Phase cycling in multiple-pulse sequence

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Why one of the rf pulses in a multiple-pulse sequence does not need to be phase cycled?

Consider a three-pulse sequence for simplicity. It is easy to show that Pulse-3 does not need to be phase cycled, for example.

Phase- modulated shifted-echo 3QMAS sequence

The receiver phase φrec is defined by
φrec = -φ1*Δp1 - φ2*Δp2 - φ3*Δp3, (Eq. 1)
where φi is the phase of Pulse-i and Δpi is the difference between the coherence order p after Pulse-i and that before Pulse-i. That is
Δp1 = p1 - p0, Δp2 = p2 - p1, and Δp3 = p3 - p2
where p0 = 0 and p3 = -1.

Since Δp1 + Δp2 + Δp3 = -1 whatever the coherence transfer pathway is, we replace Δp3 in the expression of the receiver phase (Eq. 1) by
Δp3 = -1 - Δp1 - Δp2.
As a result, the expression of the receiver phase becomes
φrec - φ3 = -(φ1 - φ3)*Δp1 - (φ2 - φ3)*Δp2, (Eq. 2)
which can be rewritten as
ψrec = -ψ1*Δp1 - ψ2*Δp2.
The phase of Pulse-3 does not appear in the definition of the receiver phase.

This result is in agreement with the well-known property in phase cycling:
"additing a constant phase to the receiver phase and to those of all the rf pulses does not change the NMR signal".

Application of this property to Eq. 1 gives Eq. 2.

Solid-state NMR bibliography for:

Aluminum-27
Antimony-121/123
Arsenic-75
Barium-135/137
Beryllium-9
Bismuth-209
Boron-11
Bromine-79/81
Calcium-43
Cesium-133
Chlorine-35/37
Chromium-53
Cobalt-59
Copper-63/65
Deuterium-2
Gallium-69/71
Germanium-73
Gold-197
Hafnium-177/179
Indium-113/115
Iodine-127
Iridium-191/193
Krypton-83
Lanthanum-139
Lithium-7
Magnesium-25
Manganese-55
Mercury-201
Molybdenum-95/97
Neon-21
Nickel-61
Niobium-93
Nitrogen-14
Osmium-189
Oxygen-17
Palladium-105
Potassium-39/41
Rhenium-185/187
Rubidium-85/87
Ruthenium-99/101
Scandium-45
Sodium-23
Strontium-87
Sulfur-33
Tantalum-181
Titanium-47/49
Vanadium-51
Xenon-131
Zinc-67
Zirconium-91
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