Diffusion measurements with the pulsed gradient nonlinear spin echo method

JOURNAL OF CHEMICAL PHYSICS VOLUME 11, NUMBER 1 MARCH 000 Diffusion measurements with the pulsed gradient nonlinear spin echo method Ioan Ardelean a) and Rainer Kimmich Universität Ulm, Sektion Kernresonanzspektroskopie, 89069 Ulm, Germany Received 5 October 1999; accepted 8 December 1999 Spin echo signal attenuation by diffusion is examined for coherence evolution in the course of ordinary pulsed gradient spin echoes and for nonlinear evolution in the presence of a spatially modulated demagnetizing field. It is shown, that, for given field gradient pulse widths or equivalently for a given gradient strength, echo attenuation by diffusion is much more efficient for nonlinear echoes than for Hahn echoes. Remarkably, in the case of nonlinear echoes the refocusing process is spoiled by diffusion not only during the gradient intervals but also thereafter. The effect of displacements occurring in the gradient intervals is enhanced according to the order of the nonlinear echo the pulse sequence is adjusted for. A second attenuation mechanism takes place after the gradient pulses due to displacements in the presence of the spatially modulated demagnetizing field. This effect even occurs when the gradient intervals are too short to contribute. A complete formalism is presented describing all features of the test experiments. It is shown that nuclear magnetic resonance diffusometry based on nonlinear echo signals permits one to measure small diffusion coefficients with moderate field gradients. Nonlinear echo experiments demonstrate that the coherence pathway dominating by far is of a purely single-quantum nature. 000 American Institute of Physics. S001-9606 00 0031-3 I. INTRODUCTION The effect of translational diffusion on Hahn spin echo pulse sequences in the presence of field gradient pulses is well known. The so-called pulsed gradient spin echo PGSE method has been reviewed in several books and articles. 1 3 This technique has become a standard method for the determination of diffusion coefficients. In the present study we consider pulsed gradients again. However, the signals to be detected arise only after nonlinear evolution effects. The method may therefore be referred to as pulsed gradient nonlinear echo PGNE experiment. The background of this sort of coherence refocusing is related to the class of multiple echoes that was first discovered already two decades ago. 3 10 In the following we will examine what peculiar features of the echo attenuation by diffusion can be expected after nonlinear evolution. A general formalism will be presented comprising both the ordinary PGSE and the unconventional PGNE scenarios. Comparative test experiments were carried out in order to demonstrate the peculiar features of a PGNE diffusometry method one could envisage on the basis of nonlinear echoes. We will pay particular attention to the question, from which evolution intervals of the coherence pathways the attenuation process originates and how efficient these mechanisms are. In particular it will be shown both in theory and experiment that nonlinear evolution after the gradient pulses contributes to attenuation even if diffusive displacements in the proper gradient pulse intervals are negligible. a On leave from Department of Physics, Technical University, 3400 Cluj- Napoca, Romania. Electronic mail: ioan.ardelean@physik.uni-ulm.de II. THEORY In the following an ensemble of spins I 1/ subject to the radio frequency RF and field gradient pulse sequences shown in Fig. 1 will be considered. We will treat them with respect to the influence of diffusion on the formation of spin echoes. It is assumed that no indirect or J coupling exists. Short-range intermolecular dipolar interactions are averaged to zero by translational diffusion on a length scale equal to the root mean squared displacement on the NMR time scale. The remaining long-range dipolar interaction cancels for spherical symmetry of the magnetization distribution in the sample, M(r). However, it comes into play after this symmetry is broken in the course of the pulse sequence. It is then represented in the form of a demagnetizing field which is known to cause a large variety of multiple-echo phenomena. 3 11 In a typical multiple-echo pulse sequence, evolution intervals occur in which the magnetization is modulated along a direction with the unit vector u s. For sine modulated magnetization distributions the demagnetizing-field reads 4,7 where B d r 0 M z s u z 1 3M s, 3 u s u z 1, and 0 is the magnetic field constant. The coordinate along the modulation direction, u s,iss r u s. The component of the demagnetizing field that is responsible for the generation of the multiple echoes is always aligned along the direction of quantization. 10 The unit vector 1 001-9606/000/11(1)/575/6/$17.00 575 000 American Institute of Physics

