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3 Multidimensional NMR Spec tro sco py ? Gerd Gemmecker, 1999
Models used for the description of NMR experiments
1. energy level diagram: only for polarisations, not for time-dependent phenomena
2. classical treatment ( B LOCH EQUATIONS ): only for isolated spins (no J coupling!)
3. vektor diagram: pictorial representation of the classical approach (same limitations)
4. quantum mechanical treatment (density matrix): rather complicated; however, using
appropriate simplifications and definitions – the product operators – a fairly easy and correct
description of most experiments is possible
3.1. B LOCH Equations
The behaviour of isolated spins can be described by classical differential equations:
d M /dt = γM (t) x B (t) - R {M(t) -M 0 } [3-1]
with M 0 being the B OLTZMANN equilibrium magnetization and R the relaxation matrix:
x y z
R =
??
?
??
?1/Τ2 0 0
0 1/Τ2 0
0 0 1/Τ1
The external magnetic field consists of the static field B 0 and the oscillating r.f. field B rf :
B(t) = B 0 + B rf [3-2]
B rf = B 1 cos( ωt + φ)e x [3-3]
The time-dependent behaviour of the magnetization vector corresponds to rotations in space (plus
relaxation), with the B x and B y components derived from r.f. pulses and B z from the static field:
dM z /dt = γB x M y - γB y M x -(M z -M 0 )/T 1 [3-4]
dM x /dt = γB y M z - γB z M y - M x /T 2 [3-5]
dM y /dt = γB z M x - γB x M z - M y /T 2 [3-6]
Product operators
To include coupling a special quantum mechanical treatment has to be chosen for description. An
operator, called the spin density matrix ρ(t), completely describes the state of a large ensemble of
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spins. All observable (and non-observable) physical values can be extracted by multiplying the
density matrix with their appropriate operator and then calculating the trace of the resulting matrix.
The time-dependent evolution of the system is calculated by unitary transformations (corresponding
to "rotations") of the density matrix operator with the proper Hamiltonian H (including r.f. pulses,
chemical shift evolution, J coupling etc.):
ρ(t') = exp{i H t} ρ(t) exp{-i H t}
(for calculations these exponential operators have to be expanded into a Taylor series).
The density operator can be written als linear combination of a set of basis operators. Two specific
bases turn out to be useful for NMR experiments:
- the real Cartesian product operators I x , I y and I z (useful for description of observable
magnetization and effects of r.f. pulses, J coupling and chemical shift) and
- the complex single-element basis set I + , I - , I α and I β (raising / lowering operators, useful for
coherence order selection / phase cycling / gradient selection).
Cartesian Product operators
Lit. O.W. S?rensen et al. (1983), Prog. NMR. Spectr. 16 , 163-192
Single spin operators
correspond to magnetization of single spins, analogous to the classical macroscopic magnetization
M x , M y , M z .
I x , I y (in-phase coherence, observable )
I z ( z polarisation, not observable)
Two-spin operators
2 I 1x I 2 z , 2 I 1y I 2 z , 2 I 1z I 2 x , 2 I 1z I 2y (antiphase coherence, not observable)
2I 1 z I 2z (longitudi nal two-spin order, not observable)
2 I 1x I 2x , 2 I 1y I 2x , 2 I 1x I 2y , 2 I 1y I 2y (multiquantum coherence, not observable)
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Sums and differences of product operators
2 I 1x I 2x + 2 I 1y I 2y = I 1 + I 2 - + I 1 - I 2 + zero-quantum coherence
2 I 1y I 2x - 2 I 1x I 2y = I 1 + I 2 - - I 1 - I 2 + ( not observable)
2 I 1x I 2x - 2 I 1y I 2y = I 1 + I 2 + + I 1 - I 2 - double-quantum coherence
2 I 1x I 2y + 2 I 1y I 2x = I 1 + I 2 + - I 1 - I 2 - ( not observable)
The single-element operators I + and I - correspond to a transition from the m z = - 1 / 2 to the m z = + 1 / 2
state and back, resp., hence "raising" and "lowering operator". P roducts of three and more operators
are also possible.
Only the operators I x and I y represent observable magnetization. However, other terms like antiphase
magnetization 2 I 1x I 2z can evolve into observable terms during the acquisition period.
