One-Dimensional NMR

The simplest NMR experiment consists of applying an rf pulse to flip the bulk mag- netisation from the longitudinal direction to the transverse plane, then observing the precession of the magnetisation about the longitudinal direction as a free induction decay (FID). Pulses are applied by passing a current, oscillating at a frequency ofωrf

through a coil arranged around the sample. As discussed in the previous chapter, ωrf

is close to the precession frequency of the nuclear species being studied, i.e. a resonant pulse. The coil is then used to detect the precession of the bulk magnetisation in the transverse plane. In order to digitise the signal, the observed signal is mixed with the spectrometer reference frequency, ωrf (approximatelyω0) such that only the resonance

offset remains, as discussed in chapter 2. It is to be noted that this is equivalent to observing the nuclei in a frame of reference rotating at the frequency ωrf.

The mixing down of the signal is performed in such a way that it is equivalent to detection of the signal in two orthogonal directions (x and y), a technique known as quadrature detection. Due to the π/2 phase difference between the signals that will be observed in orthogonal directions, these signals, are cosine and sine functions of the effective precession frequency, Ω. These signals may be regarded as the real and

3.1. PULSED FOURIER TRANSFORM NMR 39

imaginary components of the NMR signal, s(t).

s(t) = (cospΩt−isinpΩt) exp (−t/T₂)

= exp (−ipΩt) exp (−t/T2) t≥0

s(t) = 0 t≤0 (3.1)

where p is the coherence order. This expression for the NMR signal was (with the exception of the relaxation term) derived in chapter 2 (equation 2.68). Only single-spin single quantum coherences (p = ±1) are directly detectable in an NMR experiment, i.e. other coherences may exist, but do not induce a current in the coil. This can be seen by calculating the product of the raising operator (for a two spin system) with the density matrix for a system of coupled spin-½ pairs (equation 2.33). The trace of the resulting matrix (calculating the NMR signal as in equation 2.68) then contains only single-quantum coherence terms. Throughout this discussion, signals will be observed with p=−1.

In general NMR terminology, the exp (−t/T2) term in equation 3.1 describes the

loss of signal characterised by the time T2: after T2, the intensity of the FID will

have reduced to 1−e−1 ≈ 63% of the original intensity [87]. In solids, this loss of transverse magnetisation is better described as a dephasing of the initial coherence, and is typically of the order of milliseconds for the crystalline organic solids studied in later chapters of this thesis. This is very much less than the time taken for the the system to return to an equilibrium state (ρeq ∝Iˆz, as stated in equation 2.59), which

is similarly characterised by the time T1.

A more informative representation of the data is obtained by Fourier transformation of the time domain signal into the frequency domain. The spectrum, S(ω) resulting from a Fourier transform of the signal expressed in equation 3.1 is

Ω

ω

(a)

(b)

Figure 3.1: Absorptive (a) and dispersive (b) Lorentzian one-dimensional lineshapes, illustrating the difference in the width of the shapes.

where the functionsA(ω) andD(ω) are expressed as

A(ω) = 1/T2 (1/T2)2+ (ω−Ω)2 (3.3) D(ω) = ω−Ω (1/T2)2+ (ω−Ω)2 (3.4)

These expressions forA andDcorrespond toabsorptive anddispersive Lorentzian line shapes, respectively. Each shape is centred on the resonance offset frequency, Ω. Only the absorptive line shape is desired as it has a narrower width than the dispersive shape (full width at half maximum height of 1/πT2 in Hz). Examples of absorptive and

dispersive line shapes are shown in figure 3.1. Furthermore, at the frequency of the resonant offset the absorptive line shape is at its maximum value, whereas the value of the dispersive line shape is zero, making it easier to identify the central frequency of the shape. As such, in this idealised situation, the imaginary component of the spectrum would simply be discarded. However, the above analysis relies on the alignment of the magnetisation in the transverse plane lying in the x-direction, such that the real component of the signal is a cosine function, and the imaginary component a sine function. In practice the real and imaginary components of the spectrum both contain some absorptive and dispersive components, and in order to achieve a pure absorptive line shape in the final spectrum, it is necessary to take linear combinations of the real and imaginary components. This process is described as “phasing” the spectrum.

3.1. PULSED FOURIER TRANSFORM NMR 41

The importance of using quadrature detection can be seen by considering the case in which the transverse magnetisation is measured in only one direction, in this case the x-direction. The observed signal is now simply the cosine modulated component.

s(t) = cos Ωtexp(−t/T₂) = exp(−iΩt) + exp(iΩt)

2 exp(−t/T2) (3.5)

When Fourier transformed, the form of the resulting spectrum will be the same as in equation 3.2, however, the functions A(ω) andD(ω) become

A(ω) = 1 2 1/T2 (1/T2)2+ (ω−Ω)2 + 1/T2 (1/T2)2+ (ω+ Ω)2 (3.6) D(ω) = 1 2 ω−Ω (1/T2)2+ (ω−Ω)2 + ω+ Ω (1/T2)2+ (ω+ Ω)2 (3.7)

The above expressions describe a pair of peaks mirrored at ±Ω. This shows that only observing transverse magnetisation in one direction has resulted in an inability to determine the sign of the offset, i.e. there is no sign discrimination.

Finally, it is to be noted that when processing the data from a real NMR experi- ment, the spectrometer uses a discrete Fourier transform algorithm, in contrast to the analytical Fourier transform described above. This method results in a spectral width, SW, equal to the inverse of the dwell time (time between acquisition of data points). The central point of the spectrum is set at Ω = 0.