T1 and T2 relaxation

In NMR, relaxation is the process by which magnetization returns to equilibrium. Relaxation is important because it determines how long to wait between scans, how quickly signals decay, and the width of peaks in a spectrum. In the vector model relaxation is separated into T₁ relaxation along the z-axis and T₂ relaxation in the x-y plane.

The vector model is a visual depiction of the net magnetization produced by the 10²⁰ or so identical nuclei in an NMR sample. It represents the average of the magnetization produced by an ensemble of nuclei that give rise to a single peak in a spectrum. At equilibrium in a magnetic field the net magnetization of the NMR-active nuclei will be aligned with the +z-axis. After a 90^o pulse the net magnetization vector will lie in the xy-plane and will start to relax back towards the z-axis.

The figure below shows the net magnetization vector after a 90^o pulse and resolves it into two vectors; one in the xy-plane and another along the z-axis. As the magnetization relaxes towards equilibrium the vector in the xy-plane becomes smaller as the vector along the z-axis increases. Eventually, the vector in the xy-plane becomes zero and the vector on the z-axis reaches its equilibrium magnitude. The increase in net z-magnetization is known as longitudinal relaxation and occurs with time constant T₁. This process is mainly caused by the interaction of the spins with their surroundings and so is also known as spin-lattice relaxation.

The T₁ time constant is what determines how long to wait in between scans when recording an NMR spectrum. To obtain maximum signal one needs to wait until the magnetization has returned to equilibrium along the z-axis. Relaxation closely approximates an exponential process so after waiting one time constant, 63.2% (i.e. 1 - e^-1) of the magnetization is recovered. T₁s are typically in the range of 1-10 s, so a relaxation delay of two seconds is a reasonable value.

After a 90^o pulse the net magnetization vector is left rotating in the xy-plane. Since this vector is the sum of the magnetization of many different spins, if these spins experience slightly different magnetic environments (e.g. shimming or temperature variations over the sample volume) their rotation frequencies will be slightly different. This will cause the individual components of the net magnetization to spread out and a reduction in the size of the net magnetization in the xy-plane. This process is known as transverse relaxation and occurs with time constant T₂. Since it is mainly caused by the interaction of the spins it is also known as spin-spin relaxation.

The T₂ time constant determines how quickly the FID decays and thus the width of the peaks in an NMR spectrum.  Faster T₂s produce broader peaks. T₂s can range from 10s to 10ms.  Large molecules with short T₂s produce FIDs that decay rapidly and so acquisition times should be limited to prevent adding noise to the acquired signal.

The rate of T₁ and T₂ relaxation depends on the magnetic field, ω, and the correlation time of the molecule, τ_c. T₂ decreases as ωτ_c increases, but T₁ passes through a minimum around ωτ_c = 1. The T₁ minimum is what separates "small molecule" NMR from biomolecular NMR. Most small molecule samples are on the left of the T₁ minimum where T₁≈ T₂ and one does not have to worry about signals decaying too quickly. For large biomolecules where T₁ > T₂, transverse relaxation becomes the dominant factor in designing experiments.