Pictorial representations of product operators
(cf. the paper in Progr. NMR Spectrosc. by S?rensen et al.)
α
α α αα
α
αα
αβ
αβαβ αβαβ
αβ
αβαβ
β
β β ββ
β
ββ
β
β β ββ
β
ββ
I x
I 1 z I 2 z 2I
1z2zI+1z 2z
Iy
II1x2z
II1y2z
y
z
coherences
polarisations
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In the energy level diagrams for coherences, the single quantum coherences I x and I y are
symbolically depicted as black and gray arrows. Both arrows in each two-spin scheme (for the
coupling partner being α or β) belong to the same operator; in the vector diagrams these two species
either align (for in-phase coherence) or a 180° out of phase (antiphase coherence). In the NMR
spectrum, these two arrows / transitions correspond to the two lines of the dublet caused by the J
coupling between the two spins. The term 2I 1x I 2z is called antiphase coherence of spin 1 with
respect to spin 2.
For the populations, filled circles represent a population surplus, empty circles a population deficit
(with respect to an even distribution). I 1z and I 2z are polarisations of one sort of spins only, I 1z +I 2z is
the normal B OLTZMANN equilibrium state, and 2 I 1z I 2z is called longitudinal two-spin order (with the
two spins in each molecule preferentially in the same spin state).
Evolution of product operators
Chemical shift
?1 tI z
I 1x ???→ I 1x cos( ?1 t) + I 1y sin( ?1 t) [3-7]
?1 tI 1z
I 1y ???→ I 1y cos( ?1 t) - I 1x sin( ?1 t) [3-8]
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Effect of r.f. pulses
βI y
I 1z ???→ I 1z cos β + I 1x sin β [3-9]
βI y
I 1x ???→ I 1x cos β - I 1z sin β [3-10]
βI y
I 1y ???→ I 1y
The effects of x and z pulses can be determined by cyclic permutation of x , y , and z. All rotations
obey the "right-hand rule", i.e., with the thumb of the right (!) hand pointing in the direction of the
r.f. pulse, the curvature of the four other fingers indicate the direction of the rotation.
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Scalar coupling
piJtI 1z I 2z
I 1x ?????→ I 1x cos( piJt) + 2I 1y I 2z sin( piJt) [3-11]
piJtI 1z I 2z
I 1y ?????→ I 1y cos( piJt) - 2I 1x I 2z sin( piJt) [3-12]
piJtI 1z I 2z
I 1z ?????→ I 1z
(i.e., I 1z does not evolve J coupling!)
piJtI 1z I 2z
2I 1x I 2z ?????→ I 1y sin( piJt) + 2I 1x I 2z cos( piJt) [3-13]
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The antiphase term 2I 1y I 2z can be re-written using the single-element operators I α und I β:
2I 1y I 2z = I 1y I 2 α - I 1y I 2 β [3-14]
(2I z = I α - I β, I α + I β = 1; I α und I β are called polarization operators )
The antiphase state 2I 1y I 2z consists of two separate populations: for one half of the molecules in the
ensemble spin 1 is in + y coherence (when spin 2 is in the α state), for the other half spin 1 is in - y
coherence (with spin 2 in the β state); "spin 1 is in antiphase with respect to spin 2".
Such an antiphase state can develop from I 1x when spin 1 is J-coupled to spin 2. This leads to a
dublet for spin 1, i.e., it splits into two lines with an up- and downfield shift by J / 2 , depending on the
spin state of the coipling partner, spin 2. If we wait long enough ( 1 / 2 J ), then the frequency
difference of J between the dublet lines (I 1x I 2 α and I 1x I 2 β ) has brought them 180° out of phase
("antiphase"), as shown in the vector diagram.
This is an oscillation between I 1x in-phase coherence and 2I 1y I 2z antiphase coherence. The antiphase
component evolves with sin( pJt) and then refocusses back to -I 1x in-phase coherence (after t= 1 / J ).
Single-element operators
In some cases (phase cycling, gradient coherence selection) it is necessary to use operators with a
defined coherence order (Eigenstates of coherence order). Coherence order describes the changes in
quantum numbers m z caused by the coherence. A spin- 1 / 2 system (no coupling) can assume two
coherent states: a transition from α ( m z =+ 1 / 2 ) to β ( m z =- 1 / 2 ), i.e., a change (coherence order) of -1.
