## Analytical expression of W_{(4,0)} for static crystal in NMR

The second-order quadrupole interaction is related to W_{(4,0)}:

In static NMR experiment, W_{(4,0)} 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_{(4,0)} is obtained in **5 seconds**.

(* W4pas is a row-matrix with 5 columns containing the 5 nonzero eigenvalues of the EFG expressed as a 4-th rang spherical tensor, in (eq)(eq) unit *) W4pas = {{eta*eta/4, 3*eta/(2*Sqrt[7]), (9 + eta*eta/2)/Sqrt[70], 3*eta/(2*Sqrt[7]), eta*eta/4}}; (* D4 is a reduced form (5 rows x 1 column) of the 4-th order Wigner active rotation matrix *) D4 = { {Sqrt[35/2]*Sin[beta1]^4*E^(-4*I*alpha1)/8}, {Sqrt[5/2]*(-1 +7*Cos[beta1]^2)*Sin[beta1]^2*E^(-2*I*alpha1)/4}, {(3 - 30*Cos[beta1]^2 + 35*Cos[beta1]^4)/8}, {Sqrt[5/2]*(-1 +7*Cos[beta1]^2)*Sin[beta1]^2*E^(2*I*alpha1)/4}, {Sqrt[35/2]*Sin[beta1]^4*E^(4*I*alpha1)/8} }; (* W40static is an expression *) W40static = FullSimplify[W4pas.ComplexExpand[D4]]; (* suppression of the double curve brackets {{}} of W40static *) Print[W40static[[1, 1]]]; Remove[W4pas, eta, D4, alpha1, beta1, W40static];