1
Exercises for Chapters 6, 7, 8 & 9
The more difficult and challenging problems are marked with an asterisk, *
Chapter 6: Product operators
E6-1 Using the standard rotations from section 6.1.4, express the following
rotations in terms of sines and cosines:
exp exp exp exp exp exp
exp exp exp exp exp exp
-( ) ( ) -( ) -( ) ( ) - -( ) ( )
-( ) ( ) -( ) ( ) ( ) -( ) -( )
i I I i I i I I i I i I I i I
i I I i I i I I i I i I I i I
x y x z y z y z y
y x y x x x x z x
     
     1
2
1
2
Express all of the above transformations in the shorthand notation of section
6.1.5.
E6-2. Repeat the calculation in section 6.1.6 for a spin echo with the 180°
pulse about the y-axis. You should find that the magnetization refocuses onto
the ­y axis.
E6-3. Assuming that magnetization along the y-axis gives rise to an
absorption mode lineshape, draw sketches of the spectra which arise from the
following operators
I1y I2x 2I1yI2z 2I1zI2x
E6-4. Describe the following terms in words:
I1y I2z 2I1yI2z 2I1xI2x
E6-5. Give the outcome of the following rotations
I
I I
I I
I
I
I I
x
t I
x z
I I
x z
I
x
tI tI
x
J tI I
z y
J tI I
y
y y
y
z z
z z
z z
1
1 2
2
1 2
1
1
2
1 2
2
1 1
1 2
2
1 1 2 2
12 1 2
12 1 2
2
2
2





p
 
 
 
   
-  
-  
( ) +( )
-
 

Describe the outcome in words in each case.
E6-6. Consider the spin echo sequence
­  ­ 180°(x, to spin 1 and spin 2) ­  ­
applied to a two-spin system. Starting with magnetization along y, represented
by I1y, show that overall effect of the sequence is
2
I J I J I Iy y x z1 12 1 12 1 22 2 2spin echo
  - ( ) + ( )cos sin   
You should ignore the effect of offsets, which are refocused, are just consider
evolution due to coupling.
Is your result consistent with the idea that this echo sequence is
equivalent to
­ 2 ­ 180°(x, to spin 1 and spin 2)
[This calculation is rather more complex than that in section 6.4.1. You will
need the identities
cos cos sin sin cos sin2 2 22 2
     = - =and ]
E6-7. For a two-spin system, what delay, , in a spin echo sequence would
you use to achieve the following overall transformations (do not worry about
signs)? [cos /4 = sin /4 = 1/2]
I I I
I I I
I I I I
I I
y z x
z x x
x x y z
x x
2 1 2
1 2 2
1
1
2 1
1
2 1 2
1 1
2
2
2
 
 
  ( ) + ( )
  -
E6-8. Confirm by a calculation that spin echo sequence c shown on page 6­11
does not refocus the evolution of the offset of spin 1. [Start with a state I1x or I1y;
you may ignore the evolution due to coupling].
*E6-9. Express 2 1 2I Ix y in terms of raising and lowering operators: see section
6.5.2. Take the zero-quantum part of your expression and then re-write this in
terms of Cartesian operators using the procedure shown in section 6.5.2.
E6-10. Consider three coupled spins in which J23 > J12. Following section 6.6,
draw a sketch of the doublet and doublets expected for the multiplet on spin 2
and label each line with the spin states of the coupled spins, 1 and 3. Lable the
splittings, too.
Assuming that magnetization along x gives an absorption mode
lineshape, sketch the spectra from the following operators:
I I I I I I I Ix z x y z z x z2 1 2 2 3 1 2 32 2 4
E6-11. Complete the following rotations.
3
I
I I
I I I
I I
x
tI tI tI
y z
I I I
x z z
I I I
z x
J tI I
z z z
y y y
y y y
z z
2
1 2
2
1 2 3
2
1 2
2
1 1 2 2 3 3
1 2 3
1 2 3
12 1 2
2
2
2
  
     
 
 
 
