Magnetic relaxation and its uses

Any excited magnetic moment relaxes back to equilibrium, the z axis. There are two components of this relaxation for isotropic systems in the absence of chemical exchange: longitudinal or spin-lattice (T₁) and transverse or spin-spin (T₂). The rotational relaxation time (T_1ρ) is the same as T₂ unless there is chemical exchange or anisotropy. The parameter T₂* is the time constant of dephasing and is caused by a combination of relaxation and magnetic inhomogeneity. Please use our NMR service to measure NMR relaxation times.

Most relaxation times observed in routine NMR are between 0.1 and 10 seconds. Longer relaxation times are observed in the absence of oxygen for deuterated solvents, quaternary carbon signals, heavier spin ½ nuclei and in the gas phase. Shorter relaxation times are observed when there is medium to fast chemical exchange, paramagnetism and for quadrupolar nuclei. Note that the oxygen in the air is paramagnetic and speeds up relaxation even in the normal concentrations found in solution. Therefore, when investigating the relaxation properties of a compound with long relaxation times, the sample must be degassed with an inert gas or sealed under vacuum. The relaxation time is also magnetic field dependent so the relaxation time measured on a 300 MHz NMR instrument will be different than that measured on a 600 MHz instrument.

Although the relaxation time is a property of the nucleus that reflects its properties and environment, it is mostly used to decide on the length of the relaxation delay between acquisitions. For quantitative work this needs to be at least five times T₁ to achieve a 1% accuracy. For maximum sensitivity within a limited time for a regular 1D experiment, the repetition rate should be T₁ with a pulse angle of 68°. The mixing time in NOESY and EXSY experiments is dependent on T₁ while the mixing time in ROESY and TOCSY is dependent on T_1ρ. In principle, the preferred acquisition time is dependent on T₂* although in practice this is determined by observing the fid directly.

Relaxation times can be used for assignment. In general, relaxation is faster when signals are strongly coupled. For example, quaternary carbons (not attached to protons) relax slower than carbons attached to protons. This can be easily observed in the ¹³C NMR because the slower relaxing nuclei give sharper signals as shown in the fig. 1 below.

Fig. 1. Part of the ¹³C NMR spectrum of 12,14-ditbutylbenzo[g]chrysene showing the difference between proton attached and non-proton attached carbons

Under usual ¹³C NMR acquisition (not overlong relaxation delay) and processing (exponential apodization with 1 Hz line broadening) conditions as shown in the spectrum below, the carbons attached to protons appear higher than those not attached to protons due to the difference in relaxation times in addition to NOE factors (fig. 2).

Fig. 2. Aromatic Part of the ¹³C NMR spectrum of 12,14-ditbutylbenzo[g]chrysene

Relaxation is faster when there is steric hindrance and when motion is otherwise restricted. An example of the use of relaxation effects in this manner is given for the assignment of 12,14-ditbutylbenzo[g]chrysene.

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Longitudinal or spin-lattice relaxation, T₁

Longitudinal or spin-lattice relaxation (T₁) is the mechanism by which an excited magnetization vector returns to equilibrium (conventionally shown along the z axis). This is always at least slightly slower than transverse relaxation. The inversion recovery (T₁) pulse sequence (fig. 3) yields a signal of intensity . The intensity is zero at T₁ln(2) (0.693T₁) so the value of T₁ can be measured by running single experiments, changing the value of τ until a null intensity is found. If the intensity is positive then τ has to be reduced and if it is negative then τ has to be increased. The value of T₁ is 1.443τ_null. In the example below (figs. 4, 5) of the methyl of ethylbenzene (0.1%) in CDCl₃ (under vacuum), τ_null is 6.59 s so T₁ should be 9.51 s.

Fig. 3. Inversion recovery pulse sequence for measureing T₁

Fig. 4. Inversion recovery experiment for ¹H of ethylbenzene (0.1%) in CDCl₃ at 400 MHz. The residual water peak on the left relaxes slightly slower than the methyl multiplet on the right.

Fig. 5. Inversion recovery curve for the methyl protons of ethylbenzene (0.1%) in CDCl₃ at 400 MHz

However, experimental artifacts compromise the accuracy of the single scan method. For more accurate results, experiment is repeated many times with different values of τ and the resulting intensities used in a non-linear fit to find the value of T₁. In the above example, the fitted T₁ value is 10.4 s.

The relaxation of different peaks in a multiplet may appear different due to spin-coupling interactions. For most proton spectra this effect is on the order of a few percent. For greatest accuracy the peaks in a multiplet should be measured separately. The central peak(s) in a multiplet have slightly longer relaxation times than the outer peaks.

If the sample is very concentrated then the relaxation time will appear shorter than it really is due to saturation. In such a case, off-tune the probe, recalibrate the pulse-widths and repeat the experiment.

If the T₁ is very long and/or the sample has low viscosity and/or the experiment is being carried out at elevated temperatures, then convection and diffusion is likely to affect the results. The effects of convection and diffusion is evident as a biexponential decay. In the above example, the fitted curve (fig. 5) does not match the last three points well, suggesting such a problem. A double exponential decay fits better (fig. 6) indicating a T₁ closer to 15 s. It is often possible to measure relaxation times of hundreds of seconds in spinning liquid samples.

Fig. 6. Double exponential fit to the inversion recovery curve for the methyl protons of ethylbenzene (0.1%) in CDCl₃ at 400 MHz

Convection effects can be reduced by spinning the sample and using a narrower tube. In many cases such as above or in gas phase NMR it may not be possible to prevent diffusion and convection. In extreme cases, the sample should be removed from the magnet for at least T₁ which may be several hours. The sample can then be reinserted and a regular 1D acquisition with a very small pulse angle applied every minute or so for a period of a few T₁. The build-up of intensity follows the equation where τ is the time since insertion. In the case of ³He gas at a pressure of two atmospheres, the T₁ appears to be about 1 s when measured by inversion recovery due to diffusion out of the receiver coil area while the true value is a little more than 1000 s (fig. 7).

