Quadrupole
Overview
 
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
Multipole expansion
A multipole expansion is a mathematical series representing a function that depends on angles — usually the two angles on a sphere. These series are useful because they can often be truncated, meaning that only the first few terms need to be retained for a good approximation to the original...

 of a more complex structure reflecting various orders of complexity.
The zero-trace quadrupole moment tensor of a system of charges (or masses, for example) is defined as
for a discrete system with individual charges , or
for a continuous system with charge density .
Encyclopedia
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
Multipole expansion
A multipole expansion is a mathematical series representing a function that depends on angles — usually the two angles on a sphere. These series are useful because they can often be truncated, meaning that only the first few terms need to be retained for a good approximation to the original...

 of a more complex structure reflecting various orders of complexity.

Mathematical definition

The zero-trace quadrupole moment tensor of a system of charges (or masses, for example) is defined as
for a discrete system with individual charges , or
for a continuous system with charge density . The indices run over the Cartesian coordinates .

The quadrupole moment tensor has 9 components, but because of the rotational symmetry and zero-trace
Trace (linear algebra)
In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...

 property, only 5 of these are independent. As with any multipole moment, if a lower-order moment (monopole
Monopole
Monopole may refer to:*Magnetic monopole, or Dirac monopole, a hypothetical particle that may be loosely described as a magnet with only one pole, or related concepts in physics and mathematics:...

 or dipole
Dipole
In physics, there are several kinds of dipoles:*An electric dipole is a separation of positive and negative charges. The simplest example of this is a pair of electric charges of equal magnitude but opposite sign, separated by some distance. A permanent electric dipole is called an electret.*A...

 in this case) is non-zero, then the value of the quadrupole moment depends on the choice of the coordinate origin
Origin (mathematics)
In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...

. For example, a dipole
Dipole
In physics, there are several kinds of dipoles:*An electric dipole is a separation of positive and negative charges. The simplest example of this is a pair of electric charges of equal magnitude but opposite sign, separated by some distance. A permanent electric dipole is called an electret.*A...

 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 "" field, like the electric
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

 or gravitational field
Gravitational field
The gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...

, the contribution to the field's potential
Potential
*In linguistics, the potential mood*The mathematical study of potentials is known as potential theory; it is the study of harmonic functions on manifolds...

 from the quadrupole moment is:
where R is a vector with origin in the system of charges and n
is the unit vector in the direction of R. Here, is a constant that depends on the type of field, and the units being used. The factors are components of the unit vector from the point of interest to the location of the quadrupole moment.

Electric quadrupole

The simplest example of an electric quadrupole, and simultaneously the origin of the name, is: four alternating charges at the corners of a parallelogram, e.g. a square of edge length a. A slightly more complicated example is shown in the picture. There are two positive and two negative charges, arranged on the square. The monopole moment (just the total charge) of this arrangement is zero. Similarly, the dipole moment
Electric dipole moment
In physics, the electric dipole moment is a measure of the separation of positive and negative electrical charges in a system of charges, that is, a measure of the charge system's overall polarity with SI units of Coulomb-meter...

 is zero, when the coordinate origin is at the center of the picture. 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
Electric potential
In classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...

 of an electric charge quadrupole is given by
where is the electric permittivity.

Generalization: Higher multipoles

An extreme generalization ("Point octupole") would be: Eight alternating point charges at the eight corners of a parallelepiped
Parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, its definition encompasses all four concepts...

, e.g. of a cube with edge length a. The "octupole moment" of this arrangement would correspond, in the "octupole limit" , to a nonzero diagonal tensor of order three. Still higher multipoles, e.g. of order 2l, would be obtained by dipolar (quadrupolar, octupolar, ...) arrangements of point dipoles (quadrupoles, octupoles, ...), not point monopoles, of lower order, e.g. 2l-1.

Magnetic quadrupole


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.

