Spin-Lattice Relaxation Times in 'H NMR Spectroscopy DoMkl J. Wink New York University. New York, NY 10003 Sophisticated nuclear magnetic resonance applications that go heyond simple first-order chemical shift and spinspin splitting analysis are becoming more important to the working chemist ( I ) . An understanding of these methods requires a grasp of the causes and effects of several additional aspects of the NMRphenomenon (2), particularly how the excited state generated in an experiment returns to equilihrium, a process known as relaxation. The environment of a nucleus can have a profound effect on the rate of relaxation of a nuclear spin system, and thus relaxation experiments offer an additional method to probe molecular structure. This is in contrast to other forms of spectroscopy, including ultraviolet and infrared ahsorption methods, where excited states usually decay a t rates that are insensitive to the environment. Nuclear magnetic relaxation can he used for the determination of molecular structure. of molecular dvnamics (3). . .. and (with magnetic resonance imaging techniques), of tissue structure in hioloeical svstems. Relaxation is also i m ~ o r t a n t in normal NMR ~pect;oscopy, for the small energy differences that characterize NMR transitions mean they are easy to saturate; an insufficiently relaxed sample may give incorrect integrals. The best way to study relaxation phenomena is with pulsed Fourier-transform NMR techniques, taking advan-
Figure 3. Tipping of a bulk magnellraflon by a strang pulsed field HTto give a nonequilibrium spin dlslribution.
Figure 1. A bulk magnetization is me vecta sum of many small nuclear magnets.
Figure 2. Camdinate system wed to describe !he components of the bulk magnetization.
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tage of the essentially instantaneous creation of a nonequilibrium distribution of spins in the external field. The behavior of the system is described with a hulk magnetization, which is iust the vector sum of all the individual magnetic moment' (Fig. 1).The coordinate system has the i axis parallel to the hulk field Ho, and the magnetization vector has two components: M, (parallel with the z axis) and M, (lying in the xy plane a t some angle 4 with respect to the x axis-Fig. 2). At equilibrium the magnetization, Mo, is stationary and aligned with Ho so that M, = Mo and M, = 0 (Fig. 3a). Excitation by a strong pulse causes the magnetization to tip some angle R from equilibrium (Fig. 3h) so that M, = Mo cos Rand M , = Mo sin 0. The magnetization returns to equilibrium in a manner that can he depicted as the independent return of M,to Mo and M,, to zero (Fip. 4). The two relaxation proc&see both ohey simple first-order kinetics, with conveniently described rates (eqs 1 and 2) and rate laws (eqs 3 and 4, presented for the arbitrary case of a time t after a pulse of 0 degrees).
The rate laws each contain an associated time constant. Faster relaxation is expressed in terms of a shorter constant,
pulse
1% -Jr t
Figure 4. Expmmial return of Imnsverseand longitudinal magnetization after
a perturbing 0 = 90° pulse.
F i i 5. Relationship between MA0 and peak intensity as a htnction of time after an inversion pulse.
also called a relaxation time. The spin lattice relaxation time TIdescribes the recovery of M, and the spin-spin relaxation time Tz describes the decay of M,,. Each nucleus in a molecule has a unique T I and Tz. The Tz values can be very helpful in determining certain information about molecular dynamics and structure (4). Also, the line width of a resonance in a conventional frequency spectrum is inversely related to Tz, so quickly relaxing samples are often characterized by broad lines and poor resolution. Still, the spinlattice relaxation times are generally more significant for chemists because they provide a more easily measured indication of the environment of a nucleus. The Measurement of Sdn-Lattlce Relaxation Times There are several methods for the determination of T,'s, with the more sophisticated designed to remove systematic errors introduced by instrumental factors or make theacquisition of data more efficient (5).The most common method, however, remains the inversion-recovery technique (6);it is easy to program and analyze, and the results are sufficiently accurate for most applications. The inversion recovery experiment consists of a 0 = 180' pulse applied to the sample to completely invert the magnetization, so that M, = -Mo and M , = 0.The system is then allowed to relax for some time t after the initial pulse, after which M,(t) = Mo(1 - 2e-'IT$ A 6 = 90' pulse is then applied to tip the magnetization into the xy plane, so that now M, = M&). Pulsed FTNMR instruments detect M , as it decays, convert it into a digital form as a free induction decay @ID) for storage, and then transform it into a conventional intensity vs. frequency spectnun. One ends with an NMR spectrum whose peak intensities, commonly referred to by the parameter I, aredirectly proportional to M&) (Fig. 5 ) and a graph of peak intensity vs. time is then a direct analogue to a plot of M,vs. t. The relevant kinetic equation in this case is The basic data for such a "TlIR" exneriment are u s u d v presented as a stacked plot of spectra with varying t's-such a d o t is resented for I-hromo-2-methvlbutanein Fimre G. he determination of TIfrom such a plbt can be donLin any one of three ways. Since this is a first-order process, the point at which the magnetization has returned halfway to
~ i g 6.kStacked piotofdatafmrna TllRexpimem on 2-m&yl-1-bromobw tane.
