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Molecular weight Figure 5.14

out" of individual magnetic vectors in the cones is the result of slightly different effective fields experienced at each of the identical nuclei leading to slightly different precession rates. If the molecule is tumbling rapidly, the oscillation in the magnetic field is too rapid to have any effect on the phase of the individual magnetic vectors as they rotate in the x-y plane. Think of a race with a large number of exactly matched runners, each one speeding up and slowing down repeatedly with the same average speed. If the cycle of speeding up and slowing down is very rapid so that many, many cycles occur during the race, the runners will remain in a "pack" and will all cross the finish line at the same time. If the molecule is tumbling very slowly, however, one molecule might start out with a slightly faster magnetic vector and another with a slightly slower vector, and the slow vector would begin to fall behind the fast one during the recording of the FID. In the extreme case (e.g., for a solid sample) the differences in Larmor frequency do not oscillate at all and each spin is "locked" at one part of the tumbling cycle, either slower or faster than the average precession rate. This leads to very rapid loss of coherence and very short T2 values, so short that without special techniques we can not observe NMR peaks in the spectrum at all: they are "broadened" out of existence and fall into the noise baseline. Even for slowly tumbling molecules in solution, it makes a difference over the course of the FID what the orientation of the molecule was at the start of the FID and how fast the molecule is tumbling. Because the molecules are all oriented randomly at any moment, these slight oscillations in Larmor frequency are phase incoherent and there is a random distribution of tumbling frequencies at any one time during the FID, leading to loss of phase coherence. Referring back to the histogram of tumbling rates (Fig. 5.13), we see that the number of molecules tumbling at the very slow rates at the left side of the diagram increases monotonically as we go from small molecules to medium-sized molecules to large molecules. Because it is these slowly tumbling molecules that have the most dramatic "fanning out" due to random differences in vo, this explains why T2 always gets shorter (faster relaxation) as the size of the molecule increases (Fig. 5.14).

What is the significance for a simple NMR experiment of differences in T\ and T2? We are almost always using signal averaging of many FIDs to obtain a good signal to-noise ratio, and we will have to wait until Mz has recovered to near Mo before repeating the acquisition. So Ti determines how rapidly we can obtain NMR data: long Ti values will force us to wait longer to repeat the acquisition and will slow down the overall experiment. The best relaxation delay (RD or D1) is determined by the value of T1—an ideal value would be 5 x T1 for 99% recovery of z magnetization. The T2 value determines the decay rate of the FID: a short T2 corresponds to a rapidly decaying FID and a long T2 value corresponds to a long, ringing FID. The Fourier transform converts time-domain data to frequency-domain data, and because time and frequency are inversely related (s vs. s-1) there are opposite effects in an FID and a spectrum. The shorter the time duration of the FID (short T2, faster transverse relaxation), the broader the resulting peak in the spectrum after Fourier transformation. Conversely, an FID that decays slowly leads to a very sharp (narrow) peak in the spectrum. Because resolution is very important in NMR spectroscopy, especially as we study more and more complex molecules, we always want the narrowest peaks we can get. With a perfectly homogeneous magnetic field (perfectly shimmed magnet) the decay rate of the FID is determined by T2, which is determined by the molecular tumbling processes described above. A perfect world for the NMR spectroscopist would be one in which all T1s are very short and all T2s are very long. Unfortunately, this cannot happen because for any nucleus T1 is always longer than T2. We can see why this is if we consider that T1 is primarily determined by the number of molecules tumbling at the Larmor frequency (vsq), whereas T2 is primarily determined by the number of molecules tumbling at the low frequencies (vzq, Fig. 5.13). For small molecules, the numbers are nearly equal (T1 ~ T2) and for large molecules there are far more molecules tumbling at the low ("zero quantum") frequencies than at the Larmor ("single quantum") frequency (T1 » T2).

