## 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:

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:

with the convention

However, another convention is also often used (for example in the
NMR simulation program SIMPSON):

Furthermore, the Laplace equation,
V_{XX} + V_{YY} + V_{ZZ} = 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:

the largest component (最大的梯度) and the asymmetry parameter
(不對稱參數), respectively.

For SIMPSON, these two parameters become:

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:

In term of the operators (算子)
I_{+} = I_{X} + iI_{Y}
and I_{-} = I_{X} - iI_{Y},
the quadrupole interaction becomes:

### (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:

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

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:

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:

### (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

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