Chemical Shift 21,

The local field experienced by a particular nucleus, B, in an atom is typically slightly less than the external field. Theoretically, B0 is the field that would be felt by a

nucleus surrounded by no electrons, i.e., a bare atom. In reality, however, the electrons precess in their orbitals and this generates a small magnetic field, B’, which opposes B0. The field experienced by the nucleus is given by

B = B0 – B’ = B0 (1 – σ) , (2.22)

where σ is referred to as the shielding constant. The resonance frequency, ω, of the nucleus can be defined in terms of the shielding constant and is expressed as

ω!=!–γB₀(1 – σ) , (2.23) although it is often difficult to measure σ directly. It is more common to report the chemical shift, δ, of a nucleus, quoted in ppm, relative to a reference compound is given by

δ = 106ω–ωref

ωref ,

(2.24)

where ω is the resonant frequency of the specific nucleus and ωref is the resonant

frequency of a reference nucleus. Equation 2.24 can be expanded to become

δ = 106σref–σ 1 – σ_ref!≈10

6_(σ

2.9.3. The Rotating Frame

As described earlier, when a nucleus is placed into an external magnetic field, the magnetic dipole moments will align either parallel or antiparallel to the direction of the field. As there is typically an excess number of spins in the parallel (α-state), a net, or bulk magnetisation exists, represented by a bulk magnetisation vector, M0, proportional to the population difference and gyromagnetic ratios of the nuclei, and given by M0= 1 2 γ ħ ∆n0 . (2.26)

The effect of radiofrequency (rf) pulses (short bursts of intense radiation) upon this magnetisation vector can be conveniently described using the vector model.21 _{If M}

0is tipped away from the z-axis it will precess around B0 at an angular frequency, ω0. To manipulate the magnetisation vector, rf pulses are usually applied in the xy plane. A pulse is composed of an oscillating magnetic field and it can be difficult to understand its effect in the fixed, or ‘laboratory’ frame. It is more useful to transform to a frame of reference termed the ‘rotating frame’ that is itself rotating at an angular frequency,

ωrf, resulting in an effective field, Beff. The pulse can now be described as a fixed field, B1 applied usually along the x or y axes. Upon application of the pulse, the magnetisation vector nutates around the direction of B1 towards the xy plane. The angle of nutation is termed the flip angle, β, and is given by

β = γB1τp , (2.27)

where τp is the duration of the pulse. If β = 90° then the magnetisation vector will be rotated to lie in the xy plane. If the pulse is applied along the x-axis, the vector will lie

along the –y-axis, as shown in Fig. 2.11. After the application of the pulse, the magnetisation vector will then precess around Beff at a frequency

Ω = ω0 –ωrf . (2.28) It is this precession, or free induction decay (FID), that is detected (as an oscillating magnetic field) by a detector coil as the signal in a NMR experiment. Conventionally, two detectors (usually along x and y axes) are used in an approach termed ‘quadrature’ detection, to ensure that the sense of precession can be determined. 2.9.4. Relaxation

The oscillating signal detected in a NMR experiment is damped by relaxation processes. There are two distinct types of relaxation. Firstly, the bulk magnetization has a tendency to return to its equilibrium state, parallel to the z axis, and this process is referred to as longitudinal, spin-lattice, or “T1” relaxation. This relaxation is described by an exponential term and expressed as

x

y

z

x

y

z

(a)

(b)

Pulse

B

B

Figure 2.11: (a) Alignment of the bulk magnetisation vector, M0, (purple arrow) with Beff along the z direction and application of a pulse along x, (b) magnetisation is nutated into the xy plane after the pulse is applied, where they precesses about Beff.

Mz(t) = M0[1#–!exp (

–t T1)] , !

(2.29)

where Mz(t) is the z-component of the bulk magnetisation at time, t, and T1 is the relaxation time constant. The rate of longitudinal relaxation is dependent upon the nucleus and its environment, and may range from seconds to hours for solids. For example, 23Na (I = 3/2), 17O (I = 5/2) and 89Y (I = 1/2) have typical longitudinal relaxation times* on the order of seconds, minutes and hours, respectively.

In addition to longitudinal relaxation, transverse, spin-spin, or “T2” relaxation also occurs simultaneously, but is independent of longitudinal relaxation. After an rf pulse is applied along x or y axes, the bulk magnetization nutates towards the xy plane, to give Mxy. The magnetization subsequently precesses in the xy plane but the net coherent magnetization is eventually lost from this plane and becomes dephased. Individual spins will precess at different rates – some faster and some slower than others. Over time, the precession of each spin becomes sufficiently different such that no bulk magnetization is observed. This is shown in Fig. 2.12, which shows a simplified case where three spins, a, b and c, are present.

2.9.5. Free Induction Decay and Fourier Transformation22,23

In an NMR experiment, the magnetization aligned along the z direction, parallel to B0, is perturbed when a pulse is applied, such that the magnetization nutates from the z-axis into the xy plane. In the xy plane, precession then occurs (ignoring the effects of relaxation outlined in Section 2.9.5) and can be detected as a time-domain signal. This signal (the FID) can be difficult to interpret for complex systems. A Fourier transform is typically used to convert the FID into a frequency-domain signal, or a spectrum, where the FT is expressed as

* _{The timescales given are very general, and are dependent upon the surrounding} environment. Additionally, these time constants may be reduced if paramagnetic ions

Sspec(f) != SFID t cos +∞ 0 (2πft) dt , (2.30)

where Sspec(f) is the intensity of the spectrum at frequency f and SFID(t) is the amplitude of the FID at time t. The upper limit of the integral is defined as being infinite, but in practice, the calculation is only performed on data points with a magnitude greater than the noise, i.e.,the FID is damped and decays over time due to the effects of relaxation, and as such, only a finite number of data points are collected.

z a, b, c a b c View down z-axis (a) (c) (d) x 90°_x –y x (b) –y z y x x pulse

Figure 2.12: (a) Alignment of the bulk magnetisation vector (purple arrow) with B0 along the z axis, (b) application of a pulse along x to nutate the spins into the xy plane, (c) spins (a, b and c) in the xy plane immediately after the pulse and (d) dephasing of spins in the xy planes (T2 relaxation) over time.

2.10. Solid-State NMR Interactions