]> XSLT numerical search about z-filter 3QMAS sequence

## Cogwheel phase cycling parameters for spin-3/2 z-filter ±3QMAS. Contributor: Y. Millot

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 $\frac{\sqrt{\frac{3}{2}}}{\sqrt{{f\left(a+b\right)}^{2}}}\frac{\sqrt{3}}{2}$

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## Cogwheel phase cycling parameters for I = 3/2 in z-filter ±3QMAS sequence

We present the data of the 5-column table from XSLT numerical search in another table suitable for analyses.

Winding number of the third pulse C ( ${w}_{C}$) versus vertically the coherence orders ${p}_{\mathrm{AB}}$ and ${p}_{\mathrm{BC}}$ and horizontally the winding number of the second pulse B, ${w}_{B}$; the winding number of the first pulse A, ${w}_{A}$, being zero.
${p}_{\mathrm{AB}}$ ${p}_{\mathrm{BC}}$ ${w}_{B}$
12345 67891011 1213141516 17181920212223
Empty cell means no winding number is available for ${w}_{C}$. For the desired echo and antiecho coherence transfer pathways (0Q -> ±3Q -> 0Q -> -1Q), the letter T means that 23 values from 1 to 23 are available for ${w}_{C}$. This is not surprising, because the receiver phase always follows these desired pathways.
-3Q    4    8    4    8    4
-3Q    12    16    12    16    12
-3Q    20         20         20
-2Q    4    8    12    4    8
-2Q    16    20         16    20
-1Q    4    8    12    16    20
-3Q0Q    T    T    T    T    T
1Q    4    8    12    16    20
2Q    4    8    12    4    8
2Q    16    20         16    20
3Q    4    8    4    8    4
3Q    12    16    12    16    12
3Q    20         20         20
-3Q              8
-3Q              16
-2Q    6    12    6    12    6
-2Q    18         18         18
-1Q    8    16         8    16
-2Q0Q
1Q
2Q    2    4    6    8    10
2Q    14    16    18    20    22
3Q              8
3Q              16
-3Q              4
-3Q              12
-3Q              20
-2Q    8    4    12    8    4
-2Q    20    16         20    16
-1Q    12         12         12
-1Q0Q              T
1Q    20    16    12    8    4
2Q    12    12    12    12    12
3Q              4
3Q              12
3Q              20
-3Q    8    8    8    8    8
-3Q    16    16    16    16    16
-2Q    10    8    6    4    2
-2Q    22    20    18    16    14
-1Q    16    8         16    8
0Q0Q         T         T
1Q    16    8         16    8
2Q    10    8    6    4    2
2Q    22    20    18    16    14
3Q    8    8    8    8    8
3Q    16    16    16    16    16
-3Q              4
-3Q              12
-3Q              20
-2Q    12    12    12    12    12
-1Q    20    16    12    8    4
1Q0Q              T
1Q    12         12         12
2Q    8    4    12    8    4
2Q    20    16         20    16
3Q              4
3Q              12
3Q              20
-3Q              8
-3Q              16
-2Q    2    4    6    8    10
-2Q    14    16    18    20    22
-1Q
2Q0Q
1Q    8    16         8    16
2Q    6    12    6    12    6
2Q    18         18         18
3Q              8
3Q              16
-3Q    4    8    4    8    4
-3Q    12    16    12    16    12
-3Q    20         20         20
-2Q    4    8    12    4    8
-2Q    16    20         16    20
-1Q    4    8    12    16    20
3Q0Q    T    T    T    T    T
1Q    4    8    12    16    20
2Q    4    8    12    4    8
2Q    16    20         16    20
3Q    4    8    4    8    4
3Q    12    16    12    16    12
3Q    20         20         20

Recall that the winding number ${w}_{A}$ for the first pulse A has been chosen to be zero for simplicity. That of the second pulse ${w}_{B}$ (header of the table) can take any value from 1 to 23.

For a given value of ${w}_{B}$, a non-zero winding number ${w}_{C}$ of the third pulse C in the same column means that the associated coherence transfer pathway {0Q -> ${p}_{\mathrm{AB}}$Q -> ${p}_{\mathrm{BC}}$Q -> -1Q} is also observed by the cogwheel phase cycling. Conversely, if a value for the winding number of the third pulse C does not appear in this column, all the coherence transfer pathways, except for the echo and antiecho coherence transfer pathways, are canceled by the cogwheel phase cycling.

It is easy to see (shown in the MS EXCEL spreadsheet in ten steps) that the missing values for the winding number of the third pulse are in the following table:

 ${w}_{A}$ 0 0 ${w}_{B}$ 4 20 ${w}_{C}$ 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23

The winding number for the receiver phase is defined with the formula for one of the coherence transfer pathway only:

${w}_{\mathrm{Rec}}$ = -3 ${w}_{A}$ + 3 ${w}_{B}$ + 1 ${w}_{C}$ mod 24.

It is deduced from the formula for the receiver phase:

${\phi }_{\mathrm{Rec}}$ = -3 ${\phi }_{A}$ + 3 ${\phi }_{B}$ + 1 ${\phi }_{C}$ mod 24.

One set of cogwheel phase cycling is Cog24(0, 4, 19, 7). For obtaining a 90° increment for the receiver phase, we use Cog24(23, 3, 18, 6) deduced from Cog24(0, 4, 19, 7) by subtracting one to the four winding numbers then adding 24 to the first winding number.

## SIMPSON simulation with cogwheel phase cycling parameters

We provide two SIMPSON2 Tcl scripts simulating the antiecho and echo amplitude of sodium nucleus versus the duration of the first pulse.

We compare the Tcl script containing cogwheel phase cycling parameters, with the other containing the traditional coherence filtering method. The other NMR parameters are defined in these scripts, where the antiecho and echo amplitude in the coherence filtering method has been normalized to the number N (= 24) of phase cycling.

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