Biomolecular NMR spectroscopy
Exercise 6
1. Consider a weakly coupled two-spin IS-system that is subject to the pulse sequence
below known as INEPT (Insensitive Nucleus Enhanced by Polarization Transfer).
Write the form of density operator d at the time point d, when pulses are applied with
phase x unless otherwise indicated. 90-degree pulses are marked by narrow lines and
180-degree pulses with rectangles and delay  = 1/(4J). What is the signal
enhancement of S-spin compared to direct excitation?
y




d
INEPT is probably the most used method for moving magnetization (in general
coherence) from nucleus to nucleus. Once again we invoke the standard protocol to
“solve” the equation of motion step-by-step. It’s a bit tedious in particular in this case
when the system evolves under the chemical shift and the weak scalar coupling
Hamiltonians. It is important to realize that these two Hamiltonians commute which
allows you to choose the order of actions i.e. you can develop chemical shift first and
then the scalar coupling or vice versa. It will turn out that the chemical shift will
refocus in the echo of INEPT. Therefore your lazy instructor will skip to prove this and
rather leave it for you to show. (There will eventually be sixteen terms that will neatly
combine to give only one.) Only the evolution due to the scalar coupling will be
considered. Very fortunately we just obtained the result (exer. 5) prior to the last two
90-degree pulses and we should simply proceed with these to give
 I
 S
J 2 I z S z
I x
J 2 I z S z
2 y
2 x
I y 

 


 2 I x S z 
 2 I z S z 
 2 I z S y
2. In order to measure transverse relaxation of S-spin i.e. Sx extend the INEPT sequence
by an echo that will convert the antiphase 2IzSy to Sx. The result is known as the
refocused INEPT.
Later, in study of relaxation, you will find that the antiphase 2IzSy and inphase Sx are
not identical in their properties. Therefore you should be motivated to convert the
antiphase to inphase term by a simple echo.
3. Extend the refocused INEPT by series of echoes that will remove the chemical shift
and scalar coupling evolutions during which the amplitude of Sx will simply keep on
decreasing due to the transverse relaxation.
1
Very well, you probably got from the refocused-INEPT.
I
Sx
J 2 I z S z
J 2 I z S z
y
 2 I z S y 

 



 S x
Now you simply like to keep the inphase term and simply follow its amplitude falling
in steps. Your echo must now remove both chemical shift evolution and scalar coupling
evolution, i.e. you will simply spend time and monitor the decay exp(-R2).
4. Design a spin-state selective filter element, i.e. two equally long alternatives a and b, to
be added on the sequence below
y




so that either 2IzSy (a) or Sy (b) will be generated. In this way the transverse relaxation
rate has acted (approximately) equally long on both 2IzSy and Sy. Analyze the subsequent
evolution of 2IzSy and Sy under the chemical shift and scalar coupling during t1 e.g. by
making a Fourier transformation of the resulting terms. Add and subtract the signals of
the two experiments.
It is easy to see that on can create -2IzSx as in 1 by an 90-degree y-pulse. We allow this to
develop into Sy just as in the refocused INEPT. It is easy to see that on can create -2 IzSy
as in 1 by an 90-degree x-pulse. The trouble is to keep this i.e. we wish to remove
chemical shift evolution and scalar coupling evolution. We can put two 180 pulses on S
in the symmetrically in the total delay of 2 first /2 and second 3/2.
2