MathML: Quadrupole interaction in high magnetic field

Quadrupole interaction in high magnetic field

The first two terms in the Magnus expansion of the quadrupole interaction are:
the first-order quadrupole interaction (一階四極矩作用力),

H_{Q}^{(1)} = \frac{eQ}{4 I (2 I - 1) ħ} \frac{\sqrt{6}}{3} [3 I_{z}^{2} - I (I + 1)] V_{(2,0)}

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

H_{Q}^{(2)} = - \frac{1}{ω_{0}} {(\frac{eQ}{2 I (2 I - 1) ħ})}^{2} {\frac{1}{2} V_{(2, - 1)} V_{(2,1)} [4 I (I + 1) - 8 I_{z}^{2} - 1]

+ \frac{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} = \sqrt{\frac{3}{2}} eq, V_{(2, \pm 1)}^{PAS} = 0, V_{(2, \pm 2)}^{PAS} = \frac{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)} = \sqrt{\frac{8}{35}} W_{(4,0)} + \frac{1}{\sqrt{14}} W_{(2,0)} - \frac{1}{\sqrt{5}} W_{(0,0)}

V_{(2, - 2)} V_{(2,2)} = V_{(2,2)} V_{(2, - 2)} = \frac{1}{\sqrt{70}} W_{(4,0)} + \sqrt{\frac{2}{7}} W_{(2,0)} + \frac{1}{\sqrt{5}} W_{(0,0)}

where $W_{(3,0)} = W_{(1,0)} = 0$ .

The second-order quadrupole interaction becomes:

H_{Q}^{(2)} = - \frac{1}{ω_{0}} {(\frac{eQ}{2 I (2 I - 1) ħ})}^{2} {\frac{\sqrt{70}}{140} W_{(4,0)} [18 I (I + 1) - 34 I_{z}^{2} - 5]

+ \frac{\sqrt{14}}{28} W_{(2,0)} [8 I (I + 1) - 12 I_{z}^{2} - 3]

- \frac{1}{\sqrt{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:

$W_{(0,0)}^{PAS} = \frac{1}{\sqrt{5}} [2 {(V_{(2,2)}^{PAS})}^{2} + {(V_{(2,0)}^{PAS})}^{2}] = \frac{\sqrt{5}}{10} {(eq)}^{2} (η^{2} + 3)$

$W_{(2,0)}^{PAS} = \frac{\sqrt{14}}{7} [2 {(V_{(2,2)}^{PAS})}^{2} - {(V_{(2,0)}^{PAS})}^{2}] = \frac{1}{\sqrt{14}} {(eq)}^{2} (η^{2} - 3)$

$W_{(2, ± 2)}^{PAS} = \frac{4}{\sqrt{14}} V_{(2,2)}^{PAS} V_{(2,0)}^{PAS} = \sqrt{\frac{3}{7}} {(eq)}^{2} η$

$W_{(4,0)}^{PAS} = \frac{2}{\sqrt{70}} [{(V_{(2,2)}^{PAS})}^{2} + 3 {(V_{(2,0)}^{PAS})}^{2}] = \frac{1}{\sqrt{70}} {(eq)}^{2} (\frac{1}{2} η^{2} + 9)$

$W_{(4, ± 2)}^{PAS} = \frac{6}{\sqrt{42}} V_{(2,2)}^{PAS} V_{(2,0)}^{PAS} = \frac{3}{14} \sqrt{7} {(eq)}^{2} η$

$W_{(4, ± 4)}^{PAS} = {(V_{(2,2)}^{PAS})}^{2} = \frac{1}{4} {(eq)}^{2} η^{2}$

References

Wikipedia: Table of Clebsch–Gordan coefficients
Wikipedia: Wigner 3-j symbols
Wolfram MathWorld: Wigner 3j-symbol
Clebsch-Gordon or Wigner 3j coefficients for coupling angular momentum
Anthony Stone's Wigner coefficient calculator
Particle Data Group: Clebsch-Gordan coefficients (PDF file)
SO(3) Clebsch Gordan coefficients
Wikipedia: Table of spherical harmonics

Definition

One-pulse MAS

MQ-MAS

Reference

Quadrupole interaction in high magnetic field

References

Solid-state NMR bibliography for: