Helping the Beginners using NMR relaxation. Non-exponential NMR Relaxation: A Simple Computer Experiment. Vladimir I. Bakhmutov Department of Chemistry, Texas A&M University, College Station, TX 77842-3012 E-mail: bakhmoutov@mail.chem.tamu.edu INTRODUCTION T 1,2 NMR relaxation experiments play a very important role in studies of liquids and solids. 1 Generally methods of T 1,2 time determinations include measurements of signal intensities in magnetization recoveries or echo decays that are treated by convenient computer programs to calculate the T 1,2 values. In many cases, interpretations of relaxation times are mainly based on results of the above fittings. Following the phenomenological equations of Bloch, 2, 3 the longitudinal (or transverse) component of the total nuclear magnetization recovers exponentially to an equilibrium magnitude with time constant T 1 (or T 2 ). When nuclear relaxation is dominated by dipolar, quadrupolar or other interactions modulated by molecular motions, the single time constant T 1 (or T 2 ) can be obtained only for two-energy-level spin systems. In contrast, relaxation of spin systems with more than two energy levels can show non-exponential behavior even in isotropic liquids corresponding to a complex relaxation matrix. For example, the T 1 relaxation of the strongly-coupled 1 H - 19 F spin system, theoretically described by two coupled differential equations, contains crossrelaxation terms. 4 As a result, the process can become bi-exponential 4 or multiexponential, 5 corresponding to the number of relaxation components. NMR Relaxation in solids particularly in amorphous systems is not simple a priori. If each site in inhomogeneous solids relaxes exponentially with its own relaxation time, the recovery curves or echo decays, characterizing the total magnetization, will deviate from an exponential function. 6 The exception is the exponential spin-diffusion, where strongly coupled nuclear spins (for example, protons) relax due to dipolar
interaction with electron spins located in paramagnetic impurities. 7, 8 However even in this case, the spin diffusion is ineffective at short time delays in pulse sequences and the magnetization recovery is proportional to time ( 1/2 ). 9 In the presence of the, so-called, limited-spin-diffusion, the magnetization recovery again approaches an exponential law. Commonly, the non-exponential NMR relaxation can be described by the stretched exponential function, f(t) = exp(-t/t 1 ) ), where T 1 is a representative time of whole spin and takes the values between 0 and 1. 8, 10 In these terms, spin diffusion is absent at = ½, while the case ½ < < 1 can be attributed to the limited-spin-diffusion mechanism. Finally, the stretched exponential is theoretically justified as a smooth distribution of relaxation times with the large weight at short delay times. As seen, interpretations of relaxation data will strongly depend on the chosen model. The experimental approach generally-accepted in relaxation studies is based on best fittings of the signal intensities measured by the inversion-recovery or echo pulse sequences to a model function to find the fitted parameters, one of which is the relaxation time. 11 In turn, the model function applied for analyzing the non-exponential NMR relaxation is not always straightforward. It can be a sum of two or more exponentials, a sum of the Gaussian and exponential functions, or a stretched exponential. It is obvious that a larger number of varied parameters will provide better fittings. However, for example, at variations in three parameters, their reproducibility in the presence of noise contributions into the same signal intensity will be more preferable than the application of a four-parameter fitting function. 11 In addition the number of experimental points and quality of the NMR experiments can strongly affect final results. For example, the inaccurately calibrated radio-frequency pulses, themselves, can lead to a visibly non-exponential relaxation curve 1 or the relaxation behavior becomes strongly non-exponential, if the radiofrequency field applied is not homogenous (see Figure 2 in ref. 12 ). Mathematically the above task is the well-known inverse incorrect problem suffering from numerical instability or the so-called, Inverse Laplace Transform problem, solutions of which are sensitive to applied algorithms, discussed recently in a review. 13 In practice, NMR researchers treat the experimental relaxation curves mostly with the help of the convenient computing programs based on the Levenberg - Marquardt algorithm to