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This can be described by the lowering operator I - = | β>< α|, the coherent transition from the β to α
state by the raising operator I + = | α>< β| (coherence order +1).
The real Cartesian operators I x and I y correspond to mixtures of both coherence orders, ±1, although
they are more useful for directly corresponding to the observable x and y components of the
magnetization. Their relationship with the complex I ± operators is simple:
I + = I x + iI y raising operator
I x = 1 / 2 (I + + I - )
I - = I x - iI y lowering operator
I y = - i / 2 (I + - I - )
I α = 1 / 2 1 + I z polarisation operator ( α)
I z = 1 / 2 (I α - I β)
I β = 1 / 2 1 - I z polarisation operator ( β)
1 = I α + I β
The effect of r.f. pulses (here: an x pulse with flip angle ?) on single-element operators is as follows:
?x
I +/- ???→I +/- cos 2 { ?/2} + I -/+ sin 2 { ?/2} (+/- iI z sin{ ?}) [3-15]
?x
I β ???→I βcos 2 { ?/2} + I αsin 2 { ?/2} + (1/2)sin{ ?}[I + - iI - ] [3-16]
?x
I α ???→I αcos 2 { ?/2} + I βsin 2 { ?/2} - (1/2)sin{ ?}[I + - iI - ] [3-17]
Generally it is easier to calculate the effects of r.f. pulses on Cartesian operators and then use the
conversion rules to get the single-element version.
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Signal phase, In-phase and antiphase signals
For a single spin I 1 one gets the following signal during acquisition with receiver reference phase x:
I x → I x cos ( ?t ) + I y sin ( ?t)
I y → I y cos ( ?t ) - I x sin ( ?t)
These two signals are 90° out of phase (also after FT), which is indicated by one I x component
having a sine , the other one a cosine modulation.
For a spin I 1 coupled to another spin I 2 one gets the following signal during acquisition (neglecting
chemical shift evolution):
I x → I x cos ( ?t) + I y sin ( ?t) → I x cos ( ?t) cos ( piJt) + 2 I 1y I 2z cos ( ?t) sin( piJt)
+ I y sin ( ?t) cos ( piJt) - 2 I 1x I 2z sin ( ?t)
sin( piJt)
the detected x component corresponds to an in-phase dublet with splitting J, i.e., lines with intensity
1 / 2 at positions ( ? + J / 2 ) and ( ? - J / 2 ) ( piJt =
2 pi J / 2 t ).
cos α cos β = 1 / 2 [cos ( α+β) + cos ( α?β)]
2 I 1x I 2z → 2 I 1x I 2z cos ( ?t) + 2 I 1y I 2z sin ( ?t) →
2 I 1x I 2z cos ( ?t) cos( piJt) + I y cos ( ?t) sin ( piJt)
+ 2 I 1y I 2z sin ( ?t) sin( piJt) - I x sin ( ?t) sin ( piJt)
this tiem the x component corresponds to an anti-phase dublet with splitting J, i.e., lines with
intensities of 1 / 2 and - 1 / 2 at positions ( ? + J / 2 ) and ( ? - J / 2 ) :
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sin α sin β = 1 / 2 [cos ( α+β) - cos ( α?β)]
In the same way, I y leads to an in-phase dublet 90° out of phase (=dispersive) and 2 I 1y I 2z to a
dispersive anti-phase dublet.
Some applications
1. For solvent signal suppression in 1D spectra, the J ump- R eturn sequence can be used:
90°( x ) - τ - 90°( x ) - acquisition
Calculate the excitation profile with product operator formalism!
2. What happens to chemical shift evolution during this sequence, and what about J coupling?
Calculate!
