Definition

• EFG tensor
• EFG tensor
  (mathML)
• EFG tensor
  (mathML + SVG)

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• Hamiltonian in high field
• Hamiltonian in high field
  (mathML)

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• Euler angles

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• Wigner rotation matrices
• Static crystal
• Rotating crystal

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• V20 static
• Second-order quadrupole shift, V21 and V22 static
• W20 static
• W40 static

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• Second-order quadrupole shift, V20 and V22 MAS
• V20 VAS (MAS)
• W20 VAS (MAS)
• W40 VAS (MAS)

One-pulse MAS

MQ-MAS

Reference

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$${\displaystyle \frac{\sqrt{\frac{3}{2}}}{\sqrt{{f(a+b)}^2}}\frac{\sqrt{3}}{2}}$$

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Quadrupole interaction in a uniform space

Quadrupole nuclei with a half-integer spin larger than 1/2 possess a quadrupole moment Q, which interacts with the electric-field gradient (EFG) generated by their surroundings.

The coupling of Q (a property of the nucleus) with an EFG (a property of a sample) is called the quadrupole interaction (四極矩作用力).

(A) Quadrupole interaction in Cartesian tensor representation 笛卡爾張量形式

Consider a free nucleus in a uniform space, that is, the three coordinate axes x, y, and z are equivalent. The Hamiltonian representing the quadrupole interaction of this nucleus, independent of the Cartesian coordinate frame, is defined by:

$${\displaystyle ħH_Q=\frac{\mathrm{eQ}}{6I(2I-1)}\sum _{j,k=x,y,z}{V}_{\mathrm{jk}}\left[\frac{3}{2}\right(I_j{I}_k+I_k{I}_j)-I(I+1\left){\delta }_{\mathrm{jk}}\right]}$$
where V_jk are the Cartesian components of V, the EFG at the origin, which is a second-rank symmetrical tensor. In the principal-axis system of the EFG (X_PAS, Y_PAS, Z_PAS), V is diagonal:

$${\displaystyle V=\left(\begin{array}{ccc}{V}_{\mathrm{XX}}& 0& 0\\ 0& V_{\mathrm{YY}}& 0\\ 0& 0& V_{\mathrm{ZZ}}\end{array}\right)}$$
with the convention

$${\displaystyle \mid V_{\mathrm{ZZ}}\mid \ge \mid V_{\mathrm{YY}}\mid \ge \mid V_{\mathrm{XX}}\mid }$$
However, another convention is also often used (for example in the NMR simulation program SIMPSON):

$${\displaystyle \mid V_{\mathrm{ZZ}}\mid \ge \mid V_{\mathrm{XX}}\mid \ge \mid V_{\mathrm{YY}}\mid }$$

Furthermore, the Laplace equation, ${\displaystyle V_{\mathrm{ZZ}}+V_{\mathrm{YY}}+V_{\mathrm{XX}}=0}$, holds for V, because the electric field at the nucleus is produced by charges wholly external to the nucleus. Thus, only two independent parameters are required:

$${\displaystyle \left\{\begin{array}{c}\mathrm{eq}=V_{\mathrm{ZZ}}\\ \eta =\frac{V_{\mathrm{XX}}-V_{\mathrm{YY}}}{V_{\mathrm{ZZ}}}\end{array}\right.}$$
with $1\ge \eta \ge 0\text{,}$ the largest component (最大的梯度) and the asymmetry parameter (不對稱參數), respectively.

For SIMPSON, these two parameters become:

$${\displaystyle \left\{\begin{array}{c}\mathrm{eq}=V_{\mathrm{ZZ}}\\ \eta =\frac{V_{\mathrm{YY}}-V_{\mathrm{XX}}}{V_{\mathrm{ZZ}}}\end{array}\right.}$$
with $1\ge \eta \ge 0\text{.}$

In practice, the asymmetry parameter η defines the powder lineshape and eq is related to the linewidth.

The product of eq with eQ divided by Planck's constant is called the quadrupole coupling constant (Cq) 四極偶合常數.