find the best fit. Reproducing this situation, the algorithm in the EasyPlot program package has been used for treatments of a given non-exponential function by different model functions to show reliability (or unreliability) of such fittings in the absence of an independently supported relaxation model. RESULTS AND DISCUSSION Since the bi-exponential NMR relaxation, governed by short (T 1 (short)) and long (T 1 (long)) components, has been well established in many cases, 4, 11, 14 the bi-exponential has been taken as the non-exponential experimental inversion-recovery curve used for treatments by different model functions. The initial curves were obtained for different T 1 (short) / T 1 (long) cases by variations in the short (a) and long (1-a) relaxing component fractions, using an equilibrium NMR signal intensity of 50 (au). Then 100 points in the simulated curves were taken as experimental data in fitting procedures. A good fit can be accepted if it has good statistics and deviations do not exceed 5-6%, dictated by real accuracy in intensity determinations. Four cases have been analyzed at different fractions a and (1-a): T 1 (short) = 0.001 s and T 1 (long) = 1s; T 1 (short) = 0.01 s and T 1 (long) = 1s; T 1 (short) = 0.1 s and T 1 (long) = 1s; and T 1 (short) = 0.3 s and T 1 (long) = 1s. It has been found that for a 5% and T 1 (long)/t 1 (short) = 1000 or 100, the initial curves are well fitted to exponentials at good statistics resulting in the T 1 values of 0.99 0.93 s (a = 1 5%). At a = 10 95%, the best fit can be reached only by bi-exponential. The stretched exponential fit shows visibly large deviations while the three- exponential reproduces, in essence, the initial bi-exponential curve. For example, when a = 30%, T 1 (short) = 0.01 s and T 1 (long) = 1 s, the best three-exponential fit gives T 1 (1), T 1 (2) and T 1 (3) values of 0.01, 1.00 and 0.99 s with contributions of 33, 40 and 26%, respectively. Slightly different results were obtained for the case T 1 (short) = 0.1 s and T 1 (long) = 1s. Here even at a = 7% the initial curve is still treated by the exponential (Figure 1), giving T 1 = 0.91 s. As earlier, for a > 7% a good fit can be obtained by the biexponential. Even for a of 93%, the exponential fit shows large deviations in Figure 1.
Figure 1. The bi-exponential experimental curves: (o) at T 1 (short) = 0.1 s, T 1 (long) = 1 s, a = 07%, treated by the exponential (the solid line); ( ) at a = 50% treated by the stretched exponential (the solid line); ( ) - at a = 93% treated by the exponential (the solid line). However for the a area between 10 and 50%, besides the bi-exponential, a good fit is reached by the treatments with the stretched exponential model (see Figure 1, a = 50%) giving T 1 = 0.88-0.39 s at of 0.87-0.55, respectively. Under such circumstances, interpreting the collected data will depend on the relaxation model which should be established independently. For example, for a = 50% the stretched exponential curve obtained for solids at = 0.55 could correspond to the absence of spin diffusion. Then, for a > 50%, the stretched exponentials describe the initial curves very poorly. The fourth case with the commensurable short and long relaxation components (T 1 (short) = 0.3 s and T 1 (long) = 1 s) is not single-valued completely. The initial curves obtained at a < 30% can be well fitted to exponentials with T 1 of 0.95-0.81 s in Table 1.
Table 1. The initial bi-exponential curves (T 1 (short) = 0.3 s and T 1 (long) = 1s) at different short component contributions a treated by the exponential and starched exponential model functions. a(%) Exponential Stretched exponential T 1 (obs) I 0 T 1 (obs) I 0 5 10 20 30 40 50 60 70 80 90 0.95 49.6 0.91 49.2 0.81 48.6 0.73 48.1 0.65 47.8 0.57 47.8 0.50 47.9 0.44 48.2 0.39 48.7 0.34 49.3 0.82 50.2 0.89 0.73 50.1 0.86 0.65 49.8 0.84 0.57 49.6 0.83 0.50 49.3 0.84 0.44 49.2 0.86 0.38 49.2 0.89 0.34 49.5 0.94 The remarkable deviations appear only for a = 30 70% (see Figure 2, a = 70%). However they can be detected, if the number of the experimental points is sufficient (100 points in our case). In addition, the deviations from the exponential can be more pronounced in the presence of the inhomogeneous radio-frequency field, which is homogenous only at the center of the RF-coil. 12 Finally again besides the bi-exponential, a good fit to the initial curves at a = 30 90% is reached by the stretched exponential (Figure 2, a =70%) giving the T 1 and values in Table 1. As earlier, interpretations of the T 1 times will strongly depend on the model used for the fitting procedures.
