; ð7Þ

then we have

expðicBdt2Þ ¼ expðiq cos hÞ: ð8Þ

By means of the Jacobi–Anger expansion [23], Eq. (8) can be decom-posed to

expðiq cos hÞ ¼Xk¼1

k¼�1ikJkðqÞ expðikhÞ ð9Þ

where Jk(q) is the first-kind Bessel function. By substitution of Eq.(9), Eq. (5) becomes

sðt2; t1Þ ¼M0

2ifexp½ix0ðt2 þ t1Þ þ ic3Gzdþ icB0ðt2 þ t1Þ�

� exp½ix0ðt2 � t1Þ þ icGzdþ icB0ðt2 � t1Þ�g

�Xk¼1

k¼�1ikJkðqÞ expðikx0t1 þ ikcGzdþ ikcB0t1Þ ð10Þ

It is clear that there are two components (with k = �1 and k = �3)that are free of the effect of the applied gradients, but only the com-ponent with k = �1 leads to the echo

sðt2; t1Þ ¼ �M0

2J�1ðqÞ exp½iðx0 þ cB0Þðt2 � 2t1Þ� ð11Þ

So far no approximation has been used except for the presump-tion that the inhomogeneous effect is much weaker than the gradi-ent effect. In experiments, the gradient strength was 6 Gauss/cmwhich for 2 cm effective sample length amounts for 50000 Hz,while the worst inhomogeneous line width in this study was35 Hz. Hence there is no question about the presumption. In thiscase Eq. (11) can be regarded as the exact solution for the doublequantum echo acquired using the 1:2 gradients. When jqj � 1 (thiscondition is generally termed as weak demagnetization fieldapproximation or small angle approximation, and is satisfied inour experiments because the sample is 1% doped H2O in D2O),the Bessel function in Eq. (11) can be simplified and explicitly writ-ten [23]

J�1ðqÞ ¼ �J1ðqÞ � �q=2 ðq� 1Þ ð12Þ

Therefore Eq. (11) becomes

sðt2; t1Þ ¼M0q

4exp½iðx0 þ cB0Þðt2 � 2t1Þ� ð13Þ

or

sðt2; t1Þ ¼ �ðt2=12Þcl0M20 exp½ix0ðt2 � 2t1Þ� exp½icB0ðt2 � 2t1Þ� ð14Þ

Eq. (14) indicates that at the time point t2 = 2t1 the signal is free ofthe inhomogeneous effect irrespective of what distribution B0 maytake.

For a quantitative description of the inhomogeneous signals(t1, t2), Eq. (14) needs to be integrated over the inhomogeneousspace and the inhomogeneous field B0 should be explicitly known.The inhomogeneity is generally a three dimensional function ofspace and can be very complicated and consequently the NMR lineshape can also be very complicated. But on high-resolution spec-trometers the inhomogeneous line shape is normally Lorentzian.Thus, for the detected signal we need to calculate the integration

I ¼ ð1=pÞZ 1

�1exp½ix0ðt2 � 2t1Þ�

1=T 02ð1=T 02Þ

2 þ x02dx0 ð15Þ

where x0 = cB0 and T 02 is the inhomogeneous relaxation time. In fact,Eq. (15) is the reverse Fourier transformation of the Lorentzian lineshape and the result can be straightforwardly obtained.

I ¼ expð� j t2 � 2t1 j =T 02Þ ð16Þ

which is an echo. With Eq. (16) and with the inhomogeneous broad-ening factor ð1=T 02Þ but without knowing the explicit distribution ofB0 in the physical space, the observed CRAZED echoes in Figs. 2 and3 can be simulated, which will be shown in the following section.

