## Rotational Echo Adiabatic Passage DOuble Resonance

**Gullion** proposed the REAPDOR pulse sequence to recouple
the heteronuclear dipole interaction between an observed spin-1/2
and a quadrupole spin. This sequence facilitates the recoupling of
quadrupole spins in Zeeman states other
than ¦±1/2>.

This sequence combines the REDOR approach with the adiabatic
passage of TRAPDOR experiment. The dipole evolution is maintained
by a REDOR train of 180° pulse applied to the observed nuclei
^{31}P. The adiabatic passage pulse of TRAPDOR is applied
to the quadrupole nuclei ^{27}Al to less than one rotor period.

No matter the number of 180° pulses present in the dipole evolution period, there is only one adiabatic passage pulse in REAPDOR sequence.

**Scheme (1):** The ^{31}P nuclei are observed with
rotor-synchronized spin echo sequence consisting of multiple 180°
pulses. The spacing between adjacent 180° pulse is 1/2 of the rotor
period. There is no ^{31}P pulse located at the midpoint
of the dipole evolution period.

**Scheme (2):** The radio-frequency pulse in the ^{27}Al
channel is of duration in the order of 1/3 of the rotor period
and causes a partial adiabatic population inversion, allowing the
^{27}Al-^{31}P dipole interaction to be recoupled
and the ^{31}P signal dephasing to be measured.

Excellent spinning speed control is necessary for REAPDOR measurement.

Usually, the **REAPDOR fraction**, defined by

**(ΔS)/S _{0} = [signal from Scheme (1) - signal from
Scheme (2)]/signal from Scheme (1),**

is plotted versus the dipole evolution time. The interatomic distance is extracted by fitting these data with a theoretical curve.

The curve representing the normalized REAPDOR difference versus
^{27}Al offset frequency allows an estimation of the quadrupole
coupling constant of ^{27}Al.

**Goldbourt and coworkers** demonstrated the existence of a
universal REAPDOR curve,

**S _{1/2,5/2}(nDτ) = 0.63 {1 -
exp[-(3.0 nDτ)^{2}]} +
0.2 {1 - exp[-(0.7 nDτ)^{2}]}**,

which allows estimation of distances between any observable NMR nucleus and spin-5/2 nuclei, provided that the adiabaticity α is greater than 0.55. With τ = 1/ν

_{R}.

adiabacity parameter α = (ν_{1})^{2}/ν_{Q}ν_{R}

ν_{1} is the RF amplitude,

ν_{R} is the rotor spinning rate,

ν_{Q} = 3C_{Q}/[2S(2S - 1)] is the quadrupole frequency,
ν_{Q} = C_{Q}/2 for spin S = 3/2,
ν_{Q} = 3C_{Q}/20 for S = 5/2,
ν_{Q} = C_{Q}/14 for S = 7/2,

C_{Q} is the quadrupole coupling constant.

### Mathematica 5 notebook

This notebook extracts the value of the heteronuclear dipole coupling constant D from
reapdorPAl31d.csv, whose left column
contains the values of n*τ, and right column those of (ΔS)/S_{o}.

<< Statistics`NonlinearFit`; SetDirectory["C:\\Users\\pm@pascal-man.com\\Documents"]; data = Import["reapdorPAl31d.csv"]; data1 = Take[data, 6]; (* Consider only the first six couples of values in file data *) sred[d_, ntr_] := 0.63(1 - Exp[-(3.0 d ntr)^2]) + 0.2(1 - Exp[-(0.7d ntr)^2]); universalfit[init_] := NonlinearFit[data1, sred[d, ntr], ntr, {d, init}, AccuracyGoal -> Automatic, PrecisionGoal -> Automatic, ShowProgress -> True] fit = universalfit[500] tmp1 = ListPlot[data, PlotRange -> {-0.2, 1}] tmp2 = Plot[fit, {ntr, 0, 0.0024}, PlotRange -> {-0.2, 1}, Axes -> True, AxesOrigin -> Automatic, Axes -> Automatic] Show[tmp1, tmp2]

### Mathematica 5 notebook

This notebook extracts the value of D*τ from reapdorPAl31.csv, whose left column
contains the values of n, and right column those of (ΔS)/S_{o}.

<< Statistics`NonlinearFit`; SetDirectory["C:\\Users\\pm@pascal-man.com\\Documents"]; data = Import["reapdorPAl31.csv"]; data1 = Take[data, 6]; (* Consider only the first six couples of values in file data *) sred[n_, dtr_] := 0.63(1 - Exp[-(3.0 n dtr)^2]) + 0.2(1 - Exp[-(0.7n dtr)^2]); universalfit[init_] := NonlinearFit[data1, sred[n, dtr], n, {dtr, init}, AccuracyGoal -> Automatic, PrecisionGoal -> Automatic, ShowProgress -> True] fit = universalfit[0.05] tmp1 = ListPlot[data, PlotRange -> {-0.2, 1}] tmp2 = Plot[fit, {n, 0, 10}, PlotRange -> {-0.2, 1}, Axes -> True, AxesOrigin -> Automatic, Axes -> Automatic] Show[tmp1, tmp2]

### References

YouTube: Importing data into Mathematica

YouTube: Fitting data in Mathematica

YouTube: Mathematica for biologists

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