576 J. Chem. Phys., Vol. 11, No. 1, March 000 I. Ardelean and R. Kimmich After an evolution time, a second ( /) x RF pulse and immediately thereafter a second gradient pulse with an area mg are applied. As concerns diffusion and relaxation, the width of the gradient pulse and its delay after the second RF pulse are assumed to be short enough to be negligible in this respect. Anticipating unrestricted normal diffusion, the probability density for displacements along the z axis in the interval is given by 3 FIG. 1. RF and field gradient pulse sequences examined in this study. The schematic free-induction signal after the second RF pulse is expected in the absence of background gradients. On the other hand, if such additional gradients exist, echoes are generated in distinct form. The maxima are at t mt 1 with m 1,,.... The following cases are distinguished: a 0 t 1 ; b t 1 ; c 0. in this direction is u z. The field gradient applied in the pulse sequences shown in Fig. 1 is assumed to act along the external magnetic field B 0 B 0 u z. The axis of any modulation occurring in this experiment consequently coincides with the z axis. That is, u s u z, s z, and 1. In the second evolution interval of the pulse sequences given in Fig. 1, the external magnetic field is superimposed by the modulated demagnetizing field given in Eq. 1. Remembering that the demagnetizing field, Eq. 1, comes into play in Bloch equations as a vector product, MÃB d, it is obvious that the only term relevant for coherence evolution is 4,7 B d r 0 M z r u z. 3 The first field gradient pulse of the spatially constant strength G and duration is applied with a delay t G after the initial RF pulse. The width of the gradient pulse is assumed to be short enough to permit the neglect of diffusion and relaxation effects during the pulse. All RF pulses are assumed to be hard and oriented along the x axis of the rotating frame. Based on the linear Taylor expansion term of the equilibrium density operator, 3 the spin states generated by the first excitation RF pulse, ( /) x, and the first gradient pulse with an area G ) can be represented by the reduced density operator deprived from all constant terms and factors as z, I y cos z1 z I x sin z1 z. 4 I ( x,y,z) represents the components of the spin vector operator. The local rotating-frame precession phase shift is z1 z G z kz, 5 where is the gyromagnetic ratio, and k G is the wave number. All spins located in a slice at the position z experience the same phase shift. 1 p, 4 D e /(4D ), 6 where D represents the self-diffusion coefficient of the spin bearing particles. Equation 6 gives the probability density that a spin that is initially located in a reference slice at the position z will be in a slice at the position z, and vice versa. That is, spins that have diffused away from their initial position in the slice at z are replaced by spins arriving from other positions along the gradient axis during. The precession phases of the incoming spins depend on the positions z they had when the first gradient pulse was applied. Therefore the reduced density operator describing the spin state in the z slice at the time t 1 just before the second RF pulse must be written as an ensemble average over the spins residing in this slice in this instant, z,t 1 I y cos G z I x sin G z. Introducing the phase shift in analogy to Eq. 5, G, Equation 7 can be rewritten in the form z,t 1 I y cos z1 I x sin z1. The sine and cosine terms can be analyzed into sin z1 sin z1 R e i cos z1 I e i and 7 8 9 10 cos z1 cos z1 R e i sin z1 I e i. 11 The averages must be carried out on the basis of the probability density given in Eq. 6. Since Eq. 6 is even, we have R e i e i 1 and I e i 0. Equation 9 can then be rewritten as z,t 1 I y cos z1 I x sin z1 e i. 13 14 Using the probability density given in Eq. 6, the attenuation factor on the right-hand side can be evaluated according to 1,3 e i e 1/ e kd, 15 which is valid in the limit of short gradient pulses,, and ordinary unrestricted diffusion anticipated here. That is,

