Basic concepts 53,54,55

1.3.1.1 Spin number, spin angular momentum and magnetic moment

Atoms are composed of nuclei and electrons. While many spectroscopy techniques rely on the observation of electron energy transitions, NMR focuses on nuclear spin energy transitions. Nuclei possess several properties through which we can define them: mass, electric charge, magnetism and spin. To discriminate between NMR “active” or “silent” nuclei, the nuclear spin quantum number (𝐼) is the most important parameter. Indeed, nuclei with 𝐼 = 0 do not exhibit NMR properties, while for all the other nuclei we can define a spin angular momentum P, also called the nuclear spin, and defined as:

𝑃 = √𝐼(𝐼 + 1) ℏ Equation 1.4

where ℏ is the reduced Plank constant (ℏ = ℎ 2𝜋⁄ ) and 𝐼 = 1₂, 1,3₂, 2, etc. Classical angular momentum is associated to a rotating object. On the contrary, 𝑃 is an intrinsic property of the quantum particles. As in classical physics a charge in motion (e.g. associated to an angular momentum) generates a magnetic field, the intrinsic spin angular momentum leads to the generation of a magnetic moment, μ, which is proportional to 𝑃:

30 | P a g e where 𝛾 is the so-called gyromagnetic ratio, which shows a defining value for every different nuclide. 𝜇 and 𝑃 are vectors, with a defined magnitude and orientation. We can think of the individual magnetic moment of each spin in a sample as randomly oriented in normal conditions. However, when the sample is placed in a static magnetic field B0, the magnetic moments will show a tendency to align with the direction of the

field, as little magnetic bars would do. In quantum terms, the observation of the system will lead to the detection of only a discrete number of different orientations of the spins, which is equal to 2𝐼 + 1.

1.3.1.2 The differences in energetic state populations and the Larmor frequency

For nuclei with 𝐼 =1₂ (such as 1_{H and} 13_{C, the most commonly detected nuclei), there will}

be two possible orientations the spins can undertake. By convention, 𝐵₀ is always applied in the z direction, therefore, nuclei with 𝐼 =1₂ will either align with z or with –z. We call these two energy states the 𝛼 and the 𝛽 state, respectively. Usually, the population of the 𝛼 state (𝑁_𝛼) is slightly larger than the population in the 𝛽 state (𝑁_𝛽) and the ratio between the two it increases exponentially with 𝛥𝐸 (therefore it depends on γ and 𝐵₀):

𝑁_𝛼 𝑁_𝛽

= 𝑒

𝛥𝐸

𝑘𝐵𝑇 _{Equation 1.6}

where 𝑘_𝐵 is the Boltzmann constant and the difference in energy level between 𝛼 and 𝛽, 𝛥𝐸, is given by:

𝛥𝐸 = ℏ 𝛾 𝐵₀

The quantum mechanics system we are trying to describe can become complex, but a very useful way to deal with it is to depict it in classical mechanics and vectorial terms, which is accurate up to a certain level. In particular, we can think of 𝐵₀ imposing a torque on the magnetic moments 𝜇, which results in the precession of every one of the nuclear spins about the direction of the static field. The rate of the precession is expressed (in Hz) as follows:

𝜈

=

31 | P a g e In NMR, this is called the Larmor frequency of the nucleus and it also corresponds to the frequency of the electromagnetic radiation required to the nuclei to change their spin state (from 𝛼 to 𝛽 or vice versa), by acquisition of a quantum of energy (𝛥𝐸 = ℎ𝜈).

1.3.1.3 Bulk magnetisation and the perpendicular rf field B1 in the rotating frame

Now, we can imagine summing up all the single μ components (precessing at the Larmor frequency and aligned with or against 𝐵₀): the result will be a so-called “bulk” magnetisation vector 𝑀_𝑧 aligned with 𝐵₀ (Figure 1.9). The magnitude of 𝑀_𝑧 will be given by the z projection of the sum of the magnetic moments 𝜇. Formally, we would then write that:

𝑀_𝑧 ∝ 𝑁_𝛼− 𝑁_𝛽 Equation 1.9

This implies the net magnetisation on the transverse (x-y) plane to be null. As mentioned, to induce an energy transition between the 𝛼 and the 𝛽 states, the Larmor frequency of the nucleus of interest must be matched by the electromagnetic wave employed for excitation of the sample.

Figure 1.9. Bulk magnetisation as represented in the vector model. The single spin magnetic

moments rotating at the Larmor frequency in their 𝛼 and 𝛽 states (pointing above and below the x-y plane, respectively) sum up to give the bulk magnetisation vector 𝑀_𝑧, whose direction is +z and whose intensity is the projection of the sum of the μs on z. Figure adapted from 54.