An example of a magnetic quadrupole, involving permanent magnets, is depicted on the right. Electromagnet
Electromagnet
An electromagnet is a type of magnet in which the magnetic field is produced by the flow of electric current. The magnetic field disappears when the current is turned off...

s of similar conceptual design (called quadrupole magnet
Quadrupole magnet
Quadrupole magnets consist of groups of four magnets laid out so that in the multipole expansion of the field the dipole terms cancel and where the lowest significant terms in the field equations are quadrupole. Quadrupole magnets are useful as they create a magnetic field whose magnitude grows...

s) are commonly used to focus beams of charged particles
Charged particle beam
A charged particle beam is a spatially localized group of electrically charged particles that have approximately the same velocity . The kinetic energies of the particles are typically measured in keV or MeV, much larger than the energies of particles at ambient temperature...

 in particle accelerator
Particle accelerator
A particle accelerator is a device that uses electromagnetic fields to propel charged particles to high speeds and to contain them in well-defined beams. An ordinary CRT television set is a simple form of accelerator. There are two basic types: electrostatic and oscillating field accelerators.In...

s and beam transport lines, a method known as strong focusing
Strong focusing
In accelerator physics strong focusing or alternating-gradient focusing is the principle that the net effect on a particle beam of charged particles passing through alternating field gradients is to make the beam converge...

. 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
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...

 that flows in the coils of tubing wrapped around the poles.

Changing magnetic quadrupole moments produces electromagnetic radiation
Electromagnetic radiation
Electromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...

.

Gravitational quadrupole

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:
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 term falls quickly.

The mass quadrupole moment is also important in general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

 because, if it changes in time, it can produce gravitational radiation, similar to the electromagnetic radiation produced by oscillating electric or magnetic quadrupoles. (In particular, the second time derivative must be nonzero.) The mass monopole represents the total mass-energy in a system, and does not change in time—thus it gives off no radiation. Similarly, the mass dipole represents the center of mass of a system, which also does not change in time—thus it also gives off 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 black hole
Black hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...

s with equal masses orbiting each other. If we place the coordinate origin right between the two black holes, and one black hole at unit distance along the x-axis, the system will have no dipole moment. Its quadrupole moment will simply be
where M is the mass of each hole, and is the unit vector in the x-direction. As the system orbits, the x-vector will rotate, which means that it will have a nonzero second time derivative. Thus, the system will radiate gravitational waves. Energy lost in this way was indirectly detected in the Hulse–Taylor binary.

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, because GR also includes "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.

See also

  • Multipole expansion
    Multipole expansion
    A multipole expansion is a mathematical series representing a function that depends on angles — usually the two angles on a sphere. These series are useful because they can often be truncated, meaning that only the first few terms need to be retained for a good approximation to the original...

  • Multipole moments
    Multipole moments
    In mathematics, especially as applied to physics, multipole moments are the coefficients of a series expansion of a potential due to continuous or discrete sources . A multipole moment usually involves powers of the distance to the origin, as well as some angular dependence...

  • Solid harmonics
  • Axial multipole moments
    Axial multipole moments
    Axial multipole moments are a series expansionof the electric potential of acharge distribution localized close tothe origin along oneCartesian axis,denoted here as the z-axis...

  • Cylindrical multipole moments
    Cylindrical multipole moments
    Cylindrical multipole moments are the coefficients in a series expansion of a potential that varies logarithmically with the distance to a source, i.e., as \ln \ R...

  • Spherical multipole moments
    Spherical multipole moments
    Spherical multipole moments are the coefficients in a series expansionof a potential that varies inversely with the distance R to a source, i.e., as 1/R...

  • Laplace expansion
    Laplace expansion (potential)
    In physics, the Laplace expansion of a 1/r - type potential is applied to expand Newton's gravitational potential or Coulomb's electrostatic potential. In quantum mechanical calculations on atoms the expansion is used in the evaluation of integrals of the interelectronic repulsion.The Laplace...

  • Legendre polynomials
  • Quadrupole ion trap
    Quadrupole ion trap
    A quadrupole ion trap exists in both linear and 3D varieties and refers to an ion trap that uses constant DC and radio frequency oscillating AC electric fields to trap ions. It is commonly used as a component of a mass spectrometer...

  • Quadrupole mass analyzer
    Quadrupole mass analyzer
    The quadrupole mass analyzer is one type of mass analyzer used in mass spectrometry. As the name implies, it consists of 4 circular rods, set parallel to each other. In a quadrupole mass spectrometer the quadrupole is the component of the instrument responsible for filtering sample ions, based on...


External links

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