Figure 2. The bi-exponential experimental curve (o) obtained at T 1 (short) = 0.3 s (a = 70%) and T 1 (long) = 1 s ((1-a) =30%) and treated by an exponential (the dashed line) and a stretched exponential (the solid line). CONCLUSIONS This work shows that experimental relaxation curves can look exponential in spite of their more complex character. Formally such a result obtained in liquids or solids could correspond to molecular motion, described by a single correlation time. 7 The more complex motions can be revealed by the variable-temperature relaxation experiments where the visibly exponential curves transform to bi (or multi)-exponentials. 4, 15, 16 In rigid solids, the exponential relaxation curves could be explained by the spin-diffusion mechanism. However, in this case measurements at different spinning rates are necessary to confirm or decline the spin-diffusion mechanism. 8 Under some conditions, the experimental curves can be equally well fitted to bior stretched exponential. The latter is obviously more preferable due to the smaller
number of varied parameters, if the relaxation model is unknown. Then the T 1 time will describe behavior of the whole spin: (a) moving in the presence of a correlation time distribution and (b) relaxing via the limited spin diffusion mechanism if > 0.5. Finally, the represented data justifies that application of exponentials even for the non-exponential experimental curves can be useful in the context of qualitative T 1 (or T 2 ) comparisons. 17. REFERENCES 1. Bakhmutov VI. Practical NMR Relaxation for Chemists. Wiley: Chichester, 2005. 2. Bloch F. Phys. Rev. 1946; 70: 460-474. 3. Madhu PK, Kumar A. Conc. Magn. Reson. 1997; 9: 1-12. 4. Zdanowska-Fqczek M, Medycki W. Solid State Nucl. Magn. Reson. 1996; 6: 141-146. 5. Kumar A, Gracea CR, Madhu PK. Progress Nucl. Magn. Reson. Spectr. 2000; 37: 191-319. 6. Kruk D, Fujara F, Gumann P, Medycki W, Privalov AF, Tacke C. Solid State Nucl. Magn Reson. 2009; 35: 152-163. 7. Abragam A. The Principles of Nuclear Magnetism. Oxford University: London. 1961. 8. Hayashi S, Akiba E. Solid State Nucl. Magn. Reson. 1995; 4: 331-340. 9. Blumberg WE. Phys. Rev. 1960; 119: 79-84. 10. Alaimo MH, Roberts JE. Solid State Nucl. Magn. Reson. 1997; 8: 241-250. 11. Blumich S, Perlo J, Casanova F. Progress Nucl. Magn. Reson. Spectr. 2008; 52: 197-269. 12. Bakhmutov VI. J. Spectrosc. Dyn. 2013; 3: (5) 1-5. 13. Kuhlman KL. Numerical Algorithms 2012. 14. Horsewill AJ, Tomsah IBI. Solid State Nucl. Magn. Reson. 1993; 2: 61-72. 15.Lalowicz ZT, Punkkinen M, Olejniczak Z, Birczynski A, Haeberlen U. Solid State Nucl. Magn. Reson. 2002; 22: 373-393. 16. Jurga J, Woz niak-braszak A, Fojud Z, Jurga K. Solid State Nucl. Magn. Reson. 2004; 25: 47-52. 17. Xu Z, Stebbins I. Solid State Nucl. Magn. Reson. 1995; 5: 103-112.