EFG tensor for static crystal.
Contributor: Y. Millot




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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 (XPAS, YPAS, ZPAS) is described with the Euler angles α1, β1, and γ1.

Euler angles of the static magnetic field in the laboratory frame

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

analytic expression of V(2,0)

We used to defining the quadrupole coupling ωQ in the first-order quadrupole interaction by the following expression:

first-order quadrupole interaction with omegaQ

with

definition of the quadrupole coupling omegaQ

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:

definition of W(2x,0)

The expression of W(0,0) is simply:

analytical expression of W(0,0)

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

analytical expression of 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.

 

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