Quadrupole

A quadrupole or quadrapole is one of a sequence of configurations of things like electric charge or current, or gravitational mass that can exist in ideal form, but it is usually just part of a multipole expansion of a more complex structure reflecting various orders of complexity.

The quadrupole moment tensor Q is a rank-two tensor—3×3 matrix. There are several definitions, but it is normally stated in the traceless form (i.e. ${\displaystyle Q_{xx}+Q_{yy}+Q_{zz}=0}$). The quadrupole moment tensor has thus nine components, but because of transposition symmetry and the zero-trace property, in this form only five of these are independent.

For a discrete system of ${\displaystyle \ell }$ point charges or masses in the case of a gravitational quadrupole, each with charge ${\displaystyle q_{\ell }}$, or mass ${\displaystyle m_{\ell }}$, and position ${\displaystyle \mathbf {r} _{\ell }=\left(r_{x\ell },r_{y\ell },r_{z\ell }\right)}$ relative to the coordinate system origin, the components of the Q matrix are defined by:

$${\displaystyle Q_{ij}=\sum _{\ell }q_{\ell }\left(3r_{i\ell }r_{j\ell }-\left\|\mathbf {r} _{\ell }\right\|^{2}\delta _{ij}\right).}$$

The indices ${\displaystyle i,j}$ run over the Cartesian coordinates ${\displaystyle x,y,z}$ and ${\displaystyle \delta _{ij}}$ is the Kronecker delta. This means that ${\displaystyle x,y,z}$ must be equal, up to sign, to distances from the point to ${\displaystyle n}$ mutually perpendicular hyperplanes for the Kronecker delta to equal 1.

In the non-traceless form, the quadrupole moment is sometimes stated as:

$${\displaystyle Q_{ij}=\sum _{\ell }q_{\ell }r_{i\ell }r_{j\ell }}$$

with this form seeing some usage in the literature regarding the fast multipole method. Conversion between these two forms can be easily achieved using a detracing operator.