## Phase cycling for two-pulse MQ-MAS sequence

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## Two-pulse MQ-MAS phase cycling

Below are presented some phase cycling procedures for MQ-MAS with two-pulse sequence applied to half-integer quadrupole spins:
(1) in amplitude-modulated experiment involving two coherence transfer pathways,
(2) in phase-modulated experiment involving one coherence transfer pathway.

We only consider the usual case where the receiver can take four phases: 0°, 90°, 180°, and 270°.

Two approaches are used for the simulation of the echo and/or antiecho amplitudes with SIMPSON1.1.1 Tcl scripts:
(1) filtering the desired coherences (-3Q and/or 3Q after the first pulse),
(2) explicit phase cycling the pulses and the receiver.

For two-pulse 3Q-MAS experiments, these two approaches provide the same simulation data about echo and/or antiecho amplitudes.

 ``` Outline (A) Amplitude-modulated experiment for ±3Q-MAS  (1) 6-phase cycling (2) Simulation (3) Result (4) Comment (B) Phase-modulated experiment (a) -3Q-MAS for spin I = 3/2 system (1) 24-phase cycling (2) 12-phase cycling (3) Simulation (4) Result (b) +3Q-MAS for spin I = 5/2 system (1) 24-phase cycling (2) 12-phase cycling (3) Simulation (4) Result (c) -5Q-MAS for spin I = 5/2 system (1) 40-phase cycling (2) 20-phase cycling (3) Simulation (4) Result (C) Conclusion```

### (A) Amplitude-modulated experiment for ±3Q-MAS

In amplitude-modulated experiment, two coherence transfer pathways 0Q -> -3Q -> -1Q and 0Q -> 3Q ->-1Q are selected. The phase cycling is defined with the two coherence transfer pathways. That is, the phase cycling must be identical for the two coherence transfer pathways.

The total pathway phase φpath for the coherence transfer pathway 0Q -> -3Q -> -1Q is defined by:
φpath = -3*φ1 + 2*φ2 + φrec,
that for the coherence transfer pathway 0Q -> +3Q -> -1Q is defined by:
φpath = +3*φ1 - 4*φ2 + φrec,
where
φ1 is the first-pulse phase;
φ2 is the second-pulse phase;

The receiver phase is defined with the condition: φpath = 0. Therefore, we have two equations for the receiver phase with two parameters:
φrec = +3*φ1 - 2*φ2,
φrec = -3*φ1 + 4*φ2.

#### (1) 6-phase cycling

The receiver phase is defined with the condition: φ2 = 0;
the two equations for the receiver phase become:
φrec = +3*φ1,
φrec = -3*φ1.

The corresponding solution is: φ1 takes six values.

φ1 = 0° 60° 120° 180° 240° 300° 0° 0° 180°

This is the well-known phase-cycling procedure from [A. Medek, J. S. Harwood, and L. Frydman, J. Am. Chem. Soc. 117, 12779-12787 (1995)] for spin I = 3/2, 5/2, and 7/2 systems.

#### (2) Simulation

We provide SIMPSON1.1.1 Tcl scripts for simulating the echo and antiecho amplitude versus a pulse duration with two approaches:
(1) filtering the desired coherences (-3Q and 3Q after the first pulse);
(2) explicit phase cycling the pulses and the receiver.

Duration Filtering 6-phase cycling
1st-pulse al-filtering-p1 al-6-cycling-p1
2nd-pulse al-filtering-p2 al-6-cycling-p2

#### (3) Result

When the amplitudes are normalized to the number of phase cycling, these two approaches provide the same simulation data about echo and antiecho amplitude for spin I = 3/2, 5/2, and 7/2 systems.

#### (4) Comment

A 36-phase cycling for two-pulse sequence is available, but the phases of the receiver are different of 0°, 90°, 180° and 270°.

