Editing Quadrupole (section)

==Mathematical definition==
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. <math>Q_{xx} + Q_{yy} + Q_{zz} = 0</math>). The quadrupole moment tensor has thus nine components, but because of transposition symmetry and [[Trace (linear algebra)|zero-trace]] property, in this form only five of these are independent.

For a discrete system of <math>\ell</math> point charges or masses in the case of a [[Quadrupole#Gravitational quadrupole|gravitational quadrupole]], each with charge <math>q_\ell</math>, or mass <math>m_\ell</math>, and position <math>\mathbf{r}_\ell = \left(r_{x\ell}, r_{y\ell}, r_{z\ell}\right)</math> relative to the coordinate system origin,  the components of the ''Q'' matrix are defined by:

<math display="block">Q_{ij} = \sum_\ell q_\ell\left(3r_{i\ell} r_{j\ell} - \left\|\mathbf{r}_\ell \right\|^2\delta_{ij}\right).</math>

The indices <math>i,j</math> run over the [[Cartesian coordinates]] <math>x,y,z</math> and <math>\delta_{ij}</math> is the [[Kronecker delta]].  This means that <math>x,y,z</math> must be equal, up to sign, to distances from the point to <math>n</math> mutually perpendicular [[hyperplane]]s for the Kronecker delta to equal 1.

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

<math display="block">Q_{ij} = \sum_\ell q_\ell r_{i\ell} r_{j\ell}</math>

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.<ref>{{Cite journal |doi = 10.1088/0305-4470/22/20/011|title = Traceless cartesian tensor forms for spherical harmonic functions: New theorems and applications to electrostatics of dielectric media|journal = Journal of Physics A: Mathematical and General|volume = 22|issue = 20|pages = 4303–4330|year = 1989|last1 = Applequist|first1 = J.|bibcode = 1989JPhA...22.4303A}}</ref>

For a continuous system with charge density, or mass density, <math>\rho(x, y, z)</math>, the components of Q are defined by integral over the Cartesian space '''r''':<ref name="wolf_eqm">{{cite web | url=http://scienceworld.wolfram.com/physics/ElectricQuadrupoleMoment.html | title=Electric Quadrupole Moment | publisher=[[Wolfram Research]] | work=Eric Weisstein's World of Physics | access-date=May 8, 2012 | author=Weisstein, Eric}}</ref>

<math display="block">Q_{ij} = \int\, \rho(\mathbf{r})\left(3r_i r_j - \left\|\mathbf{r}\right\|^2\delta_{ij}\right)\, d^3\mathbf{r}</math>

As with any multipole moment, if a lower-order moment, [[Monopole (mathematics)|monopole]] or [[dipole]] in this case, is non-zero, then the value of the quadrupole moment depends on the choice of the [[origin (mathematics)|coordinate origin]].  For example, a [[dipole]] of two opposite-sign, same-strength point charges, which has no monopole moment, can have a nonzero quadrupole moment if the origin is shifted away from the center of the configuration exactly between the two charges; or the quadrupole moment can be reduced to zero with the origin at the center.  In contrast, if the monopole and dipole moments vanish, but the quadrupole moment does not, e.g. four same-strength charges, arranged in a square, with alternating signs, then the quadrupole moment is coordinate independent.

If each charge is the source of a "<math>1/r</math> potential" field, like the [[electric field|electric]] or [[gravitational field]], the contribution to the field's [[potential]] from the quadrupole moment is:
<math display="block">V_\text{q}(\mathbf{R}) = \frac{k}{|\mathbf{R}|^3} \sum_{i,j} \frac{1}{2} Q_{ij}\, \hat{R}_i \hat{R}_j\ ,</math>

where '''R''' is a vector with origin in the system of charges and '''R̂''' is the unit vector in the direction of '''R'''. That is to say, <math>\hat{R}_i</math> for <math>i=x,y,z</math> are the Cartesian components of the unit vector pointing from the origin to the field point. Here, <math>k</math> is a constant that depends on the type of field, and the units being used.