EFG tensor for static crystal

For static crystal, the orientation of the static magnetic field B₀ 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 α₁, β₁, and γ₁.

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 γ₁ does not appear in the above two relations because B₀ is a symmetry axis for the spin system.

Conclusion

In the above expressions, the asymmetry parameter η is associated with cos2α₁. As a result, if we add π/2 to α₁, 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 α₁. [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 α₁.