The influence of J-coupling on heteronuclear nonlinear (or multiple) spin echoes

19 October 2001

Chemical Physics Letters 347 (2001) 157±162

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The in¯uence of J-coupling on heteronuclear nonlinear (or multiple) spin echoes Elke Kossel a, Rainer Kimmich a,*, Ioan Ardelean b a b

Sektion Kernresonanzspektroskopie, Universitat Ulm, 89069 Ulm, Germany Department of Physics, Technical University, 3400 Cluj-Napoca, Romania Received 10 August 2001; in ®nal form 28 August 2001

Abstract Heteronuclear nonlinear spin echoes were investigated with respect to the in¯uence of indirect spin±spin coupling in a two-spin 1/2 system consisting of 1 H and 13 C spins. The heteronuclear nonlinear spin echo phenomenon is essentially based on the 13 C coherence evolution in the presence of the modulated demagnetizing ®eld originating from the 1 H magnetization. It is shown that the ®rst nonlinear 13 C echo amplitude oscillates as a function of the modulation period of the demagnetizing ®eld (or the pitch of the proton magnetization helix). The Fourier transform reveals a doublet with a splitting equal to the heteronuclear spin±spin coupling constant J. Ó 2001 Elsevier Science B.V. All rights reserved.

1. Introduction Multiple spin echoes [1] arise after `nonlinear' evolution of spin coherences in the presence of a modulated demagnetizing ®eld (that is the ®eld originating from the inhomogeneous spatial distribution of the magnetization). Multiple echoes are therefore often called `nonlinear echoes' in contrast to ordinary, `linear' Hahn echoes [2]. The modulation of the demagnetizing ®eld originates from the modulated z magnetization produced by a 90° pulse after coherence evolution in a gradient of the external magnetic ®eld, for instance. The nature of the equidistant nonlinear echo signals is analogous to the time domain signals corresponding to modulation side-bands well known in

*

Corresponding author. Fax: +49-731-502-3150. E-mail address: [email protected] (R. Kimmich).

radio frequency technology. A further, somewhat more distant analogy may be seen in the quantum beat echoes observed in M ossbauer spectroscopy experiments (see [3], for instance). Originally, multiple spin echoes were found and treated for the homonuclear case (e.g. solid 3 He in [1]). In this case, both the modulated z magnetization and the signal-generating nuclei refer to one and the same spin ensemble. Later it was demonstrated that nonlinear spin echoes can also be generated in heteronuclear spin systems [4,5]: The modulated z magnetization producing the pertinent modulated demagnetizing ®eld may be based on protons, whereas the signal-generating coherences are due to 13 C nuclei. In the present study, we are referring to this situation and report on peculiar e€ects due to heteronuclear J-coupling. The occurrence of the phenomenon and the dependence on various experimental parameters of nonlinear echoes can be explained using the Bloch equations modi®ed by inclusion of a modulated

0009-2614/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 1 0 0 7 - 7

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E. Kossel et al. / Chemical Physics Letters 347 (2001) 157±162

demagnetizing ®eld [1,4,5]. On the other hand, a product operator formalism was suggested [6±11] reproducing the same results in a simple straightforward way. The modi®ed Bloch equation method has the advantage that relaxation and di€usion e€ects can be accounted for relatively easily. A problem is, however, that complicated nonlinear di€erential equations must be solved. A further restriction is that J-coupling, the phenomenon of interest in the present study, cannot be treated on this basis. Refraining from detailed discussions of relaxation and di€usion e€ects, we therefore employ the spin operator formalism used before [6±11]. The demagnetizing ®eld represents long-range dipolar couplings in just the same way as the magnetic ¯ux density B0 of a permanent magnet collectively stands for the long-range dipolar couplings of the sample nuclear spins with the unpaired electrons in the magnet poles. Of course, instead of treating them in ®eld form, these interactions could also be accounted for in principle on the basis of the dipolar interaction Hamiltonian expressed by spin operators. The diculty of such an attempt would however be how to represent the spatial modulation of the z magnetization in operator form. A spatially modulated z magnetization means that the populations of spin-up and spin-down states are modulated in the same way. The ordinary spin operator formalism of dipolar coupling anticipates negligible di€erences of the thermal equilibrium populations, so that all nonlinear coherence evolution e€ects would be eliminated a priori in such an approach. We therefore apply the conventional demagnetizing ®eld representation of long-range dipolar couplings. It may be of interest to note that the pseudocontact shift [12] is another example where Boltzmann population di€erences are crucial. In this Letter, we report on nonlinear spin echo experiments with 13 C coherences evolving in the presence of a modulated 1 H z magnetization. The two spin species belong to the same molecule and, hence, are subject to heteronuclear scalar coupling characterized by the J-coupling constant. In the theoretical treatment, di€usion e€ects will be neglected, and the attenuation by relaxation will be

