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CTP of DOF-COSY:

CTP of NOESY (pathway 1):

Designing of pulse program:

1. Design the pulse program to excite desired coherence.2. Get ride of unwanted coherence.3. Optimize pulse program design.

Coherence transfer pathway. Phase cycling. Gradient selection

Coherence order:

(Rotate 2, double quantum. P = 2)

(Order : p = ± 1)

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 = -1For the operator: 2I1xI2x we have:2I1xI2x = 2x ½(I1+ + I1-) x ½(I2+ + I2-) = ½(I1+I2+ + I1-I2-) + ½(I1+I2- + I1-I2+)

The double quantum part, ½(I1+I2+ + I1-I2-) can be rewritten as:

Similar the zero quantum part can be rewritten as:

½(I1+I2- + I1-I2+) = ½ (2I1xI2x – 2I1yI2y)

P = +2

P = -2 P = 0

P = 0

(Pure double quantum state)

(Pure zero quantum state)

Evolution under offsets:

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

Review on product operator formalism:

1. At thermal equilibrium: I = Iz

2. Effect of a pulse (Rotation): exp(-iIa)(old operator)exp(iIa) = 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)

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.

Evolution under coupling:

Hamiltonian: HJ = 2J12I1zI2z

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 !!!

2D-NOESY of two spins w/ no J-coupling:

Consider two non-J-coupled spin system:1. 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:

7. Detection during t2:

The y-magnetization =Let A1

(2) = FT[cos1t2] is the absorption signal at 1 in F2 and A2(2) = FT[os2t2] as the absor

ption 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)cos1t1 and fcos1t1, respectively.

FT w.r.t. t1 gives:

where A11 = FT[cost] 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), (F1, F2) = (2, 1), and (F1, F2) = (2, 2)

The diagonal peaks will have intensity (1-f) and the off-diagonal peaks will have intensities f, where f is the fraction magnetization transferred, which is usually < 5%.

(Diagonal) (Cross peak)

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:

(p=0, unobservable)

(p=0 or ±2)(unobservable)

(In-phase, dispersive)

(Anti-phase)(Single quantum coh.)

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.

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)

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

- NOESY

Pathway 1: At t1:

After the second X-pulse only the IY term contributeand it becomes cost1IX, 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:

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 asFollowing thru the rest of the pulse sequence gives the following phase modulated observable signal:

If we choose the p=+1 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.

Pathway 2:

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.

- 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 -pxk 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

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: ?

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.

Tricks: 1. 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.

Shifting the whole sequence – CYCLOPS.The overall CTP is 0 -1, or p = -1. Thus, any pulse sequence can be cycled as a group toselect p = -1 with the CYCLOPS sequence with both pulse and receiver cycle through0o, 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 = 90o2 = 180o

3 = 270o

P 0 1 2 3R 0 2 0 2

P 0 1 2 3R 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

(HMQC)

P-type (+1, +2):

N-type (-1, +2):

To generate pure P- or N-type magnetization one can combine as follow: