Relaxation times and Linewidths in NMR Spectra
We have already established that a bulk sample containing numerous spin ½ nuclei in a static magnetic field possesses a net magnetisation in the z direction (as defined by the direction of the magnetic field). Application of a brief electromagnetic pulse in the xy plane produces a magnetisation component in the xy plane which decays with time.
There are two reasons that this xy component shrinks with time, and they both reflect the fact that the system is not in equilibrium. The equilibrium situation is the one the system was in initially, with the magnetisation vector aligned parallel to the static magnetic field, in the z direction, and the tendency of the system is to return to this state. The process of returning to equilibrium is called spin relaxation.
The magnetisation vector in the xy plane immediately after a 90º pulse has equal numbers of α and β spins, but at equilibrium the populations of the two states are in a Boltzmann distribution, with more of the lower energy state than the upper. The populations of the two states revert to their equilibrium values exponentially, and as they do so the z component of the magnetisation returns to its equilibrium value, MEqm. The reversion of the z component to its equilibrium value is also exponential, and obeys the following equation:
where T1 is a constant with dimensions of time known as the longitudinal relaxation constant. T1 depends upon the experimental conditions, e.g. the identity of the nuclei involved.
The relaxation process involves giving up energy to the surroundings (the ‘lattice‘) so is termed spin-lattice relaxation, and T1 is commonly referred to as the spin-lattice relaxation time.
Spin-lattice relaxation is caused by the random motion of molecules within the sample. This motion generates fluctuating local magnetic fields as the magnetic nuclei tumble in solution, and some components of these fluctuating fields will be at the resonance frequency of the nuclei. These components of the magnetic fields can induce transitions between the two different states, allowing them to return to their equilibrium populations.
A second aspect of spin relaxation is the loss of coherence in the xy component of the magnetisation vector as the spins precess at different rates around the z axis. (Recall that the different rates of precession arise because of the slightly different Larmor frequencies of nuclei at different positions in a molecule.) The component of the magnetisation vector in the xy plane is large immediately after a 90º pulse, as all of the spins are oriented together. However, the different rates of precession lead to the spins spreading out around the z axis until they are uniformly distributed. This is the equilibrium situation, and the component of the magnetisation in the xy plane is now zero. The randomisation of spin directions occurs exponentially, with a time constant called the transverse relaxation time, T2:
This relaxation is termed spin-spin relaxation, as it involves the relative orientations of the spins. Thus T2 is sometimes referred to as the spin-spin relaxation time.
Spin-spin relaxation is a contributor to the broadening of spectral lines, as it reduces the lifetime of the excited state. Given that the y component of the magnetisation decays according to the above equation, then the width of a spectral line at half height is given by the following relation:
The interconversion of different conformations of a molecule can alter the appearance of a spectrum if magnetic nuclei are shifted between different magnetic environments by the interconversion. One of the simplest examples of such a molecule is dimethylnitrosamine, which can interconvert between two identical conformers as shown:
The resonance frequency of a methyl group depends upon whether it is cis or trans to the nitroso (N=O) group. At low temperatures, when the rate of interconversion is slow, two equally intense resonance lines are indeed observed in the 1H spectrum of the molecule. As the temperature (and thus the rate of conformational exchange) is increased, these two lines first broaden, then shift towards each other until eventually they merge into one wide, flat-topped resonance. If the rate of exchange is increased still further, a sharp resonance at the mean of the two original resonance frequencies is produced.
The maximum broadening occurs when the lifetime, τ, of a conformation gives rise to a linewidth that is comparable to the difference, δ ν, between the resonance frequencies for the two positions. The two lines are then so broad that they blur into each other to form one very broad line.
Coalescence of the two lines into one occurs when the following equation is satisfied:
Note that a similar reason lies behind the fact that the hydroxyl protons in alcohols rarely show any splitting or split the resonances of any other nuclei due to spin-spin coupling. This is because in many solvents, rapid proton exchange between the hydroxyl group and the solvent can take place. This occurs without regard for the spin of the protons involved, so the hydroxyl proton may switch rapidly between an α and a β spin, meaning that any splitting it might cause is averaged out over time. The exchange is usually rapid enough that no splitting can be detected in the spectrum.
When we speak of processes being slow or rapid in NMR, we are talking on the NMR timescale, which means relative to the difference in resonance frequencies between the exchanging nuclei, δ ν. (Properly, since frequencies have units of s-1, the NMR timescale is actually determined by the reciprocal of δ ν.)