accounted for in a semi-empirical way as suggested in [7,13]. 2. Theory The heteronuclear nonlinear spin echo experiment to be described in the following is schematically shown in Fig. 1. This pulse sequence was applied to an AX spin system consisting of the two spin species I and S both with the spin quantum number 1/2. The weak coupling limit is assumed, so that the indirect coupling constant obeys J  Dm, where Dm represents the di€erence between the resonance frequencies of the two spin species. In this limit the Hamiltonian describing the indirect spin±spin interaction is given by [6] H ˆ hJIz Sz ;

…1†

where h is the Planck constant. In the high-temperature approximation, the initial spin state of isolated spin pairs just before the ®rst rf (radio frequency) pulse (denoted by the time 0 ) can be described by the reduced density operator [9] deprived by all constant terms which cannot lead to the signal, r…0 † ˆ cI Iz ‡ cS Sz :

…2†

The quantities cI and cS represent two constants which can be determined from the condition that before the ®rst rf pulse the magnetization components are M I …0 † ˆ cI hN Trfr…0 †Iz gez ˆ M0I ez ;

…3†

Fig. 1. Pulse sequence used for the investigation of the in¯uence of J-coupling on heteronuclear nonlinear spin echoes.

E. Kossel et al. / Chemical Physics Letters 347 (2001) 157±162

and S

M …0 † ˆ cS  hN Trfr…0 †Sz gez ˆ

M0S ez ;

…4†

respectively. In the above expressions, M0I and M0S represent the two equilibrium magnetizations of the two spin species, cI , cS are the two magnetogyric ratios, N is the number of spin pairs in the unit volume, and ez is the unit vector along the z axis. The pulse sequence shown in Fig. 1 refers to selective rf pulses that selectively excite only one spin species at a time. The ®rst rf pulse generates single-quantum coherences of the I spins which evolve in the ®rst evolution interval, s1 , under the in¯uence of a constant gradient G, assumed to be aligned along the z direction. Furthermore, Jcoupling to the S-spins and transverse relaxation is e€ective. Demagnetizing-®eld e€ects can safely be neglected in this ®rst interval (for instance see [1,4,7,8,10,11]). Ignoring attenuation by relaxation for the moment, we obtain the spin density operator just after the second rf pulse on the I spins as r…s‡ 1 † ˆ cI f‰Ix sin…XI s1 †

Iz cos…XI s1 †Š cos…pJ s1 †

‰2Iz Sz sin…XI s1 † ‡ 2Ix Sz cos…XI s1 †Š  sin…pJ s1 †g ‡ cS Sz ;

…5†

where XI ˆ DxI ‡ cI Gz, and DxI is the frequency shift of the I spins in the doubly rotating frame of reference. Analyzing Eq. (5), it is obvious that operator terms like 2Ix Sz and 2Iz Sz cannot generate any signal with this pulse sequence. The only terms of interest in the frame of our experiment refer to single-quantum coherences and longitudinal magnetization. Retaining only such terms and introducing a semi-empirical relaxation attenuation factor as suggested in [7,13], the density operator can be written as r…s‡ 1 † ˆ cI ‰Ix sin…XI s1 †

Iz cos…XI s1 †Še

 cos…pJ s1 † ‡ cS Sz ;

s1 =T2I

…6†

where T2I represents the transverse relaxation time of the I spins. For long evolution intervals s1 , an unmodulated magnetization component along the z direction can appear moreover as a result of

