Conclusion about cogwheel phase cycling
The XSLT numerical search of winding numbers for the selection of a single coherence transfer pathway suggests us the following procedure:
It is simpler to keep constant the values of two winding numbers and search for the missing value of the other winding number among its allowed values in [1, N-1].
Application to other half-integer quadrupole spins
In 2005, Goldbourt and Madhu established the cogwheel winding numbers for the four half-integer quadrupole spins:
| Spin | Experiment | *Cogwheel solution CogN() |
Agreement |
|---|---|---|---|
| 3/2 | +3QMAS | Cog23(-3, 0, 1, 11) | yes |
| 5/2 | +3QMAS | Cog51(-5, 0, 3, 21) | yes |
| +5QMAS | Cog57(-5, 0, 1, 27) | yes | |
| 7/2 | +3QMAS | Cog87(-7, 0, 5, 31) | yes |
| +5QMAS | Cog97(-7, 0, 3, 41) | yes | |
| +7QMAS | Cog107(-7, 0, 1, 51) | yes | |
| 9/2 | +3QMAS | Cog131(-9, 0, 7, 41) | yes |
| +5QMAS | Cog145(-9, 0, 5, 55) | yes | |
| +7QMAS | Cog159(-9, 0, 3, 69) | Cog159(103, 0, 1, -83) | |
| +9QMAS | Cog173(-9, 0, 1, 83) | yes | |
| *These solutions are provided by A. Goldbourt and P. K. Madhu, Annu. Rep. NMR Spectrosc. 54, 81-153 (2005). | |||
In order to reduce the size of the XML modelling file with large N value, we provide a new XML modelling of the pulse sequence with a Java file called XmlDataCogwheelwA.java (and its PDF), in which = 0, being a constant, and can take any value from 1 to N - 1.
We proceed in the same way as for the Java file XmlData.java. After compilation and execution, a new XML file also called mqmas.xml is generated. We copy it onto WINDOWS desktop. With the help of the previous mqmas.xsl file on WINDOWS desktop, the execution of mqmas.xml via the default web browser provides a new 5-column table of smaller size.
We copy the data of this 5-column table to a MS EXCEL worksheet. Then we delete the rows related to the desired coherence transfer pathway. In our case, the rows concern with the antiecho coherence transfer pathway of a spin I = 3/2 system, that is, = 3 and = 1.
Finally, we search for the missing value in [1, 22] of the winding number in the second column of the shorten 5-column table. In the present case, the number 20 is missing. Since N = 23, this missing value is equivalent to -3 as found by Goldbourt and Madhu.
Among the cogwheel solutions in the above table, we do not agree only with the +7QMAS experiment of a spin I = 9/2 system.
Again we use SIMPSON simulations to support our results about the +7QMAS experiment of a spin I = 9/2 system. The echo amplitude in the coherence filtering method has been normalized to the number N (= 159) of phase cycling:
| 1st-pulse duration (μs) |
Cog159(-9, 0, 3, 69) | Cog159(103, 0, 1, -83) | Coherence filtering |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 0.25 | 1.53730229e-08 | 9.3240744e-09 | 1.1429838e-15 |
| 0.50 | -1.7651111e-07 | 9.39863831e-09 | 1.5419748e-11 |
| 0.75 | -6.81181402e-06 | 1.23277433e-08 | 3.2870394e-09 |
| 1.00 | -8.35658681e-05 | 1.55568299e-07 | 1.4721821e-07 |
| 1.25 | -0.000566299195 | 2.75039619e-06 | 2.74301448e-06 |
| 1.50 | -0.00262916943 | 2.88025484e-05 | 2.87963396e-05 |
| 1.75 | -0.00935361546 | 0.000200319003 | 0.000200314105 |
| 2.00 | -0.0271997264 | 0.00101787756 | 0.00101787405 |
| 2.25 | -0.0673036154 | 0.00402646379 | 0.00402646171 |
| 2.50 | -0.145609547 | 0.0129595087 | 0.012959508 |
| 2.75 | -0.280992955 | 0.035050097 | 0.0350500976 |
| 3.00 | -0.4914322556 | 0.0816375703 | 0.0816375723 |
| 3.25 | -0.789389504 | 0.166972306 | 0.166972309 |
| 3.50 | -1.17793231 | 0.30473034 | 0.304730344 |
| 3.75 | -1.64844027 | 0.503102221 | 0.503102226 |
| 4.00 | -2.17970447 | 0.76055247 | 0.760552475 |
Due to the value of N = 159 used in SIMPSON1.1.1 Tcl scripts, the echo amplitudes from Cog159(103, 0, 1, -83) and from coherence filtering agree with an accuracy of 10-8 only for long pulse durations.