90°( x ) - τ - 180°( x ) - τ -
90°( x ) - τ - 180°( y ) - τ -
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A simple 2D experiment (COSY)
Let's calculate the result of this two-pulse COSY sequence for a single spin I:
yy
The first 90°y pulse creates transverse magnetization, which then evolves under the influence of
chemical shift and J coupling during the delay t 1 , after the second 90° y pulse we get:
90° y ?t 1 90° y
I z ??→ I x ??→ I x cos( ?t 1 ) ??→ ? I z cos( ?t 1 )
+ I y sin( ?t 1 ) + I y sin( ?t 1 )
During the acquisition time t 2 th e first component is not detectable (polarization I z ), the other ( bold )
term is a coherence which will evolve during t 2 as follows:
?t 2
I y sin( ?t 1 ) ??→ I y sin( ?t 1 ) cos( ?t 2 )
- I x sin( ?t 1 ) sin( ?t 2 )
If we compare this to a normal 1D spectrum (let's call the acquisition time again t 2 )
90° y ?t 1
I z ??→ I x ??→ I x cos( ?t 2 )
+ I y sin( ?t 2 )
we see that both signals correspond the real and imaginary part of a precession with frequency ?
during the acquisition time t 2 . The only difference between the 1D and the COSY spectrum – besides
a 90° phase shift (for the 1D, the absorptive and dispersive components are I x and I y , for the COSY
they are I y and -I x , resp.) – is the factor sin( Wt 1 ) in the COSY terms. So far, this is just a constant
with a value somewhere between -1 and +1, depending on the chosen t 1 value (and on ?, of course).
However, the result of the COSY sequence contains very similar factors for t 1 and t 2 , i.e., a sine or
cosine modulation with the argument ?t i . Instead of a t 2 FT, we could also perform a FT in t 1 .
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Of course, we have only data for a single t 1 value (the one we chose in the COSY sequence), but for
a FT we need to know a whole series of values of the oscillatory function, as we do for t 2 (all the
TD2 time domain data points sampled during the acquisition time, for t 2 =0, t 2 =DW, t 2 =2DW, …
t 2 =AQ 2 ).
We can re-run the COSY sequence with a different setting for t 1 , and another one, etc., starting from
t 1 =0, then t 1 =DW, t 1 =2DW, … t 1 =AQ 1 (TD1 different t 1 values). Our complete data set now consists
of TD1 FIDs (with TD2 data points each):
We can now perform a "normal" FT along t 2 , converting the series of FIDs into a series of spectra,
which are all identical (with a signal at ?), except for a modulation with sin( Wt 1 ) along the t 1
dimension:
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If we now read out single columns from our 2D data matrix, then we will generally get a pseudo-FID
A sin( Wt 1 ) for each column, with A=0 where there is no signal in the F2 dimension (the frequency
dimension generated by the t 2 -FT) and with A ≠0 where we have our signal (at ? in F2).
From these pseudo-FIDs we can again generate a frequency spectrum by FT (now along the t 1
dimension), and will get a signal at the frequency ? in this F1 dimension – in the column at ? in F2:
This is a 2D COSY spectrum! – although not very interesting, since it contains just one diagonal
peak (=with identical chemical shift ? in both dimensions), no more information than a simple 1D
spectrum.
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In order to achieve a distinction between positive and negative ? values ( ?=0 is in the center of
each dimension), a complex FT also in the indirect F1 dimension is required, i.e., the sine and the
cosine component. From our pulse sequence we get only
I y sin( ?t 1 ) cos( ?t 2 ) - I x sin( ?t 1 ) sin( ?t 2 )
These are the two (real and imaginary) components for a t 2 value, but only the sine component in t 1 .
We have to re-run our complete set of TD1 t 1 increments with a slightly modified pulse sequence,
with the first pulse phase shifted by 90°:
90° -x ?t 1 90° y ?t 2
I z ?→ I y ?→ I y cos( ?t 1 ) ?→ I y cos( ?t 1 ) ?→ I y cos( ?t 1 ) cos( ?t 2 ) - I x cos( ?t 1 ) sin( ?t 2 )
-I x sin( ?t 1 ) + I z sin( ?t 1 ) + I z sin( ?t 1 )
We now get exactly the same terms as for the first COSY pulse sequence, only with a cosine
modulation. Combining the two data sets, we get four data points for each t 1 /t 2 combination:
I y sin( ?t 1 ) cos( ?t 2 ) - I x sin( ?t 1 ) sin( ?t 2 )
I y cos( ?t 1 ) cos( ?t 2 ) - I x cos( ?t 1 ) sin( ?t 2 )
This is called a hypercomplex data point, and a 2D matrix of such data points contains all sine/cosine
combinations needed for a hypercomplex 2D-FT, yielding the frequency ? including the correct sign
in both dimensions. There are different ways to acquire the equivalent of a hypercomplex data set:
- the S TATES (-R UBEN -H ABERKORN ) method outlined here (first pulse phase x/y for each t 1 point)
- the TPPI (Time Proportional Phase Incrementation), which is analogous to the R EDFIELD trick
for single-channel acquisition (only real data points in t 1 , but twice as many, with half the time
increment
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- echo-antiecho quadrature detection, which does not sample the real and imaginary (I x and I y )
components separately, but rather I + and I - (by coherence selction through phase cycling or
gradients) – which can then be easily converted into I x and I y by the computer
Magnitude mode spectra
Alternatively, one can only select either I + or I - during t 1 (i.e., only one data point per t 1 value), again
by phase cycling or gradients. This corresponds to either the spectrum or its mirror image in the
indirect dimension, so no quad images will occur. However, according to I + = I x + iI y / I - = I x - iI y
these are complex components with no pure phase. The resulting spectrum cannot be phased to pure
absorptive lineschapes in F1 and hence has to be displayed in absolute value/magnitude or power
mode.