159

longitudinal relaxation. This unmodulated magnetization does however not in¯uence the echo amplitude, and will hence be disregarded in the following. During the spoiling interval, sspoil , the transverse component of the magnetization is dephased and further attenuated by relaxation. The third rf pulse acts on the S spins and transfers the reduced density operator into r…s1 ‡ sspoil † ˆ

cI Iz e

s1 =T2I

e

sspoil =T1I

cos…XI s1 †

 cos…pJ s1 † ‡ cS Sy ;

…7†

where spin±lattice relaxation of the I spins was taken into account by the longitudinal relaxation time T1I . The ®rst term on the right-hand side of Eq. (7) produces a demagnetizing-®eld component in¯uencing the evolution of the S spin coherences. Neglecting here and in the following the contribution to the demagnetizing ®eld by the S spin magnetization itself, the e€ective demagnetizing ®eld can be expressed as [4] 2 Bd …r† ˆ l0 MzI ez : 3

…8†

Apart from spin±lattice relaxation, the z magnetization will not be changed in the subsequent interval. The demagnetizing ®eld e€ective for the evolution of S spin coherences thus reads Bd …z; t† ˆ

2 I I l0 M0I e s1 =T2 e sspoil =T1 e 3  cos…XI s1 † cos…pJ s1 †:

t=T1I

…9†

The corresponding phase shift picked up on these grounds by the S spin coherences is given by Z t ud …z; t† ˆ cS Bd …z; t0 † dt0 0

ˆ

n…t† cos…DxI s1 ‡ cI Gzs1 †;

…10†

where 2 I I n…t† ˆ l0 cS M0I e s1 =T2 e sspoil =T1 3 h i I  T1I 1 e t=T1 cos…pJ s1 †:

…11†

Taking into account the combined in¯uences of J-coupling, the ®eld gradient, and the demagne-

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tizing ®eld, the S spin coherences evolve according to rS …s1 ‡ sspoil ‡ t†  ˆ cS f Sx sin…XS t ‡ ud …z; t††  ‡ Sy cos…XS t ‡ ud …z; t†† cos…pJt†  ‡ 2Sy Iz sin…XS t ‡ ud …z; t††  2Sx Iz cos…XS t ‡ ud …z; t†† sin…pJt†g:

…12†

Since there are no further rf pulses, the antiphase single-quantum coherences 2Sy Iz and 2Sx Iz cannot generate any detectable signal. The corresponding terms in the reduced density operator can consequently be omitted. For the in phase S spin singlequantum coherences merely terms of the operators Sx and Sy are mattering. The complex transverse S spin magnetization thus becomes MxS ‡ iMyS ˆ N c5 2TrfrS …s1 ‡ sspoil ‡ t†…Sx ‡ iSy †g ˆ iM0S e

t=T2S

cos…pJt† exp… i…DxS ‡ cS Gz†t†

 exp…in…t†cos…DxI ‡ cS Gz†s1 †:

…13†

An …t ˆ n…cI =cS †s1 † ˆ M0S Jn …nn…cI =cS †s1 †  exp… …n…cI =cS †s1 =T2S ††  cos…pJn…cI =cS †s1 †: …16† In the limit of small demagnetizing ®elds, or equivalently for n…t†  1, the Bessel functions of integer order can be approximated as [14]  n 1 n Jn …n†  : …17† n! 2 The amplitude of the ®rst nonlinear echo …n ˆ 1†, is then proportional to  M0I M0S I  T1 1 exp… ……cI =cS †s1 =T1I †† 3 I …s1 =T2I † e exp… ……cI =cS †s1 =T2S ††e …sspoil =T1 †   c  cos…pJ s1 † cos pJ I s1 : …18† cS

A1 …s1 † ˆ cS l0

From this result, one concludes that the Fourier transform of the echo amplitude with respect to s1 leads to two peaks separated by a splitting J from each other.

