Faster Relaxation Time Measurements in Solids

T₁ relaxation times are typically measured with the inversion recovery technique. In this method the magnetization is inverted and its recovery is monitored as a function of time. For nuclei with long T₁'s, the measurements are very time consuming as a recycle delay of at least five times T₁ must be used between scans. Typical T₁'s for ¹³C in the solid state range from several seconds to tens of minutes, so their direct measurement via the inversion recovery method could be prohibitively long.

High resolution ¹³C solid state NMR spectra of solids are routinely measured with cross polarization and magic angle spinning (CPMAS) in order to take advantage of the signal enhancement due to magnetization transfer between the abundant protons and isotopically dilute ¹³C nuclei. Additionally, the recycle delay needed for this measurement depends on the T_1ρ of the protons rather than the T₁ of the ¹³C. Proton T_1ρ's are typically shorter than ¹³C T₁'s by at least an order of magnitude, so many more scans can be collected per unit data collection time compared to a direct one-pulse measurement.

One might think that ¹³C T₁'s can simply be measured with cross polarization using a simple inversion recovery scheme by applying a 90° pulse to the ¹³C spins immediately following the contact pulse and then following their recovery over time. This method would have both the advantages of signal enhancement due to CP and more scans per unit time. The problem however, is that the enhanced magnetization of the inverted spins relaxes back to its unenhanced Boltzmann value and not its enhanced value. So, in order to measure the T₁, the direct ¹³C magnetization would have to be measured first (without CP) which would be very time consuming. This difficulty can be eliminated with the pulse sequence introduced by Torchia in 1978^* shown in the figure below.
This sequence uses a simple two step phase cycle to subtract out the effect of the direct ¹³C Boltzmann magnetization. The first part of the sequence uses a (90°_-y) pulse to return the CP enhanced magnetization to the z axis. The decay of the enhanced magnetization down to its Boltzmann value is followed using a (90°_x) pulse with detection of signals on the -y axis. The second part of the sequence uses a (90°_y) pulse to put the CP enhanced magnetization on the -z axis. The recovery of the enhanced inverted magnetization back to its equilibrium Boltzmann value is followed using a (90°_x) pulse with detection of signals on the y axis. The addition of the first and second parts of the experiment by way of the phase cycle allows for a simple calculation of the ¹³C T₁ with both the advantages of CP enhancement and the ability to collect more scans per unit time. An illustration of this method is shown in the figure below where the ¹³C T₁'s of glycine were measured. (The small peak in the spectrum is a spinning sideband of the carbonyl carbon)

* D.A. Torchia, J. Mag. Res. 30, 613, (1978).

T1 Anisotropy

In the solid state, in the absence of magic angle spinning, the frequency of NMR lines depends on the orientation of the molecules with respect to the static magnetic field. For powdered samples, all orientations are represented in the sample and one obtains a broad envelope of peaks resulting from all possible orientations. Such broad resonance are called powder patterns and are said to be anisotropic. The frequency is not necessarily the only orientation dependant parameter. In some cases, the T₁ relaxation time also depends on the orientation of the molecules with respect to the magnetic field. In such cases the T₁ is said to be anisotropic. In contrast to the NMR resonances in solution which are characterized with a single T₁, the powder pattern can be characterized with many different T₁ relaxation times. Furthermore, the presence or absence of anisotropy in the T₁ can help discriminate between certain types of molecular motion. An example of an anisotropic T₁ is illustrated in the figure below for the wide line ²H inversion recovery spectra of acetone-d6 trapped in an organic inclusion compound. The line shape indicates that the acetone molecules undergo both fast methyl group rotation and fast two-fold flips about he carbonyl bond. One can see that the entire powder pattern does not have the same T₁ as the line shapes are a function of the inversion recovery delay, tau. The T₁ depends on the frequency within the powder pattern which in turn depends on the orientation of the molecules with respect to the magnetic field.

T2 vs T2*

The T₂ relaxation time is the exponential decay constant for transverse magnetization (i.e. magnetization in the xy plane). In principle, one should be able to measure the T₂ relaxation time by applying a 90 degree pulse to create transverse magnetization and measuring the decay constant of the FID. In reality however, the decay rate of the FID is also affected by such things as magnetic field homogeneity, unresolved coupling, temperature gradients.....etc. Because of these effects, the decay constant of the FID is called T₂* rather than T₂. T₂* is an instrumentally dependant parameter and it determines the line width of an NMR resonance. T₂, on the other hand, is a physically meaningful parameter independent of field inhomogeneity, J coupling and other factors. It is measured with a 90-tau-180-tau-FID pulse sequence as a function of tau. T₂ is always greater than or equal to T₂*. The figure below compares T₂ to T₂* for the proton resonance of CHCl₃ for the lineshape sample in a reasonably well shimmed 300 MHz magnet. The line shape specifications were 0.3 Hz (at 50%), 2.9 Hz (at 0.55 %) and 6.2 Hz (at 0.11%). Even in a well shimmed magnet, the T₂ for CHCl₃ is nearly 19 times longer than the T₂*.

T1 Measurements and Estimation

T1 relaxation time measurements are usually done with a simple 180 -tau -90, inversion recovery pulse sequence (see figure). Tau is varied from a small value to a large value and a nonlinear regression is carried out to fit the best T1 value.These measurements can be very time consuming. One can get a reasonable estimate of the T1 much more quickly. Follow these simple steps:

1. Call up the pulse sequence "t1ir1d" (Bruker) or "s2pul" (Varian).

2. Set p1 and p2 (Bruker) or PW and P1 (Varian) to the 90 degree and 180 degree pulses , respectively.

Set the recycle delay, d1 (Bruker and Varian) to something you believe is much longer than the T1.

3. Set tau to a very small value (3 microseconds for example). Tau is d7 on a Bruker spectrometer or d2 on a Varian spectrometer.

4. Collect a spectrum and phase it such that all peaks are negative (one scan is often enough for protons). Store the phase correction.

5. Repeat step 3. increasing d7 (Bruker) or d2 (Varian) until the peak of interest is nulled. If the peak is negative, tau is too short. If it is positive, tau is too long.

6. The T1 of the peak of interest is the tau value for the null divided by the natural log of 2.