### (B) Phase-modulated experiment

In some 2D experiments such as the double resonance MQ-HETCOR experiment, one coherence transfer pathway is required. For spin I = 3/2 system, the coherence transfer pathway 0Q -> -3Q -> -1Q is needed. For spin I = 5/2 system, the coherence transfer pathway 0Q -> 3Q -> -1Q is needed.

In phase-modulated experiment, phase cycling is defined with the desired coherence transfer pathway.

### (a) -3Q-MAS for spin I = 3/2 system

Two phase cycling procedures are possible: one involving 24-phase cycling from [S. H. Wang, S. M. De Paul, and L. M. Bull, J. Magn. Reson. 125, 364-368 (1997)] and one involving 12-phase cycling [S. P. Brown and S. Wimperis, J. Magn. Reson. 128, 42-61 (1997)].

#### (1) 24-phase cycling

The total pathway phase φpath for the coherence transfer pathway 0Q -> -3Q -> -1Q is defined by:
φpath = -3*φ1 + 2*φ2 + φrec.

The receiver phase is defined with the conditions: φpath = 0;
therefore φrec = 3*φ1 - 2*φ2.
φ1 takes six values: 0°, 60°, 120°, 180°, 240°, and 300°; φ2 takes four values: 0°, 90°, 180°, and 270°.

φ1 = 0° 60° 120° 180° 240° 300° {0°}X6 {90°}X6 {180°}X6 {270°}X6 {0° 180°}X3 {180° 0°}X3

If we reduce the number of phases of the second pulse from four to two as in the following table:

φ1 = 0° 60° 120° 180° 240° 300° {0°}X6 {180°}X6 0° 180°

the two coherence transfer pathways 0Q -> -3Q -> -1Q and 0Q -> 3Q ->-1Q are selected. This is due to the fact that alternating the phase of the second pulse in a two-pulse spin-echo sequence does not change the sign of the echo and that of the antiecho.

#### (2) 12-phase cycling

The total pathway phase φpath for the coherence transfer pathway 0Q -> -3Q -> -1Q is defined by:
φpath = -3*φ1 + 2*φ2 + φrec.

The receiver phase is defined with the conditions: φpath = 0 and φ2 = 0;
therefore φrec = 3*φ1.
To select only the desired coherence transfer pathway, φ1 takes 12 values.

φ1 = 0° 30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330° 0° 0° 90° 180° 270°

#### (3) Simulation

We provide SIMPSON1.1.1 Tcl scripts for simulating the echo amplitude versus a pulse duration with two approaches:
(1) filtering the desired coherences (-3Q after the first pulse);
(2) explicit phase cycling the pulses and the receiver.

Duration Filtering 24-ph cycling 12-ph cycling
1st-pulse na-filt-p1 na-24-cycl-p1 na-12-cycl-p1
2nd-pulse na-filt-p2 na-24-cycl-p2 na-12-cycl-p2

#### (4) Result

When the amplitudes are normalized to the highest number of phase cycling, these two approaches provide the same simulation data about -3Q echo amplitude for spin I = 3/2 system and about -3Q antiecho amplitude for spin I = 5/2 system.

### (b) +3Q-MAS for spin I = 5/2 system

Two phase cycling procedures are possible: one involving 24-phase cycling from [C. Fernandez, C. Morais, J. Rocha, and M. Pruski, Solid State NMR 21, 61-70 (2002)] and one involving 12-phase cycling [S. P. Brown and S. Wimperis, J. Magn. Reson. 128, 42-61 (1997)].

#### (1) 24-phase cycling

The total pathway phase φpath for the coherence transfer pathway 0Q -> +3Q -> -1Q is defined by:
φpath = 3*φ1 - 4*φ2 + φrec.

The receiver phase is defined with the condition: φpath = 0;
therefore φrec = -3*φ1 + 4*φ2.
φ1 takes six values: 0°, 60°, 120°, 180°, 240°, and 300°; φ2 takes four values: 0°, 90°, 180°, and 270°.

φ1 = 0° 60° 120° 180° 240° 300° {0°}X6 {90°}X6 {180°}X6 {270°}X6 0° 180°

If we reduce the number of phases of the second pulse from four to two as in the following table:

φ1 = 0° 60° 120° 180° 240° 300° {0°}X6 {180°}X6 0° 180°

