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| {{Distinguish2|[[two-port network]], which is sometimes called '''quadripole'''}}
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| 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.
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| ==Mathematical definition==
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| The '''quadrupole moment tensor''' Q is a rank-two [[tensor]] (3x3 matrix) and is [[traceless]] (i.e. <math>Q_{xx}+Q_{yy}+Q_{zz}=0</math>). The quadrupole moment tensor has thus 9 components, but because of the rotational symmetry and [[Trace (linear algebra)|zero-trace]] property, only 5 of these are independent.
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| For a discrete system of point charges (or masses in the case of a [[Quadrupole#Gravitational_quadrupole|gravitational quadrupole]]), each with charge <math>q_{l}</math> (or mass <math>m_{l}</math>) and position <math>\vec{r_l}=(r_{xl},r_{yl},r_{zl})</math> relative to the coordinate system origin, the components of the Q matrix are defined by:
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| <math>Q_{ij}=\sum_l q_l(3r_{il} \cdot r_{jl}-r_l^2\delta_{ij})</math>.
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| 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]].
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| 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 | accessdate=May 8, 2012 | author=Weisstein, Eric}}</ref>
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| <math>Q_{ij}=\int\, \rho(3r_i r_j-r^2\delta_{ij})\, d^3\bold{r}</math>
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| 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.
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| If each charge is the source of a "<math>1/r</math>" field, like the [[electric field|electric]] or [[gravitational field]], the contribution to the field's [[potential]] from the quadrupole moment is:
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| :<math>V_q(\mathbf{R})=\frac{k}{|\mathbf{R}|^3} \sum_{i,j} Q_{ij}\, n_i n_j\ ,</math>
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| where '''R''' is a vector with origin in the system of charges and '''n'''
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| is the unit vector in the direction of '''R'''. Here, <math>k</math> is a constant that depends on the type of field, and the units being used. The factors <math> n_i, n_j </math> are components of the unit vector from the point of interest to the location of the quadrupole moment.
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| ==Electric quadrupole {{Anchor|electric quadrupole}}==
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| [[File:QuadrupoleContour.jpg|right|thumb|200px|Contour plot of the [[equipotential surface]]s of an electric quadrupole field.]]
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| The simplest 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, the [[Electric dipole moment|dipole moment]] is zero, regardless to 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. The [[electric potential]] of an electric charge quadrupole is given by <ref>{{Cite book| author= Jackson, John David| year=1975|title= Classical Electrodynamics| publisher =John Wiley & Sons| isbn= 0-471-43132-X}}</ref>
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| :<math>V_q(\mathbf{R})=\frac{1}{4\pi \epsilon_0} \frac{1}{|\mathbf{R}|^3} \sum_{i,j} Q_{ij}\, n_i n_j\ ,</math>
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| where <math>\epsilon_0</math> is the [[electric permittivity]].
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| {{Clear}}
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| ==Generalization: Higher multipoles==
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| An extreme generalization ("Point octupole") would be: Eight alternating point charges at the eight corners of a [[parallelepiped]], e.g. of a cube with edge length ''a''. The "octupole moment" of this arrangement would correspond, in the "octupole limit" <math>\lim_{a\to 0;\,a^3\cdot Q\to\rm{const.}}</math>, to a nonzero diagonal tensor of order three. Still higher multipoles, e.g. of order 2<sup>l</sup>, would be obtained by dipolar (quadrupolar, octupolar, ...) arrangements of point dipoles (quadrupoles, octupoles, ...), not point monopoles, of lower order, e.g. 2<sup>l-1</sup>.
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| ==Magnetic quadrupole==
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| [[File:VFPt quadrupole coils 1.svg|left|thumb|Coils producing a quadrupole field.]]
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| [[File:Magnetic quadrupole moment.svg|right|thumb|Schematic [[quadrupole magnet]] ("''four-pole''").]]
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| {{See also|Quadrupole magnet}}
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| All known magnetic sources give dipole fields. However, to make a magnetic quadrupole it is possible to place 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.
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| An example of a magnetic quadrupole, involving permanent magnets, is depicted on the right. [[Electromagnet]]s of similar conceptual design (called [[quadrupole magnet]]s) are commonly used to focus [[Charged particle beam|beams of charged particles]] in [[particle accelerator]]s and beam transport lines, a method known as [[strong focusing]]. The quadrupole-dipole intersect can be found by multiplying the spin of the unpaired nucleon by its parent atom. There are four steel pole tips, two opposing magnetic north poles and two opposing magnetic south poles. The steel is magnetized by a large [[electric current]] that flows in the coils of tubing wrapped around the poles.
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| Changing magnetic quadrupole moments produces [[electromagnetic radiation]].
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| {{Clear}}
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| ==Gravitational quadrupole==
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| The mass quadrupole is very analogous to the electric charge quadrupole, where the charge density is simply replaced by the mass density. The gravitational potential is then expressed as:
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| :<math>V_q(\mathbf{R})=G \frac{1}{2} \frac{1}{|\mathbf{R}|^3} \sum_{i,j} Q_{ij}\, n_i n_j\ .</math> | |
| 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 <math>\frac{1}{|\mathbf{R}|^3}</math> term falls quickly.
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| 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. 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.<ref>{{Cite journal| last = Thorne | first = Kip S. | journal = Reviews of Modern Physics | title = Multipole Expansions of Gravitational Radiation |date=April 1980 | volume = 52 | issue = 2 | pages = 299–339 | doi = 10.1103/RevModPhys.52.299 | bibcode=1980RvMP...52..299T}}</ref>
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| 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 hole]]s). 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
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| :<math>Q_{ij}=M(3x_i x_j-\delta_{ij})\ ,</math>
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| where '''M''' is the mass of each point, and <math>x_i</math> are components of the (unit) position vector of one of the points. As they orbit, this '''x'''-vector will rotate, which means that it will have a nonzero first, and also the 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 inferred in the changing period of the [[Hulse–Taylor binary pulsar]], a pulsar in orbit with another neutron star of similar mass.
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| 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 "[[Gravitomagnetism|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.
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| ==See also==
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| * [[Multipole expansion]]
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| * [[Multipole moments]]
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| * [[Solid harmonics]]
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| * [[Axial multipole moments]]
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| * [[Cylindrical multipole moments]]
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| * [[Spherical multipole moments]]
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| * [[Laplace expansion (potential)|Laplace expansion]]
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| * [[Legendre polynomials]]
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| * [[Quadrupole ion trap]]
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| * [[Quadrupole mass analyzer]]
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| ==References==
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| <references />
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| ==External links==
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| * [http://scienceworld.wolfram.com/physics/MultipoleExpansion.html Multipole expansion]
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| [[Category:Electromagnetism]]
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| [[Category:Gravitation]]
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