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13– Solution State NMR
Experiments: COSY and
Coherence Pathway Selection
As we saw in the previous chapter, in solution state
NMR, we can describe any pulse sequence by using
product operator formalism. In this chapter, we will
continue to use this formalism to understand a
fundamental two-dimensional sequence, namely the
COSY pulse sequence.
13.1 COrrelation SpectroscopY
Although the COSY pulse sequence looks deceptively
simple, it cannot be understood using classical vector
diagrams.
90 90
t1 t2
x x
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So let us use product operator formalism, and let us
begin by assuming that we have a two-spin system,
coupled via the scalar coupling J12. We start with
the equilibrium magnetization
I1z + I2z (13.1)
and apply the first 90o pulse along x:
I1z90Ix
−→ I1zcos(90) − I1ysin(90)
I1z90Ix
−→ −I1y (13.2)
and likewise for the second spin, which we will ignore
for the moment (i.e. we will not write equations for
this one).
Next we have an evolution period t1 during which
chemical shift evolution and evolution due to scalar
coupling occurs, taking the latter one first (order
doesn’t matter) we get:
−I1y2πJ12t1I1z
I2z
−→ −I1y cos(πJ12t1)
+ 2I1xI2zsin(πJ12t1)
−I1ycos(πJ12t1)Ω1t1Iz
−→ −I1y cos(πJ12t1)cos(Ω1t1)
+ I1xcos(πJ12t1)sin(Ω1t1)
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2I1xI2zsin(πJ12t1)Ω1t1Iz
−→ 2I1xI2z sin(πJ12t1)cos(Ω1t1)
+2I1yI2z sin(πJ12t1)sin(Ω1t1)
(13.3)
i.e. we have four terms, two in single-spin product
operators, and two double-spin product operators.
To keep track of the terms, we will make the
following abbreviations:
cos(πJ12t1)cos(Ω1t1) = cJ1cΩ1
cos(πJ12t1)sin(Ω1t1) = cJ1sΩ1
sin(πJ12t1)cos(Ω1t1) = sJ1cΩ1
sin(πJ12t1)sin(Ω1t1) = sJ1sΩ1
(13.4)
where the subscript 1 refers to t1 here and not spin 1.
Now applying the second 90o pulse to the terms in
equation 13.3, we get
− I1ycJ1cΩ1
90Ix
−→ −I1zcJ1cΩ1
I1xcJ1sΩ1
90Ix
−→ I1xcJ1sΩ1
2I1xI2zsJ1cΩ1
90Ix
−→ −2I1xI2ysJ1cΩ1
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2I1yI2zsJ1sΩ1
90Ix
−→ −2I1zI2ysJ1sΩ1
(13.5)
NOTE: The pulse also affects spin 2, which is
why in the third and fourth equation I2z is converted
into I2y.
These equations can be summarized in terms of a
flow chart to show that starting from I1z, we get a
number of different terms:I1z
I1x
I1x
2I1xI2z
2I1xI2z
−I
2I1yI2z
1xI2y−2I −2I
−I
−I
−I1y
1y
1y
1z I 2y1z
The terms in green are not observable, so we do not
need to consider their evolution during t2. Therefore
keeping only the observable terms, we can write out
the resulting evolution due to chemical shift and
scalar coupling:
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for I1x
I1xcJ1sΩ1
Ω1t2Iz
−→ I1xcJ1sΩ1cΩ2 + I1ycJ1sΩ1sΩ2
I1xcJ1sΩ1cΩ2
2πJ12t2I1zI2z
−→ I1xcJ1sΩ1cΩ2cJ2
+2I1yI2zcJ1sΩ1cΩ2sJ2
I1ycJ1sΩ1sΩ2
2πJ12t2I1zI2z
−→ I1ycJ1sΩ1cΩ2cJ2
−2I1xI2zcJ1sΩ1cΩ2sJ2
(13.6)
One can write the equations for the −2I1zI2y term
as well, such that overall the COSY sequence
produces:I1z
I1x
I1x
2I1xI2z
2I1xI2z
I1x
2I1yI2z
I1y
−2I I 2y1z
−2I I 2y1z I1z2I 2x
−I1xI2y−2I
−I
−I
−I1y
1y
1y
1z
−2I −2I I 2y1z I2y
I1x 2I1yI2z I1yI2x I1z2I 2x
1x 2zI
where the terms in red are diagonal peaks (since the
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magnetization belongs to spin 1 during both t1 and
t2, the terms in blue are cross-peaks (since the
magnetization shifts to spin 2), and the other terms
are antiphase unobservable magnetization. Note that
a similar tree can be written down starting from I2z.