This was quite popular many years ago, when data storage and processing capacity were limited.
However, the S/N is only 1 / 2 (ca. 71 %) of a S TATES or TPPI spectrum of equal measuring time
and digital resolution , because the r.f. pulses create I x = 1 / 2 (I + + I - ) or I y = - i / 2 (I + - I - ) at the
beginning of t 1 , so only 50 % of the signal is selected by the phase cycle or gradients. Worse even,
the very unfavourable magnitude or power mode lineshapes greatly reduce the spectral resolution,
even with optimized apodization functions.
Some types of spectra, however, cannot be phase corrected because of J coupling evolution during
the pulse sequence, resulting in a mixture of absorptive/dispersive in-phase/antiphase signals
(Relayed-COSY, long-range COSY). One can get the S/N advantage of the phase sensitive version
(i.e., acquired in S TATES or TPPI mode), but still has to convert the spectrum to absolute value mode
after the complex FT.
A COSY with crosspeaks
Let's calculate the result of the COSY sequence for two coupled spins I 1 and I 2 :
90° y ?t 1 piJt 1
I 1z ??→ I 1x ??→ I 1x cos( ?1 t 1 ) ??→ I 1x cos( ?1 t 1 ) cos( piJt 1 ) + 2I 1y I 2 z cos( ?1 t 1 ) sin( piJt 1 )
+ I 1y sin( ?1 t 1 ) + I 1y sin( ?1 t 1 )
cos( piJt 1 ) - 2I 1x I 2 z sin ( ?1 t 1 ) sin( piJt 1 )
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after the second 90° y pulse these four terms are converted into
90° y
??→ ?I 1z cos( ?1 t 1 ) cos( piJt 1 ) + 2I 1y I 2 x cos( ?1 t 1 ) sin( piJt 1 )
+ I 1y sin( ?1 t 1 ) cos( piJt 1 ) + 2I 1z I 2x sin ( ?1 t 1 ) sin( piJt 1 )
What signal will be detected now during the acquisition time t 2 ?
- the first two components are not detectable (polarization I 1z and MQC 2I 1y I 2 x )
- the other two ( bold ) terms are spin 1 in-phase coherence and spin 2 antiphase coherence, which
will evolve during t 2 as follows (shown without the sine and cosine terms from t 1 ):
?t 2 piJt 2
I 1y (…) ??→ I 1y cos( ?1 t 2 ) ??→ I 1y cos( ?1 t 2 ) cos( piJt 2 ) - 2I 1x I 2 z cos( ?1 t 2 ) sin( piJt 2 )
-I 1x s in( ?1 t 2 ) - I 1x sin( ?1 t 2 ) cos( piJt 2 ) - 2I 1y I 2 z
sin ( ?1 t 2 ) sin( piJt 2 )
The two detectable in-phase components are, in full length,
I 1y sin( ?1 t 1 ) cos( piJt 1 ) cos ( ?1 t 2 ) cos( piJt 2 ) - I 1x sin( ?1 t 1 ) cos( piJt 1 ) sin( ?1 t 2 ) cos( piJt 2 )
From the second SQC term 2I 1z I 2x sin ( W1 t 1 ) sin( pJt 1 ) present at the beginning of t 2 we get:
?t 1 piJt 1
2I 1z I 2x (…) ?→ 2I 1z I 2x cos( ?2 t 2 ) ?→ 2I 1z I 2x cos( ?2 t 2 ) cos( piJt 2 ) + I 2y cos( ?2 t 2 ) sin( piJt 2 )
+ 2I 1z I 2y sin( ?2 t 2 ) + 2I 1z I 2y sin( ?2 t 2 ) cos( piJt 2 ) - I 2 x
sin ( ?2 t 2 ) sin( piJt 2 )
(since this is a spin-2 coherence, it will evolve chemical shift of spin 2, ?2 !).