( ) + +( )
( ) + +( )



22
4
1 2
2
1 2 3
2
1
2 2
23 2 3
23 2 3
12 1 2 13 1 3
I I
I I I
I
z x
J tI I
z y z
J tI I
x
J tI I tJ I I
z z
z z
z z z z


 
 
 
   
Chapter 7: Two-dimensional NMR
E7-1. Sketch the COSY spectra you would expect from the following
arrangements of spins. In the diagrams, a line represents a coupling. Assume
that the spins have well separated shifts; do not concern yourself with the details
of the multiplet structures of the cross- and diagonal-peaks.
A B C A B C
A B C
E7-2. Sketch labelled two-dimensional spectra which have peaks arising from
the following transfer processes
frequencies in F1 / Hz frequencies in F2 / Hz
a 30 Transferred to 30
b 30 and 60 transferred to 30
c 60 transferred to 30 and 60
d 30 transferred to 20, 30 and 60
e 30 and 60 transferred to 30 and 60
E7-3. What would the diagonal-peak multiplet of a COSY spectrum of two
spins look like if we assigned the absorption mode lineshape in F2 to
magnetization along x and the absorption mode lineshape in F1 to sine
modulated data in t1?
What would the cross-peak multiplet look like with these assignments?
E7-4. The smallest coupling that will gives rise to a discernible cross-peak in a
COSY spectra depends on both the linewidth and the signal-to-noise ratio of the
spectrum. Explain this observation.
4
*E7-5. Complete the analysis of the DQF COSY spectrum by showing in
detail that both the cross and diagonal-peak multiplets have the same lineshape
and are in anti-phase in both dimensions. Start from the expression in the
middle of page 7­10, section 7.4.21. [You will need the identity
cos sin sin sinA B B A B A= +( ) + -( )[ ]1
2 ].
*E7-6. Consider the COSY spectrum for a three-spin system. Start with
magnetization just on spin 1. The effect of the first pulse is
I Iz
I I I
y
x x x
1
2
1
1 2 3( ) + +( )
  -
Then, only the offset of spin 1 has an effect
-   - +I t I t Iy
t I
y x
z
1 1 1 1 1 1 1
1 1 1
 cos sin
Only the term in I1x leads to cross- and diagonal peaks, so consider this term
only from now on.
First allow it to evolve under the coupling to spin 2 and then the coupling to spin
3
sin1 1 1
2 212 1 1 2 13 1 1 3
t I x
J t I I J t I Iz z z z 
   
Then, consider the effect of a 90°(x) pulse applied to all three spins. After this
pulse, you should find one term which represents a diagonal-peak multiplet, one
which represents a cross-peak multiplet between spin 1 and spin 2, and one
which represents a cross-peak multiplet between spin 1 and spin 3. What does
the fourth term represent?
[More difficult] Determine the form of the cross-peak multiplets, using the
approach adopted in section 6.4.1. Sketch the multiplets for the case J12  J23 >
J13. [You will need the identity
sin sin cos cosA B A B A B= +( ) - -( )[ ]1
2 ]
*E7-7. The pulse sequence for two-dimensional TOCSY (total correlation
spectroscopy) is shown below
t1 isotropic mixing
t2

The mixing time, of length , is a period of isotropic mixing. This is a multiple-
pulse sequence which results in the transfer of in-phase magnetization from one
spin to another. In a two spin system the mixing goes as follows:
I J I J Ix x x1
2
12 1
2
12 2
isotropic mixing for time 
     +cos sin
We can assume that all terms other than I1x do not survive the isotropic mixing
sequence, and so can be ignored.
Predict the form of the two-dimensional TOCSY spectrum for a two-spin
system. What is the value of  which gives the strongest cross peaks? For this
optimum value of , what happens to the diagonal peaks? Can you think of any
5
advantages that TOCSY might have over COSY?
E7-8. Repeat the analysis for the HMQC experiment , section 7.4.3.1, with
the phase of the first spin-2 (carbon-13) pulse set to ­x rather than +x. Confirm
that the observable signals present at the end of the sequence do indeed change
sign.
E7-9. Why must the phase of the second spin-1 (proton) 90° pulse in the
HSQC sequence, section 7.4.3.2, be y rather than x?
E7-10. Below is shown the pulse sequence for the HETCOR (heteronuclear
correlation) experiment
1
H
13
C
t1
A C
t2

