]> MathML: Quadrupole interaction in high magnetic field

## 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),

${H}_{Q}^{\left(1\right)}=\frac{\mathrm{eQ}}{4I\left(2I-1\right)ħ}\frac{\sqrt{6}}{3}\left[3{I}_{z}^{2}-I\left(I+1\right)\right]{V}_{\left(2,0\right)}$
and (2) the second-order quadrupole interaction (二階四極矩作用力),

${H}_{Q}^{\left(2\right)}=-\frac{1}{{\omega }_{0}}{\left(\frac{\mathrm{eQ}}{2I\left(2I-1\right)ħ}\right)}^{2}\left\{\frac{1}{2}{V}_{\left(2,-1\right)}{V}_{\left(2,1\right)}\left[4I\left(I+1\right)-8{I}_{z}^{2}-1\right]$ $+\frac{1}{2}{V}_{\left(2,-2\right)}{V}_{\left(2,2\right)}\left[2I\left(I+1\right)-2{I}_{z}^{2}-1\right]\right\}{I}_{z}$
with ${\omega }_{0}=\gamma {B}_{0}$
where ${V}_{\left(2,i\right)}$ are the spherical components of the second-rank EFG tensor. In its principal-axis system, the components of this tensor are:

${V}_{\left(2,0\right)}^{\mathrm{PAS}}=\sqrt{\frac{3}{2}}\mathrm{eq},{V}_{\left(2,±1\right)}^{\mathrm{PAS}}=0,{V}_{\left(2,±2\right)}^{\mathrm{PAS}}=\frac{1}{2}\mathrm{eq}\eta$

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 ${V}_{\left(2,-2\right)}{V}_{\left(2,2\right)}={V}_{\left(2,2\right)}{V}_{\left(2,-2\right)}$ and ${V}_{\left(2,-1\right)}{V}_{\left(2,1\right)}={V}_{\left(2,1\right)}{V}_{\left(2,-1\right)}$, we simply have
${V}_{\left(2,-1\right)}{V}_{\left(2,1\right)}={V}_{\left(2,1\right)}{V}_{\left(2,-1\right)}=\sqrt{\frac{8}{35}}{W}_{\left(4,0\right)}+\frac{1}{\sqrt{14}}{W}_{\left(2,0\right)}-\frac{1}{\sqrt{5}}{W}_{\left(0,0\right)}$ ${V}_{\left(2,-2\right)}{V}_{\left(2,2\right)}={V}_{\left(2,2\right)}{V}_{\left(2,-2\right)}=\frac{1}{\sqrt{70}}{W}_{\left(4,0\right)}+\sqrt{\frac{2}{7}}{W}_{\left(2,0\right)}+\frac{1}{\sqrt{5}}{W}_{\left(0,0\right)}$
where ${W}_{\left(3,0\right)}={W}_{\left(1,0\right)}=0$.

${H}_{Q}^{\left(2\right)}=-\frac{1}{{\omega }_{0}}{\left(\frac{\mathrm{eQ}}{2I\left(2I-1\right)ħ}\right)}^{2}\left\{\frac{\sqrt{70}}{140}{W}_{\left(4,0\right)}\left[18I\left(I+1\right)-34{I}_{z}^{2}-5\right]$ $+\frac{\sqrt{14}}{28}{W}_{\left(2,0\right)}\left[8I\left(I+1\right)-12{I}_{z}^{2}-3\right]$ $-\frac{1}{\sqrt{5}}{W}_{\left(0,0\right)}\left[I\left(I+1\right)-3{I}_{z}^{2}\right]\right\}{I}_{z}$

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

• ${W}_{\left(0,0\right)}^{\mathrm{PAS}}=\frac{1}{\sqrt{5}}\left[2{\left({V}_{\left(2,2\right)}^{\mathrm{PAS}}\right)}^{2}+{\left({V}_{\left(2,0\right)}^{\mathrm{PAS}}\right)}^{2}\right]=\frac{\sqrt{5}}{10}{\left(\mathrm{eq}\right)}^{2}\left({\eta }^{2}+3\right)$
• ${W}_{\left(2,0\right)}^{\mathrm{PAS}}=\frac{\sqrt{14}}{7}\left[2{\left({V}_{\left(2,2\right)}^{\mathrm{PAS}}\right)}^{2}-{\left({V}_{\left(2,0\right)}^{\mathrm{PAS}}\right)}^{2}\right]=\frac{1}{\sqrt{14}}{\left(\mathrm{eq}\right)}^{2}\left({\eta }^{2}-3\right)$
• ${W}_{\left(2,±2\right)}^{\mathrm{PAS}}=\frac{4}{\sqrt{14}}{V}_{\left(2,2\right)}^{\mathrm{PAS}}{V}_{\left(2,0\right)}^{\mathrm{PAS}}=\sqrt{\frac{3}{7}}{\left(\mathrm{eq}\right)}^{2}\eta$
• ${W}_{\left(4,0\right)}^{\mathrm{PAS}}=\frac{2}{\sqrt{70}}\left[{\left({V}_{\left(2,2\right)}^{\mathrm{PAS}}\right)}^{2}+3{\left({V}_{\left(2,0\right)}^{\mathrm{PAS}}\right)}^{2}\right]=\frac{1}{\sqrt{70}}{\left(\mathrm{eq}\right)}^{2}\left(\frac{1}{2}{\eta }^{2}+9\right)$
• ${W}_{\left(4,±2\right)}^{\mathrm{PAS}}=\frac{6}{\sqrt{42}}{V}_{\left(2,2\right)}^{\mathrm{PAS}}{V}_{\left(2,0\right)}^{\mathrm{PAS}}=\frac{3}{14}\sqrt{7}{\left(\mathrm{eq}\right)}^{2}\eta$
• ${W}_{\left(4,±4\right)}^{\mathrm{PAS}}={\left({V}_{\left(2,2\right)}^{\mathrm{PAS}}\right)}^{2}=\frac{1}{4}{\left(\mathrm{eq}\right)}^{2}{\eta }^{2}$

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