Spin, and the Nuclear Magnetic Resonance Effect

NMR is the result of a quantum effect called ‘nuclear spin’. Nuclear spin arises from the subatomic particles making up the nucleus, which also have a characteristic ‘spin’ number associated with them. This spin effect is attributable to all subatomic particles, however, some have an overall spin of zero.

This thesis will not probe the exact nature of quantum spin or its causes, nor will it outline in any detail the workings of the NMR spectrometer. However, a brief analysis of spin

and the NMR effect will enable a greater understanding of the information presented later in the chapter.

Total nuclear spin can take only integer or half integer values (quanta) which are always multiples of I = ½. Nuclei with spin of zero (I = 0) do not exhibit an NMR resonance and so do not feature in any NMR experiment. Most elements contain at least one isotope with I > 0, enabling analysis by NMR spectroscopy. The most abundant nucleus in the universe, and in most organic/biological chemistry, is the hydrogen nucleus (1

H), with I = ½. Of note is also 13

C, which gives an NMR signal (I = ½) but is only present at 1.1% natural abundance, and so measurements of it are not as sensitive as 1

H-NMR spectroscopy.

Possessing a nuclear spin in effect gives the nucleus an angular momentum vector, which, combined with the charge in the nucleus, gives them a weak magnetic field (µ – the nuclear magnetic dipole moment), enabling them to respond to external magnetic fields in the same manner as macroscopic magnets do. Moreover, the nucleus will precess with a frequency known as the Lamor frequency.

For any given proton, the nuclear magnetic dipole moment (µ) is given by Equation 3.1, where m_p is the mass of the proton, e is the charge on one electron and ħ is the Planck constant:

µ = _2meħ

p = 5.051 × 10

-27 _JT-1 _{Equation 3.1}

In the presence of a large external magnetic field (B¯), µ orients in accordance, and for a nucleus with spin = I, there are 2I + 1 possible orientations (for a proton, I = ½, and the number of orientations is 2, ±½). The energies of these orientations (m₁) are given by Equation 3.2:

Where g is the nuclear g-factor: an experimentally determined characteristic of the nucleus. If g is a positive number, then the nucleus is aligned with the external field, if g is negative, then the nucleus is aligned against the field. In the presence of a large external magnetic field, two energy levels are formed, and there is an energy gap given by the difference of these two levels:

ΔE = g1µ1B¯ Equation 3.3

If a photon can be introduced with energy that corresponds to the energy gap, as determined by the external magnetic field, then the nucleus can be induced to ‘flip’ from spin up to spin down. This gives rise to the resonance effect.

In any given sample, however, there are a large number of nuclei present, and given the same chemical environment, they will resonate with the same frequency [Figure 3.1 (a)]. Hence, individual μ vectors can be combined into a larger summed vector, named the ‘bulk magnetisation vector’, ‘M’ which is further described in the diagram Figure 3.1 (b).

Figure 3.1 The combination of individual µ’s (a) to give a bulk magnetization vector M (b).

The response produced and measured by the NMR spectrometer is a result of the bulk magnetization vector, and analysis of the pulse sequences can also be thought of as acting on the bulk magnetization vector. As RF pulses are applied to the sample in the spectrometer, M rotates along the axis of the RF pulse, changing the direction by an angle which depends on

the duration of the RF pulse. At the appropriate duration, this results in the magnetization lying in the x’-y’ plane, known as transverse magnetization. As this magnetization returns to the ground state of M₀, it undergoes relaxation, that is, it loses the energy that was added to the system [Figure 3.2].

Figure 3.2 Bulk magnetization vector after a 90° pulse (a), and the resulting relaxation process back to the Z'-axis (b). The M vector along each axis is shown.

There are two ways a nucleus can ‘relax’ – or ‘lose’ that energy back to the environment. The first is via longitudinal relaxation, the process by which the population of nuclei return to the ground energy state from which they came. This process gives rise to the spiral seen in Figure 3.2 (b), and also is the process from which the NMR signal, or Free Induction Decay (FID) is produced. This relaxation occurs via an exponential function, with an exponential constant called T₁.

The second relaxation process, called transverse relaxation, arises from magnetic field inhomogeneity, the proximity of other I = ½ nuclei which can interact with the nucleus, and the molecular dynamics of the species in solution. This also fits an exponential decay function, and is given the constant T₂.