2

2

2

2
B
This sequence is closely related to HSQC, but differs in that the signal is
observed on carbon-13, rather than being transferred back to proton for
observation. Like HSQC and HMQC the resulting spectrum shows cross peaks
whose co-ordinates are the shifts of directly attached carbon-13 proton pairs.
However, in contrast to these sequences, in HETCOR the proton shift
is in F1 and the carbon-13 shift is in F2 . In the early days of two-dimensional
NMR this was a popular sequence for shift correlation as it is less demanding of
the spectrometer; there are no strong signals from protons not coupled to
carbon-13 to suppress.
We shall assume that spin 1 is proton, and spin 2 is carbon-13. During period
A, t1, the offset of spin 1 evolves but the coupling between spins 1 and 2 is
refocused by the centrally placed 180° pulse. During period B the coupling
evolves, but the offset is refocused. The optimum value for the time  is
1/(2J12), as this leads to complete conversion into anti-phase. The two 90°
pulses transfer the anti-phase magnetization to spin 2.
During period C the anti-phase magnetization rephases (the offset is refocused)
and if  is 1/(2J12) the signal is purely in-phase at the start of t2.
Make an informal analysis of this sequence, along the lines of that given in
section 7.4.3.2, and hence predict the form of the spectrum. In the first instance
assume that  is set to its optimum value. Then, make the analysis slightly
more complex and show that for an arbitrary value of  the signal intensity goes
as sin2
J12.
Does altering the phase of the second spin-1 (proton) 90° pulse from x to y
make any difference to the spectrum?
[Harder] What happens to carbon-13 magnetization, I2z, present at the beginning
of the sequence? How could the contribution from this be removed?
6
Chapter 8: Relaxation
E8-1. In an inversion-recovery experiment the following peak heights (S,
arbitrary units) were measured as a function of the delay, t, in the sequence:
t / s 0.1 0.5 0.9 1.3 1.7 2.1 2.5 2.9
S ­98.8 ­3.4 52.2 82.5 102.7 115.2 120.7 125.1
The peak height after a single 90° pulse was measured as 130.0 Use a graphical
method to analyse these data and hence determine a value for the longitudinal
relaxation rate constant and the corresponding value of the relaxation time, T1.
E8-2. In an experiment to estimate T1 using the sequence [180° ­  ­ 90°]
acquire three peaks in the spectrum were observed to go through a null at 0.5,
0.6 and 0.8 s respectively. Estimate T1 for each of these resonances.
A solvent resonance was still inverted after a delay of 1.5 s; what does this tell
you about the relaxation time of the solvent?
*E8-3. Using the diagram at the top of page 8­5, write down expressions for
dn1/dt, dn2/dt etc. in terms of the rate constants W and the populations ni. [Do
this without looking at the expressions given on page 8­6 and then check
carefully to see that you have the correct expressions].
*E8-4. Imagine a modified experiment, designed to record a transient NOE
enhancement, in which rather than spin S being inverted at the beginning of the
experiment, it is saturated. The initial conditions are thus
I I Sz z z0 0 00
( ) = ( ) =
Using these starting conditions rather than those of Eq. [16] on page 8-10, show
that in the initial rate limit the NOE enhancement builds up at a rate proportional
to IS rather than 2IS. You should use the method given in Section 8.4.1 for
your analysis.
Without detailed calculation, sketch a graph, analogous to that given on page 8-
12, for the behaviour of Iz and Sz for these new initial conditions as a function of
mixing time.
E8-5. Why is it that in a two spin system the size of transient NOE
enhancements depends on RI , RS and IS, whereas in a steady state experiment
the enhancement only depends on RI and IS? [Spin S is the target].
In a particular two-spin system, S relaxes quickly and I relaxes slowly. Which
experiment would you choose in order to measure the NOE enhancement
between these two spins? Include in your answer an explanation of which spin
you would irradiate.
7
E8-6. For the molecule shown on the right, a transient
NOE experiment in which HB is inverted gave equal
initial NOE build-up rates on HA and HC. If HA was
inverted the initial build-up rate on HB was the same as in
the first experiment; no enhancement is seen of HC. In
steady state experiments, irradiation of HB gave equal
enhancements on HA and HB. However, irradiation of
HA gave a much smaller enhancement on HB than for the
case where HB was the irradiated spin and the
enhancement was observed on HA. Explain.
Z
HA
HB
HC
X
Y
E8-7. What do you understand by the terms correlation time and spectral
density? Why are these quantities important in determining NMR relaxation rate
constants?
E8-8. The simplest form of the spectral density, J(), is the Lorentzian:
J 

