The NMR analogy

An analogy can be made between the relaxation processes in a two level system involving the relaxation of a spin 1/2 system following application of a magnetic field pulse and evolution of the wavefunction in two level system following excitation by an electromagnetic field. We begin by describing the dynamics of a two level (e.g. spin-1/2) system in an external magnetic field (B0) following a 90-degree radiofrequency pulse (B1). The analogy will then be developed by observing that population relaxation (T1) and loss of coherence (T2) are characteristic relaxation times in both kinds of spectroscopy.

The two level system in NMR is produced by the application of a static magnetic field B0. There is a net alignment of spins in the presence of the external field B0 and this produces a net magnetization M0. M0 results from the fact that more spins are aligned with the field than against the field.

The NMR experiment uses a pulse of radiofrequency radiation orthogonal to the static field B0 shown in purple in the Figure. This is illustrated below. 

The above diagram shows how the magnetization is rotated into the x,y plane by the B1 pulse. In the example shown the duration of the B1 pulse is just long enough to cause a rotation of the magnetization vector into the x,y plane. This is called a 90o or p/2 pulse. Once in the x,y plane the magnetization vector M0 begins to rotate due to the static field. In addition to rotating the magnetization vector begins to dephase. That is different nuclei precess at different rates and there is a spreading in the in-plane component of the magnetization.

The red arrows show a spreading of the magnetization in the x,y plane. In NMR spectroscopy this process occurs through mutual spin flips and is known as spin-spin relaxation. It is described by the Bloch equations as a single exponential process. This is also known as the transverse relaxation process. We can think of the transverse relaxation process as a loss of coherence. Initially, all of the spins have the same phase with respect to one another. After dephasing they have lost this phase relationship or coherence. Eventually, the spins will be completely randomized in the x,y plane. The time constant for this process is T2.

In addition to transverse relaxation there is a longitudinal relaxation process that results in the return of alignment of the magnetization along the z-axis. The characteristic time for this process is T1. Again the T1 time represents a time constant for an exponential process.

We can also think of longitudinal relaxation as change in population. As the spins return to their equilibrium alignment the z magnetization grows. After the longitudinal relaxation process is complete, the spins have returned to their equilibrium distribution. The population relaxation process in NMR is referred to as a spin-lattice relaxation. Spin-lattice relaxation is almost always slower than spin-spin relaxation and so the T2 process dominates.

Experimentally, the observed magnetization will decrease with due to both T1 and T2 processes. The time-dependent magnetization is measured by the receiver coil along y. The observed magnetization will decrease due to both dephasing processes that tend to smear out the magnetization in the x,y plane and population relaxation processes that tend to tip the magnetization up towards z.

 

The figures demonstrate both processes. The shorter purple vector shown in the x,y plane for the T2 process indicates that the spreading out of spin in the x,y plane (dephasing) results in a smaller magnetization vector in the plane. The black vector represents the initial vector after the p/2 pulse. The blue vector in the bottom part of the Figure shows how magnetization begins to tip out of the x,y plane and have a z projection. In reality, both of the these processes occur simultaneously and we have an overall decay called T2. The relationship between T1 and T2 is: 

where the time denoted T2* is the pure dephasing time. If we include the precession about the static magnetic field at frequency wL we have the following expression.

We specify the y-coordinate here because the receiver coil detects the magnetization along y. Thus, as the vector rotates above the signal at the coil oscillates at the rotation frequency (which is wL) and decays exponentially due to both the transverse (T2) and longitudinal (T1) processes.

This is known as the free-induction decay (FID). The Fourier-transform of this is a Lorentzian function centered at wL.

The optical analogy

In optical spectroscopy we can think of the same two types of processes occuring. After excitation from the ground state to the excited state the coherence or phase relationship of the ground state and excited state wavefunctions will be maintained with a time constant T2. The population will relax from the excited state back to the ground state with a time constant T1. The relationship between T1, T2* and T2 is the same as in NMR spectroscopy. The FID in optical spectroscopy is an extremely rapid oscillation of the transition dipole moment produced by a very short electromagnetic field pulse. The optical FID is not observed directly except in femtosecond spectroscopy experiments. However, the same Fourier transform relationship exists and so in principle the frequency and linewidth of an observed absorption band are determined by the frequency and decay of a damped sinuisoidal function such as that shown above. In optical spectroscopy as in NMR it is frequently the coherence time, T2 that dominates the observed linewidth. In other words, coherence between the ground and excited states that is impressed by the applied electromagnetic field pulse decays much more rapidly than the radiative and non-radiative decay processes of the molecule that contribute to population relaxation.