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The introduction o f the basic concepts above facilitates an understanding o f the quantum spectroscopic phenomenon occurring in NMR. However, using only the

model described o f the quantum a- and (3-states it is difficult to appreciate the behaviour o f the macroscopic nuclear magnetisation during the course o f an NMR experiment. At this point, it is useful to introduce the more intuitive model o f an NMR experiment which is provided by the ‘classical formalism’. The classical formalism cannot be applied directly to the description o f a single nucleus since atomic phenomena do not behave classically. Rather, a quantum mechanical approach would be required to describe the non-Newtonian behaviour o f an individual nucleus during an NMR experiment. In NMR, the rather abstract ‘product operator formalism’ can be used for such a description; however, the classical formalism is preferred here for the sake o f simplicity. However, since the signal actually observed in an NMR experiment is derived from a large ensemble o f nuclei it is seen to behave in a classical manner. Thus, despite some limitations concerning the description o f multidimensional NMR experiments, the classical formalism can be an extremely useful descriptive tool and is sufficient to provide considerable insight into the evolution with time o f the net nuclear magnetisation during an NMR experiment.

The primary value o f the classical formalism is that the ‘bulk magnetisation’ derived from all the nuclear spins (magnetic moments, jp in the NMR sample can be represented by a vector. There exists a bulk magnetisation due to the population difference between the two nuclear spin states o f a spin Vz nucleus in an applied magnetic field. At equilibrium, an excess o f nuclear spins populate the lower energy state (a), parallel to the applied magnetic field ( B q) , such that there will be a resulting

macroscopic longitudinal magnetisation (M^ = ILji ) parallel to the applied magnetic

Figure 2-1: Basic principles of N M R — at equilibrium, an excess of nuclear pins aligns parallel with the applied magneticfield, Bg. A s a result, there is a net magnetisation, along the axis. The lack ofphase coherence results in the absence of any net magnetisation in the XY-plane.

field ( Bq) (Figure 2-1). The transverse components o f the various magnetic moments

will rotate in the XY-plane with no preferential phase. These magnetic moments are randomly distributed in the XY-plane such that total = My = 0.

The bulk magnetisation vector (Mq) can be resolved into contributions along the X, Y and Z axes. The phenomenologically formulated Bloch equations use this approach to describe the motion o f the bulk magnetisation vector with time. Precursors to the full Bloch equations describe the motion o f Mq with respect to time under the influence o f Bq and in the absence o f any additional exciting field (i.e. at equilibrium) as follows:

dMz/dt = 0,

dM x/dt = Y Bo My and dMy/dt = -y Bq Equations NMR-2

Equation NM R-2 describes the precessive motion o f Mq at thermal equilibrium. In order to progress to a description o f the effects on Mq o f a pulsed NMR experiment it is useful to consider a simplified system in which there is only one magnetically active nucleus. Then, if a magnetic field (B^) oscillating with an angular frequency equal to the Larmor frequency CO is applied along the X-axis (e.g. by a pulse o f an appropriate radio-frequency radiation), the perpendicular bulk magnetisation vector will be affected and will begin to precess about Bj in the YZ-plane. The visualisation o f the effect o f this pulse can be simplified by the introducing the following concepts. Firstly, the magnetic component o f a radio- frequency field that is linearly polarised along the X-axis can be expressed as a cosine function o f co, the angular frequency o f the field and (j) the phase o f the field. This expression can be decomposed into two circularly polarised fields in the XY-plane rotating in opposite directions about the Z-axis (Cavanagh et al., 1996). One component o f the magnetic field B^ rotates at the Larmor frequency co, while the other is rotating in an opposite direction, i.e. has a frequency o f -C O . Only the co component o f Bj rotating synchronously with the resonant Larmor frequency interacts significantly with the bulk magnetisation; the counter-rotating nonresonant field influences the nuclear spins only very slightly and so can be neglected.

Modified versions o f the precursor Bloch equations (Equations NMR-2 above) can thus be formulated in order to describe the rate o f change with time o f

Mx, My and in the presence o f an exciting radio-frequency field 2Bj acting along the X-axis:

dM x/dt = Y [ Bq My - BiyMz ] and dMy/dt = -y [ Bq Mx + B^x M% ]

dM z/dt = - y [ B^y Mx - Bjx My ] Equation NMR-3

- where the X and Y components o f the exciting resonant component o f the radio- frequency field 2Bj which rotates with the Larmor frequency (co) and precesses about the Z-axis are given by Bjx = B^ cos (cot) and B^y = - B^ sin (cot).

An understanding o f the system can be significantly simplified by the definition o f a reference frame rotating with the frequency co, called the rotating frame. In this rotating frame, the cause o f the precession about the Z-axis ( B q ) is

removed. This makes it easier to appreciate that the radio-frequency pulse which

Figure 2-2: The appUcation of a 9(7^ pulse (BJ tips the bulk magnetisation into the XY-plane.

r x -M

9 Ox p u ls e

- z

- Z

generates B^ along the X-axis makes it possible to move the bulk magnetisation by precession about the X-axis. This precession is induced by the same mechanism as was described for the motion o f the bulk magnetisation under the influence o f Bq. The rate o f this new precessive motion depends on the angular frequency o f Bj. It is

possible to calculate the duration o f a pulse o f a specific angular frequency required in order to move the bulk magnetisation through precisely 90° about the X-axis. Thus, it is possible to move the magnetisation from the Z-axis to the Y-axis by applying a pulse Bj along the X-axis {Figure 2-2). At this point the B, pulse is terminated and, returning from the rotating frame to the laboratory frame, the bulk magnetisation along the Y-axis resumes a precession solely due to the effect o f Bq. The result is a precession o f the bulk magnetisation in the XY-plane, again with the Larmor frequency. This oscillating magnetisation in the XY-plane will induce an alternating voltage in the receiver coils o f the NMR spectrometer set in this plane. The observable NMR signal is this voltage, a time-dependent sinusoidal function with the Larmor frequency ca In practice, the NMR signal is seen to decay exponentially in less than a few seconds, due to relaxation processes which are described below.