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 has thus nine components, but because of transposition symmetry and zero-trace property, in this form only five of these are independent.
relative to the coordinate system origin, the components of the Q matrix are defined by: The indices
mutually perpendicular hyperplanes for the Kronecker delta to equal 1.
Conversion between these two forms can be easily achieved using a detracing operator.
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.
A simple example of an electric quadrupole consists of alternating positive and negative charges, arranged on the corners of a square.
Similarly, the dipole moment is zero, regardless of the coordinate origin that has been chosen.
A consequence of this is that a quadrupole in a uniform field experiences neither a net force nor a net torque, although it can experience a net force or torque in a non-uniform field depending on the field gradients at the different charge sites.
tensor itself, such that: which makes more explicit the connection to Legendre polynomials which result from the multipole expansion, namely here
[6] An electric field constructed using four metal rods with an applied voltage forms the basis for the quadrupole mass analyzer, in which the electric field separates ions based on their mass-to-charge ratio (m/z).
Such a configuration cancels the dipole moment and gives a quadrupole moment, and its field will decrease at large distances faster than that of a dipole.
Electromagnets of similar conceptual design (called quadrupole magnets) are commonly used to focus beams of charged particles in particle accelerators and beam transport lines, a method known as strong focusing.
As a given quadrupole configuration deflects charged particles in one direction and focuses them in another, by using alternating quadrupole magnets a particle beam can be made to focus in the direction of travel.
The steel is magnetized by a large electric current that flows in the coils of tubing wrapped around the poles.
A changing magnetic quadrupole moment produces electromagnetic radiation.
The mass quadrupole is analogous to the electric charge quadrupole, where the charge density is simply replaced by the mass density and a negative sign is added because the masses are always positive and the force is attractive.
The gravitational potential is then expressed as: For example, because the Earth is rotating, it is oblate (flattened at the poles).
While the contribution to the Earth's gravitational field from this quadrupole is extremely important for artificial satellites close to Earth, it is less important for the Moon because the
The mass quadrupole moment is also important in general relativity because, if it changes in time, it can produce gravitational radiation, similar to the electromagnetic radiation produced by oscillating electric or magnetic dipoles and higher multipoles.
The mass monopole represents the total mass-energy in a system, which is conserved—thus it gives off no radiation.
The mass quadrupole, however, can change in time, and is the lowest-order contribution to gravitational radiation.
[11] The simplest and most important example of a radiating system is a pair of mass points with equal masses orbiting each other on a circular orbit, an approximation to e.g. special case of binary black holes.
Since the dipole moment is constant, we can for convenience place the coordinate origin right between the two points.
Then the dipole moment will be zero, and if we also scale the coordinates so that the points are at unit distance from the center, in opposite direction, the system's quadrupole moment will then simply be where M is the mass of each point,
Energy lost in this way was first observed in the changing period of the Hulse–Taylor binary, a pulsar in orbit with another neutron star of similar mass.
Just as electric charge and current multipoles contribute to the electromagnetic field, mass and mass-current multipoles contribute to the gravitational field in general relativity, causing the so-called gravitomagnetic effects.
Changing mass-current multipoles can also give off gravitational radiation.
However, contributions from the current multipoles will typically be much smaller than that of the mass quadrupole.
, would be obtained by dipolar (quadrupolar, octopolar, ...) arrangements of point dipoles (quadrupoles, octopoles, ...), not point monopoles, of lower order, e.g.,