equilibrium-the half-lifewill occur at a characteristic time tlfi related to the value of TI: Simple curve fitting can be used to obtain more accurate values for T Irequired for any quantitative analysis. A nonlinear fit of the data to eq 5 to obtain 10 and T is the most rigorous approach. One can also perform a least-squares fit of a linearized version of eq 5:
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I 0 can also he approximated
by choosing a very short (say 10 me) delay for the first data point. The intensity of this point of the inverse oflo and isgood is usually within a few for an initial calculation.
Mechanism of Nuclear Magnetlc Relaxatlon The excited state in NMR systems can he very long-lived. Simple spontaneous emission of a photon or phonon (a quantum of vibrational energy) of the appropriate frequency rarely occurs in NMR systems. Instead the environment must provide an oscillatine magnetic field of the correct frequency w ~ t h i nthe aornple to stimulate relaxation. The magnetic fields that can stimulate nuclear relaxation arise fromlocal magnetic moments present in molecules and atoms. The rate of any spin-lattice relaxation (the inverse of the associated relaxation time) has the general form, 1lT1RI = (61,)' fb,)
(8)
Here bl- is the local mametic field. and (61.3 is its mean square&alue, present toreflect the' aver&&agnetic field felt hv some nucleus A due to some nearbv fluctuatine field. The iunction f(7,) is knownasa correlation function &d the timer, is the associated correlation time. These describe the efficiehq of the interaction of the magnetic field bl, with the individual nucleus. The overall relaxation rate is sum of all individual interactions that arise in one of six different ways. Some are intramolecular, some are intermolecular, and one is intrannclear. Only the three mechanisms that are important in common applications will he discussed in detail. Di~ole-Di~ole Interaction Nuclear Magnetic This is rather easy to depict, for it involves the local fields of two nuclear spins interacting throwh space intramolecularly or, less of& intermolecdarly. 6 is the principle form of relaxation for nuclei with spin 112 nuclei (including protons or '3C nuclei) in the ahsince of electronic pargagnetism (see [b] below). The mean square of the field felt at nucleus A due to a dipole a t B is (here the parameter I refers to the nuclear spin quantum number of the nucleus under examination):
he affected hv local motions. This occurs. for examnle. & in. the protons of a methyl group of an organic or inorganic methyl ester, where the CH3 group is extremely uncrowded by the adjacent oxygen. Long TI'S may occur in such situations hecause the correlation needed for relaxation becomes verv inefficient. (b) Electron Dipole Relaxation
The other important magnetic dipole in chemistry is produced by an unpaired electronic spin. The local field is identical in its form to the nuclear dipole interaction except that an unpaired electron has a magnetic moment much larger than any nucleus. Thus, unpaired electrons can severelv affect the relaxation of an NMR sample. Since the effect also goes as the inverse sixth power of distance, nuclei on the periphery of a molecule (especially protons) are nsually relaxed more, except where specialchemical effects such as complexation occur. ~lecirondipole relaxation can be a help or a hindrance in NMH spectra. Sometimes it is a good idea deliberately to add a paramagnetic "relaxation agent" such as tris(acety1acetnnate)chromium to a sample whose long relaxation times interfere with rapid data collection. On the other hand, accurate measurement of nuclear dipoldipole relaxation usually requires that all paramagnetic substances (including oxygen in the air) he removed or excluded from the sample.