 
( ) =
+
2
1 2
c
c
2
Describe how this spectral density varies with both  and c. For a given
frequency, 0, at what correlation time is the spectral density a maximum?
Show how this form of the spectral density leads to the expectation that, for a
given Larmor frequency, T1 will have a minimum value at a certain value of the
correlation time, c.
E8-9. Suppose that he Larmor frequency (for proton) is 800 MHz. What
correlation time will give the minimum value for T1? What kind of molecule
might have such a correlation time?
E8-10. Explain why the NOE enhancements observed in small molecules are
positive whereas those observed for large molecules are negative.
E8-11. Explain how it is possible for the sign of an NOE enhancement to
change when the magnetic field strength used by the spectrometer is changed.
E8-12. What is transverse relaxation and how it is different from longitudinal
relaxation? Explain why it is that the rate constant for transverse relaxation
increases with increasing correlation times, whereas that for longitudinal
relaxation goes through a maximum.
8
Chapter 9: Coherence selection: phase cycling and gradient pulses
E9-1. (a) Show, using vector diagrams like those of section 9.1.6, that in a
pulse-acquire experiment a phase cycle in which the pulse goes x, y, ­x, ­y and
in which the receiver phase is fixed leads to no signal after four transients have
been co-added.
(b) In a simple spin echo sequence
90° ­  ­ 180° ­  ­
the EXORCYCLE sequence involves cycling the 180° pulse x, y, ­x, ­y and the
receiver x, ­x, x, ­x. Suppose that, by accident, the 180° pulse has been omitted.
Use vector pictures to show that the four step phase cycle cancels all the signal.
(c) In the simple echo sequence, suppose that there is some z-magnetization
present at the end of the first  delay; also suppose that the 180° pulse is
imperfect so that some of the z-magnetization is made transverse. Show that the
four steps of EXORCYCLE cancels the signal arising from this magnetization.
*E9-2. (a) For the INEPT pulse sequence of section 9.1.8, confirm with
product operator calculations that: [You should ignore the evolution of offsets
as this is refocused by the spin echo; assume that the spin echo delay is 1/(2JIS)].
(i) the sign of the signal transferred from I to S is altered by changing the phase
of the second I spin 90° pulse from y to ­y;
(ii) the signs of both the transferred signal and the signal originating from
equilibrium S spin magnetization, Sz, are altered by changing the phase of the
first S spin 90° pulse by 180°.
On the basis of your answers to (i) and (ii), suggest a suitable phase cycle,
different to that given in the notes, for eliminating the contribution from the
equilibrium S spin magnetization.
(b) Imagine that in the INEPT sequence the first I spin 180° pulse is cycled x, y,
­x, ­y. Without detailed calculations, deduce the effect of this cycle on the
transferred signal and hence determine a suitable phase cycle for the receiver
[hint - this 180° pulse is just forming a spin echo]. Does your cycle eliminate
the contribution from the equilibrium S spin magnetization?
(c) Suppose now that the first S spin 180° pulse is cycled x, y, ­x, ­y; what
effect does this have on the signal transferred from I to S?
E9-3. Determine the coherence order or orders of each of the following
operators [you will need to express Ix and Iy in terms of the raising and lowering
operators, see section 9.3.1]
I I I I I I I I I I I I Iz x y x z x x y y1 2 1 2 3 1 1 1 2 1 2 1 24 2 2 2+ - + + +( )
In a heteronuclear system a coherence order can be assigned to each spin
9
separately. If I and S represent different nuclei, assign separate coherence orders
for the I and S spins to the following operators
I S I S I Sx y x z x x2 2
E9-4. (a) Consider the phase cycle devised in section 9.5.1 which was
