## EFG tensor for rotating crystal

For rotating crystal, two axis-systems are involved. One orientates the static
magnetic field B_{0} in the rotor-axis system (轉子坐標系, x_{VAS}, y_{VAS},
z_{VAS}). The second orientates the rotor in the principal-axis system (主軸坐標系) of
the EFG tensor (X_{PAS}, Y_{PAS}, Z_{PAS}).

In fact, NMR rotor performs positive or right-hand rotation
around its z_{VAS} axis. Since the rotor spinning rate
ω_{rot} is a positive number by definition,
a minus sign must appear with the first Euler angle, that is,
-ω_{rot}*t in the above figure.
[see O. N. Antzutkin; **Sideband manipulation in magic-angle-spinning
nuclear magnetic resonance**, *Prog. Nucl. Magn. Reson.
Spectrosc.*** 35**, 203-266 (1999)]

**As a result, analytical expressions containing the rotor spinning
rate (in quadrupole interaction web pages of this site)
must change its sign.**

## (A) Variable-Angle-Spinning experiment 變角旋轉

### (1) V_{(2,0)} tensor component

The component V_{(2,0)} in the observation frame is
related to the eigen values of **V** by the
following two relations:

and

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

with

### (2) W_{(2,0)} tensor component

The component W_{(2,0)} in the
observation frame is related to the eigen values of
**W** by the following two relations:

and

We also provide a Mathematica notebook that calculates
the component W_{(2,0)}
of the fourth-rank spherical tensor **W**,
whose final expression is:

with

### (3) W_{(4,0)} tensor component

The component W_{(4,0)} in the
observation frame is related to the eigen values of
**W** by the following two relations:

and

We also provide a Mathematica notebook that calculates
the component W_{(4,0)} of
the fourth-rank spherical tensor **W**, whose final
expression is:

with

## (B) Magic-Angle-Spinning experiment 魔角旋轉

In practice, the angle θ_{VAS} of the rotor
is the magic-angle with:

θ_{VAS} = θ_{MAS} = arccos{sqrt(1/3)}.

### (1) V_{(2,0)} tensor component

The component V_{(2,0)} has the following form:

with

### (2) W_{(2,0)} tensor component

The component W_{(2,0)} has the following form:

with

### (3) W_{(4,0)} tensor component

The component W_{(4,0)} has the following form:

with

## (C) Conclusion

In the above expressions, the asymmetry parameter η is
associated with cos2α or with sin2α. As a result,
if we add π/2 to α, **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. [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.