3.3. Simulation of the double quantum CRAZED echoes

By taking into account the transverse relaxation in both t2 andt1, Eq. (14) can be changed to

sðt2; t1Þ ¼ Kt2 cos½x0ðt2 � 2t1Þ� exp½�ðt2 þ 2t1Þ=T2�� exp½� j t2 � 2t1 j =T 02� ð17Þ

where K ¼ �ð1=12Þcl0M20. Eq. (17) can be directly used in numerical

simulation.The simulated results using MATHCAD 2001 (www.math-

cad.com) and using Eq. (17) are presented in Fig. 4A and B, whereit was assumed that t1 was varied (t1 = 10, 40 and 80 ms for the top,middle and bottom echoes, respectively) and K = 1, x0 = 400 p,T2 = 160 ms. The only difference between the calculations ofFig. 4A and B was the inhomogeneous relaxation time T 02. ForFig. 4A, T 02 ¼ 10 ms, and for Fig. 4B, T 02 ¼ 160 ms. The calculatedechoes in Fig. 4 are in good agreement with the experimental

0 0.1 0.2 0.3 0.4 0.5

Acquisition Time (sec)

A

0 0.1 0.2 0.3 0.4 0.5

Acqusition Time (sec)

B

Fig. 4. Calculated DQ CRAZED echoes based on Eq. (17) with K = 1, x0 = 400p, T2 = 120 ms and varied t1 (top echoes: 10 ms, middle echoes: 40 ms and bottom echoes: 80 ms).The homogeneous relaxation time was assumed to be 150 ms and the inhomogeneous relaxation time was assumed to be 10 ms in (A) and 200 ms in (B).

36 B. Jiang et al. / Chemical Physics 351 (2008) 33–36

observations in Figs. 2 and 3 and the fitting parameters(T 02 ¼ 10 ms for Fig. 4A and T 02 ¼ 160 ms for Fig. 4B) are also in goodagreement with experiments (T 02 was found to be 9.5 ms for Fig. 2and 147 ms for Fig. 3). The asymmetric feature exhibited in the topechoes in both Figs. 2 and 3 has also been successfully simulated. Itis evident that the change in inhomogeneity strongly affects theDQ CRAZED echo shape, in the same way as in conventional spinechoes. The simulation also indicates that the DQ CRAZED echogenerally appears at t2 = 2t1, except when t1 is very short undergood homogeneous conditions (see the top echoes in Fig. 3 and 4B).

It should be pointed out that Eqs. (15)–(17) are based on theassumption of a Lorentzian inhomogeneous broadening, but inexperiments this might not be exactly true. The inhomogeneityin experiments could be more complicated than Lorentzian lineshape. Therefore, one should not be surprised to see small discrep-ancies between the experimental echoes and the simulated echoes,for example, the echoes in Fig. 4B are sharper than the echoes inFig. 3.

A more general DQ CRAZED echo should be expressed as Eq. (2)where the inhomogeneous line shape is g (x0). Then different echoshape can be calculated when g(x0) is known. However, the echoposition is independent of the inhomogeneous line shape, whichalways appears at t2 = 2t1.

Since g(x0) is generally not Lorentzian, the Fourier transforma-tion of Eq. (2) is not straightforward and numerical calculationhas to be performed. But it is expected that the Fourier transforma-tion of Eq. (2) is different from the Fourier transformation of Eq.(1). This difference may account for the discrepancy found be-tween the simulated line shape and the experimental line shapein Ref. [12], where the discrepancy was (improperly) assumed tobe due to the radiation damping effects.

It should also be pointed out that the double quantum CRAZEDecho is about 100 times weaker than the 90–90 Hahn echo (datanot shown) for the sample used in this study which contains only1% doped H2O in D2O. The magnetization at equilibrium is consider-ably weaker than pure water and weaker CRAZED echo is expected.

4. Summary

In summary, we have demonstrated, both in theory and inexperiments, that the double quantum CRAZED signal has all fea-

tures of a double quantum echo, which is capable of refocusingthe inhomogeneous effect. We have presented a new formula,Eq. (2) for the DQ CRAZED signal in the time domain by includingthe inhomogeneous broadening effect and the double quantumfeature of the relaxation in t1, which can be used to simulatethe observed CRAZED echoes once the inhomogeneous line shapeis known. Although the CRAZED experiments in inhomogeneousfields have been discussed many times in literature, this is thefirst time that the inhomogeneous effect is quantitativelydescribed.

Acknowledgments

This work was support by the National Natural Science Founda-tion of China under the Grant numbers NSFC (20620140104,20635040).

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