Eq. (13) can be rewritten as MxS ‡ iMyS ˆ

S M0S e t=T2

cos…pJt†

‡1 X

3. Experiments i

n‡1

Jn …n…t††

nˆ 1

 exp…i…nDxI s1

DxS t†† exp…i…ncI s1

cS t†Gz†; …14†

where we have used the Jacobi/Anger identity [14] ein cos…a† ˆ

‡1 X

in Jn …n†eina ;

…15†

nˆ 1

which is based on Bessel functions of integer order, J …n†. Since the recorded signal consists of contributions from all z positions in the sample, it vanishes except for times when the exponential function in Eq. (13) becomes independent of z. That is (nonlinear) echoes will appear at these times. The condition is cS t ˆ ncI s1 . At times t ˆ n…cI =cS †s1 , the S magnetization is refocused, and spin echoes are produced. The amplitude of the nth order echo is proportional to

The above theory was tested with 13 C enriched formic acid with deuterated hydroxy groups as a two-spin 12 system. The remaining protons refer to the I spins, whereas the 13 C spins are to be identi®ed with S spins. The hydrogens in the hydroxy groups were exchanged by deuterium using the method described in [16]. The obtained deuterated formic acid has a purity of more than 95%. The 13 C content of the formic acid was enriched to be 99%. The relaxation times of the formic acid protons were measured to be T1I ˆ 1:12 s and T2I ˆ 105 ms. The 13 C spin±lattice relaxation time was found to be T1S ˆ 5:5 s. The transverse 13 C relaxation time was estimated to be of the order 800 ms. The experiments have been performed on a Bruker DPX 400 MHz spectrometer at room temperature. The sample consisted of approximately 0.5 ml formic acid and was contained in a NMR sample tube with a diameter of 5 mm. A small constant z-gradient was produced by inten-

E. Kossel et al. / Chemical Physics Letters 347 (2001) 157±162

tional misadjustment of the shimming currents. The gradient strength of about 0.001 T m 1 was small enough to neglect any di€usion e€ects. Sample shape e€ects were also negligible. The spoiling interval sspoil between the second rf pulse (acting on protons) and the third rf pulse (acting on carbons) was set to 50 ms. The 90° pulse lengths for protons and 13 C were 15 and 45 ls, respectively. Four transients were accumulated. The repetition time was 60 s, so that relaxation of the S spins was complete before each scan. The total duration of the experiment was 17 h and 13 min. The ®rst nonlinear echo, which appears at the time t ˆ ccSI s1 ˆ 4s1 after the 90° pulse in the 13 C channel (see Fig. 1), was recorded as a function of the interval s1 . This interval was varied in 256 steps from 10 to 137.5 ms. The results are shown in Fig. 2. The predicted oscillations due to J-coupling are obvious. The Fourier transform of this curve with respect to s1 is shown in Fig. 3. The resolution depends only on the increment of s1 . As expected, the line splitting is 218:75  3:9 Hz. According to the formula derived in the previous section, this value corresponds to the J-coupling constant of the protons directly bound to carbons in formic acid. The literature value of 222 Hz [15] is almost perfectly reproduced.

Fig. 2. Amplitude of the ®rst heteronuclear nonlinear S spin echo versus the evolution time, s1 , between the two proton pulses (see Fig. 1). The sample was 13 C enriched and hydroxy deuterated formic acid. The oscillations are due to J-coupling between the 1 H and 13 C spins.

161

Fig. 3. Fourier transform of the curve shown in Fig. 2. The Jcoupling constant corresponds to the doublet splitting.

