Phase-modulated split-t1 MQ-MAS phase cycling

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 or antiecho amplitudes with SIMPSON1.1.1 Tcl scripts:
(1) filtering the desired coherences;
(2) explicit phase cycling the pulses and the receiver.

These two approaches are applied to the two practical cases:
(1) the three pulses have identical and strong amplitudes;
(2) the first two pulses have strong amplitudes and the third has a weak amplitude.
These two approaches provide the same simulation data about echo or antiecho amplitudes whatever the nature (strong or not) of the third pulse, if we choose to select the coherence transfer pathway only by phase cycling. They become approximation if the amplitudes of the pulses are also involved for selecting the coherence transfer pathway.

Phase cycling for three-pulse split-t1 sequences is identical to that associated with phase-modulated three-pulse shifted-echo MQ-MAS experiments. The various phase cycling data for spin I = 3/2 are given in three-pulse shifted-echo 3Q-MAS experiments.

The phase cycling procedure for split-t1 +3QMAS applied to a spin I = 3/2 is provided by:

1. Brown and Wimperis, Two-dimensional MQ-MAS NMR of quadrupolar nuclei: a comparison of methods, J. Magn. Reson. 128, 42-61 (1997).
2. Goldbourt and coworkers, High resolution heteronuclear correlation NMR spectroscopy between quadrupolar nuclei and protons in the solid state, J. Magn. Reson. 169, 342-350 (2004).

Therefore, we focus on spin I = 5/2 only.

Outline
(A) Three strong amplitude pulses
(A.1) -3Q-MAS for spin I = 5/2 system
    (1) 96-phase cycling
    (2) Simulation
    (3) Result
(A.2) +3Q-MAS for spin I = 5/2 system
    (1) 96-phase cycling
    (2) Simulation
    (3) Result
(A.3) -5Q-MAS for spin I = 5/2 system
    (1) 160-phase cycling
    (2) Simulation
    (3) Result
(A.4) +5Q-MAS for spin I = 5/2 system
    (1) 160-phase cycling
    (2) Simulation
    (3) Result
(B) Two strong amplitude pulses and one soft pulse
(B.1) +3Q-MAS for spin I = 5/2 system
    (1) x-phase cycling
    (2) Simulation
    (3) Result
(B.2) +5Q-MAS for spin I = 5/2 system
    (1) x-phase cycling
    (2) Simulation
    (3) Result
(C) Conclusion

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

(A) Three strong amplitude pulses

(A.1) -3Q-MAS for spin I = 5/2 system

(1) 96-phase cycling

The total pathway phase φ_path for the antiecho coherence transfer pathway 0Q -> -3Q -> +1Q -> -1Q is defined by:
φ_path = -3*φ1 + 4*φ2 - 2*φ3 + φ_rec.

The receiver phase is defined with the conditions: φ_path = 0 and φ2 = 0 ;
therefore φ_rec = 3*φ1 + 2*φ3.
φ3 takes eight values so that 2*φ3 is manifold of 90°; this allows the most efficiency for filtering the coherences by the third pulse. φ1 takes 12 values so that 3*φ1 is manifold of 90°.

This 96-phase cycling is identical to that described in phase modulated shifted-echo sequence.

This phase cycling procedure is also provided by [Brown and Wimperis, Two-dimensional MQ-MAS NMR of quadrupolar nuclei: a comparison of methods, J. Magn. Reson. 128, 42-61 (1997)].

(2) Simulation

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

(3) Result

When the amplitudes are normalized to the number of phase cycling, these two approaches provide the same simulation data about antiecho amplitude for spin I = 5/2 systems. For spin I = 7/2 systems, this 96-phase cycling also allows the coherence transfer pathway 0Q -> -3Q -> -7Q -> -1Q to be detected.

(A.2) +3Q-MAS for spin I = 5/2 system

(1) 96-phase cycling

The total pathway phase φ_path for the echo coherence transfer pathway 0Q -> +3Q -> +1Q -> -1Q is defined by:
φ_path = 3*φ1 - 2*φ2 - 2*φ3 + φ_rec.

The receiver phase is defined with the conditions: φ_path = 0 and φ2 = 0 ;
therefore φ_rec = -3*φ1 + 2*φ3.
φ3 takes eight values so that 2*φ3 is manifold of 90°; this allows the most efficiency for filtering the coherences by the third pulse. φ1 takes 12 values so that 3*φ1 is manifold of 90°.

