http://www.nmr.sinica.edu.tw/~thh/nmr_core_facility_training_cours.html

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Course web page
http//www.nmr.sinica.edu.tw/thh/nmr_core_facilit
y_training_cours.html
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• Designing of pulse program
• Design the pulse program to excite desired
  coherence.
• Get ride of unwanted coherence.
• Optimize pulse program design.
• ? Coherence transfer pathway.
• ? Phase cycling.
• ? Gradient selection

CTP of NOESY (pathway 1)
CTP of DOF-COSY
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Coherence order
(Order p 1)
(Rotate 2?, double quantum. P 2)
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• Coherence order Only single quantum coherences
  are observables
• Single quantum coherences (p 1) Ix, Iy,
  2I1zI2y, T1yI2z . etc
• Zero quantum coherence Iz, I1z2z
• Raising and lowering operators I ½(Ix
  iIy) I- 1/2 (Ix i-Iy)
• Coherence order of I is p 1 and that of
  I- is p -1
• Ix ½(I I-) Iy 1/2i (I - I-) are both
  mixed states contain order
• p 1 and p -1
• For the operator 2I1xI2x we have
• 2I1xI2x 2x ½(I1 I1-) x ½(I2 I2-)
  ½(I1I2 I1-I2-) ½(I1I2- I1-I2)

P 2
P -2
P 0
P 0
(Pure double quantum state)
(Pure zero quantum state)
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Evolution under offsets
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Phase-shifted pulses Let the initial state of
order p as ?(p) and the final state of order p
after a pulse as ?(p). The effect of the radio
frequency pulse can be written as where Uo is
the unitary transformation. If the rf pulse is
applied along an axis having a phase angle ?
w.r.t. the X-axis the effect is to rotate the
unitary matrix by Thus,
But Thus,
where and Therefore if the rf
pulse is phase shifted by ? the coherence will
acquire a phase of
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• Review on product operator
  formalism
• 1. At thermal equilibrium I Iz
• 2. Effect of a pulse (Rotation)
• exp(-i?Ia)(old operator)exp(i?Ia) cos? (Old
  operator) sin? (new operator)
• 3. Evolution during t1
• (free precession) (rotation w.r.t. Z-axis)

- Iy for ?1tp 90o
Product operator for two spins Cannot be
treated by vector model Two pure spin states
I1x, I1y, I1z and I2x, I2y, I2z I1x and I2x are
two absorption mode signals and I1y and I2y are
two dispersion mode signals. These are all
observables (Classical vectors)
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Coupled two spins Each spin splits into two spins
Anti-phase magnetization 2I1xI2z, 2I1yI2z,
2I1zI2x, 2I1zI2y (Single quantum
coherence) (Not observable)
Double quantum coherence 2I1xI2x, 2I1xI2y,
2I1yI2x, 2I1yI2y (Not observable) Zero quantum
coherence I1zI2z (Not directly
observable) Including an unit vector, E, there
are a total of 16 product operators in a
weakly-coupled two-spin system. Understand the
operation of these 16 operators is essential to
understand multi-dimensional NMR expts.
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Evolution under coupling Hamiltonian HJ
2?J12I1zI2z Causes inter-conversion of
in-phase and anti-phase magnetization according
to the Diagram, i.e. in-phase ? anti-phase and
anti-phase ? in-phase according to the rules
Must have only one component in the X-Y plane !!!
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2D-NOESY of two spins w/ no J-coupling

• Consider two non-J-coupled spin system
• Before pulse Itotal
• Let us focus on spin 1 first
• 2. 90o pulse (Rotation)

3. t1 evolution 4. Second 90o pulse 5.
Mixing time Only term with Iz can transfer
energy thru chemical exchange. Other terms
will be ignored. This term is frequency labelled
(Dep. on ?1 and t1). Assume a fraction of f
is lost due to exchange. Then after mixing time
(No relaxation)
6. Second 90o pulse
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7. Detection during t2

• The y-magnetization
• Let A1(2) FTcos?1t2 is the absorption
  signal at ?1 in F2 and A2(2) FTos?2t2 as the
  absorption mode signal at ?2 in F2. Thus, the
  y-magnetization becomes
• Thus, FT w.r.t. t2 give two peaks at ?1 and ?2
  and the amplitudes of these two peaks are
  modulated by (1-f)cos?1t1 and fcos?1t1,
  respectively.
• FT w.r.t. t1 gives
  
• where A11 FTcos?t is the absorption mode
  signal at ?1 in F1.
• Starting from spin 1 we observe two peaks at
  (F1, F2) (?1, ?1) and (F1, F2) (?1, ?2)
• ? Similarly, if we start at spin 2 we will get
  another two peaks at (F1, F2) (?2, ?2) and
• (F1, F2) (?2, ?1)
• ? Thus, the final spectrum will contain four
  peaks at (F1, F2) (?1, ?1), (F1, F2) (?1,
  ?2),

