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Why do spin-$\frac{1}{2}$ nuclei have zero electric quadrupole moment? How does this come about, and how can one tell in general whether a spin-$j$ nucleus can have a nonzero quadrupole (or higher multipole) moment?

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  • $\begingroup$ I'm pretty certain you can show that $\langle\frac{1}{2}|Q|\frac{1}{2}\rangle = 0$, where $Q$ is the quadrupole operator. However the details have slipped from what I laughingly refer to as my memory. This might give you enough clues to Google the answer. $\endgroup$ – John Rennie Dec 14 '14 at 10:56
  • $\begingroup$ As far as I've googled, it looks not that simple. I found no credible answer. $\endgroup$ – xiaohuamao Dec 18 '14 at 8:18
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    $\begingroup$ Related: physics.stackexchange.com/a/244110/44126 $\endgroup$ – rob Mar 18 '16 at 2:49
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The answer is: The Wigner-Eckart theorem.

In plain old angular momentum coupling (see Clebsch-Gordan coefficients) we learn that if a system consists of a spin-1/2 subsystem and a spin-2 subsystem, then the total system altogether can be spin-5/2 or spin-3/2, no other value. The rules are

  1. $|j_1-j_2| \leq j_\mathrm{total} \leq j_1+j_2$
  2. $j_1+j_2+j_\mathrm{total}$ is an integer.

Well, the Wigner-Eckart theorem teaches us that similar rules apply involving operators. The electric quadrupole operator $Q$ can be expressed as a rank-2 spherical tensor operator. When it acts on a spin-1/2 system, the resulting spin can only be $\frac{5}{2}$ or $\frac{3}{2}$. So if you're calculating $\langle\frac{1}{2}| Q|\frac{1}{2}\rangle$, you can group it like $\langle \frac{1}{2}| \;\;\; Q |\frac{1}{2}\rangle$. In other words, this is the inner product of a spin-$\frac{1}{2}$ system with $Q|\frac{1}{2}\rangle$ (which is spin $\frac{3}{2}$ or $\frac{5}{2}$). So they're orthogonal; the inner product is zero. $\langle\frac{1}{2}| Q |\frac{1}{2}\rangle=0$.

By similar logic, dipole moments (electric or magnetic) require spin ≥ 1/2, quadrupole moments require spin ≥ 1, octupole moments require spin ≥ 3/2, etc. (Dipole operators are rank 1, quadrupole are rank 2, octupole are rank 3, etc.)

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