This 96-phase cycling is identical to that described in phase modulated shifted-echo sequence.

This phase cycling procedure is also provided by:

1. Mildner and coworkers, Rotationally induced triple quantum coherence excitation in MAS NMR spectroscopy of I = 5/2 spins, Chem. Phys. Lett. 301, 389-394 (1999).
2. Logan and coworkers, Investigations of low-amplitude radio frequency pulses at and away from rotary resonance conditions for I = 5/2 Nuclei, Solid State NMR 22, 97-109 (2002).
3. Goldbourt and coworkers, Characterization of aluminum species in alumina multilayer grafted MCM-41 using ²⁷Al FAM(II)-MQMAS NMR, J. Phys. Chem. B 107, 724-731 (2003).
4. Goldbourt and Madhu, Multiple-quantum magic-angle spinning: High-resolution solid-state NMR of half-integer spin quadrupolar nuclei, Annu. Rep. NMR Spectrosc. 54, 81-153 (2005).

(2) 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 and +1Q after the second pulse);
(2) explicit phase cycling the pulses and the receiver.

These simulated amplitudes allow us to compare the efficiency of the two coherence transfer pathways: 0Q -> -3Q -> +1Q -> -1Q and 0Q -> +3Q -> +1Q -> -1Q.

(3) Result

When the amplitudes are normalized to the number of phase cycling, these two approaches provide the same simulation data about echo amplitude for spin I = 5/2 systems. For spin I = 7/2 systems, this 96-phase cycling also allows the coherence transfer pathway 0Q -> +3Q -> -7Q -> -1Q to be detected.

(A.3) -5Q-MAS for spin I = 5/2 system

(1) 160-phase cycling

The total pathway phase φ_path for the echo coherence transfer pathway 0Q -> -5Q -> +1Q -> -1Q is defined by:
φ_path = -5*φ1 + 6*φ2 - 2*φ3 + φ_rec.

The receiver phase is defined with the conditions: φ_path = 0 and φ2 = 0 ;
therefore φ_rec = 5*φ1 + 2*φ3.
φ3 takes eight values so that 2*φ3 is manifold of 90°; this allows the most efficiency for filtering the coherences by the third pulse. φ1 takes 20 values so that 5*φ1 is manifold of 90°.

(2) 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 and +1Q after the second pulse);
(2) explicit phase cycling the pulses and the receiver.

(3) Result

When the amplitudes are normalized to the number of phase cycling, these two approaches provide the same simulation data about echo amplitude for spin I = 5/2 systems. For spin I = 7/2 systems, this 160-phase cycling also allows the coherence transfer pathway 0Q -> -5Q -> -7Q -> -1Q to be detected.

(A.4) +5Q-MAS for spin I = 5/2 system

(1) 160-phase cycling

The total pathway phase φ_path for the coherence transfer pathway 0Q -> +5Q -> +1Q -> -1Q is defined by:
φ_path = 5*φ1 - 4*φ2 - 2*φ3 + φ_rec.

The receiver phase is defined with the conditions: φ_path = 0 and φ2 = 0;
therefore φ_rec = -5*φ1 + 2*φ3.
φ3 takes eight values so that 2*φ3 is manifold of 90°; this allows the most efficiency for filtering the coherences by the third pulse. φ1 takes 20 values so that 5*φ1 is manifold of 90°.

This phase cycling procedure is provided by:

