Quadrupole interaction in high field.
Contributor: Y. Millot




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Quadrupole interaction in high magnetic field

The first two terms in the Magnus expansion of the quadrupole interaction are:

(1) the first-order quadrupole interaction (一階四極矩作用力),

first-order quadrupole interaction

and (2) the second-order quadrupole interaction (二階四極矩作用力),

second-order quadrupole interaction

where V(2,i) are the spherical components of the second-rank EFG tensor. In its principal-axis system, the components of this tensor are:

EFG eigen values in spherical tensor representation

The direct products of the EFG second-rank irreducible tensors can be expressed as higher rank spherical tensors W using the Clebsch-Gordan coefficients. Since V(2,2)V(2,-2) = V(2,-2)V(2,2) and V(2,1)V(2,-1) = V(2,-1)V(2,1), we simply have

Clebsch-Gordan coefficients relating V-tensor to W-tensor

where W(3,0) = W(1,0) = 0.

The second-order quadrupole interaction becomes:

second-order quadrupole interaction with W-tensor

The eigen values of the fourth-rank EFG tensor W in the spherical tensor representation are:

EFG eigen values of W-tensor

References

 

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