Basic Concepts
T₂ - Relaxation time

Spin-Spin relaxation or transverse relaxation - T₂
Spin-Spin relaxation mechanisms
Scalar relaxation
Measurement of T ₂ relaxation process | Hahn echo | Carr-Purcell-Meiboom-Gill sequence| Diffusion

Spin-Spin Relaxation time T₂ (transverse)

The existence of relaxation implies that an NMR line must have a width. The smallest width can be estimated from the uncertainty principle. Since the average lifetime of the upper state cannot exceed T₁, this energy level must be
broadened to the extent: h/T₁. This means that line width at half-height of the NMR line must be at least: 1/T₁.

In Solution NMR, very often T₂ and T₁ are equal (small/medium molecules and fast tumbling rate).
For solids, T₁ is usually much larger than T₂. The very fast spin-spin relaxation time provide very broad signals.

There are other processes that can increase the line width substantially over the expected value extracted from T₁ analysis. T₂ represents the lifetime of the signal in the transverse plane (XY plane) and it is this relaxation time that is responsible for the line width. The "true" line width on an NMR signal depends on the relaxation time T₂ (line width at half-height=1/T₂).

In fact, by measuring the experimental line width, one can in principle determine the T₂ relaxation time. Unfortunately, the experimental line width depends also on the inhomogeneity from the magnetic field. The experimental relaxation rate, extracted from the line width, is called 1/T₂* (1/T₂*=1/T₂ + 1/T₂(inhomogeneity)). The inhomogeneity factor is more critical for nuclei with higher frequency (more critical for proton than for Carbon or Phosphorus). The reason behind this statement can be explained as follows: imagine that identical nuclei (identical chemical shift) in a NMR tube are submitted to slightly different field (inhomogeneity from the magnet). The resonance frequency for each individual nuclei is described by the equation: . If the field vary through the NMR tube, so do the observed frequency, broadening the NMR line. This broadening would affect more the proton (higher ) than lower frequency nuclei. The broadening is in fact directly proportional to the frequency.

Spin-Spin relaxation mechanisms

The same mechanism that are active in Spin-Lattice are active for spin-spin relaxation. You can consult them on the T₁ description. We will describe here the mechanisms that can bring extra broadening to the peak.

Scalar relaxation (dynamic NMR and scalar coupling with quadrupolar nuclei)

Scalar relaxation occurs when two spins interact through bond (electron mediated) - J coupling.

Spin I can feel fluctuating in the field from spin S in two ways:

The scalar coupling J can be time dependent due to chemical exchange
If nuclei S is jumping in and out of a site in which it is coupled to spin I, splitting will collapse at high rate. At intermediate rate, the line will broaden due to partial scalar coupling (this can be observed on exchangeable protons like OH or NH)
the spin S can be time dependent due to rapid relaxation of spin S (like in quadrupolar nuclei)

Relaxation induced by Quadrupolar nuclei

If the relaxation time of the quadrupolar nuclei is rapid (1/T₁(S) >> 2*pi*J), nuclei I will not couple to the quadrupolar nuclei.
e.g. ¹H next to ¹⁴N or ¹¹B (T₁ => 10-100 msec) => produce broadened lines
e.g. ¹H next to ³⁵Cl (T₁ => 1 usec) => insignificant broadening

Scalar coupling with rapidly relaxing quadrupolar nucleus can be determined based on T₁ and T₂ analysis.

Measurement of T₂ relaxation process

In non-viscous liquids, usually T₂ = T₁. But some process like scalar coupling with quadrupolar nuclei, chemical exchange, interaction with a paramagnetic center, can accelerate the T₂ relaxation such that T₂ becomes shorter than T₁.

In principle T₂ can be obtained by measuring the signal width at half-height (line-width = (pi * T₂)^-1

However the line width for non-viscous liquids is most often dominated by field inhomogeneity. Fortunately, the dephasing of spins isochromats resulting from field inhomogeneity is a reversible process: it can be refocused by using a 180 degree pulse inserted in the center of an evolution time.

Hahn Spin-Echo

The Hahn echo is constituted first by a 90 degree pulse that flips the magnetization in the XY plane.
During the first tau delay, the magnetization evolve according to it's chemical shift (and field inhomogeneity)
Then a 180 pulse is applied. This 180 degree pulse inverts the magnetization. (This pulse can be applied along the X or the Y axis).
Following the inversion pulse, another tau delay is applied. During this delay, the magnetization refocus:

for example, if the nuclei with chemical shift 1 have had the time to precess clockwise by 15 degree during the first tau delay, it will also precess by the same amount after the inversion pulse as the identity of the nuclei have not been changed.
Similarly, if field inhomogeneity is present for the first tau delay, the same inhomogeneity is present after the inversion pulse and will influence the precession in the same manner allowing us to refocus this unwanted effect at the end of the second delay.
This will give rise to an echo after the 2*tau delays. The size of this echo will only be affected by the spin-spin relaxation processes.

At the end of the second delay, the echo will be lined up:

along the Y axis if the 180 degree pulse was applied along the Y axis,
along the -Y axis if the 180 degree pulse was applied along the X axis.

Carr-Purcell-Meiboom-Gill sequence

The Carr-Purcell-Meiboom-Gill (CPMG) sequence shown above is derived from the Hahn spin-echo sequence. This sequence is equipped with a "built-in" procedure to self-correct pulse accuracy error. For a description of the first half of the sequence, look above in the Hahn echo section. In the picture above, only the first shift is shown but with field inhomogeneity. The letters f and s means that those nuclei affected by the inhomogeneity of the magnet precess faster and slower than the chemical shift respectively.

If the first inversion pulse applied is shorter (e.g. 175 degree) than a 180 degree pulse, a systematic error is introduced in the measurement. The echo will form above the XY plane (e.g. 5 degree) and therefore the signal will be smaller than expected.
To correct that error, instead of sampling immediately the echo, a third tau delay is introduced, during which, the magnetization evolve as before but slightly above the XY plane (see figure above).
If the second inversion pulse, also shorter than a 180 degree pulse (e.g. 175 degree), is applied, as the magnetization is already above the plane, this shorter inversion pulse will put the magnetization exactly in the XY plane.
At the end of the last tau delay, the echo will form exactly in the XY plane self correcting the pulse error!

The following animation.presents those two echo techniques discussing the effect of scalar coupling.

Measuring diffusion

This experiment can also be used to measure diffusion in the NMR tube by using gradients.
To have more information about the gradient version of that experiment
To learn more about this topic, you can consult the following animation

| Chemistry | NMR internal site | NMR on Avance spectrometer ]

Basic Concepts T2 - Relaxation time

Spin-Spin Relaxation time T2 (transverse)