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There are several groups actively working to measure the electric dipole moment of the electron. Is it possible for the electron to have higher order multipole moments, like quadrupole?

The reason for this question is an observation in Gottfried & Yans "Quantum Mechanics: Fundamentals" book (pg. 125) that all observables for a spin 1/2 system are either scalar or vector. Thus it would seem that higher order multipole moments are forbidden for an electron. However, that statement is about an isolated, non-relativistic electron, so I could imagine that there might be tiny corrections due to field theory effects in the same way that the electron EDM is expected to be generated.

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No, cannot, and Yes, it can...

A) it is possible for the quantum object with spin J=1/2, to have an "intrinsic" quadrupole deformation of the internal charged distribution. The real example is a strongly deformed J=1/2 nucleus Cf-251, with half-life approx 900 years. You may ask Berkeley scientists, they know about the internal = "intrinsic" deformation of Cf-251 enough.

B) however, if You tried to measure the quadrupole moment of Cf-251, via spectroscopic NMR technique, in the crystal lattice with large electric field gradients, You would Not observe any quadrupole moment of Cf-251 spectroscopically. Nevertheless, all scientists understanding the physics of deformed nuclei will confirm: Cf-251 is a strongly deformed nucleus.

C) if the electric field around the "frozen electron" deviates from the spherical (expected) distribution (that means: if it has "intrinsic" quadrupole moment) then, due to J=1/2 spin of the electron this cannot be experimentally detected using standard spectroscopic methods. Quantum state with Jz = +/- (1/2), makes the "intrinsic" quadrupole moment unobservable, and this applies to any superposition of +1/2 and -1/2 states, so for any quantum state of J=1/2 quantum object.

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