designed to select p = ­3: the pulse phase goes 0, 90, 180, 270 and the receiver
phase goes 0, 270, 180, 90. Complete the following table and use it to show that
such a cycle cancels signals arising from a pathway with p = 0.
step
pulse
phase
phase shift experienced
by pathway with p = 0
equivalent
phase
rx. phase for
p = ­3
difference
1 0 0
2 90 270
3 180 180
4 270 90
Construct a similar table to show that a pathway with p = ­1 is cancelled, but
that one with p = +5 is selected by this cycle.
(b) Bodenhausen et al. have introduced a notation in which the sequence of
possible p values is written out in a line; the values of p which are selected by
the cycle are put into bold print, and those that are rejected are put into
parenthesis, viz (1). Use this notation to describe the pathways selected and
rejected by the cycle given above for pathways with p between ­5 and +5 [the
fate of several pathways is given in section 9.5.1, you have worked out two
more in part (a) and you may also assume that the pathways with p = ­5, ­4,
­2, 3 and 4 are rejected]. Confirm that, as expected for this four-step cycle, the
selected values of p are separated by 4.
(c) Complete the following table for a three-step cycle designed to select p =
+1.
step pulse
phase
phase shift experienced by
pathway with p = +1
equivalent
phase
rx. phase
1 0
2 120°
4 240°
(d) Without drawing up further tables, use the general rules of section 9.5.2 to
show that, in Bodenhausen's notation, the selectivity of the cycle devised in (c)
can be written:
­2 (­1) (0) 1 (2)
(e) Use Bodenhausen's notation to describe the selectivity of a 6 step cycle
designed to select p = +1; consider p values in the range ­6 to +6.
E9-5. Draw coherence transfer pathways for (a) four-quantum filtered COSY
10
[use the sequence in section 9.5.5.4 as a model]; (b) a 180° pulse used to refocus
double-quantum coherence; (c) N-type NOESY.
E9-6. Write down four-step phase cycles to select (a) p = ­1 and (b) p =
+2. Suppose that the cycles are applied to different pulses. Combine them to
give a 16-step cycle as was done in section 9.5.4; give the required receiver
phase shifts.
*E9-7. The CTP and pulse sequence of triple-quantum filtered COSY are
t1
t2
0
­1
­2
­3
1
2
3
(a) Write down the values of p brought about by (i) the first pulse, (iii) the
second pulse, (iii) the first and second pulses acting together and considered as a
group, (iv) the last pulse.
(b) Imagine the spin system under consideration is able to support multiple
quantum coherence up to and including p = 4. Consider a phase cycle in which
the first two pulses are cycled as a group and, having in mind your answer to (a)
(iii), use the notation of Bodenhausen to indicate which pathways need to be
selected and which blocked. Hence argue that the required phase cycle must
have 6 steps. Draw up such a six-step phase cycle and confirm that the
proposed sequence of pulse and receiver phases does indeed discriminate against
filtration through double-quantum coherence.
(c) Consider an alternative phase cycle in which just the phase of the last pulse
is cycled. As in (b), use Bodenhausen's notation to describe the wanted and
unwanted values of p. Devise a suitable phase cycle to select the required
values of p.
(d) Try to devise a cycle which involves shifting just the phase of the second
pulse.
E9-8. (a) Write down a 16 step cycle which selects the pathways shown in
the double quantum spectroscopy pulse sequence of section 9.5.9.1; include in
your cycle double-quantum selection and EXORCYCLE phase cycling of the
180° pulse.
(b) Write down an 8 step cycle which selects the pathway for NOESY shown
in section 9.5.9.2; include in your cycle explicit axial peak suppression steps.
*E9-9. The following sequence is one designed to measure the relaxation-
induced decay of double-quantum coherence as a function of the time 2
11
0
­1
­2
1
2
 
