Clarification on electric quadrupole moment definition

[user1139]

Homework Statement

Clarification on electric quadrupole moment definition

Relevant Equations

The equations are as given below

I have encountered two (?) definitions of the electric quadrupole moment. They are:

$$Q_{ij}=\frac{1}{2}\int \rho(\vec{x}')x'_i x'_j\,\mathrm{d}^3x'$$

and

$$Q_{ij}=\int (3x'_i x'_j-\delta_{ij}x'^2)\rho(\vec{x}')\,\mathrm{d}^3x'$$

I am trying to study radiation arising from the electric quadrupole. I am confused since sometimes the first definition is used while other times, the second. As such, which definition should be used and why?

 

[Twigg]

These two definitions arise for different forms of the multipole expansion. The second expression (called the "traceless" quadrupole moment because it's trace is 0) is obtained by expanding a potential in terms of spherical harmonics. The first variant must use some basis other than spherical harmonics to expand the Green's function. Whatever the case may be, the only requirement is that you keep your multipole moments consistent with the basis in which you expand your Green's function. Just don't mix and match.

Some relevant wikipedia pages:
https://en.wikipedia.org/wiki/Quadrupole
https://en.wikipedia.org/wiki/Multipole_expansion

What are you using the quadrupole moment for? Is there a particular equation you pulled out of a book or paper? If so, folks might be able to tell you which definition to use.

 

[user1139]

These two definitions arise for different forms of the multipole expansion. The second expression (called the "traceless" quadrupole moment because it's trace is 0) is obtained by expanding a potential in terms of spherical harmonics. The first variant must use some basis other than spherical harmonics to expand the Green's function. Whatever the case may be, the only requirement is that you keep your multipole moments consistent with the basis in which you expand your Green's function. Just don't mix and match.

Some relevant wikipedia pages:
https://en.wikipedia.org/wiki/Quadrupole
https://en.wikipedia.org/wiki/Multipole_expansion

What are you using the quadrupole moment for? Is there a particular equation you pulled out of a book or paper? If so, folks might be able to tell you which definition to use.

I am trying to find the radiation from 2 like charges rotating about the axis at the origin. It seems to me that both definitions were plausible, hence, I was wondering which one should I use.

 

[Twigg]

I would stick to the traceless definition. I believe spherical harmonics have the symmetries that best fit your problem.

 

[user1139]

I would stick to the traceless definition. I believe spherical harmonics have the symmetries that best fit your problem.

This is a silly question but, the electric quadrupole moment calculated from both are not the same. Will this become a problem or is it simply just a matter of the derivation taken?

 

[Twigg]

They are different. You can't interchange them. It depends on how you're tackling the problem. I recommend using the usual multipole expansion in terms of spherical harmonics. See Jackson chapter 9 section 7 for an example (or Jackson chapter 4 section 1 for the electrostatic case which is much easier to wrap your head around before tackling radiation). This will outline where the traceless quadrupole moment comes from. I couldn't tell you how to derive the other definition, other than the obscure comment on wikipedia that it's used for the fast multipole method.

Edit: The electrostatic case is also covered in Griffiths chapter 4.