Quadrupole

Not to be confused withtwo-port network,which is sometimes calledquadripole.

Aquadrupoleorquadrapoleis one of a sequence of configurations of things likeelectric chargeorcurrent,orgravitational massthat can exist in ideal form, but it is usually just part of amultipole expansionof a more complex structure reflecting various orders of complexity.

Mathematical definition

Thequadrupole moment tensorQis a rank-twotensor—3×3 matrix. There are several definitions, but it is normally stated in thetracelessform (i.e.$Q_{xx}+Q_{yy}+Q_{zz}=0$). The quadrupole moment tensor has thus nine components, but because of transposition symmetry andzero-traceproperty, in this form only five of these are independent.

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

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

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

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

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

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

For a continuous system with charge density, or mass density,$\rho (x,y,z)$,the components of Q are defined by integral over the Cartesian spacer:

$Q_{ij}=\int \,\rho (\mathbf {r} )\left(3r_{i}r_{j}-\left\|\mathbf {r} \right\|^{2}\delta _{ij}\right)\,d^{3}\mathbf {r}$

As with any multipole moment, if a lower-order moment,monopoleordipolein this case, is non-zero, then the value of the quadrupole moment depends on the choice of thecoordinate origin.For example, adipoleof 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 "$1/r$potential "field, like theelectricorgravitational field,the contribution to the field'spotentialfrom the quadrupole moment is:

$V_{\text{q}}(\mathbf {R} )={\frac {k}{|\mathbf {R} |^{3}}}\sum _{i,j}{\frac {1}{2}}Q_{ij}\,{\hat {R}}_{i}{\hat {R}}_{j}\,$

whereRis a vector with origin in the system of charges andR̂is the unit vector in the direction ofR.That is to say,${\hat {R}}_{i}$for$i=x,y,z$are the Cartesian components of the unit vector pointing from the origin to the field point. Here,$k$is a constant that depends on the type of field, and the units being used.

Electric quadrupole

Contour plot of theequipotential surfacesof an electric quadrupole field

A simple example of an electric quadrupole consists of alternating positive and negative charges, arranged on the corners of a square. The monopole moment (just the total charge) of this arrangement is zero. Similarly, thedipole momentis zero, regardless of the coordinate origin that has been chosen. But the quadrupole moment of the arrangement in the diagram cannot be reduced to zero, regardless of where we place the coordinate origin. Theelectric potentialof an electric charge quadrupole is given by

$V_{\text{q}}(\mathbf {R} )={\frac {1}{4\pi \varepsilon _{0}}}{\frac {1}{|\mathbf {R} |^{3}}}\sum _{i,j}{\frac {1}{2}}Q_{ij}\,{\hat {R}}_{i}{\hat {R}}_{j}\,$

where$\varepsilon _{0}$is theelectric permittivity,and$Q_{ij}$follows the definition above.

Alternatively, other sourcesinclude the factor of one half in the$Q_{ij}$tensor itself, such that:

$Q_{ij}=\int \,\rho (\mathbf {r} )\left({\frac {3}{2}}r_{i}r_{j}-{\frac {1}{2}}\left\|\mathbf {r} \right\|^{2}\delta _{ij}\right)\,d^{3}\mathbf {r}$,and

$V_{\text{q}}(\mathbf {R} )={\frac {1}{4\pi \varepsilon _{0}}}{\frac {1}{|\mathbf {R} |^{3}}}\sum _{i,j}Q_{ij}\,{\hat {R}}_{i}{\hat {R}}_{j}\,$

which makes more explicit the connection toLegendre polynomialswhich result from the multipole expansion, namely here${\textstyle P_{2}(x)={\frac {3}{2}}x^{2}-{\frac {1}{2}}.}$

Generalization: higher multipoles

An extreme generalization ( "pointoctopole") would be: Eight alternating point charges at the eight corners of aparallelepiped,e.g., of a cube with edge lengtha.The "octopole moment" of this arrangement would correspond, in the "octopole limit"${\textstyle \lim _{a\to 0}{a^{3}\cdot Q}\to {\text{const. }}}$to a nonzero diagonal tensor of order three. Still higher multipoles, e.g. of order$2^{\ell }$,would be obtained by dipolar (quadrupolar, octopolar,...) arrangements of point dipoles (quadrupoles, octopoles,...), not point monopoles, of lower order, e.g.,$2^{\ell -1}$.

Magnetic quadrupole

Coils producing a quadrupole field

Schematicquadrupole magnet( "four-pole")

See also:Quadrupole magnet

All known magnetic sources give dipole fields. However, it is possible to make a magnetic quadrupole by placing four identical bar magnets perpendicular to each other such that the north pole of one is next to the south of the other. 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.

An example of a magnetic quadrupole, involving permanent magnets, is depicted on the right.Electromagnetsof similar conceptual design (calledquadrupole magnets) are commonly used to focusbeams of charged particlesinparticle acceleratorsand beam transport lines, a method known asstrong focusing.There are four steel pole tips, two opposing magnetic north poles and two opposing magnetic south poles. The steel is magnetized by a largeelectric currentthat flows in the coils of tubing wrapped around the poles.

A changing magnetic quadrupole moment produceselectromagnetic radiation.

Gravitational quadrupole

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:

$V_{\text{q}}(\mathbf {R} )=-{\frac {G}{2|\mathbf {R} |^{3}}}\sum _{i,j}Q_{ij}\,{\hat {R}}_{i}{\hat {R}}_{j}\.$

For example, because the Earth is rotating, it is oblate (flattened at the poles). This gives it a nonzero quadrupole moment. 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${1}/{|\mathbf {R} |^{3}}$term falls quickly.

The mass quadrupole moment is also important ingeneral relativitybecause, if it changes in time, it can producegravitational radiation,similar to the electromagnetic radiation produced by oscillating electric or magnetic dipoles and higher multipoles. However, only quadrupole and higher moments can radiate gravitationally. The mass monopole represents the total mass-energy in a system, which is conserved—thus it gives off no radiation. Similarly, the mass dipole corresponds to the center of mass of a system and its first derivative represents momentum which is also a conserved quantity so the mass dipole also emits no radiation. The mass quadrupole, however, can change in time, and is the lowest-order contribution to gravitational radiation.

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 binaryblack 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

$Q_{ij}=M\left(3x_{i}x_{j}-|\mathbf {x} |^{2}\delta _{ij}\right)$

whereMis the mass of each point, and$x_{i}$are components of the (unit) position vector of one of the points. As they orbit, thisx-vector will rotate, which means that it will have a non-zero first, and also a non-zero second time derivative (this is of course true regardless the choice of the coordinate system). Therefore, the system will radiate gravitational waves. Energy lost in this way was first observed in the changing period of theHulse–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-calledgravitomagneticeffects. 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.

See also

• Multipole expansion
• Multipole moments
• Solid harmonics
• Axial multipole moments
• Cylindrical multipole moments
• Spherical multipole moments
• Laplace expansion
• Legendre polynomials
• Quadrupole ion trap
• Quadrupole mass analyzer
• Multipolar exchange interaction
• Star quad cable
• Magnetic lens
• Quadrupole formula

External links

• Multipole expansion