1. Brown and Wimperis, Two-dimensional MQ-MAS NMR of quadrupolar nuclei: a comparison of methods, J. Magn. Reson. 128, 42-61 (1997).
2. Brown and coworkers, ²⁷Al MQ-MAS NMR study of the thermal transformation between the microporous aluminum methylphosphonates AlMePO-β and AlMePO-α, J. Phys. Chem. B 103, 812-817 (1999).
3. Mildner and coworkers, 2D five quantum MAS NMR using rotationally induced coherence transfer, Chem. Phys. Lett. 306, 297-302 (1999).
4. Logan and coworkers, Investigations of low-amplitude radio frequency pulses at and away from rotary resonance conditions for I = 5/2 Nuclei, Solid State NMR 22, 97-109 (2002).
5. Goldbourt and Madhu, Multiple-Quantum Magic Angle Spinning: High-resolution solid state NMR spectroscopy of half-integer quadrupolar nuclei, Monatsh. Chem. 133, 1497-1534 (2002).
6. Goldbourt and coworkers, Characterization of aluminum species in alumina multilayer grafted MCM-41 using ²⁷Al FAM(II)-MQMAS NMR, J. Phys. Chem. B 107, 724-731 (2003).
7. Bräuniger and coworkers, Efficient 5QMAS NMR of spin-5/2 nuclei: use of fast amplitude-modulated radio-frequency pulses and cogwheel phase cycling, J. Magn. Reson. 163, 64-72 (2003).
8. Goldbourt and Madhu, Multiple-quantum magic-angle spinning: High-resolution solid-state NMR of half-integer spin quadrupolar nuclei, Annu. Rep. NMR Spectrosc. 54, 81-153 (2005).

(2) Simulation

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

(3) Result

When the amplitudes are normalized to the number of phase cycling, these two approaches provide the same simulation date about antiecho amplitude for spin I = 5/2 systems. For spin I = 7/2 systems, this 160-phase cycling also allows the coherence transfer pathway 0Q -> +5Q -> -7Q -> -1Q to be detected.

(B) Two strong amplitude pulses and one soft pulse

The first two pulses of the sequence have strong amplitudes whereas the third pulse has a weak amplitude.

(B.1) +3Q-MAS for spin I = 5/2 system

We discuss the echo coherence transfer pathway for a spin I = 5/2 system.

When the third pulse has a weak amplitude, a four-phase cycling is applied to this pulse instead of eight-phase cycling in the strong amplitude case. As a result, the number of phase cycling reduces to 48.

(1) 48-phase cycling

The following 48-phase cycling is applied:

This phase cycling procedure is also provided by [Gu and Power, Improved quantitation in 3QMAS of spin 5/2 nuclei by RF power modulation of FAM-II, Solid State NMR 27, 192-199 (2005)].

This 48-phase cycling selects not only the desired coherence transfer pathway
0Q -> +3Q -> +1Q -> -1Q,
but also the two coherence transfer pathways
0Q -> +3Q -> -3Q -> -1Q and 0Q -> +3Q -> +5Q -> -1Q,
as shown in the following figure:

(2) Simulation

We provide SIMPSON1.1.1 Tcl scripts for simulating the echo amplitude versus a pulse duration with two approaches:
(1) filtering the coherences (+3Q after the first pulse, and +5Q, +1Q and -3Q after the second pulse);
(2) explicit phase cycling the pulses and the receiver.

(3) Result

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

(B.2) +5Q-MAS for spin I = 5/2 system

When the third pulse has a weak amplitude, a four-phase cycling is applied to this pulse instead of eight-phase cycling in the strong amplitude case. As a result, the number of phase cycling reduces to 80.

(1) 80-phase cycling

The following phase cycling is applied:

This 80-phase cycling selects not only the desired coherence transfer pathway
0Q -> +5Q -> +1Q -> -1Q,
but also the two coherence transfer pathways
0Q -> +5Q -> +5Q -> -1Q and 0Q -> +5Q -> -3Q -> -1Q,
as shown in the following figure:

(2) Simulation

We provide SIMPSON1.1.1 Tcl scripts for simulating the antiecho amplitude versus a pulse duration with two approaches:
(1) filtering the coherences (+5Q after the first pulse, and +5Q, +1Q and -3Q after the second pulse);
(2) explicit phase cycling the pulses and the receiver.

(3) Result

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

(C) Conclusion

In split-t1 experiments for I = 5/2 systems, two cases are presented for the phase cycling:

1. the three pulses have strong and identical amplitudes - the desired coherence transfer pathway is selected by phase cycling;
2. the first two pulses have strong amplitudes and the third pulse has a weak amplitude - not only the desired coherence transfer pathway is selected, but additional coherence transfer pathways are also selected by the simplified phase cycling and the third pulse.

In the two cases, the variation of the echo or antiecho amplitudes versus a pulse duration can be simulated with coherence filtering method and phase cycling method. But the coherence filtering method provides the results in much shorter time.