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Inversion - recovery

Inversion - recovery

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Spectrum

Figure 5.15

of T1 by curve fitting of the data to an exponential function. The phase correction parameters are first set using a simple 90° pulse acquisition (starting with equilibrium magnetization, along +z) and then applied to a series of inversion-recovery spectra acquired with increasing values of the delay t. For t = 0 we should see an upside-down spectrum, with each peak at its maximum height but inverted. As the delay is increased each peak will become less intense, pass through zero, and finally become positive. At very long t delays the spectrum should look just like a normal spectrum. For each signal in the spectrum, corresponding to a unique position in the sample molecule, the recovery of z magnetization will obey the first equation and the signal intensity after the 90° pulse will follow the second equation as a function of t:

Z magnetization: Mz(t) = Mo + (Mz(0) - Mo) e-T/Tl = Mo + (-Mo - Mo) e-T/Tl = Mo (1 - 2e-T/Tl)

Signal intensity (peak height): I(t) = Iinf (1 - 2e-t/T1)

where Iinf is the signal intensity for very long values of t where Mz has recovered all the way to Mo. We will use peak height as a surrogate for peak area because peak width should not change for a given peak as a function of t .A simple way to estimate T1 without making a plot is to look for a null or near-null condition for a given peak:

I (To) = (1 - 2e-T/T1) Iinf = 0; 1 - 2e-T/T1 = 0; 1 = 2e-T/T1; 0.5 = e-T/T1

Here we use to to indicate the delay time at the null point and set the intensity I(to) to zero. To eliminate the exponential, we take the natural logarithm of both sides of the equation: ln(ex) = x. This is the same as saying that the half-life of T1 relaxation occurs at the time 0.693 times T1. For more accurate T1 measurement, a line can be fitted to a plot of log(I) versus, time, or even better a nonlinear least-squares fit can be performed directly on the time course data of I(t) vs. t, with Iinf and T1 as parameters to be adjusted.

ln(0.5) = -to/Ti; ln(2) = to/T1; Tx = to/ln(2) = to/0.693

75 MHz C inversion-recovery

Figure 5.16

75 MHz C inversion-recovery

Figure 5.16

The data from a 13 C inversion-recovery experiment on sucrose in D2O is shown in Figure 5.16. The experiment was run on a Varian Unity-300, and the data were acquired in an array, a back-to-back series of FIDs acquired one after the other by varying a single parameter—in this case the recovery delay t. The spectra are plotted side-by-side in a "horizontal stacked plot" with the t values ranging from 0.0 (left) to 2.1 s (right). In each spectrum we see the anomeric carbons fructose-2 (quaternary, □) and glucose-1 (CH, ♦) furthest downfield, followed by seven peaks representing singly oxygenated CH carbons (CHOH), and at the upfield end three peaks representing singly oxygenated CH2 carbons (CH2OH). We see that the "CHOH" carbons and C-g1 all pass through zero (Mz = 0) at t = 0.2 s, which means that 1/ = 0.2 s for these carbons (Mz = 0 is halfway between Mz = —Mo, the starting value, and Mz = +Mo, the equilibrium value). Since t/ = 0.693 T1, we have as a rough estimate T1 = 0.2/0.693 = 0.29 s for all of the CH ("methine") carbons. To get more accurate values, the peak heights can be measured and plotted against time, fitting the data to the equation I = /inf (1-2 e-t/T1). This gives T1 values of 2.0 s for the quaternary carbon C-f2 and 0.30 s for the anomeric CH carbon C-g1 (Fig. 5.16, inset). There is a very dramatic difference between having no directly bonded protons (C-f2) and having one directly bonded proton (all CH carbons). The directly bonded protons provide a strong oscillating magnetic field to the 13C nucleus as the molecule tumbles, and those molecules tumbling at the Larmor frequency (75 MHz) are stimulated to relax from the 5 state to the a state. A slight difference in T1 can be observed between the CH carbons and the CH2 carbons: at 0.2 s the three CH2OH carbons at the upfield end of the spectrum are beginning to rise from the noise as positive peaks, whereas the C-g1 and CHOH carbons are still nulled. This means that t/ is a bit shorter for the CH2OH carbons, and T1 = t//0.693 is also shorter. This is because two protons at close range are a bit more effective than one at stimulating T1 relaxation.

Figure 5.17 shows the 1H inversion-recovery experiment for sucrose. Recovery values of 0, 0.1, 0.2, 0.3, 0.5, 0.75, 1.0, and 2.0 s were used, plotting the whole 1H spectrum for each experiment. In the analysis of these data, we use the proton assignments for sucrose derived

