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
(
(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 Vjk 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 (XPAS, YPAS, ZPAS), 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, , 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:
with
the largest component (
For SIMPSON, these two parameters become:
with
![powder lineshape and asymmetry parameter](https://www.pascal-man.com/tensor-quadrupole-interaction/image/eta.gif)
In practice, the asymmetry parameter
The product of eq with eQ divided by Planck's constant is called
the quadrupole coupling constant (Cq)
In the principal-axis system of V, the quadrupole interaction takes the form:
In term of the operators I+ = IX + iIY and I- = IX - iIY, 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+ = Ix + iIy and I- = Ix - iIy. 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
- University of Tennessee, Knoxville: The Wigner-Eckart theorem
- Chemistry Libretexts: Quadrupolar Coupling
(D) Mathematica-5 notebook
- Double dot product of spherical basis tensors tkq:trs
- Cartesian and spherical tensors in NMR Hamiltonian