Practically, as the frequencies typically associated with nuclear spin transitions fall in the range of radio waves (tens to hundreds of MHz), for typical magnets used in NMR spectroscopy, the excitation is done applying a radiofrequency pulse oscillating at around the Larmor frequency (passing current via a coil surrounding the sample). This produces a magnetic field, 𝐵₁, perpendicular to 𝐵₀ (the direction of 𝐵₁ can be modulated by the coil properties). To help visualising the effects of the oscillating rf pulses on the bulk magnetisation rotating along z in the presence of 𝐵₀, the formalism of the rotating frame of reference was introduced (Figure 1.10).

32 | P a g e In the laboratory frame the x, y, z coordinates are viewed as static, while the rotating frame of reference uses coordinates x’, y’, z’ which are rotating with the frequency of the B1 field of the radiofrequency pulse . If the magnetic field introduced by the rf pulse

oscillates in the x axis (x-pulse), it can formally be decomposed into two counter-rotating components oscillating in the transverse plane with frequencies ± 𝜈₀ (Figure 1.10) (where 𝜈₀ is expected to be the frequency for NMR condition to be satisfied). In the rotating frame of reference, one of the rf field vectors is then static and the second rotates with 2𝜈₀ in opposite (-) direction. Being this now far from the Larmor frequency, it can be ignored. Therefore, in the rotating frame, time dependency of the rf field is removed and both 𝑀_𝑧 and 𝐵₁ are static and perpendicular to each other.

Figure 1.10. Representation of laboratory and rotating frame of reference. Figure adapted from

54.

1.3.1.4 Chemical shift and the fundamental pulse-acquire NMR experiment

We already argued that the resonance frequency of a given nucleus depends on its gyromagnetic ratio (𝛾) and applied external field 𝐵₀, and we mentioned that the local environment of the nuclei in the molecule (namely surrounding electrons of any neighbouring nucleus) has a large effect too. Electrons circulating in their orbitals can be seen as currents passing through wires and producing small (local) magnetic fields (𝐵’) in the opposite direction to 𝐵₀. Hence, the effective (local) magnetic field (𝐵) felt by the nucleus is slightly smaller than the applied external magnetic field (𝐵₀) and is given by the equation:

𝐵 = 𝐵₀− 𝐵′ _{= 𝐵}

0(1 − 𝜌) Equation 1.10

33 | P a g e The electron density surrounding the atoms in any molecule is typically rather complex and depends on several factors including formation of hydrogen bonds, presence of electronegative nuclei in close proximity, unpaired electrons or through-space interactions within molecular clusters. Experimentally, the chemical shift (𝛿) is measured as a frequency difference of the analysed nuclei (say resonating at 𝜈_𝑖) with respect to one reference frequency. If we imagine the reference frequency to be on- resonance with the rotating frame (𝜈₀), the other sets of nuclei can either move slower or faster, and consequently will have a smaller or larger δ. The chemical shift of each set of nuclei is defined in the ppm (part per million) scale as:

𝛿_𝑖 = 106 𝜈𝑖−𝜈0

𝜈₀ Equation 1.11

and it is independent from the strength of the field 𝐵₀.

We have now all the tools to describe the fundamental pulse-acquire experiment, consisting in turning on an rf irradiation of given amplitude for a given time. As already 𝐵₀ did on 𝜇 (inducing precession), the rf field imposes a torque on 𝑀_𝑧 rotating it perpendicularly to the direction of the 𝐵₁. This drives the bulk magnetisation vector away from its equilibrium position (from the z-axis towards the x-y plane, according to the “motor rule”). The 90° pulse, bringing the magnetisation on to the x-y plane, equalises the spin populations of the 𝛼 and 𝛽 states, that is, it makes 𝑀_𝑧 null. A 180° pulse, instead, which will bring 𝑀_𝑧 to –z, does invert the populations. All these elements (and many more) are useful to manipulate the magnetisation in pulse sequences way more complicated than the pulse-acquire. For that purpose, it is also important to know that if the rf field is applied for a time 𝑡p, the flip or nutation angle of the pulse (𝜃)

through which the magnetisation is rotated is found as:

𝜃 = 𝛾

2𝜋𝐵1𝑡𝑝 degrees Equation 1.12

Far from its equilibrium state by any 𝜃 angle, the bulk magnetisation will tend to recover its +z position by precessing around z. As we have seen, different sets of nuclei will precess at different frequencies, each of them inducing a weak oscillating voltage in the coil (now acting as a receiver coil rather than a transmitter coil). This small signal will be amplified and processed (Fourier transformed from the detected time domain to the

34 | P a g e frequency domain) and the result will be a frequency spectrum containing as many “peaks” as many are the sets of nuclei experimenting different electronic environment (i.e. having different chemical shifts).

We have explained very briefly the use of rf pulses to perturb the equilibrium between the energy states of the nuclear spins (generating sets of signals on the receiver coil, and allowing to detect an NMR spectrum of the sample). Of deepest interest for the techniques used and developed in the present thesis, though, are the complicated processes of re-establishing the equilibrium state after perturbation. These processes are called relaxation, and to them we dedicate the following sub-section.