from the homonuclear decoupling experiment (see section 5.10 below). The residual water peak (HOD in the D2O solvent, labeled "H") does not show any recovery at all: at 18 Da it is very small and has a very long T1. At the right edge of each spectrum is the singlet peak for the added methanol reference (CH3OH, 32 Da, labeled "M"), which has recovered only about 30% of the way from -Mo to Mo after 2 s, corresponding to a T1 value of about 6 s (e-2 0/6 0 = 0.72). The T1 values for sucrose (C12H22O12, 358 Da) are all much shorter, in keeping with the trend for small molecules of decreasing T1 with increasing molecular size. Site-specific assignments for all of the sucrose resonances are determined below using the homonuclear decoupling experiment. These allow us to look at differences in T1 relaxation within the sucrose molecule. The fastest relaxing protons are the CH2OH protons: H-g6 and H-f6 (0) and H-fi (O). These can be seen rising above the baseline in the t = 0.2 s spectrum as two peaks (H-g6 and H-f6 on the left-hand side, H-fi on the right-hand side). We can estimate the time they cross zero (t/) as about 0.18 s, corresponding to a T1 value of 0.18/0.693 = 0.26 s. The farthest downfield peak, H-g1 (□), crosses zero between 0.3 and 0.5 s, so if we use t/ = 0.4 s we can estimate T1 = 0.4/0.693 = 0.58 s. In the t = 0.5 s experiment, we can see that the H-f3 doublet and the H-g3 triplet are negative, whereas the H-f5/g5 multiplet, the H-f4 triplet, the H-g2 double doublet, and the H-g4 triplet are all very close to zero. Thus T1 can be estimated to be 0.5/0.693 = 0.72 s for all five of these resonances. H-f3 and H-g3 are the last of the sucrose resonances to cross zero with t/ of about 0.75 s and T1 = 0.75/0.693 = 1.1 s:

g1 HOD f3 f4 f5/g5 g6/f6 g3 f1 g2 g4 MeOH

0.58 long 1.1 0.72 0.72 0.26 1.1 0.26 0.72 0.72 6.0 s

Most of these differences can be explained by the number and proximity of nearby protons in the sucrose structure. H-f3 has only one vicinal neighbor, whereas H-g3's two vicinal neighbors are both far away in a fixed anti relationship. This may explain their relatively slow relaxation. H-g1 relaxes faster than the "pack" of -CH-O protons (f4, f5, g5, g2, and g4), perhaps due to its proximity to the H-f1 CH2OH group. This is not evident in the structure diagram, but the strong NOE observed between these two resonances (see below) suggests that a conformation that places them close together may dominate. Finally, the CH2O protons f1, f6, and g6 relax the fastest because of the very close proximity of a geminal proton in a fixed geometric relationship. In all of these cases, we can see how molecular tumbling rotates the vector between a pair of protons, causing each one to experience an oscillating magnetic field due to the nuclear magnet of the other. For those molecules that happen to be tumbling at the Larmor frequency at any particular moment, transitions are stimulated that allow them to return to the equilibrium (Boltzmann) distribution between the aligned (a) and disaligned (ft) states.

What kind of information can we gain from T1 values? First of all, regardless of any physical interpretation, we need to have some idea of Ti values to set up any kind of repetitive scanning experiment because the relaxation delay must allow for reestablishing the equilibrium Boltzmann population distribution before starting the next scan. In the real world we usually do not wait as long as 5T1, and we get a "steady state" for each spin where it recovers only partially and then gets hit again by the pulse. This means that positions in the molecule with long T1 values, such as quaternary carbons, will have lower intensity peaks than positions with short T1. Usually this does not affect proton integration because all protons relax fairly rapidly, but at higher Bo field strengths (600-900 MHz), even the proton T1 values can be long enough that you need relaxation delays of 2 or 3 s to get accurate integration.

To understand the physical meaning of T1 values, we need to consider the mechanism of longitudinal relaxation. As we saw above, the relaxation rate depends primarily on: (1) the number, type, and proximity of nearby nuclear magnets (especially the strong 1H magnets) within the molecular structure and (2) the percentage of molecules tumbling at the Larmor frequency. In a practical sense, a few rules of thumb can be stated. For 13 C nuclei, the effect of directly bonded protons far outweighs the effect of any more distant nuclei, so that the relaxation rate depends pretty much on the number of attached protons: CH3 groups have the shortest T1 values, followed by CH2 groups and CH groups. Quaternary carbons, with no directly attached protons, have very long T1 values because they experience only very weak oscillating magnetic fields and therefore relax very slowly. In fact, the relaxation delay in 13 C experiments is dominated by the need to allow time for the quaternary carbons to relax. The majority of protons in organic molecules are bound to 12C rather than 13C, and therefore experience no intense oscillating magnetic fields from directly bound atoms because 12C nuclei are not magnetic. Protons are affected weakly by a large number of other protons in their immediate vicinity. For small molecules, local motions of flexible groups within the molecule can decrease the rotational correlation time, tc, leading to longer T1 values than the rigid portions of the molecule. In a sense, the flexible portions are behaving like smaller molecules (independent pieces) and thus have a wider distribution of reorientation rates and a smaller percentage moving at the Larmor frequency.

Unpaired electrons produce much stronger magnetic fields than nuclei, so in paramagnetic molecules this effect dominates and greatly reduces T1 values. These strong magnetic fields have a much longer "reach" (20-30 A), so even paramagnetic ions in solution can increase longitudinal relaxation rates (reduce T1) of non-paramagnetic molecules. Transition metal ions such as Cr(III) or Cu(II) can be used in this way to shorten 13C T1 values and speed up data acquisition. In medical NMR imaging, T1 values of water vary greatly in different tissue types due to differing degrees of association with large biological molecules and membranes, providing contrast in the images. Injectable contrast agents in MRI consist of T1-increasing paramagnetic ions complexed to ligands with specific affinities for tissues (e.g., tumors).

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