HOME

TheInfoList



OR:

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 A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, \R^3. Similarly ...
of a more complex structure reflecting various orders of complexity.


Mathematical definition

The quadrupole moment tensor ''Q'' is a rank-two
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
—3×3 matrix. There are several definitions, but it is normally stated in the
traceless In linear algebra, the trace of a square matrix , denoted , is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of . The trace is only defined for a square matrix (). It can be proved that the trace o ...
form (i.e. Q_ + Q_ + Q_ = 0). 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. For a discrete system of \ell point charges or masses in the case of a gravitational quadrupole, each with charge q_\ell, or mass m_\ell, and position \vec_\ell = \left(r_, r_, r_\right) relative to the coordinate system origin, the components of the ''Q'' matrix are defined by: : Q_ = \sum_\ell q_\ell\left(3r_ r_ - \left\, \vec_\ell \right\, ^2\delta_\right). The indices i,j run over the
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
x,y,z and \delta_ is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...
. This means that x,y,z must be equal, up to sign, to distances from the point to n mutually perpendicular
hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
s for the Kronecker delta to equal 1. In the non-traceless form, the quadrupole moment is sometimes stated as: : Q_ = \sum_\ell q_\ell r_ r_ with this form seeing some usage in the literature regarding the
fast multipole method __NOTOC__ The fast multipole method (FMM) is a numerical technique that was developed to speed up the calculation of long-ranged forces in the ''n''-body problem. It does this by expanding the system Green's function using a multipole expansion, ...
. 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 space r: : Q_ = \int\, \rho(\mathbf)\left(3r_i r_j - \left\, \vec\right\, ^2\delta_\right)\, d^3\mathbf As with any multipole moment, if a lower-order moment, monopole or
dipole In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways: *An electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple example of this system i ...
in this case, is non-zero, then the value of the quadrupole moment depends on the choice of the
coordinate origin 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 physical problems, the choice of origin is often arbitrar ...
. For example, a
dipole In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways: *An electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple example of this system i ...
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 "1/r potential" field, like the
electric Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by ...
or
gravitational field In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...
, the contribution to the field's
potential Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple re ...
from the quadrupole moment is: : V_\text(\mathbf) = \frac \sum_ \frac Q_\, \hat_i \hat_j\ , where R is a vector with origin in the system of charges and R̂ is the unit vector in the direction of R. That is to say, \hat_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

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, the dipole moment is 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. The
electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
of an electric charge quadrupole is given by :V_\text(\mathbf) = \frac \frac \sum_ \frac Q_\, \hat_i \hat_j\ , where \varepsilon_0 is the
electric permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' ( epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
, and Q_ follows the definition above. Alternatively, other sources include the factor of one half in the Q_ tensor itself, such that: : Q_ = \int\, \rho(\mathbf)\left(\fracr_i r_j - \frac\left\, \vec\right\, ^2\delta_\right)\, d^3\mathbf , and :V_\text(\mathbf) = \frac \frac \sum_ Q_\, \hat_i \hat_j\ , which makes more explicit the connection to
Legendre polynomials In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applicat ...
which result from the
multipole expansion A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, \R^3. Similarly ...
, namely here P_2(x)=\fracx^2 - \frac.


Generalization: higher multipoles

An extreme generalization ("point octopole") would be: Eight alternating point charges at the eight corners of a
parallelepiped In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidea ...
, e.g., of a cube with edge length ''a''. The "octopole moment" of this arrangement would correspond, in the "octopole limit" \lim_\to\text to a nonzero diagonal tensor of order three. Still higher multipoles, e.g. of order 2^, would be obtained by dipolar (quadrupolar, octopolar, ...) arrangements of point dipoles (quadrupoles, octopoles, ...), not point monopoles, of lower order, e.g., 2^.


