Two-Dimensional NMR

In order to perform more sophisticated NMR experiments, for example to identify correlations between nuclei or probe coherences that may not be directly observed, it is possible to use experiments with more than one time dimension [88]. While the experiments performed in this work used no more than two dimensions, it is in princi- ple possible to extend this approach further to higher numbers of dimensions [89, 90]. Fourier transformation of a 2D time-domain data set results in a spectrum with peaks in two frequency dimensions. 2D experiments are a commonly used technique to probe specific interactions within a sample. For example, a heteronuclear correlation exper- iment will excite one nuclear species, allow that coherence to evolve during the first

t2

t1

1H

π

2

π

2

(a)

(b)

(c)

Figure 3.2: (a) Pulse sequence diagram for a simple two dimensional NMR experiment. Coherence transfer pathway diagrams for (b) a phase modulated experiment and (c) an amplitude modulated experiment.

time delay, then transfer the magnetisation to a second nuclear species before acquiring a FID. The resulting 2D spectrum will consist of peaks at the resonances of the pairs of nuclei linked by the interaction used to transfer the magnetisation.

A pulse sequence for a simple generalised 2D experiment is shown in figure 3.2. The first pulse (or sequence of pulses) creates transverse magnetisation, which is allowed to evolve during t1. This time period is incremented in successive experiments, for

example it may have a duration of 0 µs during the first experiment, 10 µs during the second experiment, etc. The second pulse (or sequence) converts the magnetisation to an observable coherence, which is then acquired (using quadrature detection as in the 1D case) in t2. In this discussion, it will be assumed that during t1and t2, the

system evolves only under the resonance offset Ω. Whereas in the one dimensional case, a pure absorptive line shape is simply achieved by combining real and imaginary components of the FID, in the two dimensional case, achieving absorptive line shapes in both dimensions, with sign discrimination, is more challenging and requires more complex methods.

Experiments may be designed using either a phase modulated or amplitude modu- lated scheme. This depends upon the coherence transfer pathway, which is manipulated by phase cycling as will be described in section 3.1.3. In the case of the phase mod- ulated scheme, the form of the two dimensional signal as a function of t1 and t2 will

3.1. PULSED FOURIER TRANSFORM NMR 43

be

s(t1, t2) = exp(−iΩt1) exp(−t1/T2(1)) exp(iΩt2) exp(−t2/T2(2)) (3.8)

which is simply analogous to the one dimensional NMR signal described in equation 3.1. This expression for the signal may be Fourier transformed in both dimensions

s(t1, ω2) = exp(−iΩt1) exp(−t1/T2(1))(A + 2 −iD + 2) (3.9) s(ω1, ω2) = (A−1 −iD − 1 )(A+2 −iD2+) = (A−₁A+₂ −D−₁D₂+)−i(A−₁D₂++D−₁A+₂) (3.10)

In the above expression, the plus and minus superscripts refer to the sign of the offset frequency. It is clear that the phase modulated sequence has achieved sign discrimina- tion as all terms in theω1 dimension are negative, and all terms in theω2 dimension are

positive. However the spectrum contains a mixture of absorptive and dispersive com- ponents. This results in peaks taking on a phase twist line shape. The contribution of dispersive elements in these shapes leads to similar problems to the purely dispersive one dimensional line shape, notably the large width of the lines. An example of the real and imaginary components of a phase twist line shape are shown in figure 3.3(a) and (b).

It is possible to achieve pure absorptive two dimensional peaks using an amplitude modulated scheme. The evolution under both p = 1 andp=−1 coherences duringt1

results in a signal of the form

s(t1, t2) = (exp(−iΩt1) + exp(iΩt1)) exp(−t1/T2(1)) exp(iΩt2) exp(−t2/T2(2))

= 2 cos(Ωt1) exp(−t1/T₂(1)) exp(iΩt2) exp(−t2/T₂(2)) (3.11)

Performing sequential Fourier transforms on the signal will not achieve the desired ab- sorptive spectrum. Instead, a hypercomplex Fourier transform is required in which the real and imaginary components are separated before performing the Fourier transform

Figure 3.3: Two-dimensional NMR line shapes. (a) real and (b) imaginary components of a phase twist line shape, containing both absorptive and dispersive contributions (plotted using the real and imaginary components of equation 3.10, respectively). (c) pure absorptive line shape, as would result from the States procedure (plotted using the real component of the expression in equation 3.19).

3.1. PULSED FOURIER TRANSFORM NMR 45

int1.

s(t1, ω2) = 2 cos(Ωt1) exp(−t1/T2(1))(A+2 −iD2+)

= (exp(−iΩt1) + exp(iΩt1)) exp(−t1/T2(1))(A+2 −iD+2) (3.12)

The Fourier transforms in t1 of the real and imaginary components of equation 3.12

are then performed separately

s(t1, ω2)Re= (exp(−iΩt1) + exp(iΩt1)) exp(−t1/T2(1))A+2

s(ω1, ω2)Re= (A−₁ +A+₁)A+₂ −i(D−₁ +D₁+)A+₂ (3.13)

s(t1, ω2)Im= (exp(−iΩt1) + exp(iΩt1)) exp(−t1/T2(1))D + 2 s(ω1, ω2)Im= (A+1 +A − 1)D+2 −i(D1++D − 1)D2+ (3.14)

The real component of the Fourier transform of the real component is now purely absorptive. However, sign discrimination has been lost in the ω1 dimension. Sign

discrimination may be restored by the use of the States method [91]. In addition to recording the amplitude modulated experiment, which results in a signal of a cosine form int1, a second experiment is performed with the phase of the first pulse shifted by

π/2|p| wherep is the order of the coherence that evolves during t1. In the case of the

simple 2D experiment under consideration, |p|= 1, and this second signal is therefore shifted in phase by π/2 so as to give sine modulation int1.

ssin(t1, t2) = h exp −iΩt1+ π 2 −exp i Ωt1+ π 2 i exp(−t1/T₂(1)) ×exp(−iΩt₂) exp(−t₂/T₂(2)) = 2 cosΩt1+ π 2

exp(−t₁/T₂(1)) exp(iΩt2) exp(−t2/T2(2))

= 2 sin(Ωt1) exp(−t1/T2(1)) exp(iΩt2) exp(−t2/T2(2)) (3.15)

Note that this sine modulated signal is denoted ssin. The cosine modulated signal in

transformed in t2.

scos(t1, ω2) = 2 cos(Ωt1) exp(−t1/T2(1))(A+2 −iD+2) (3.16)

ssin(t1, ω2) = 2 sin(Ωt1) exp(−t1/T2(1))(A+2 −iD+2) (3.17)

The real components of bothscos andssinare then combined (the dispersive imaginary

components are discarded) to form the real and imaginary components respectively, of a new “States signal”, sStates:

sStates(t1, ω2) = Re(scos(t1, ω2)) +iRe(ssin(t1, ω2))

= 2 exp(Ωt1) exp(t1/T₂(1))A+₂ (3.18)

A final Fourier transform int1 is then applied:

sStates(ω1, ω2) = 2(A+₁ −iD+₁)A+₂

= 2A+₁A+₂ −2iA+₂D+₁ (3.19)

The real component of the above expression represents a spectrum with purely absorp- tive line shapes and sign discrimination in both ω1 and ω2 dimensions. An example of

a purely absorptive 2D line shape is shown in figure 3.3(c).