A quadrupole or quadrapole is one of a sequence of configurations of things like
electric charge
Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
or
current
Currents, Current or The Current may refer to:
Science and technology
* Current (fluid), the flow of a liquid or a gas
** Air current, a flow of air
** Ocean current, a current in the ocean
*** Rip current, a kind of water current
** Current (hydr ...
, 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. Multipo ...
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 associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
—3×3 matrix. There are several definitions, but it is normally stated in the
traceless form (i.e.
). 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
point charges or masses in the case of a
gravitational quadrupole, each with charge
, or mass
, and position
relative to the coordinate system origin, the components of the ''Q'' matrix are defined by:
The indices
run over the
Cartesian coordinates
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
and
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
must be equal, up to sign, to distances from the point to
mutually perpendicular
hyperplane
In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...
s for the Kronecker delta to equal 1.
In the non-traceless form, the quadrupole moment is sometimes stated as:
with this form seeing some usage in the literature regarding the
fast 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,
, the components of Q are defined by integral over the Cartesian space r:
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 moment, electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple ...
in this case, is non-zero, then the value of the quadrupole moment depends on the choice of the
coordinate origin. For example, a
dipole
In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways:
* An electric dipole moment, electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple ...
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 "
potential" field, like the
electric
Electricity is the set of physical phenomena associated with the presence and motion of matter possessing an electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by Maxwel ...
or
gravitational field
In physics, a gravitational field or gravitational acceleration field is a vector field used to explain the influences that a body extends into the space around itself. A gravitational field is used to explain gravitational phenomena, such as ...
, 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 r ...
from the quadrupole moment is:
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,
for
are the Cartesian components of the unit vector pointing from the origin to the field point. Here,
is a constant that depends on the type of field, and the units being used.
Examples
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—the total charge—of this arrangement is zero. 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. As opposed to the monopole and dipole moments, 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 (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as electric potential energy per unit of electric charge. More precisely, electric potential is the amount of work (physic ...
of an electric charge quadrupole is given by
where
is the
electric permittivity, and
follows the definition above.
Alternatively, other sources include the factor of one half in the
tensor itself, such that:
and
which makes more explicit the connection to
Legendre polynomials
In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and t ...
which result from the multipole expansion, namely here
In atomic nuclei the electric quadrupole moment is used as a measure of the nucleus' obliquity, with the quadrupole moment in the nucleus given by
where
is the position within the nucleus and
gives the charge density at
.
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).
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 (likely copper) wound into a electromagnetic coil, coil. A current through the wire creates a magnetic ...
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 electric charge, charged particles to very high speeds and energies to contain them in well-defined particle beam, beams. Small accelerators are used for fundamental ...
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 ...
. 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. 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 flow of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is defined as the net rate of flow of electric charge through a surface. The moving particles are called charge c ...
that flows in the coils of tubing wrapped around the poles.
A changing magnetic quadrupole moment produces
electromagnetic radiation
In physics, electromagnetic radiation (EMR) is a self-propagating wave of the electromagnetic field that carries momentum and radiant energy through space. It encompasses a broad spectrum, classified by frequency or its inverse, wavelength ...
.
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:
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, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
because, if it changes in time, it can produce
gravitational radiation
Gravitational waves are oscillations of the gravitational field that travel through space at the speed of light; they are generated by the relative motion of gravitating masses. They were proposed by Oliver Heaviside in 1893 and then later by ...
, 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. Because only the mass multipole moments above the dipole moment can contribute to gravitational radiation, gravitational radiation is significantly weaker in magnitude to electromagnetic radiation. The magnitude of the gravitational radiation, or gravitational wave, can be described by
Einstein's quadrupole formula.
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 massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
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
where M is the mass of each point,
are components of the (unit) position vector of one of the points, and
is the Kronecker delta. As they orbit, this x-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 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.
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.
Three equiva ...
, e.g., of a cube with edge length ''a''. The "octopole moment" of this arrangement would correspond, in the "octopole limit"
to a nonzero diagonal tensor of order three. Still higher multipoles, e.g. of order
, would be obtained by dipolar (quadrupolar, octopolar, ...) arrangements of point dipoles (quadrupoles, octopoles, ...), not point monopoles, of lower order, e.g.,
.
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. Multipo ...
*
Solid harmonics
*
Axial multipole moments
*
Cylindrical multipole moments
*
Spherical multipole moments
*
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 . Spe ...
*
Legendre polynomials
In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and t ...
*
Quadrupole ion trap
*
Quadrupole mass analyzer
*
Multipolar exchange interaction
*
Star quad cable
*
Magnetic lens
*
Quadrupole formula
References
{{Reflist
External links
Multipole expansion
Electromagnetism
Gravity
Moment (physics)