The 180° pulse placed in the middle of the double-quantum period is used to
refocus evolution due to offsets and inhomogeneous line broadening.
Devise a suitable phase cycle for the second 180° pulse bearing in mind that the
180° pulse may be imperfect. [hint: are four steps sufficient?]
E9-10. Use the formula given in section 9.6.2 for the overall decay of
magnetization during a gradient
2
Gtrmax
to calculate how long a gradient is needed to dephase magnetization to (a) 10%
and (b) 1% of its initial value assuming that: G = 0.1 T m­1
(10 G
cm­1
), rmax = 0.005 m (0.5 cm) and  = 2.8 × 108
rad s­1
. [Put all the quantities
in SI units].
E9-11. Imagine that two gradients, G1 and G2, are placed before and after a
radiofrequency pulse. For each of the following ratios between the two
gradients, identify two coherence transfer pathways which will be refocused:
(a) G1:G2 = 1:1 (b) G1:G2 = ­1:1 (c) G1:G2 = 1:­2 (d) G1:G2 = 0.5:1
E9-12. Determine the ratio G1:G2 needed to select the following pathways:
(a) p = 2  1 (b) p = 3  1 (c) p = ­3  1
(d) p = ­1  1 and p = 1  ­1 (e) p = 0  1
Comment on the way in which case (e) differs from all the others.
E9-13. Consider a gradient selected N-type DQF COSY
t1
G1
G2
G3
t2
0
­1
­2
1
2
We use three gradients; G1 in t1 so as to select p = +1 during t1; G2 to select
double quantum during the filter delay and G3 to refocus prior to acquisition.
(a) Show that the pathway shown, which can de denoted 012­1, is
12
selected by gradients in the ratio G1:G2:G3 = 1:1:3.
(b) Show that this set of gradients also selects the pathway
0­14­1. What kind of spectrum does such a pathway give rise to?
(c) Consider gradients in the ratios G1:G2:G3 (i) 1: 1
2 :2 and (ii) 1: 1
3 : 5
3 . Show
that these combination select the DQ filtered pathway desired. In each case, give
another possible pathway which has p = 1 during t1 and p = ­1 during t2 that
these gradient combinations select.
(d) In the light of (b) and (c), consider the utility of the gradient ratio
1.0 : 0.8 : 2.6
*E9-14.
Devise a gradient selected version of the triple-quantum filtered COSY
experiment, whose basic pulse sequence and CTP was given in E9-7. Your
sequence should include recommendations for the relative size of the gradients
used. The resulting spectrum must have pure phase (i.e. p = 1 must be
preserved in t1) and phase errors due to the evolution of offsets during the
gradients must be removed.
How would you expect the sensitivity of your sequence to compare with its
phase-cycled counterpart?
*E9-1. A possible sequence for P/N selected HMQC is
t2
1
0
­1
t1
1
0
­1
a
b
13
C
1H
pI
pS
g
G1 G2
  
The intention is to recombine P- and N-type spectra so as to obtain absorption
mode spectra.
(a) Draw CTPs for the P-type and N-type versions of this experiment [refer to
section 9.6.7.3 for some hints].
(b) What is the purpose of the two 180° pulses a and b, and why are such
pulses needed for both proton and carbon-13?
(c) Given that H/C = 4, what ratios of gradients are needed for the P-type and
the N-type spectra?
(d) Compare this sequence with that given in section 9.6.7.3, pointing out any
advantages and disadvantages that each has.