Phase Cycling and Gradient Pulses - The James Keeler Group

RF
g
2
1
0
–1
–2
t 1
t 2
τ1
τ 2
Simple pulse sequence for
DQF COSY with gradient
selection and retention of
symmetrical pathways in t 1 .
The second gradient is twice
the area of the first.
(a)
RF
g
(b)
RF
g
τ
τ
τ
180°
τ
τ
180°
τ
τ
In multi-dimensional NMR the uncompensated evolution of offsets during
gradient pulses has disastrous effects on the spectra. This is illustrated here for
the DQF COSY experiment, for which a pulse sequence using the gradient
pulses is shown opposite. The problem with this sequence is that the double
quantum coherence will evolve during delay τ 1
. Normally, the delay between
the last two pulses would be a few microseconds at most, during which the
evolution is negligible. However, once we need to apply a gradient the delay
will need to be of the order of milliseconds and significant evolution takes
place. The same considerations apply to the second gradient.
We can investigate the effect of this evolution by analysing the result for a
two-spin system. In the calculation, it will be assumed that only the indicated
pathway survives and that the spatially dependent part of the evolution due to
the gradients can be ignored as ultimately it is refocused. The coherence with
order of + 2 present during τ 1
evolves according to
( ( ) )
Ω1τ1I1z
+ Ω2τ1I2z
I I ⎯⎯⎯⎯ ⎯ →I I exp – i Ω + Ω τ ,
1+ 2+
1+ 2+
1 2 1
where Ω 1
and Ω 2
are the offsets of spins 1 and 2, respectively. After the final
90° pulse and the second gradient the observable terms on spin 1 are
[ ]
( ( ) ) +
exp – i Ω + Ω τ cosΩτ 2I I sinΩτ 2I I [14]
i
2 1 2 1 1 2 1x 2z 1 2 1y 2z
where it has been assumed that τ 2
is sufficiently short that evolution of the
coupling can be ignored. It is clearly seen from Eqn. [14] that, due to the
evolution during τ 2
, the multiplet observed in the F 2
dimension will be a
mixture of dispersion and absorption anti-phase contributions. In addition,
there is an overall phase shift due to the evolution during τ 1
. The phase
correction needed to restore this multiplet to absorption depends on both the
frequency in F 2
and the double-quantum frequency during the first gradient.
Thus, no single linear frequency dependent phase correction could phase
correct a spectrum containing many multiplets. The need to control these phase
errors is plain.
The general way to minimise these problems is to associate each gradient
with a refocusing pulse, as shown opposite. In sequence (a) the gradient is
placed in one of the delays of a spin echo; the evolution of the offset during the
first delay τ is refocused during the second delay τ. So, overall there are no
phase errors due to the evolution of the offset.
An alternative is the sequence (b). Here, as in (a), the offset is refocused
over the whole sequence. The first gradient results in the usually spatially
dependent phase and then the 180° pulse changes the sign of the coherence
order. As the second gradient is opposite to the first, it causes further
dephasing; effectively, it is as if a gradient of length 2τ is applied. Sequence
(b) will give the same dephasing effect as (a) if each gradient in (b) is of
duration τ/2; the overall sequence will then be of duration τ. If losses due to
relaxation are a problem, then clearly sequence (b) is to be preferred as it takes
less time than (a).
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