## Second-order quadrupole interaction for static crystal. Contributor: R. Hajjar

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## Analytical expression of W(2,0) for static crystal in NMR

The second-order quadrupole interaction is related to W(2,0):

In static NMR experiment, W(2,0) is defined by:

Its analytical expression can be determined as follows:

(1) Select and copy the following green lines; then paste them into a cell of Mathematica, a software for numerical and symbolic calculations.

(2) Press "Ctrl-A" for select all; then press "Shift-enter" for evaluate cells.
(Or in the menu bar, select Kernel > Evaluation > Evaluate Cells)

Using Mathematica-5 running with a 3-GHz processor, the analytical expression of W(2,0) is obtained in 5 seconds.

```(* W2pas is a row-matrix with 3 columns
containing the 3 nonzero eigenvalues of the EFG
expressed as a 2-nd rang spherical tensor, in (eq)(eq) unit *)

W2pas = {{Sqrt[3/7]*eta, (eta*eta - 3)/Sqrt[14], Sqrt[3/7]*eta}};

(* D2 is a reduced form (3 rows x 1 column) of the 2-nd order
Wigner active rotation matrix *)

D2 = {

{Sqrt[3/8]*Sin[beta1]^2*E^(-2*I*alpha1)},

{(-1 + 3*Cos[beta1]^2)/2},

{Sqrt[3/8]*Sin[beta1]^2*E^(2*I*alpha1)}

};

(* W20static is an expression *)
W20static = FullSimplify[W2pas.ComplexExpand[D2]];

(* suppression of the double curve brackets {{}} of W20static *)
Print[W20static[[1, 1]]];

Remove[W2pas, eta, D2, alpha1, beta1, W20static];
```

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