Magnetic quadrupole

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.
Electromagnet An electromagnet is a type of magnet in which the magnetic field is produced by an electric current. Electromagnets usually consist of wire wound into a coil. A current through the wire creates a magnetic field which is concentrated in the ...
s of similar conceptual design (called
quadrupole magnet Quadrupole magnets, abbreviated as Q-magnets, consist of groups of four magnets laid out so that in the planar multipole expansion of the field, the dipole terms cancel and where the lowest significant terms in the field equations are quadrupole. ...
s) are commonly used to focus beams of charged particles in
particle accelerator A particle accelerator is a machine that uses electromagnetic fields to propel charged particles to very high speeds and energies, and to contain them in well-defined beams. Large accelerators are used for fundamental research in particle ...
s and beam transport lines, a method known as
strong focusing In accelerator physics strong focusing or alternating-gradient focusing is the principle that, using sets of multiple electromagnets, it is possible to make a particle beam simultaneously converge in both directions perpendicular to the direction ...
. 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 An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving pa ...
that flows in the coils of tubing wrapped around the poles. A changing magnetic quadrupole moment produces
electromagnetic radiation In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic field, electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, inf ...
.


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(\mathbf) = -\frac \sum_ Q_\, \hat_i \hat_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 \frac term falls quickly. The mass quadrupole moment is also important in
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
because, if it changes in time, it can produce
gravitational radiation Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that propagate as waves outward from their source at the speed of light. They were first proposed by Oliver Heaviside in 1 ...
, 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 binary
black hole A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...
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 :Q_ = M\left(3x_i x_j - , \mathbf, ^2 \delta_\right) where M is the mass of each point, and x_i 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 The Hulse–Taylor binary is a binary star system composed of a neutron star and a pulsar (known as PSR B1913+16, PSR J1915+1606 or PSR 1913+16) which orbit around their common center of mass. It is the first binary pulsar ever discovere ...
, 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 Gravitoelectromagnetism, abbreviated GEM, refers to a set of formal analogies between the equations for electromagnetism and relativistic gravitation; specifically: between Maxwell's field equations and an approximation, valid under certain c ...
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 A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, \R^3. Similarly ...
*
Multipole moments A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, \R^3. Similarly ...
*
Solid harmonics In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions \mathbb^3 \to \mathbb. There are two kinds: the ''regular solid harmonics'' R^m_\ell(\mathbf), wh ...
*
Axial multipole moments Axial multipole moments are a series expansion of the electric potential of a charge distribution localized close to the origin along one Cartesian axis, denoted here as the ''z''-axis. However, the axial multipole expansion can also be applied ...
*
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. Such potentials arise in the electric potential of long line charges, and the anal ...
*
Spherical multipole moments Spherical multipole moments are the coefficients in a series expansion of a potential that varies inversely with the distance R to a source, ''i.e.'', as 1/''R''. Examples of such potentials are the electric potential, the magnetic potential ...
*
Laplace expansion In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression of the determinant of an matrix as a weighted sum of minors, which are the determinants of some submatrices of . Spec ...
*
Legendre polynomials In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applicat ...
*
Quadrupole ion trap A quadrupole ion trap or paul trap is a type of ion trap that uses dynamic electric fields to trap charged particles. They are also called radio frequency (RF) traps or Paul traps in honor of Wolfgang Paul, who invented the device and shared the ...
*
Quadrupole mass analyzer The quadrupole mass analyzer, originally conceived by Nobel Laureate Wolfgang Paul and his student Helmut Steinwedel, also known as quadrupole mass filter, is one type of mass analyzer used in mass spectrometry. As the name implies, it consists o ...
*
Multipolar exchange interaction Magnetic materials with strong spin-orbit interaction, such as: LaFeAsO, PrFe4P12, YbRu2Ge2, UO2, NpO2, Ce1−xLaxB6, URu2Si2 and many other compounds, are found to have magnetic ordering constituted by high rank multipoles, e.g. quadruple, octopl ...
*
Star quad cable Star-quad cable is a four-conductor cable that has a special quadrupole geometry which provides magnetic immunity when used in a balanced line. Four conductors are used to carry the two legs of the balanced line. All four conductors must be an ...
*
Magnetic lens thumb thumb A subtype of a magnetic lens ( quadrupole magnet) in the Maier-Leibnitz laboratory, Munich A magnetic lens is a device for the focusing or deflection of moving charged particles, such as electrons or ions, by use of the magnetic Lor ...
*
Quadrupole formula In general relativity, the quadrupole formula describes the rate at which gravitational waves are emitted from a system of masses based on the change of the (mass) quadrupole moment. The formula reads : \bar_(t,r) = \frac \ddot_(t-r/c), where \ba ...


References

{{Reflist


External links


Multipole expansion
Electromagnetism Gravity Moment (physics)