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• kelly0303
In summary: It's possible that the nucleus has a quadrupole deformation in the ground state, but its ground state has spin 0.
kelly0303
Hello! I am confused about the definition of the quadrupole moment in nuclei. One definition I found, in Wong, says that the quadrupole moment of a nucleus is given (ignoring some numerical constant) by: $$<J,M=J|r^2Y_{20}|J,M=J>$$ so the expectation value of a second order spherical harmonic (times ##r^2##) in the state with maximum M of the nucleus. However, given that we have a second order tensor (##Y_{20}##), for a nucleus with ##J=0## we can't have a quadrupole moment (this follows from the addition rules of angular momentum). However, there are several nuclei that have a (pretty big) quadrupole deformation in the ground state (for example they show a clear rotational spectrum, which is possible only if the nucleus is deformed), yet their ground state has ##J=0## (for example Hf). How is this possible, shouldn't a ##J=0## state indicate a spherical nucleus? Can someone explain this to me? Thank you!

mfb said:
Hello! My example (the one from Wong) is Hf 170. But my question was general, not only about that one particular nucleus. In general, even-even nuclei have spin zero in the ground state. Yet nuclei in between filled shells are known to have quadrupole deformations. But as I mentioned in the post, this seems like a contradiction to me. So I am missing something.

mfb said:
https://periodictable.com/Isotopes/072.170/index.p.html
Still no quadrupole moment according to the same source.
Clicking around I don't find any even/even nucleus with quadrupole moment.
That's weird. It has a quadrupole, as it has a spectrum specific to quadrupole deformation.

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Aren't these excited states with ##J\neq 0##?

mfb said:
Aren't these excited states with ##J\neq 0##?
As far as I understand from Wong (I might be wrong tho) this spectrum is a rotational one (the energies increase as J(J+1)). In order to have that spectrum, the nucleus must be deformed (a spherical nucleus doesn't have rotational energy in QM). So the ground state must be deformed in order to develop this spectrum. Yet the ground state has spin 0, which means that the state is spherical. So there is a clear contradiction. Of course I am miss-understanding something, but I am not sure what.

But it's not in the ground state when it's rotating.

mfb said:
But it's not in the ground state when it's rotating.
That's true, but the ground state needs to be deformed in order to set it in motion i.e. give it a rotation quanta. Assuming that the shape of the nucleus doesn't change (which is what Wong is assuming in his derivations) the shape of the nucleus in the ground state and in any excited rotational state should be the same (at a point it might be more convenient to start vibrating and thus change its shape, rather than add one more rotational quanta, but that is another story). So in order to get the nucleus to rotate and keep it rotating, its shape in the ground state must have (at least) a quadrupole. What am I missing here?

I think 2nd order perturbation theory would work.

## 1. What is quadrupole deformation in nuclei?

Quadrupole deformation in nuclei refers to the distortion or elongation of the shape of atomic nuclei, specifically the deviation from a spherical shape. This deformation is caused by the distribution of protons and neutrons within the nucleus.

## 2. How is quadrupole deformation measured?

Quadrupole deformation is measured using a parameter called quadrupole moment, which is a measure of the asymmetry in the distribution of charge or mass within the nucleus. This can be measured through experiments such as electron scattering or gamma-ray spectroscopy.

## 3. What causes quadrupole deformation in nuclei?

The main cause of quadrupole deformation in nuclei is the repulsive electrostatic force between protons. As more protons are added to the nucleus, the repulsion becomes stronger, causing the nucleus to elongate and become deformed.

## 4. How does quadrupole deformation affect the properties of nuclei?

Quadrupole deformation affects the energy levels and stability of nuclei. Deformed nuclei have different energy levels compared to spherical nuclei, which can affect their stability and decay modes. This deformation also plays a role in nuclear reactions and nuclear fission processes.

## 5. Can quadrupole deformation be observed in all nuclei?

No, not all nuclei exhibit quadrupole deformation. It is more commonly observed in heavier nuclei with a larger number of protons and neutrons. The degree of deformation also varies among different nuclei, with some being more spherical and others more elongated.

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