(Diagonal)
(Cross peak)
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7.4. 2D experiments using coherence transfer
through J-coupling
7.4.1. COSY After 1st 90o pulse t1
evolution J-coupling Effect of the second
pulse
(p0, unobservable)
(p0 or 2) (unobservable)
(In-phase, dispersive)
(Anti-phase) (Single quantum coh.)
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The third term can be rewritten as Thus, it
gives rise to two dispersive peaks at ?1 ?J12
in F1 dimension Similar behavior will be
observed in the F2 dimension, Thus it give a
double dispersive line shape as shown below.
The 4th term can be rewritten as Two
absorption peaks of opposite signs (anti-phase)
at ?1 ?J12 in F1 dimension will be observed.
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Similar anti-phase behavior will be observed in
F2 dimension, thus multiplying F1 and F2
dimensions together we will observe the
characteristic anti-phase square array.
? Use double-quantum filtered COSY (DQF-COSY)
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Double-quantum filtered-COSY (DQF-COSY) Using
phase cycling to select only the double quantum
term (2) can be converted to single quantum for
observation. (Thus, double quantum-filtered)
P 2
P -2
P 0
P 0
Rewrite the double quantum term as
The effect of the last 90o pulse
Anti-phase absorption diagonal peak
Anti-phase absorption cross peak
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- NOESY
Pathway 1 At t1 After the second X-pulse
only the IY term contribute and it becomes
cos?t1IX, which is also the only observable
after the third X-pulse. Thus, the signal
detected at t2 0 is

During detection only the I- term is observable.
Thus, the final signal is a amplitude modulated
signal of the form
Pathway 2
If, instead we choose the CTP shown on the right
then at t1 0 Iy If we keep
only the I- term of IY During t1 the
magnetization evolve as
Following thru the rest of the pulse sequence
gives the following phase modulated observable
signal
If we choose the p1 CTP we will again observe
phase modulated signal
Note if we choose p -1 or 1 the signal
strength is only half that of the
amplitude modulated CTP.
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Pathway selection by phase cycling Select only
coherence transferred from p 2 to -1, or ?p
- 3. Phase shift caused by the pulse sequence
will - ?p - (-3)?. For ? 90o ?p - 270o.
If we wish to select ?p - 3 CTP the receiver
need to phase shifted accordingly, i.e. set the
receiver phase as 0o, 270o, 1800 and -90o at each
cycle. The same pulse sequence will cause ?p 2
coherent to shift - ?p - (-2)? 180o, shown on
the following table.
With the receiver set at 0, 270, 180 and 90 the -
?p - 2 CTP will cancel each other.
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- ?p - 1 coherence Phase change -(-1)?
90o

• Same as - ?p - 3
• Both - ?p - 3 and 1 coherences are preserved
  but - ?p - 2 coherence is
• not observed.
• It can be shown that this four step phase
  cycling scheme will select any pathway
• with - ?p - 3 4n where n 1, 2, 3 .

General rules For a phase cycling scheme with
To select a change in coherence order, ?p, the
receiver phase is set to -?px?k for each step and
the resulting signals are summed. The cycle will
also select changes in coherence order ?p nN
where n 1, 2, The highest coherence
order that can be generated for a system with m
coupled spins one-half nuclei is m
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Refocusing pulses A 180o pulse simply change
the sign of the coherence order, i.e. p 1 ? p
-1, or ?p -2. Likewise, for p -1 ? p 1
CTP ?p 2.
Two step EXORCYCLE 180o pulse 0o, 180o
Receiver 0o, 0o select all even ?p. Selection
of double quantum (p 2 ? p -2) pulse 450
(8 steps) Receiver ?
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Combined phase cycles To select the CTP shown on
the left the first pulse select ?p 1 and the
second pulse select ?p 2. then we need to
consider each pulse separately and combine them
together. If we use is a 4 step cycle for each
pulse the combined phase cycle will take 16
steps, as shown below.
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• Tricks
• The first pulse The first pulse usually generate
  coherence order p 1. If retaining
• both of these coherence orders is
  acceptable one needs not to cycle the first pulse.

2. Grouping pulse together Devise a phase
cycling scheme to select for double quantum
coherence, p 2.
Method 1 Focus only on the last step to select
for p 1 ? p 2 and/or p
-1 ? p 2. Method 2 Consider the sequence
as a whole and select for CTP p
0 ? p 2 or ?p 2.
3. The last pulse Since only p -1 is
observable one needs not to worry about other
coherence orders that may be generated by the
last pulse even though they may be
generated.
Example DQF-COSY 1. Grouping Group the first
two pulses to achieve CTP 0 ? 2, i.e. ?p
2. This has been discussed above the result
is pulse 0o, 90o, 180o, 270o receiver
0o, 180o, 0o, 180o. 2. Focus on the last pulse
and select ?p 1 and ?p - 3 since only p
-1 is observable. Luckily, the following
scheme select both CTP pulse 0o, 900, 180o,
270o Receiver 0o, 270o, 180o and 90o.
Axial peaks (F1 0, F2) Causes 1. Recovery
due to T1 relaxation (T1 noise) 2. Imperfect 90o
pulse Remedy Two-step phase cycling pulse, 0o,
180o and receiver 00, 180o to select ?p 1.
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Shifting the whole sequence CYCLOPS. The
overall CTP is 0 ? -1, or ?p -1. Thus, any
pulse sequence can be cycled as a group to select
?p -1 with the CYCLOPS sequence with both pulse
and receiver cycle through 0o, 90o, 180o and 270o.
Equivalent cycles For a given pulse there are
several alternative phase cycling schemes to
achieve the same results. For example, the
DQF-COSY sequence
If, instead, we wish to keep the receiver at
fixed phase we can the alter the third pulse as
follow two cycles to achieve the same consequence.
Denote 0 0o 1 90o 2 180o 3 270o
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(HMQC)
P 0 1 2 3 R 0 2 0 2
P 0 1 2 3 R 0 0 0 0
P1 0 1 2 3 2 3 0 1 p2 0 1 2 3
0 1 2 3 R 0 0 0 0 2 2 2 2
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P-type (?1, ?2)
N-type (-?1, ?2)
To generate pure P- or N-type magnetization one
can combine as follow