J. Chem. Phys., Vol. 11, No. 1, March 000 Nonlinear spin echo method 577 the exponent of the echo attenuation factor depends quadratically on the field gradient and linearly on the diffusion coefficient. In the free-evolution interval t 1 comprising t G and see Fig. 1, the coherences are further and independently attenuated by transverse relaxation which is accounted for a posteriori in a phenomenological way. 9,1 The attenuation factor by transverse relaxation is exp t 1 /T, so that Eq. 14 becomes z,t 1 I y cos z1 I x sin z1 e kd e t 1 /T. 16 Note that the delay t G is only relevant for transverse relaxation, but irrelevant for diffusive attenuation because we anticipate here that field inhomogeneities are absent in the intervals before and after the field gradient pulses. The second RF pulse ( /) x followed by the second gradient pulse mg converts Eq. 16 into z,t 1 I z cos z1 e t 1 /T e D( G) I x cos z I y sin z sin z1 e t 1 /T e D( G). 17 z (z) m G z represents the local phase shift caused by the second gradient pulse with the area mg where m 1,,3,.... Theinteger m corresponds to the order of the selected echo. The spin density operator given in Eq. 17 is composed of a longitudinal and a transverse spin operator component, long trans. The spatially modulated longitudinal component produces a modulated demagnetizing field according to Eq. 3. This modulated field influences the further evolution of the coherences represented by the transverse component. After the second gradient pulse no external field gradient is applied apart from the weak and therefore negligible background inhomogeneity of the magnet. Displacements of spins therefore are phase insensitive in this respect. However, the effect of the spatially modulated demagnetizing field contribution on the coherences is subject to diffusion. Displacements of the spins attenuate the spatial modulation and, hence, the evolution of spin coherences. Let us first consider the longitudinal component of the spin density operator in Eq. 17 which is attenuated by translational displacements. Remembering that spins in the z slice are replaced by spins displaced from other positions so that an ensemble average of the cosine function must be carried out in complete analogy to Eq. 11, we find for the density operator effective in the z slice at a time t after the second RF pulse, long z,t 1 t I z cos z1 e t 1 /T e D( G) ( t ). 18 The demagnetizing field in the z slice produced by this component is given by 3 B d z,t 1 t 0 M 0 e t 1 /T e D( G) ( t ) cos G z. 19 The above expression for the demagnetizing field refers to a fixed space position z. However, the actual value of the demagnetizing field seen by a reference spin in the course of the t interval also depends on the displacement of the reference spin. That is, the demagnetizing field relevant at the time t 1 t refers to the position z instead of z expected in the case of a static reference spin. We are dealing with the diffusion of two independent ensembles of spins: Those which produce the demagnetizing field, and those whose transverse components are to be observed. The modulation of the demagnetizing field must therefore be averaged over all displacements of the spins to be observed. The evolution of the transverse spin components takes place according to the effective demagnetizing field B deff z,t 1 t 0 M 0 e t 1 /T e D( G) ( t ) cos G z, 0 where the ensemble average refers to the displacements. It can be carried out analogously to Eq. 11. Owing to the demagnetizing field, Eq. 0, the total additional phase shift, adopted by the coherences evolving during the time t after the second RF pulse is where d z,t 1 t 0 t B deff z,t 1 t dt t 1 t cos G z, t 1 t 0 M 0 e t 1 /T e D( G) 1 1 e D( G) t. D G The transverse component at a time t after the second RF pulse reads then trans z,t 1 t e (t 1 t )/T e D( G) I x cos z d I y sin z d sin z1. 3 The complex transverse magnetization of the z slice corresponding to the reduced density operator given in Eq. 3 thus is M z,t 1 t M 0 Tr trans z,t 1 t I x ii y M 0 i e t 1 t /T e D( G) e i[ z z1 ] e i[ z z1 ] e i (t 1 t )cos z1. 4 This expression can be rewritten in a more convenient form using the Bessel function expansion 4,13 e i cos n i n J n e in. 5 Here, J n ( ), represents Bessel functions of integer order. Making use of the above equations in Eq. 4 we find