In the principal-axis system of V, the quadrupole interaction takes the form:

$${\displaystyle ħH_Q=\frac{e^2\mathrm{qQ}}{4I(2I-1)}[3I_Z^2-I(I+1)+\eta (I_X^2-I_Y^2\left)\right]}$$
In term of the operators I₊ = I_X + iI_Y and I_- = I_X - iI_Y, the quadrupole interaction becomes:

$${\displaystyle ħH_Q=\frac{e^2\mathrm{qQ}}{4I(2I-1)}[3I_Z^2-I(I+1)+\frac{1}{2}\eta (I_{+}^2+I_{-}^2\left)\right]}$$

(B) Quadrupole interaction in spherical tensor representation 球張量形式

The passage from one coordinate frame to another is more conveniently realized if the quadrupole interaction of a free nucleus is expressed as a product of second-rank irreducible spherical tensors:

$${\displaystyle H_Q=\frac{\mathrm{eQ}}{2I(2I-1)ħ}\sum _{q=-2}^2{(-1)}^q{V}_{(2,-q)}{T}^{(2,q)}}$$

In any Cartesian coordinate frame, the spherical tensor and Cartesian tensor components of V and T are related by:

${\displaystyle V_{\left(\mathrm{2,0}\right)}=\frac{\sqrt{6}}{2}{V}_{\mathrm{zz}}}$	${\displaystyle T^{(2,0)}=\frac{\sqrt{6}}{6}\{3I_z^2-I(I+1\left)\right\}}$
${\displaystyle V_{\left(\mathrm{2,1}\right)}=-V_{\mathrm{xz}}-i{V}_{\mathrm{yz}}}$	${\displaystyle T^{(2,1)}=-\frac{1}{2}(I_z{I}_{+}+I_{+}{I}_z)=-\frac{1}{2}{I}_{+}(2I_z+1)}$
${\displaystyle V_{(2,-1)}=V_{\mathrm{xz}}-i{V}_{\mathrm{yz}}}$	${\displaystyle T^{(2,-1)}=\frac{1}{2}(I_z{I}_{-}+I_{-}{I}_z)=\frac{1}{2}{I}_{-}(2I_z-1)}$
${\displaystyle V_{\left(\mathrm{2,2}\right)}=\frac{1}{2}(V_{\mathrm{xx}}-V_{\mathrm{yy}})+i{V}_{\mathrm{xy}}}$	${\displaystyle T^{(2,2)}=\frac{1}{2}{I}_{+}^2}$
${\displaystyle V_{(2,-2)}=\frac{1}{2}(V_{\mathrm{xx}}-V_{\mathrm{yy}})-i{V}_{\mathrm{xy}}}$	${\displaystyle T^{(2,-2)}=\frac{1}{2}{I}_{-}^2}$

with I₊ = I_x + iI_y and I_- = I_x - iI_y. These two operators are different from those used previously despite the same notation. The numerical factors in the components of V and T depend on the authors. The spherical tensor representation of the quadrupole interaction becomes:

$${\displaystyle H_Q=\frac{\mathrm{eQ}}{2I(2I-1)ħ}\left\{\frac{\sqrt{6}}{6}\right[3I_z^2-I(I+1)]V_{\left(\mathrm{2,0}\right)}}$$
$${\displaystyle +\frac{1}{2}{I}_{+}(2I_z+1)V_{(2,-1)}-\frac{1}{2}{I}_{-}(2I_z-1)V_{(2,1)}}$$
$${\displaystyle +\frac{1}{2}{I}_{+}^2{V}_{(2,-2)}+\frac{1}{2}{I}_{-}^2{V}_{(2,2)}\}}$$

Expressing the latter Hamiltonian in the principal-axis system of the EFG tensor and comparing the result with the last expression of quadrupole interaction in Cartesian tensor section yield the spherical tensor components of V in the principal-axis system:

$${\displaystyle V_{(2,0)}^{\mathrm{PAS}}=\sqrt{\frac{3}{2}}\mathrm{eq},V_{(2,\pm 1)}^{\mathrm{PAS}}=0,V_{(2,\pm 2)}^{\mathrm{PAS}}=\frac{1}{2}\mathrm{eq}\eta }$$

(C) References

• Wikipedia Covariance and contravariance of vectors
• Wikipedia: Tensor
• Roy McWeeny: Tensor Techniques in Physics – a concise introduction
• Kees Dullemond and Kasper Peeters: Introduction to tensor calculus
• R. A. Sharipov: Quick Introduction to Tensor Analysis
  

  Rule 5.3. For any double indexed array with indices on the same level (both upper or both lower) the first index is a row number, while the second index is a column number. If indices are on different levels (one upper and one lower), then the upper index is a row number, while lower one is a column number.

• Michael Fowler, University of Virginia: Tensor operators
• Berkeley: Irreducible tensor operators and the Wigner-Eckart theorem
• Ivan Deutsch, University of New Mexico: Irreducible tensor operators and the Wigner-Eckart theorem
• University of Tennessee, Knoxville: The Wigner-Eckart theorem

(D) Mathematica-5 notebook

• Double dot product of spherical basis tensors t_kq:t_rs
• Cartesian and spherical tensors in NMR Hamiltonian