The two detectable in-phase components are in full length:
I 2y sin ( ?1 t 1 ) sin( piJt 1 ) cos( ?2 t 2 ) sin( piJt 2 ) - I 2x sin ( ?1 t 1 ) sin( piJt 1 ) sin ( ?2 t 2 ) sin( piJt 2 )
According to the rules for in-phase and antiphase terms, we can now easily figure out what the 2D
spectrum will look like, remembering that the harmonics with t 1 in the argument describe the signal
in the F1 domain (after t 1 -FT), and the ones with t 2 correspond to the signals look in F2.
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1. I 1y sin( ?1 t 1 ) cos( piJt 1 ) cos ( ?1 t 2 ) cos( piJt 2 ) - I 1x sin( ?1 t 1 ) cos( piJt 1 ) sin( ?1 t 2 ) cos( piJt 2 )
This is an in-phase dublet signal at ?1 in F2 and at ?1 in F1, i.e., a diagonal peak again (as entioned
above, so far we have only recorded one component in t 1 , and we will have to repeat the whole 2D
experiment with a 90° shifted first r.f. pulse to get hypercomplex data points.
diagonal peak / F1 multiplett
sin( ?1 t 1 )cos( piJt 1 ) = 1 / 2 {sin( ?1 + piJ)t 1 + sin( ?1 - piJ)t 1 }
(dispersive)
diagonal peak / F2 multiplett
cos( ?1 t 1 )cos( piJt 1 ) = 1 / 2 {cos( ?1 - piJ)t 1 + cos( ?1 + piJ)t 1 } (absorptive)
2. I 2y sin ( ?1 t 1 ) sin( piJt 1 ) cos( ?2 t 2 ) sin( piJt 2 ) - I 2x sin ( ?1 t 1 ) sin( piJt 1 ) sin ( ?2 t 2 ) sin( piJt 2 )
This describes again the absorptive ( I 2x ) and dispersive parts ( I 2y ) of a signal at frequency ?1 in F1
and at frequency ?2 in F2, i.e., a cross-peak with different resonance frequencies in the two
dimensions. Furthermore, it is an antiphase signal with respect to the J 12 coupling in both
dimensions.
cross-peak / F1 /
sin( ?1 t 1 )sin( piJt 1 ) = 1 / 2 {-cos( ?1 + piJ)t 1 + cos( ?1 - piJ)t 1 }
(absorptive)
cross-peak / F2
cos( ?2 t 2 )sin( piJt 2 ) = 1 / 2 {sin( ?2 + piJ)t 2 - sin( ?2 - piJ)t 2 }
(dispersive)
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If we compare the expressions for the diagonal peak and the cross-peak, we can see that they are 90°
out of phase relativ to each other in both dimensions. E.g., for the diagonal peak we have an I x
component which is dispersive in F1 and F2, while from the cross-peak we get an I x component that
is absorptive (and vice versa for the I y components). So, no matter what phase correction we choose
in each of the two (independently phase corrected) dimensions, always one of the two signals will be
dispersive.
Two ways of phase correcting a 2D COSY spectrum: diagonal peaks in-phase absorptive, cross-
peaks antiphase dispersive (left); or diagonal peaks in-phase dispersive, cross-peaks antiphase
absorptive (right).
No matter what phase corrrection is chosen, the dispersive tails always tend to obscure near-by
cross-peaks. Absolute value processing is no real solution either, since now the dispersive
components of both the diagonal and the cross-peaks contribute to the spectrum. Only the
employment of apodization functions with rigorous resolution enhancement and line narrowing
characteristics can yield a reasonable spectrum (although at the cost of losing S/N).