578 J. Chem. Phys., Vol. 11, No. 1, March 000 I. Ardelean and R. Kimmich The choice of the integer m determining the weight of the second gradient pulse permits one to choose the order of the signal to be detected. If the second gradient pulse matches the first one, i.e., m 1, the conventional 90/90 spin echo SE is produced with a modified amplitude. For the time moments t after the second RF pulse and/or for sufficiently strong gradient pulses, for which the condition (t 1 t ) 1 holds the Bessel functions can be approximated as J n 1 n! n. 8 In this limit the spin echo amplitude can consequently be written as FIG.. Signal amplitude as a function of t after the second gradient pulse. The data have been calculated on the basis of Eq. 7 neglecting background gradients. The following gradient pulse parameters were assumed: width ms; strength G 0.0 T/m; gradient pulse spacing 100 ms. The RF pulse interval was set to t 1 10 ms. In the computations a diffusion coefficient D.5 10 9 m /s and longitudinal and transverse relaxation times T 1 T 3 s were assumed. The value of the equilibrium magnetization in the sample was M 0 0.031 as expected for water at 93 K in a field of 9.4 T. The data for the gradient strengths used in our experiments justify the approximation given in Eq. 8. SE, spin echo; DF, demagnetizing field; MSE, multiple spin echo of order m. M z,t 1 t M 0 e t 1 t /T e D( G) i n 1 J n t 1 t e i[(m n 1)z G e i(m n 1)z G. 6 This equation becomes independent of the position if the conditions n m 1 and n m 1 are fulfilled. That is, coherences and, hence, free-induction signals exist in this case. The amplitude of the signal is given by A m t 1 t M 0 e t 1 t /T e D( G) i m J m 1 t 1 t J m 1 t 1 t, 7 and is modulated with time according to the Bessel functions starting with a slow increase until a maximum is reached and then a decrease see Fig.. In case a steady field gradient exists e.g., due to the inhomogeneity of the magnet, this function acts as the envelope of the distinct echoes occurring then. Remarkably, the signal is attenuated by diffusion even in the limit of a vanishing interval. The attenuation is then determined by the coefficient (t) given in Eq.. In other words, the signals described by Eq. 7 sense the molecular displacements after the conventional diffusion interval. If a non-negligible steady gradient would exist in addition, a further diffusive attenuation would have to be taken into account, of course. A SE t 1 t i M 0 e t 1 t /T e D( G). 9 In the presence of a weak background inhomogeneity, a distinct echo maximum appears at t t 1. In contrast to the nonlinear signals expected for m 1, the 90/90 spin echo, Eq. 9, is solely attenuated by displacements occurring in the gradient interval, i.e., predominantly in, in agreement to the Stejskal/Tanner treatment of the PGSE experiment. 14 That is, no diffusive damping occurs in the limit when displacements in the gradient intervals become negligible. In the case of a second gradient pulse twice as strong as the first one, i.e., m, a detectable signal occurs with the amplitude given by Eq. 7. In the presence of a background field gradient, a distinct echo will be refocused with a maximum at t t 1 after the second RF pulse. According to Eqs. 7 and 8 the amplitude is predicted as A MSE 3t 1 1 4 0M 0 e 4t 1 /T e D( G) 1 e 4D( G) t 1. 30 D G Here diffusion effects contribute in two different ways. On the one hand, displacements during attenuate the signal amplitude in the conventional way. Additional attenuation is then expected for diffusion in the presence of the modulated demagnetizing field according to the function (t 1 ). This demagnetizing-field attenuation effect occurs even if 0. Likewise, higher-order signals can be generated by choosing larger m values. III. EXPERIMENT The experiments were carried out with undegassed distilled water samples at 98 K on a Bruker DSX400 NMR spectrometer. The equilibrium magnetization at B 0 9.4Tis high enough to permit the detection of nonlinear echoes. The field gradient pulses were produced with a microimaging gradient unit. The RF and gradient pulse sequence is shown in Fig. 1. The width of a / pulse amounted 1 s. The gradient strength was in the range 0.048 0. T/m. Experiments were carried out with different intervals and different orders m. The width of the gradient pulses,, was either or 4 ms, so that diffusion effects during these pulses can safely be neglected. In the experiment corresponding to Fig.