(c) Electric Ouadrupole Relaxation Nuclear magnetic moments come in a variety of sizes and, if you will, shapes. The most important nuclei in NMR spectroscopy have a spin of 112. Many others, including 2H and 14N. have a s ~ i nof 1 or more and nossess an electric quadrupole moment resulting from a nonspherical charge distribution within the nucleus. The rate of relaxation is proportional to the quadrupole coupling constant, e2Qqlh (where Q is the electric quadrnpole moment and q is the electric field gradient). The rate of relaxation is thus dependent on the size of Q, which can he quite large, especially in heavier nuclei, and q, which depends on the asymmetry of the molecule. This means that, except for highly symmetric systems (e.g., for 14Nin NH4), quadrupolar nuclei relax very quickly and often have very broad lines.
(a Other Mechanisms There are three i m ~ o r t a nvariables: t the distance ran .- between A and B and the values of their gyromagnetic ratios y~ and r a . where the evromaenetic ratio is a measure of the size of the-nuclear mGnetism. The relaxation rate R1 will increase quickly with decreasing distance and with larger y's. Thus the relaxation time TI will be shortest for nuclei with laree 7's that are near each other. The relatively large nuclear magnetic moment of the proton, alone with its abundance in most organic molecules, means that it is the dominant source of &lear magnetic dipole relaxation. Thus, in 'H NMR spectroscopy the relaxation times are often closelv related to the numher of eeminal protons (Fig. 6),and in 13C spectra the TI'S are letermined largely by the numher of attached protons. The correlation time in this case depends on the rate of molecular tumbling and intramolecular rotations. The derivation of expressions for correlation times is in the province of statistical mechanics, outside of the scope of this article. Nonetheless, two very important considerations must he mentioned: the closer the correlation time is to the resonance frequency (ahout 108 Hz), the more efficient is the relaxation. Thus, medium-sized molecules with molecular weights around 300-700 usually have shorter relaxation times. Small, rapidly tumbling molecules such as l-hromo-2methylpropane or large, slowly moving molecules such as proteins, on the other hand, usually have rather long TI'S. Also. the correlation time varies within a molecule and can 812
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A workine chemist rarelv has to consider the other mechanisms for relaxation: spin rotation, chemical shift anisotroPV. ... or scalar couplins. There are situations where thev are important, and the reader is referred to the specified references as well as the relevant chapter in Wasvlichen's review (7)and Shaw's text (lb) for further information. Appllcatlons of Relaxation Tlmes There are several ways relaxation phenomena can he used to extract additional information a-bout moleculear structure, as discussed t~elow.But it is also important to note that relaxation plays a crucial role in more common methods of NMR spectrum analysis. Nuclei must have the time to relax fullv for accurate determination of snectrum inteerals. If the delay hetween pulses is too short; insufficientrelaxation causes the partial saturation of a resonance, and the resulting decrease in its (relative) intensity may mean the integral is too small (8). In severe cases, the resonance may not he ohserved a t all! Accurate integrals and reliable peak ohservation require a delay of -5Tl between pulses. The distance dependence of nuclear dipole-dipole relaxation means it is often possible to make certain inferences about molecular struct&e on the basis of TI'S. A recent example in organic spectroscopy comes from Chazin and Colehrook (9). For asuhstituted qninoline, as in 1, adramatic increase in the TI of 7'hydrogen occurs on methoxylation; this strongly suggests the CH3 group lies away from the 7'
hydrogen, consistent with a 5'-cis conformation of methoxyl groups.