4. Discussion and conclusions The pulse sequence shown in Fig. 1 strongly resembles the pulse sequence used for heteronuclear shift correlation spectroscopy [17]. However, a closer look at the pulse sequences reveals the di€erences between the two experiments: In the heteronuclear shift correlation experiment, the FID of the S spins is recorded directly after the second pulse on the I spins, and no gradient is present. The whole two-dimensional spectrum is Fourier transformed with respect to s1 and t resulting in six peaks at the positions ‰…mI  J2†; …mS  J2†Š; ‰0; mS  J2Š. In the experiment we are proposing here the amplitude of the ®rst nonlinear echo is evaluated as a function of the proton preparation interval s1 . The amplitude of this echo is Fourier transformed with respect to s1 resulting in a doublet with a splitting J. The heteronuclear nonlinear echo experiment discussed here di€ers from the homonuclear case in the sense that J-coupling as additional interaction matters for the echo intensity. We have shown how J-coupling in¯uences the signal and that Jcoupling leads to oscillations of the signal with respect to the proton evolution time s1 . For a twospin 12 system there are two frequencies contributing to the oscillations. Their di€erences correspond to the J-coupling constant. Therefore these oscillations may be used for the determination of the Jcoupling constant. The accuracy of this method is only limited by the number and the increment of the time intervals s1 .

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E. Kossel et al. / Chemical Physics Letters 347 (2001) 157±162

The theoretical predictions presented in this Letter were tested experimentally with partly deuterated and 13 C enriched formic acid. In accordance with the theory, the J-coupling constant between the 13 C spin and the directly bound 1 H spins could be determined from an analysis of the oscillations of the ®rst nonlinear echo with respect to s1 . After 22 years, the investigation of nonlinear echoes is still a timely research ®eld. With the recent developements in high-®eld NMR spectrometers, the experimental work in this area becomes less and less demanding. At the large magnetic ¯ux densities as they are ubiquitous in modern NMR laboratories, nonlinear evolution e€ects may appear in quite unexpected cases. It is therefore important to know the phenomena that can arise on these grounds in principle. On the other hand, it can be expected that some techniques based on nonlinear echoes will ®nd their way into the library of standard experiments. Until now there has been only little work on heteronuclear nonlinear echoes. Since J-coupling is always present in these experiments, our aim was to show how it contributes to the signal and to develop a pulse sequence which enables the determination of the coupling constant from nonlinear echoes. The pulse sequence presented in this Letter turned out to be suitable for this purpose. Acknowledgements The authors thank Dr.Uwe Beginn from the Department of Organic Chemistry at the Univer-

sitat Ulm for deuterating the formic acid. Financial support by the Alexander von Humboldt foundation and the Deutsche Forschungsgemeinschaft is gratefully acknowledged. References [1] G. Deville, M. Bernier, J.M. Delrieux, Phys. Rev. B 19 (1979) 5666. [2] E. Hahn, Phys. Rev. 80 (1950) 580. [3] H. Jex, A. Ludwig, F.J. Hartmann, E. Gerdau, O. Leupold, Europhys. Lett. 40 (1997) 317. [4] R. Bowtell, J. Magn. Res. 100 (1992) 1. [5] I. Ardelean, E. Kossel, R. Kimmich, J. Chem. Phys. 114 (2001) 8520. [6] R. Kimmich, NMR Tomography, Di€usometry, Relaxometry, Springer, Berlin, 1997. [7] I. Ardelean, R. Kimmich, S. Stapf, D.E. Demco, J. Magn. Reson. 127 (1997) 217. [8] R. Kimmich, I. Ardelean, Y.Ya. Lin, S. Ahn, W.S. Warren, J. Chem. Phys. 111 (1999) 6501. [9] R. Kimmich, I. Ardelean, J. Chem. Phys. 110 (1999) 3708. [10] I. Ardelean, R. Kimmich, J. Chem. Phys. 112 (2000) 5275. [11] I. Ardelean, R. Kimmich, Chem. Phys. Lett. 320 (2000) 81. [12] I. Bertini, C. Luchinat, NMR of Paramagnetic Substances, Coord. Chem. Rev., vol. 150, Elsevier, Amsterdam, 1996. [13] P. Mutzenhardt, D. Canet, J. Chem. Phys. 105 (1996) 4405. [14] P.M. Morse, H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York, 1953. [15] N. Muller, J. Chem. Phys. 36 (1962) 359. [16] G.A. Ropp, C.E. Melton, J. Am. Chem. Soc. 80 (1958) 3509. [17] D. Shaw, Fourier Transform N.M.R. Spectroscopy, second ed., Elsevier, Amsterdam, 1984.