Fig. 7. Signal buildup of ³He gas as a function of time after insertion into the magnet on a 400 MHz instrument (304 MHz ³He). The T₁ is just over 1000 s.

The long relaxation delay required by the inversion recovery experiment is a problem when sensitivity is low and when the relaxation time is long or when there is moderate convection as in the example of ethylbenzene. In such cases, the DESPOT method (driven equilibrium single pulse observation of T₁ also known as the variable nutation angle method) can be used. The intensity is plotted as a relation of pulse angle (fig. 8).

Fig. 8. DESPOT curve for the methyl protons of ethylbenzene (0.1%) in CDCl₃ at 400 MHz

Two or more regular 1D experiments are carried out with different pulse angles. There is no need for a long relaxation delay but the experiment must start with dummy scans that continue for at least 5T₁. The pulse angle must be calibrated accurately taking into account the rise time of the transmitter, typically 0.8 μs. A plot (fig. 9) of I cosec(φ) against I cot(φ) gives a straight line of slope where φ is the pulse angle and T_r is the time between acquisitions. In the case of the methyl of ethylbenzene, the T₁ is 17.4 s.

Fig. 9. Linear DESPOT plot for the methyl protons of ethylbenzene (0.1%) in CDCl₃ at 400 MHz

If only two angles are used then the relaxation time can be expressed analytically as . If the two angles, φ₁ and φ₂, are 45° and 90° respectively then this simplfies to .

In the case of the methyl protons of ethylbenzene, I₁ = 3.4469, I₂ = 3.1118 this gives at T₁ of 15.0 s which is reasonably close for a two point experiment and certainly good enough if the purpose is to select a reasonable relaxation delay in another experiment.

In anisotropic systems, the spin-lattice relaxation also becomes anisotripic. The isotropic component is then termed the Zeeman relaxation whose time constant is T_1Z. There is also a dipolar relaxation, T_1D, and, for quadrupolar nuclei, a quadrupolar relaxation T_1Q. As this website is mainly concerned with isotropic solution NMR, the anisotropy of T₁ will not be discussed further.

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Transverse or spin-spin relaxation, T₂

Tranverse or spin-spin relaxation (T₂) is the mechanism by which the excited magnetization vector (conventionally shown in the x-y plane) decays. This is always at least slightly faster than longitudinal relaxation. The magnitude of the magnetic moment in the x-y plane decays according to . The CPMG (Carr-Purcell-Meiboom-Gill) (T₂) experiment (fig. 10) yields signals of intensity where τ is the total evolution time (2Δ). This pulse sequence is best for singlets but is very sensitive to instrumental phase and enviromental instability for long relaxation times.

Fig. 10. Spin-echo pulse sequence for measuring T₂

For multiplets a multipulse sequence (fig. 11) that compromises between a T₂ and a T_1ρ measurement can be used while suppressing the coupling artifacts. The mixing time, τ, is 4nΔ. The value of Δ in the pulse sequence should be much shorter than the reciprocal coupling constant, , but long enough that the sample should not heat up significantly. The use of the inserted pulse train is to suppress the effect of coupling at the expense of providing a combination of the T₂ and T_1ρ value. In most cases of isotropic liquids, T₂ is very similar to T_1ρ. A value of Δ of 10 ms is usually appropriate, much smaller than the reciprocal coupling constant, not overheating the sample and keeping the T_1ρ to a minimum. The experiment is repeated many times with different values of τ and the resulting intensities used to find the value of T₂. Like the simple T₂ pulse sequence this one is still very sensitive to environmental changes during acquisition, especially for long relaxation times. If the sample is very concentrated then the relaxation time will appear shorter than it really is due to saturation. In such a case, off-tune the probe, recalibrate the pulse widths and repeat the experiment.

Fig. 11. Spin-echo pulse sequence for measuring a mixture of T₂/T_1ρ

Figs. 12 and 13 show the measurement of the combined T₂/T_1ρ for the methyl protons of ethylbenzene. This was measuremed to be 16.5 s.

Fig. 12. Spin-echo experiment for the proton spectrum of ethylbenzene (0.1%) in CDCl₃ at 400 MHz. The residual water peak on the left relaxes faster than the methyl on the right.

Fig. 13. Exponential decay curve for the methyl protons of ethylbenzene (0.1%) in CDCl₃ at 400 MHz.

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Rotating-frame spin-lattice relaxation, T_1ρ

Rotating-frame spin-lattice relaxation, T_1ρ, is usually the same as T₂ for isotropic solutions but may deviate when there is chemical exchange. It is measured using a spin-lock pulse sequence (fig. 14) to yield a signal intensity of where τ is the spin-lock pulse length. The experiment is repeated many times with different values of τ and the resulting intensities used to find the value of T_1ρ. If the sample is very concentrated then the relaxation time will appear shorter than it really is due to saturation. In such a case, off-tune the probe, recalibrate the pulse widths and repeat the experiment

Fig. 14. Pulse sequence for measuring T_1ρ

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Dephasing time, T₂*

The dephasing time, T₂*, also known as the effective transverse relaxation time, is a combination of transverse relaxation and magnetic field inhomogeneity. In a perfectly homogeneous magnetic field T₂* = T₂ but T₂* is shorter when the field is inhomogeneous, i. e., when the shimming is not perfect. The dephasing time is the reciprocal of π times the line-width at half height (w_½): .

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