The process of refocusing a coherence which has been dephased by a gradient pulse is inhibited if the spins move either during or between the defocusing and refocusing gradients. Such movement alters the magnetic field experienced by the spins so that the phase acquired during the refocusing gradient is not exactly opposite to that acquired during the defocusing gradient.
In liquids there is a translational diffusion of both solute and solvent which causes such movement at a rate which is fast enough to cause significant effects on NMR experiments using gradient pulses. As diffusion is a random process we expect to see a smooth attenuation of the intensity of the refocused signal as the diffusion contribution increases. These effects have been known and exploited to measure diffusion constants since the very earliest days of NMR.
(a)
g
δ
∆
δ
(b)
t
1t
2(c)
t
1t
2τ
τ
Figure 21. (a) A spin echo sequence used to measure diffusion rates (see text); (b) and (c) are alternative ways of implementing gradients into a COSY spectrum.
An analysis of the simple spin echo sequence, shown in Fig. 21 (a), illustrates very well the way in which diffusion affects refocusing. Note that the two gradient pulses can be placed anywhere in the intervals τ either side of the 180° pulse. For a single uncoupled resonance, the intensity of the
observed signal, S, expressed as a fraction of the signal intensity in the absence of a gradient, S0 is given by
S
S0 G D
2 2 2
3
=exp−γ τ ∆−τ [40] where D is the diffusion constant, ∆ is the time between the start of the two
gradient pulses and τ is the duration of the gradient pulses; relaxation has been ignored. For a given pair of gradient pulses it is diffusion during the interval between the two pulses, ∆, which determines the attenuation of the echo. The gradients are used to label the magnetization with a spatially dependent phase, and then to refocus it. The stronger the gradient the more rapidly the phase varies across the sample and thus the more rapidly the echo will be attenuated. This is the physical interpretation of the term
γ2
G2τ2 in Eqn. [40].
Diffusion constants generally decrease as the molecular mass increases. A small molecule, such as water, will diffuse up to twenty times faster than a protein with molecular weight 20,000. Table 1 shows the loss in intensity due to diffusion for typical gradient pulse pair of 2 ms duration and of strength 10 G⋅cm–1 for a small, medium and large sized molecule; data is given for ∆ = 2 ms and ∆ = 100 ms. It is seen that even for the most rapidly diffusing molecules the loss of intensity is rather small for ∆ = 2 ms, but becomes significant for longer delays. For large molecules, the effect is small in all cases.
Table I : Fraction of Transverse Magnetization Refocused After a Spin Echo with Gradient Refocusinga
∆/msb small moleculec medium sized
moleculed
macro moleculee
2 0.99 1.00 1.00
100 0.55 0.88 0.97
a Calculated for the pulse sequence of Fig. 21 (a) for two gradients of
strength
10 G⋅cm–1and duration, τ, 2 ms; relaxation is ignored.
b As defined in Fig. 21 (a).
c Diffusion constant, D, taken as that for water, which is 2.1 × 10–9
m2 s–1 at ambient temperatures.
d Diffusion constant taken as 0.46 × 10–9
m2 s–1.
e Diffusion constant taken as 0.12 × 10–9
4.3.5.1 Minimisation of Diffusion Losses
The foregoing discussion makes it clear that in order to minimise intensity losses due to diffusion the product of the strength and durations of the gradient pulses, G2τ2, should be kept as small as is consistent with achieving the required level of suppression. In addition, a gradient pulse pair should be separated by the shortest time within the limits imposed by the pulse sequence. This condition applies to gradient pairs the first of which is responsible for dephasing, and the second for rephasing. Once the coherence is rephased the time that elapses before further gradient pairs is irrelevant from the point of view of diffusion losses.
In two-dimensional NMR diffusion can lead to line broadening in the F1 dimension if t1 intervenes between a gradient pair. Consider the two alternative pulse sequences for recording a simple N-type COSY spectrum shown in Fig. 21 (b) and (c). In (b) the gradient pair are separated by the very short time of the final pulse, thus keeping the diffusion induced losses to an absolute minimum. In (c) the two gradients are separated by the incrementable time t1; as this increases the losses due to diffusion will also increase resulting in an extra decay of the signal in t1. The extra line broadening due to this decay can be estimated from Eqn. [40], with ∆ = t1, as γ2G2τ2D/π Hz. For a pair of 2 ms gradients of strength 10 G⋅cm–1 this amounts §+]LQWKHFDVHRIDVPDOOPROHFXOH
This effect by which diffusion causes an extra line broadening in the F1 dimension is usually described as diffusion weighting. Generally it is possible to avoid this effect by careful placing of the gradients. For example, the sequences in Fig. 21 (b) and (c) are in every other respect equivalent, thus there is no reason not to chose (b). It should be emphasised that diffusion weighting occurs only when t1 intervenes between the dephasing and refocusing gradients.