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Quadrupole interaction in high field.
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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),

H Q ( 1 ) = eQ 4 I ( 2 I - 1 ) ħ 6 3 [ 3 I z 2 - I ( I + 1 ) ] V ( 2,0 )
and (2) the second-order quadrupole interaction (二階四極矩作用力),

H Q ( 2 ) = - 1 ω 0 ( eQ 2 I ( 2 I - 1 ) ħ ) 2 { 1 2 V ( 2 , - 1 ) V ( 2,1 ) [ 4 I ( I + 1 ) - 8 I z 2 - 1 ] + 1 2 V ( 2 , - 2 ) V ( 2,2 ) [ 2 I ( I + 1 ) - 2 I z 2 - 1 ] } I z
with ω 0 = γ B 0
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:

V ( 2 , 0 ) PAS = 3 2 eq , V ( 2 , ± 1 ) PAS = 0 , V ( 2 , ± 2 ) PAS = 1 2 eq η

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
V ( 2 , - 1 ) V ( 2,1 ) = V ( 2,1 ) V ( 2 , - 1 ) = 8 35 W ( 4,0 ) + 1 14 W ( 2,0 ) - 1 5 W ( 0,0 ) V ( 2 , - 2 ) V ( 2,2 ) = V ( 2,2 ) V ( 2 , - 2 ) = 1 70 W ( 4,0 ) + 2 7 W ( 2,0 ) + 1 5 W ( 0,0 )
where W ( 3,0 ) = W ( 1,0 ) = 0 .

The second-order quadrupole interaction becomes:

H Q ( 2 ) = - 1 ω 0 ( eQ 2 I ( 2 I - 1 ) ħ ) 2 { 70 140 W ( 4,0 ) [ 18 I ( I + 1 ) - 34 I z 2 - 5 ] + 14 28 W ( 2,0 ) [ 8 I ( I + 1 ) - 12 I z 2 - 3 ] - 1 5 W ( 0,0 ) [ I ( I + 1 ) - 3 I z 2 ] } I z

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

  • W ( 0,0 ) PAS = 1 5 [ 2 ( V ( 2,2 ) PAS ) 2 + ( V ( 2,0 ) PAS ) 2 ] = 5 10 ( eq ) 2 ( η 2 + 3 )
  • W ( 2,0 ) PAS = 14 7 [ 2 ( V ( 2,2 ) PAS ) 2 - ( V ( 2,0 ) PAS ) 2 ] = 1 14 ( eq ) 2 ( η 2 - 3 )
  • W ( 2 , ± 2 ) PAS = 4 14 V ( 2,2 ) PAS V ( 2,0 ) PAS = 3 7 ( eq ) 2 η
  • W ( 4,0 ) PAS = 2 70 [ ( V ( 2,2 ) PAS ) 2 + 3 ( V ( 2,0 ) PAS ) 2 ] = 1 70 ( eq ) 2 ( 1 2 η 2 + 9 )
  • W ( 4 , ± 2 ) PAS = 6 42 V ( 2,2 ) PAS V ( 2,0 ) PAS = 3 14 7 ( eq ) 2 η
  • W ( 4 , ± 4 ) PAS = ( V ( 2,2 ) PAS ) 2 = 1 4 ( eq ) 2 η 2

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