## Definition of cogwheel phase cycling parameters

### 1. Introduction

In 2002, **Levitt** and coworkers introduced the cogwheel
phase cycling in NMR to reduce the number of steps for coherence transfer
pathway selection. The various parameters involved in this approach require
a numerical search. We present a procedure using XML (eXtensible Markup
Language) for modelling the pulse sequence with cogwheel phase cycling and XSLT
(eXtensible Stylesheet Language Transformation) for numerical search of these
parameters.

### 2. Split-t1 MQMAS sequence

We illustrate our web-tool approach with the well-studied sequence: the selection of the antiecho coherence transfer pathway in a three-pulse split-t1 MQMAS sequence applied to a spin I = 3/2 system.

In cogwheel phase cycling, the phases of the three pulses (A, B, and C) are incremented simultaneously as:

where ${w}_{A}$, ${w}_{B}$, and ${w}_{C}$ are integers called winding numbers, N is the number of steps in the phase cycling, and m is the phase counter (N - 1 ≥ m ≥ 0). These definitions are used by Levitt and coworkers.

The signal S(p) from a general coherence transfer pathway p = {0Q -> ${p}_{\mathrm{AB}}$Q -> ${p}_{\mathrm{BC}}$Q -> -1Q} is given by:

The desired coherence transfer pathway is ${p}^{0}$ = {0Q -> ${p}_{\mathrm{AB}}^{0}$Q -> ${p}_{\mathrm{BC}}^{0}$Q -> -1Q}. That is, ${p}_{\mathrm{AB}}^{0}$ = 3 and ${p}_{\mathrm{BC}}^{0}$ = 1 for this MQMAS sequence.

The signal S(p) is not zero if the following condition is fulfilled:

The main difficulty about the cogwheel phase cycling is the numerical search about the number of steps N and the winding numbers. Fortunately, several rules have been established by Levitt and coworkers in 2004.