Phase Cycling and Gradient Pulses - The James Keeler Group

Using this, the effect of the phase shifted pulse on the initial state σ ( p)
can be
written
U σ
φ
( p)
– 1
Uφ
( ) ( )
( )
−
( ) ( )
p
– 1
= exp −iφFz
U0exp iφFz
σ exp iφFz
U0
exp iφFz
The central three terms can be simplified by application of Eq. [2]
giving
( ) ( − ) = ( )
p
– 1
exp iφF σ exp iφF U exp ipφ σ
z
( ) ( p)
z 0
( ) ( )
( ) ( )
p
p
Uφσ U – 1
φ
= ( pφ) − φFz
U σ U – 1
exp i exp i
0 0
exp iφFz
The central three terms can, from Eq. [5], be replaced by σ ( p' ) to give
( ) ( )
φ
z
p
p
U σ U – 1
= exp( ipφ) exp −iφF σ '
exp iφF
φ
Finally, Eq. [5] is applied again to give
( ) ( )
( ) – ( p'
)
φ
p 1
U σ U exp i pφ exp –ip'
φ σ
φ
= ( ) ( )
Defining ∆p = (p' – p) as the change is coherence order, this simplifies to
z
[7]
p – 1
p'
U σ U exp – i∆ pφ σ
[8]
φ
( ) ( )
φ
= ( )
Equation [8] says that if the phase of a pulse which is causing a change in
coherence order of ∆p is shifted by φ the coherence will acquire a phase label
(–∆p φ). It is this property which enables us to separate different changes in
coherence order from one another by altering the phase of the pulse.
In the discussion so far it has been assumed that U φ
represents a single pulse.
However, any sequence of pulses and delays can be represented by a single
unitary transformation, so Eq. [8] applies equally well to the effect of phase
shifting all of the pulses in such a sequence. We will see that this property is
often of use in writing phase cycles.
If a series of phase shifted pulses (or pulse sandwiches) are applied a phase
(–∆p φ) is acquired from each. The total phase is found by adding up these
individual contributions. In an NMR experiment this total phase affects the
signal which is recorded at the end of the sequence, even though the phase shift
may have been acquired earlier in the pulse sequence. These phase shifts are,
so to speak, carried forward.
9.3.4 Coherence transfer pathways diagrams
In designing a multiple-pulse NMR experiment the intention is to have specific
orders of coherence present at various points in the sequence. One way of
indicating this is to use a coherence transfer pathway (CTP) diagram along
with the timing diagram for the pulse sequence. An example of shown below,
which gives the pulse sequence and CTP for the DQF COSY experiment.
9–12