Quadrupole

A quadrupole or quadrapole is one of a sequence of configurations of—for example—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.

Mathematical definition

The quadrupole moment tensor Q is a rank-two tensor (3x3 matrix) and is traceless (i.e. ). The quadrupole moment tensor has thus 9 components, but because of the symmetry and zero-trace property, only 5 of these are independent.

For a discrete system of point charges (or masses in the case of a gravitational quadrupole), each with charge (or mass ) and position relative to the coordinate system origin, the components of the Q matrix are defined by:

.

The indices run over the Cartesian coordinates and is the Kronecker delta.

For a continuous system with charge density (or mass density) , the components of Q are defined by integral over the Cartesian space r:

As with any multipole moment, if a lower-order moment (monopole or dipole in this case) is non-zero, then the value of the quadrupole moment depends on the choice of the 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.