the two coherence transfer pathways 0Q -> -3Q -> -1Q and 0Q -> 3Q ->-1Q are selected. This is due to the fact that alternating the phase of the second pulse in a two-pulse spin-echo sequence does not change the sign of the echo and that of the antiecho.

#### (2) 12-phase cycling

The total pathway phase φpath for the coherence transfer pathway 0Q -> +3Q -> -1Q is defined by:
φpath = 3*φ1 - 4*φ2 + φrec.

The receiver phase is defined with the conditions: φpath = 0 and φ2 = 0;
therefore φrec = -3*φ1.
To select only the desired coherence transfer pathway, φ1 takes 12 values.

φ1 = 0° 30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330° 0° 0° 270° 180° 90°

#### (3) Simulation

We provide SIMPSON1.1.1 Tcl scripts for simulating the echo amplitude versus a pulse duration with two approaches:
(1) filtering the desired coherences (3Q after the first pulse);
(2) explicit phase cycling the pulses and the receiver.

Duration Filtering 24-ph cycling 12-ph cycling
1st-pulse al-filt-p1 al-24-cycl-p1 al-12-cycl-p1
2nd-pulse al-filt-p2 al-24-cycl-p2 al-12-cycl-p2

#### (4) Result

When the amplitudes are normalized to the highest number of phase cycling, these two approaches provide the same simulation data about 3Q echo amplitude for spin I = 5/2 system and about 3Q antiecho amplitude for spin I = 3/2 system.

### (c) -5Q-MAS for spin I = 5/2 system

A phase cycling procedure involving 40-phase cycling is provided by [L. Delevoye and coworkers, Solid State NMR 22, 501-512 (2002)].

#### (1) 40-phase cycling

The total pathway phase φpath for the coherence transfer pathway 0Q -> -5Q -> -1Q is defined by:
φpath = -5*φ1 + 4*φ2 + φrec.

The receiver phase is defined with the condition: φpath = 0;
therefore φrec = 5*φ1 - 4*φ2.
φ1 takes ten values: 0°, 36°, 72°, 108°, 144°, 180°, 216°, 252°, 288°, and 324°; φ2 takes four values: 0°, 90°, 180°, and 270°.

φ1 = 0° 36° 72° 108° 144° 180° 216° 252° 288° 324° {0°}X10 {90°}X10 {180°}X10 {270°}X10 0° 180°

#### (2) 20-phase cycling

The total pathway phase φpath for the coherence transfer pathway 0Q -> -5Q -> -1Q is defined by:
φpath = -5*φ1 + 4*φ2 + φrec.

The receiver phase is defined with the condition: φpath = 0 and φ2 = 0;
therefore φrec = 5*φ1.
φ1 takes 20 values: 0°, 18°, 36°, 54°, 72°, 90°, 108°, 126°, 144°, 162°, 180°, 198°, 216°, 234°, 252°, 270°, 288°, 306°, 324°, and 342°.

φ1 = 0° 18° 36° 54° 72° 90° 108° 126° 144° 162° 180° 198° 216° 234° 252° 270° 288° 306° 324° 342° 0° 0° 90° 180° 270°

#### (3) Simulation

We provide SIMPSON1.1.1 Tcl scripts for simulating the echo amplitude versus a pulse duration with two approaches:
(1) filtering the desired coherences (-5Q after the first pulse);
(2) explicit phase cycling the pulses and the receiver.

Duration Filtering 40-ph cycling 20-ph cycling
1st-pulse al-filt-5Q-p1 al-40-cycl-5Q-p1 al-20-cycl-5Q-p1
2nd-pulse al-filt-5Q-p2 al-40-cycl-5Q-p2 al-20-cycl-5Q-p2

#### (4) Result

When the amplitudes are normalized to the highest number of phase cycling, these two approaches provide the same simulation data about -5Q echo amplitude for spin I = 5/2 system.

### (C) Conclusion

For two-pulse MQ-MAS sequences, filtering the desired coherences or phase cycling the pulses and the receiver provide the same simulation data about echo and/or antiecho amplitudes.

Reducing the number of phases allows unwanted coherence transfer pathways to be detected.

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