J. Chem. Phys., Vol. 11, No. 1, March 000 Nonlinear spin echo method 579 nonlinear echo (m ) are displayed as a function of the squared gradient strength for gradient pulse delays ms and 100 ms. The diffusion coefficient was fitted to the Hahn echo data on the basis of Eq. 9 as D.5 10 9 m /s. On this basis, all data for the nonlinear echo with m could be described on the basis of Eq. 30 without further fitting. The Hahn echo data show no attenuation by diffusion in the limit of small gradient pulse spacings ( ms, whereas the amplitudes of the nonlinear echo do so. This demonstrates that diffusion in the presence of a spatially modulated demagnetizing field causes signal damping by diffusion. V. DISCUSSION FIG. 3. Normalized amplitudes of the Hahn echo (m 1) and the first nonlinear echo (m ) measured in water as a function of the squared gradient strength. The gradient pulse spacing was 100 ms, the pulse width ms. The solid curves were calculated on the basis of Eqs. 9 and 30, respectively, for D.5 10 9 m /s. 1 c, the gradient pulses were placed before and after the second RF pulse with a spacing of only 1 ms. In this case, diffusion between the gradient pulses can safely be neglected too. The number of accumulated signals was 8, the repetition time 0 s. The background gradient of the magnet was of the order 0.1 mt/m. IV. RESULTS Figures 3 and 4 show experimental data recorded with the pulse sequence displayed in Fig. 1. The normalized amplitudes of the modified Hahn echo (m 1) and of the first FIG. 4. Normalized amplitudes of the Hahn echo (m 1) and the first nonlinear echo (m ) measured in water as a function of the squared gradient strength. The gradient pulse spacing was ms upper two curves and 100 ms lower data set. The gradient pulse width was ms. The solid curve for the case of a Hahn echo, m 1 and 100 ms, was fitted to the data. The resulting value for the diffusion coefficient was D.5 10 9 m /s.. Using this value, the solid curve for the case of the first nonlinear echo, m and ms, was calculated without further fit on the basis of Eq. 30. For comparison, the Hahn echo (m 1) data for a gradient pulse interval ms is also shown. No attenuation by diffusion occurs, whereas the first nonlinear echo (m ) is strongly damped. In this study we have treated and tested the pulsed gradient nonlinear echo experiment for diffusion measurements. The effect of diffusion on the amplitude of nonlinear echo signals remarkably deviates from the case of ordinary Hahn echoes in the presence of pulsed field gradients. Hahn echoes are solely attenuated by displacements in the gradient interval. The signal intensity of nonlinear echoes is also affected by displacements in this interval. However, the attenuation effect is much stronger as demonstrated above. This behavior reminds of multiple-quantum coherence transfer variants of the pulsed gradient spin echo experiment 15 in which the dephasing effect of the gradients is also enhanced with increasing order of the multiple-quantum coherence excited. However, the experimental procedure is quite different here. No preparation of coherences other than single-quantum coherences as produced by an ordinary / RF pulse is needed. As demonstrated by the experiments, the coherence pathway dominating by far is of a single-quantum nature throughout. 3 That is, the PGNE sequence is as simple as the standard Stejskal/Tanner PGSE experiment. 14 There is another peculiar feature of the PGNE experiment. With the PGSE method, echo attenuation by diffusion vanishes if displacements in the gradient pulse intervals, become negligible. This in particular applies to the limit, if is short enough. However, in the PGNE experiment, echo attenuation by diffusion takes place even under such conditions. That is, displacements occurring after the gradient pulse intervals matter for echo attenuation. The reason is that the origin of the signal generation process, i.e., evolution in a spatially modulated demagnetizing field, is sensitive to spin displacements. Actually, this peculiar effect was already discovered by Robyr and Bowtell 4 in their single gradient pulse experiment see the treatment in the Appendix. With the PGNE method, we have two superimposed attenuation mechanisms due to displacements in the gradient intervals on the one hand, and due to displacements after the pulses on the other. Exploiting both effects together permits one to measure small diffusion coefficients with moderate gradients. The formalism presented in this paper describes all phenomena observed so far. Provided that diffusion is normal, experimental PGNE data can reliably be evaluated with the formulas for echo attenuation derived above. Thus a new tool for diffusion studies has become available.

