## 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];