Therefore, in summary, the COSY sequence produces
a set of observable and unobservable terms, which
can be represented using product operator vectors
(see following page - note that the rotation
convention in this case is different to the one we
normally use, so all terms will have opposite sign).
The observable terms are modulated by cos and sin
functions, and therefore, form in-phase or anti-phase
doublets in t1 and t2, i.e.
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ref: MLev, p. 397-398.
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In the case of spectral overlap, such peak shapes can
make interpretation very difficult. By using a
double-quantum filtered COSY (DQF-COSY), both
the diagonal and cross-peaks can be made to be
absorptive, thus removing this problem (see
additional handout).
A look at the “tree” of product operators generated
above leads to the idea that we can select for a
branch of this tree by choosing the phase of our
pulses appropriately. This leads us to the idea of
phase cycling and coherence pathway selection.
13.2 Coherences
In the previous chapter, we mentioned that for a
single spin, the density operators and Hamiltonians
are expressed in terms of cartesian base operators
(1/2, Ix, Iy, Iz). We can also write these operators in
terms of a shift operator basis: (1/2, I+, I−, Iz),
where I+ and I− are defined in the usual way.
We can define a parameter p, called coherence order,
which is defined as the number of I+ operators minus
the number of I− operators in a density operator.
Examples:
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Product operator p
I+
1 I+
2 +2
I−1 I+
2 0
I+
1 I2z +1
I+
2 +1
I1zI−
2 -1
Given this definition, we see that the equilibrium
state I1z we had above has a p = 0. For each
experiment, we can follow the experiment by writing
a coherence transfer pathway diagram. For the
COSY sequence above, we can write it as
90 90
t1 t2
01
2
−1−2
x x
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whereas for a DQF-COSY, we have:
90 90
t1
01
2
−1−2
t2
90xx x
How can we rationalize that a 90x pulse changes the
quantum number by ±1? We can do so by
remembering the terms that we derived above,
namely I1z becomes −I1y after the first π/2 pulse. If
we examine these two product operators in matrix
form we can see what is happening:
I1z =1
2
1 0 0 0
0 1 0 0
0 0 −1 0
0 0 0 −1
90Ix
−→ (13.7)
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−I1y =1
2i
0 0 −1 0
0 0 0 −1
1 0 0 0
0 1 0 0
(13.8)
or pictorially,
ββ
βα
αα
αβ βα
αα
αβ
ββ
For a more extensive look at coherence and different
aspects of it, I recommend Malcolm Levitt’s book.
13.3 Phase Cycling
There are two “levels” of phase cycling, A) one
where we change the phase of the pulses with every
scan and sum up the signal, and the other, B) where
we change one phase in the sequence with every (or
every second) t1 increment.
Pulse Phases In the first case, the basic idea is
to change the phase of one or more pulses from scan
to scan:
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90 90
t1 t2
x x, −x
This is like the example given in the previous
chapter, where we varied the phase of certain pulses
and the receiver to add and subtract different
signals. For this sequence, what we have is:
I1z
I1x
I1x
2I1xI2z
2I1xI2z
I1z
I1x
I1x
2I1xI2z
2I1xI2z 2I1y 2z
1x 2y −2I
−I
−I
−I1y
1y
1y
1z I 2y
90x
90−xI 2I
I
1zI
I
I−I
2I1y 2z
1x 2y−2I −2I
−I
−I
−I1y
1y
1y
1z I 2y1z
90x
90x
first scan:
second scan:
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So summing up the operators after two scans, the
green terms cancel out. This is just to illustrate the
concept.
States and TPPI In 2D experiments, the
signal often consists of a combination of absorptive
and dispersive terms.
Often, we would only like to keep the absorptive
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component as this gives us a much nicer peak shape.
To do this we can use States or Time Proportional
Phase Incrementation (TPPI):
t2
t2
t2
t2
t1=2∆
t2 t2
t2
t2
t2
t2
∆t1=
∆t1=
t1=2∆
t1=0
t1=∆
t1=0
x
−x
−y
x
y
States
t1=0
x
y
−x
−y
x
t1=3∆
t1=4∆
TPPI
The general approach on how to use States is
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described in Malcolm Levitt’s book, p. 118-119 and
involves a judicious combination/processing of the
signals:
For more details on coherence pathways, phase
cycling and States/TPPI see the excellent lecture
notes by James Keeler at:
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http://www-keeler.ch.cam.ac.uk/lectures/.