580 J. Chem. Phys., Vol. 11, No. 1, March 000 I. Ardelean and R. Kimmich ACKNOWLEDGMENTS Financial support by the Alexander von Humboldt foundation, the Deutsche Forschungsgemeinschaft, and the Volkswagen-Stiftung is gratefully acknowledged. APPENDIX: SINGLE-GRADIENT PULSE EXPERIMENT In Ref. 4, Robyr and Bowtell have proposed a singlegradient pulse experiment for diffusion measurements. The sequence may be represented by / x G x signal. A1 The coherences are refocused by the demagnetizing field so that no second refocusing gradient is needed. In this reference, the results were explained on the basis of a Bloch equation treatment. In the following, we show that the above formalism is suitable as well. The same experimental conditions as in Sec. II are assumed. The reduced spin density operator describing the spins in a z slice at the beginning of the period when the demagnetizing field becomes modulated, that is, after the application of the above pulse sequence, can be written as 0 I x sin G z I y cos cos G z I z sin cos G z. A A longitudinal component causing the demagnetizing field and a transverse component evolving in this demagnetizing field are generated. The demagnetizing field is influenced by diffusion analogously to Eq. 0. The corresponding phase shift can be written as d z,t 0 M 0 sin 1 e D( G) t cos G z D G t cos G z. A3 Here we have neglected relaxation effects as in Ref. 4. The signal a time t after the second RF pulse is proportional to M z,t im 0 cos e i d (z,t) cos G z M 0 e i d (z,t) sin G z. A4 Analyzing this in terms of a Bessel function expansion, Eq. 5, readily shows that only the first expansion term can contribute. The amplitude of the corresponding echo is A t M 0 cos J 1 t sin 1 e D( G) t 0 M 0. A5 4 D G In the limit (t) 1, the expression obtained in Ref. 4 is reproduced. 1 J. Kărger, H. Pfeifer, and W. Heink, Adv. Magn. Reson. 1, 1 1988. P. T. Callaghan, Principles of Nuclear Magnetic Resonance Microscopy Oxford University Press, Oxford, 1991. 3 R. Kimmich, NMR Tomography, Diffusometry, Relaxometry Springer- Verlag, Berlin, 1997. 4 G. Deville, M. Bernier, and J. M. Delrieux, Phys. Rev. B 19, 5666 1979. 5 D. Einzel, G. Eska, Y. Hirayoshi, T. Kopp, and P. Wölfle, Phys. Rev. Lett. 53, 31 1984. 6 W. Dürr, D. Hentschel, R. Ladebek, R. Oppelt, and A. Oppelt, Abstracts of the 8th Annual Meeting Society of Magnetic Resonance in Medicine, 1989, p. 1173 unpublished. 7 R. Bowtell, R. M. Bowley, and P. Glover, J. Magn. Reson. 88, 643 1990. 8 I. Ardelean, S. Stapf, D. E. Demco, and R. Kimmich, J. Magn. Reson. 14, 506 1997. 9 I. Ardelean, R. Kimmich, S. Stapf, and D. E. Demco, J. Magn. Reson. 17, 17 1997. 10 R. Kimmich, I. Ardelean, Y. Ya. Lin, S. Ahn, and W. S. Warren, J. Chem. Phys. 110, 3708 1999. 11 A. Jerschow, Chem. Phys. Lett. 96, 466 1998. 1 D. E. Demco, A. Johansson, and J. Tegenfeldt, J. Magn. Reson., Ser. A 110, 183 1994. 13 P. M. Morse and H. Feshbach, Methods of Theoretical Physics McGraw Hill, New York, 1953. 14 E. O. Stejskal and J. E. Tanner, J. Chem. Phys. 4, 88 1965. 15 J. F. Martin, L. S. Selwyn, R. R. Vold, and R. L. Vold, J. Chem. Phys. 76, 63 198. 16 D. Zax and A. Pines, J. Chem. Phys. 78, 6333 1983. 17 L. E. Kay and J. H. Prestegard, J. Magn. Reson. 67, 103 1986. 18 T. J. Norwood, J. Magn. Reson. 99, 08 199. 19 L. van Dam, B. Andreasson, and L. Nordenskiöld, Chem. Phys. Lett. 6, 737 1996. 0 P. Mutzenhardt and D. Canet, J. Chem. Phys. 105, 4405 1996. 1 R. S. Luo, M. Liu, and X. A. Mao, Meas. Sci. Technol. 9, 1347 1998. M. Liu, X. A. Mao, C. Ye, J. K. Nicholson, and J. C. Lindon, Mol. Phys. 93, 913 1998. 3 R. Kimmich and I. Ardelean, J. Chem. Phys. 110, 3708 1999. 4 P. Robyr and R. Bowtell, J. Magn. Reson., Ser. A 11, 06 1996.