Qualitative understanding of electrical quadrapole moment

[blaisem]

Hello, as a chemist I sometimes run into mechanisms or reactions that are explained through the use of an electric quadrapole moment. I've spent some time looking at various sources, and I'd like to summarize what I believe it is. Hopefully someone here can provide feedback on my understanding.

Any charge distribution (electric field) is completely described by its multipole expansion. The quadrapole moment is simply the third term in this expansion, after a monopole and dipole.

Basically, when we, as chemists, refer to a dipole moment in a molecule, we are really only partially describing the electric field present because we are limiting ourselves to one term. This often works--at least approximately--because the dipole moment is a relatively highly contributing term in the multipole expansion.

This leads me to some questions on under what conditions a term in the multipole expansion, like the quadrapole moment, is zero or non-zero:

1. Since all the pictures of a quadrapole moment show four point charges, I am wondering if that is a minimum requirement for a non-zero quadrapole moment. For example, if I have two unlike point charges in space, can I have both non-zero dipole and quadrapole moments?

2. If two charges are opposite and equal, what terms would I expect to be non-zero in its multipole expansion? Right now I would expect only the monopole and dipole terms to be non-zero. Although I have to admit I don't really understand how a monopole here fits in since there are two charges; I only assume it because the dipole moment is non-zero.

Thanks so much for any advice and patience :)

 

[VantagePoint72]

Your summary of your understanding all sounds good.

1. Since all the pictures of a quadrapole moment show four point charges, I am wondering if that is a minimum requirement for a non-zero quadrapole moment. For example, if I have two unlike point charges in space, can I have both non-zero dipole and quadrapole moments?

A four-point charge distribution is the minimum needed to have both a non-zero quadrupole moment and zero monopole and dipole (according to any choice of coordinate origin) moments. Thus it's called an "elementary quadrupole". However, you don't need an elementary quadrupole to have non-zero quadrupole moment as long as you don't mind having non-zero monopole and dipole moments too. So, to answer your question, yes, two unlike charges separated in space have quadrupole (and higher) moments. This is what we call a "physical dipole". Often in EM we talk instead of a "pure" or "ideal" dipole moment in which the charge separation is taken to vanish (while holding the dipole moment constant). Pure dipoles do not have quadrupole, etc., moments.

2. If two charges are opposite and equal, what terms would I expect to be non-zero in its multipole expansion? Right now I would expect only the monopole and dipole terms to be non-zero. Although I have to admit I don't really understand how a monopole here fits in since there are two charges; I only assume it because the dipole moment is non-zero.

Keep in mind that moments can be coordinate dependent. Monopole moment is coordinate independent; however, unless the monopole moment is zero, the dipole moment depends on your choice of origin—and so on with the higher moments. So, in saying a particular moment is non-zero, we mean, "There is a coordinate system in which it is non-zero, though you might (if the lower moments are non-zero) be able to make it vanish by a judicious choice of origin." Two equal and opposite charges separated by some distance are a physical dipole. The monopole moment is zero because the total charge is zero. The dipole moment is non-zero (and its value is origin independent), as are the higher order moments (though their values are origin dependent). The higher order moments vanish in the limit of zero separation, with the charge magnitude increased to keep the dipole moment constant.

 

[blaisem]

Thank you LastOneStanding, this is exactly what I was looking for :)

One further question for anyone available: Why does a spherically symmetric charge have zero quadrupole moment?

I am using this link as a reference (but I did not understand the explanation used there): http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elequad.html#c1

I think I am a little confused about the meaning of the array of charges in space. Taking the elementary quadrupole moment shown in the hyperphysics link, it is reduced to a point charge at a distance. I thought all point charges were spherically symmetrical, so this would mean this non-zero value elementary quadrupole moment should also be spherically symmetrical and therefore have zero value.

The elementary quadrupole has three symmetrical planes. What would an example of a spherically symmetrical charge distribution be, such that the quadrupole moment is zero?

Thank you!

 

[VantagePoint72]

Thank you LastOneStanding, this is exactly what I was looking for :)

One further question for anyone available: Why does a spherically symmetric charge have zero quadrupole moment?

I am using this link as a reference (but I did not understand the explanation used there): http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elequad.html#c1

The website is being a bit sloppy about the importance of coordinates I highlighted above. The argument they are making is that a spherical charge distribution produces an electric potential that is identical to that of a point charge (as shown by Gauss's law). A point charge at the origin is a pure monopole. You can see this just by looking at Coulomb's law for a point charge at the origin and comparing it to multipole expansion. All the higher moments, including quadrupole, are zero. Hence, the website's explanation is that, since this is identical to the expression for the potential due to a spherical charge distribution, such a distribution also has zero quadrupole moment—when the distribution is centred on the origin. That last point is what they're a bit sloppy about. If you move a point charge away from the origin, it is no longer a pure monopole in that coordinate system. The monopole moment is unchanged (since it's just the total charge) but it gains higher moments, including quadrupole. The same goes for a spherical charge distribution moved away from the origin.

I think I am a little confused about the meaning of the array of charges in space. Taking the elementary quadrupole moment shown in the hyperphysics link, it is reduced to a point charge at a distance. I thought all point charges were spherically symmetrical, so this would mean this non-zero value elementary quadrupole moment should also be spherically symmetrical and therefore have zero value.

The difference is that a spherical charge distribution is exactly equivalent to a point charge while an elementary quadrupole is only approximately a (neutral) point at large distances. It is the quadrupole (and higher) moments that are precisely why the equivalence is only approximate.

 

[blaisem]

Awesome. Thanks so much again! Those have been great explanations for me that have made a lot of sense. I definitely feel on better footing now when dealing with quadrupoles.