## EFG tensor for static crystal

For static crystal, the orientation of the static magnetic
field B_{0} in the principal-axis system (主軸坐標系) of the EFG tensor
(X_{PAS}, Y_{PAS}, Z_{PAS}) is described
with the Euler angles α_{1}, β_{1},
and γ_{1}.

We provide a Mathematica notebook that calculates
the component V_{(2,0)}
of the second-rank spherical tensor **V**:

We used to defining the **quadrupole coupling**
ω_{Q} in the first-order quadrupole interaction
by the following expression:

with

The quadrupole coupling is defined experimentally by
**half** the frequency separating two consecutive
absorption lines of a single crystal. It is half that
used by A. Abragam.

On the other hand, the two components W_{(2,0)}
and W_{(4,0)} of the fourth-rank spherical tensor
**W** are obtained using the following expression:

The expression of W_{(0,0)} is simply:

We also provide Mathematica notebooks for calculating
the two components W_{(2,0)} and
W_{(4,0)}:

The third Euler angle γ_{1} does not appear
in the above two relations because B_{0} is a symmetry axis
for the spin system.

## Conclusion

In the above expressions, the asymmetry parameter η is
associated with cos2α_{1}. As a result, if we add
π/2 to α_{1}, **η will change to -η
**. In other words, the passage from our
convention for η to that used in the simulation program
SIMPSON and vice-versa is the addition of π/2 to the Euler angle
α_{1}. [see J. M. Koons, E. Hughes, H. M. Cho, and P. D. Ellis;
**Extracting multitensor solid-state NMR parameters from lineshapes**,
*J. Magn. Reson. A* **114**, 12-23 (1995)]

However, if a software allows us to change the sign of η,
we should use this possibility instead of adding π/2 to the
Euler angle α_{1}.