Phase Cycling and Gradient Pulses - The James Keeler Group

1
0
–1
–y
t 1 τ m
t 2
and I –
1
cosΩt ( I + I )
9–14
2 1
+ −
Of these operators, only I –
leads to an observable signal, as this corresponds to
p = –1. Allowing I –
to evolve in t 2
gives
( ) −
cosΩt exp iΩt I
1
2 1 2
The final detected signal can be written as
1
S t , t cosΩt exp iΩt
C
( )= ( )
1 2
2 1 2
This signal is said to be amplitude modulated in t 1
; it is so called because the
evolution during t 1
gives rise, via the cosine term, to a modulation of the
amplitude of the observed signal.
The situation changes if we select a different pathway, as shown opposite.
Here, only coherence order –1 is preserved during t 1
. At the start of t 1
the
operator present is –I y
which can be written
( )
1
−
2i
I+ − I−
Now, in accordance with the CTP, we select only the I –
term. During t 1
this
evolves to give
( ) −
exp iΩt I
1
2i
1
Following through the rest of the pulse sequence as before gives the following
observable signal
( )= ( ) ( )
1
SP t1, t2
exp iΩt exp iΩt
4 1 2
This signal is said to be phase modulated in t 1
; it is so called because the
evolution during t 1
gives rise, via exponential term, to a modulation of the
phase of the observed signal. If we had chosen to select p = +1 during t 1
the
signal would have been
( )= ( ) ( )
1
SN t1, t2
exp –iΩt exp iΩt
4 1 2
which is also phase modulated, except in the opposite sense. Note that in either
case the phase modulated signal is one half of the size of the amplitude
modulated signal, because only one of the two pathways has been selected.
Although these results have been derived for the NOESY sequence, they are
in fact general for any two-dimensional experiment. Summarising, we find
• If a single coherence order is present during t 1
the result is phase
modulation in t 1
. The phase modulation can be of the form exp(iΩt 1
) or
exp(–iΩt 1
) depending on the sign of the coherence order present.
• If both coherence orders ±p are selected during t 1
, the result is amplitude
modulation in t 1
; selecting both orders in this way is called preserving
symmetrical pathways.
9.4.2 Frequency discrimination
The amplitude modulated signal contains no information about the sign of Ω,