These arguments only attempt to capture some general trends. The math leads to a precise dependence of T1 and T2 on the tumbling rates at three frequencies: vzq (difference between the Larmor frequencies of the two nuclei that are close in the molecule), vsq (the Larmor frequency of the nucleus being observed), and vdq (the sum of the two Larmor frequencies). For a pair of protons interacting within a molecule, vzq is on the order of Hz or kHz because vo differs only slightly due to chemical shift differences, and vdq is essentially twice the Larmor frequency (Fig. 5.13). We will come back to this topic and give some more detailed numbers when we look at the NOE and the effect of molecular size on the sign and magnitude of the NOE. Another detail we have ignored is that the term "tumbling rate" implies that molecules rotate at a constant rate in solution as they would in the gas phase and that they behave as rigid bodies. In fact, the v we use in the spectral density function J(v) is really an instantaneous "reorientation rate" describing how rapidly the molecule is changing its orientation with respect to the Bo direction as it is bumped and shoved around by solvent molecules. Futhermore, this reorientation rate is the rate of reorientation of the H-H (or C-H) vector with respect to Bo, which may not be the same as the motion of molecule as a whole. This is an advantage because we can look at local flexibility and conformational change within a large molecule such as a protein by studying relaxation rates at different locations within the molecule. The reorientation of a particular relationship between two nuclei is determined by the motion of the molecule as a whole as well as the sometimes much faster motion of the two nuclei within the molecule. It is this ability to study molecular dynamics at many different timescales (ms, ^s, ns, etc.) that makes solution-state NMR a powerful tool in biology.

5.7.3 Other Relaxation Mechanisms: CSA

There are several other ways besides the dipole-dipole mechanism by which spins can be induced to drop down and reestablish the Boltzmann equilibrium, but we will look at only one. Recall that the chemical shift of a spin within a molecule actually depends on the orientation of the molecule with respect to the magnetic field Bo (Chapter 2, Section 2.6.2). In some cases (e.g., aromatic rings or amide bonds) this variation (chemical shift anisotropy or CSA) can be quite large, and in other cases (e.g., a CH group in a saturated hydrocarbon environment) there is very little dependence on orientation. As far as the NMR spectrum is concerned, the rapid tumbling of a molecule in a solution causes this variation to blur so that on the NMR timescale (roughly milliseconds) only a single, sharp peak is observed at a chemical shift that is the average over all orientations. But chemical shift is nothing more than a perturbation of the magnetic field strength experienced by a nucleus (Beff), so as the molecule tumbles and samples various orientations the Beff field at the nucleus is modulated in a sinusoidal fashion at a rate equal to the tumbling rate of the molecule and with an amplitude proportional to the amount of chemical-shift dependence on orientation (the CSA). Like the oscillating magnetic fields produced by the through-space interaction of the magnetic dipoles of a pair of nuclei (dipole-dipole relaxation), the oscillating magnetic field resulting from CSA can also induce transitions and lead to NMR relaxation. The dependence on molecular tumbling rate and molecular size is exactly the same as that described above for the dipole-dipole effect.

5.8 INVERSION-RECOVERY: MEASUREMENT OF T1 VALUES

The inversion-recovery method is a convenient way to measure T\ values of both XH and 13C nuclei. In a moderately complex molecule (15-30 carbons), the T1 values of all positions in the molecule can be determined simultaneously, with spectral overlap the only limitation. The method is a multiple-pulse experiment in which net magnetization of the sample nuclei is first inverted with a 180° pulse ("inversion") and then allowed to relax along the z axis with the characteristic time constant T1 ("recovery"). The effect of the 180° pulse is to interchange all of the spins between the upper and lower energy levels, so that now the higher energy spin state has a slight excess of population and the lower energy spin state has a slightly depleted population. This causes the net magnetization vector to be turned upside-down so that Mz now equals —Mo. Recovery begins immediately according to the exponential law, with characteristic rate R1 = 1/T1. Because z magnetization is not a directly observable quantity, the recovery period is followed by a 90° pulse that "samples" or "reads" the z magnetization by converting it into observable x-y magnetization (Fig. 5.15 ).

Notice how we diagram a multiple-pulse NMR experiment: the horizontal axis represents time and the vertical axis represents RF amplitude for pulses. The times and amplitudes are not drawn to scale—they are just cartoon representations. 90° pulses are shown as half the width of 180° pulses, and recording of the FID is shown as a decaying signal. Each RF channel is labeled according to the nucleus being irradiated (pulses) and/or observed (FID).

The magnitude of the FID signal that results from this x-y magnetization (and the peak height in the spectrum) should be directly proportional to the sample's z magnetization just before the 90° pulse. By repeating the experiment with different time delays after the 180° pulse, we can monitor this return of z magnetization to equilibrium and determine the value

INVERSION-RECOVERY: MEASUREMENT OF T1 VALUES

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