Spin-lattice relaxation times are also a very effective wav to determine the structure of polyhydride complexes of thk transition metals. Several of these have been shown to contain a weakened but intact H-H bond. and the closeness of the two protons leads to very efficient &ear dipole-dipole relaxation (10).For examole. . . there are two hvdride resonances in the low-temperature l H NMR spectr&ofiridium 2, studied by complexes [IrH3(PPh3)2(7,8-henzoquinolate)]+
(notably X-ray) fail to distinguish different tissue well. MRI methods are very sensitive to the presence of tumor tissue, which has much longer Ti's than the surrounding "normal" tissue. Finally, certain paramagnetic agents-in particular lanthanide complexes-serve to increase the contrast of MRI data hy dramatically shortening the relaxation times of the tissues they penetrate. Conclusions There is much that can be learned from normal NMR spectra-essentially a presentation of the energy states of nuclei in an external marnetic field. But the time dimension. which we enter when we examine the kinetics of nuclear relaxation, provides additional information about the environment of the nucleus that cannot be discerned in information about chemical shift and spin-spin coupling. This can he as simple as the number of geminal hydrogens on a carbon or something as complicated as the existence of tumor cells deep within a patient. Other NMR applications, such as the nuclear Overhauser effect spectroscopy, also have relaxation as part of their framework. Their proper use requires an understanding of relaxation first. Literature Cited
Crabtree et al. (11). One has a TI value at 500 MHz and -80 OC of 390 ms,the other, just 30 ms. The latter resonance is assigned to two H's bound to the metal and to each other, while the former is ascribed to a simple terminal H on the metal. -.
A very different use of relaxation times comes uo in maenetic resonance imaging (MRI) (12).Information about the spatial di~tributionof Ti's and Tl's within a sample can be obtained with sophisticated pulse sequences and-computational techniques. The utility of MRI arises from the subtle but consistent variation of relaxation times for water protons among soft biological tissues. This permits an image of the tissue-essentially a tomographic plot of relaxation times-to be obtained and analyzed where other methods
1. (a) Backer. E. D. H k h Reaolurion NMR: Academic: New Ymk, 1980. lbl Shaw, D. Fourier Tmnsfom N.M. R. Spectmscopy. 2nd ed.; Elmvier: New York, 1984 2. SchwarU, L. J. J. Chem. Educ. 198865.959. 3. (a) Hdl, L. D. Chem. Sor. Re". 1975,4,101. (bl Derome, A. E. Modern NMR Techniques far Chemistry Rosearch: Pergamon: Oxford. 1987. 4. la) Sandrtflm. J. Dynomic NMR Spectroscopy; Academic: New York, 1982. lb) Inngaki, F;Miyazaaa,T. Pmg. NMR Spectres. 1980.14,67. 5. Ejchaut, A; Oleski, P.: Wroblcaraki,K. J.Mopn. Reson. 1986,68,207. 6. (a) Void, R. L.; Weugh, J. S.; Klein, M. P.; Phelpa, D. E. J. Chem. Phys. 1966.48,3831. (b) Reeman, R.; Hill. H. D. W. J. Chem. Phys. 1969. 51, 3140. (4A related, and accasionlrliy useful, technique iaprogrrsaivcsaturation: Freeman, R.:Hill, H.D. W. J Chem. Phyr. 1971.54.3367. 7. Wasylichen, R.E. In NMR Spectroscopy Techniques; Dyboaraki. C.; Lichter, R. L., Eds: Dekker: New York. 1967:Chapter2. 8. Rabensfoin. D. L. J. Chrm. Edue. 1981.61.909. 9. C h i n , w. J.; Colebrook, L. D.con. J. cham. 19k.64.2220. 10. (el Crabtree, R. H.:Hamilton, D. 0.Ado. Orgonomat. Chem, 1988, 28, 299. (b)
Bautim,M.T.;Earl,K.A.:Maltby,P.A.;Monis,R.H.;Sehwitzer,C.T.;Mk.A.(I Am. Chem. Sm. 1988,110,7031. 11. Clabtroe,R. H.:Lavin, M.;Eonneviot.L.J.Am. Chem.Soc. 1986,108,4032. 12. la) Kcsn, D. M.: Smith, M. A. Mopnefic R p ~ o w n c eImogiw: Willisma and W i ~ i n a : Baltimore, 1986. lbl Randal, J. E. Toch. Re". 1988,9111) 58. (cl M a n , J. L. Science 1987,U8.886.
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