# Pulse Techniques in NMR

In the early days of NMR spectroscopy, it was common to hold the magnetic field at a constant strength and then irradiate the sample with highly monochromatic electromagnetic radiation (i.e. electromagnetic radiation spanning only a small frequency range). This radiation could then be swept through different frequencies, and the frequencies at which nuclei in the sample came into resonance could be noted. However, this was a time-consuming process, with much time being wasted scanning through frequencies at which no nuclei resonated.

Today, this type of scanning technique has been superseded by pulse techniques, in which a pulse of radiation spanning a wide frequency range is applied to the sample to equilibrate the populations of α and β spins. The frequencies of the radiation emitted as the populations relax back to their equilibrium values may then be monitored.

It will be useful to go into this in slightly greater detail. The fact that an angular momentum can be represented by a vector should be familiar. The idea is that the length of the vector is the same as the magnitude of the angular momentum. The direction of the vector is such that the components of the arrow on each Cartesian axis are equal to the components of the angular momentum in that direction.
The value of the quantum number I specifies the magnitude of the spin angular momentum, and thus the length of the vector representing it.

We will consider a sample containing many identical spin ½ nuclei, and for I = ½ the vector is of length / 2. The value of mI specifies the component of the vector on the z axis, and thus specifies the angle that the vector must make with the z axis. For I = ½ , mI = ±½ and thus by elementary trigonometry it is possible to show that the vector must make an angle of approximately 55º with the z axis.

The uncertainty principle does not allow simultaneous specification of the x and y components of the angular momentum as well, so we are unable to specify the direction of the vector any more precisely. Thus we must imagine the vector as lying somewhere on a cone around the z axis. From our knowledge of the z component of the angular momentum, we know that the half angle of the cone (the angle between the z axis and the vector) is approximately 55º:

When no magnetic field is present, the sample contains equal numbers of α and β spins with the vectors lying at random angles on their cones. We picture the vectors as being stationary in this situation. The magnetisation, M, of the sample, is given by its net nuclear magnetic moment, which in this case is zero.

Upon the application of a magnetic field, B, two changes occur. Firstly, the two orientations α and β change in energy so that they are no longer degenerate. If γ > 0 then it is the α spins which move to low energy and the β spins which move to high energy (vice versa if γ < 0). In the vector model, we now picture the vectors as sweeping round their cones (or precessing) at a rate equal to their Larmor frequency.

Secondly,  the populations of the two spin states adjust to take into account the fact that they are no longer equal in energy. Assuming that the magnetogyric ratio, γ > 0 , and thus that the α spins are lower in energy than the β, then there will be fewer β spins than α at equilibrium. Now, since the difference in energy between the two states is very small compared to the thermal energy kT, the population imbalance will be very small, but it will still exist.

From this model, it should be apparent that there will be zero magnetic moment in the xy plane (the z axis being defined by the magnetic field direction) – the precession of the magnetic moment around the z axis means that components in the xy plane cancel out completely. However, since there is a small imbalance in the populations of the α and β states, there will be a net magnetisation along the z axis. This can also be represented by a vector, M, with length equal to the magnitude of the magnetisation. This vector M thus points along the z axis and has a length proportional to the population imbalance (as it is the difference in populations that means the components of magnetisation in the z direction do not cancel out).

We have now to consider what happens when an circularly polarised electromagnetic field is applied in the xy plane. We will designate the strength of this field as BA.

Note that unlike other forms of spectroscopy in which the electric part of the field interacts with the sample, here it is the magnetic component of the field, which interacts with the nuclear magnetic moment.

The field must fall into the radiofrequency part of the spectrum if it is to be of the correct frequency to stimulate transitions between the two states. Suppose that we choose the frequency of the field to be the Larmor frequency of the nuclei. (Choosing the frequency of the applied field to be that at which the nuclei resonate.) The nuclei now effectively experience a steady field of strength BA, as the rotating magnetic field is rotating at the same frequency as the precessing nuclear magnetic moments.

Under the influence of this effectively steady magnetic field, the magnetisation vector, that originally pointed in the z direction, begins to precess (rotate) about the direction of BA at a rate proportional to BA. Now, since the direction of BA lies somewhere in the xy plane, the rotation of the magnetisation vector about this axis will eventually bring the magnetisation vector itself into the xy plane.

It is possible to apply a pulse of radiation of the correct duration to cause the magnetisation vector to precess into the xy plane and then remain stationary in this plane (once the pulse has ceased). Such a pulse is termed a 90º pulse (or a π/2 pulse). The duration of such a pulse is dependent upon the strength of the magnetic field, but is typically of the order of microseconds.

Now, once the pulse has ceased, to a stationary external observer (such as the detector in the NMR apparatus) the magnetisation vector will now be rotating in the xy plane at the Larmor frequency under the influence of the static magnetic field B. This rotating magnetisation induces a signal in the detector in the NMR apparatus.

Over time, the individual spins move out of step with each other (they do not all precess at precisely the same rate because of their different chemical shifts, which cause them to resonate and precess at slightly different frequencies). Signals are obtained for each of the different Larmor frequencies of the nuclei in the sample, and they may be extracted from the detected signal by means of a mathematical technique called a Fourier transform. The randomisation of the xy component of the magnetisation means that over time the xy component averages to zero, and the magnetisation vector aligns itself with the z axis, as it was prior to the application of the pulse.

Note that it is not necessary to know the Larmor frequencies of the nuclei prior to the experiment, or even to apply a pulse at precisely the Larmor frequency of any of the nuclei. The range of Larmor frequencies for a particular isotope due to chemical shifts is typically very much smaller than the strength of the applied polarised magnetic field. Thus the field will shift the magnetisation vectors of all the nuclei that resonate around that frequency into the xy plane, regardless of their precise resonance frequency.
The applied pulse will simultaneously excite all nuclei of a given isotope
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