Phase Cycling and Gradient Pulses - The James Keeler Group

and then further transformation with respect to t 1
gives
1
SP ( F1,
F2)= 4
A1+ + iD1+
A2 iD2
[ ][ + ]
The real part of this spectrum is
1
Re { SP( F1,
F2)
}= A
+
A − D
+
D
[ ]
4 1 2 1 2
The quantity in the square brackets on the right represents a phase-twist
lineshape at F 1
= +Ω, F 2
= Ω
Perspective view and contour plot of the phase-twist lineshape. Negative contours are shown dashed.
This lineshape is an inextricable mixture of absorption and dispersion, and it is
very undesirable for high-resolution NMR. So, although a phase modulated
signal gives us frequency discrimination, which is desirable, it also results in a
phase-twist lineshape, which is not.
The time domain signal for the amplitude modulated data set can be written
as
1
S t , t cos Ωt exp iΩt exp t T exp t T
C
( )= ( ) ( ) ( − ) ( − )
1 2
2 1 2 1 2 2 2
Fourier transformation with respect to t 2
gives
which can be rewritten as
1
S t , F cos Ωt exp t T A iD
C
( 1 2)= 2 ( 1) ( −
1 2) [ 2
+
2]
[ ] −
( )= ( )+ ( − )
( )[ + ]
1
SC t1, F2
exp iΩt exp iΩt exp t T A iD
4 1 1 1 2 2 2
Fourier transformation with respect to t 1
gives, in the real part of the spectrum
{ ( )}= [ ]+
4 [ 1 2 1 2]
1
1
Re SC F1, F2
4
A1+ A2 – D1+
D2
A
−A – D
–
D
This corresponds to two phase-twist lineshapes, one at F 1
= +Ω, F 2
= Ω and the
other at F 1
= –Ω, F 2
= Ω; the lack of frequency discrimination is evident.
Further, the undesirable phase-twist lineshape is again present.
The lineshape can be restored to the absorption mode by discarding the
imaginary part of the time domain signal after the transformation with respect
to t 2
, i.e. by taking the real part
{ C( 1 2)
}=
2 ( 1) ( −
1 2)
2
1
Re S t , F cos Ωt exp t T A
Subsequent transformation with respect to t 1
gives, in the real part
9–16