Quadrupole: Difference between revisions

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{{Short description|Arrangement that creates a quadrupole field of some sort}}

A '''quadrupole''' is one of a sequence of configurations of electric charge or gravitational mass that can exist in ideal form, but it usually just part of a [[power series]] representation of a more complex structure reflecting various orders of complexity.

{{Distinguish|text=[[two-port network]], which is sometimes called '''quadripole'''}}

A '''quadrupole''' or '''quadrapole''' is one of a sequence of configurations of things like [[electric charge]] or [[electric current|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.

== Electric multipoles ==

==Mathematical definition==

In the theory of [[electricity]] there are two signs of charge. The net total charge is the ''monopole moment.'' If there are charges of two signs, separated, for example along some line, then there is a ''[[dipole]] moment'' along that line. If there is no net charge, the [[lines of force]] resemble those of bar [[magnet]], though they are electric, not magnetic. If there are charges of both signs, but separated in a more complicated way, an electric quadrupole, but not a monopole or dipole may be present. A simple configuration of that kind could be modeled with an insulating weathervane having letters N, E, S, and W as indicators. If there are, for example, four equal charges present, two of each sign, with the two positive charges on the N and S letters, and the two negative on the E and W letters, we have an electric quadrupole but no dipole or monopole! Configurations of electric charges of dipole and higher multipolar nature that change in time radiate [[electromagnetic radiation]], whose character is named ''dipole'', ''quadrupole'', etc. according to a specific pattern generated by such sources. (There is no monopole radiation due to the conservation of electric charge, or [[Gauss' Law]].) Time changes can be exemplified by oscillating or by rotating configurations.

The '''quadrupole moment tensor''' ''Q'' is a rank-two [[tensor]]—3×3 matrix. There are several definitions, but it is normally stated in the [[traceless]] form (i.e. <math>Q_{xx} + Q_{yy} + Q_{zz} = 0</math>). The quadrupole moment tensor has thus nine components, but because of transposition symmetry and the [[Trace (linear algebra)|zero-trace]] property, in this form only five of these are independent.

For a discrete system of <math>\ell</math> point charges or masses in the case of a [[Quadrupole#Gravitational quadrupole|gravitational quadrupole]], each with charge <math>q_\ell</math>, or mass <math>m_\ell</math>, and position <math>\mathbf{r}_\ell = \left(r_{x\ell}, r_{y\ell}, r_{z\ell}\right)</math> relative to the coordinate system origin, the components of the ''Q'' matrix are defined by:

==Magnetic multipoles==

<math display="block">Q_{ij} = \sum_\ell q_\ell\left(3r_{i\ell} r_{j\ell} - \left\|\mathbf{r}_\ell \right\|^2\delta_{ij}\right).</math>

Because [[magnetic monopoles]] have never been observed (though there was one contested observation), they are often assumed not to exist; certainly they cannot at the present time [[2005]] be made in the laboratory or put to any known use. Thus, the magnetic sequence, though similar to the electric one, begins with the [[dipole]] case. In terms of the above weathervane example, we would have a single bar magnet as a [[dipole]]. To make a quadrupole we could take two identical bar magnets, putting the North pole of one on North and its opposite end on East [sic], while the other magnet would have its North pole on South [sic] and its South pole on West [sic]. Such a configuration would have no dipole moment, and its field can be proven to decrease at large distances faster than that of a dipole - see below. Again, a changing [[dipole]] or quadrupole moment will lead to the production of [[electromagnetic radiation]].

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]]. This means that <math>x,y,z</math> must be equal, up to sign, to distances from the point to <math>n</math> mutually perpendicular [[hyperplane]]s for the Kronecker delta to equal 1.

==Gravitational multipoles==

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

The situation here is similar in some ways to the magnetic case and in one way different. We do have gravitational monopole configurations and field; they are very commonly represented by ideal, stationary, spherically symmetric suns, planets, and so on. But, because no negative gravitational mass has ever been found, we have no possibility for a dipole moment. A gravitational quadrupole can be represented by two massive balls (say, lead) on opposite ends of a light rod, or, more simply, just as a long massive rod or a thin massive disk. A football shaped (prolate) or spheroidal (oblate) mass has a quadrupole moment. For example, the Earth is flattened at the poles, so it has a quadrupole moment. If a quadrupole (or higher order [[multipole]]) of mass rotates or oscillates (in vibration) it will emit [[gravitational radiation]].

<math display="block">Q_{ij} = \sum_\ell q_\ell r_{i\ell} r_{j\ell}</math>

==Distance dependence of multipole fields ==

The sequence ''monopole, dipole, quadrupole.'' can be extended to higher orders. A simple quadrupole is constructed by placing two opposing dipoles near each other, an octupole by adding another opposing quadrupole displaced from the one just made, and so on. The static fields of electric and magnetic [[multipole]]s fall off more and more rapidly as one moves away to an increasing radius ''r'' from the center. A monopole field (as for a single electric charge, or a single mass in [[Isaac Newton]]'s [[law of universal gravitation]], falls off with the [[inverse square law]]. A dipole field falls off as the cube of the distance, an octupole as the inverse fourth power, and so on.

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.<ref>{{Cite journal |doi = 10.1088/0305-4470/22/20/011|title = Traceless cartesian tensor forms for spherical harmonic functions: New theorems and applications to electrostatics of dielectric media|journal = Journal of Physics A: Mathematical and General|volume = 22|issue = 20|pages = 4303–4330|year = 1989|last1 = Applequist|first1 = J.|bibcode = 1989JPhA...22.4303A}}</ref>

When varying configurations of charges, currents, or masses are present, however, radiation is generally produced, and the radiation field falls off only as the first power of the distance, at large distances.

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 | access-date=May 8, 2012 | author=Weisstein, Eric}}</ref>

* [http://scienceworld.wolfram.com/physics/MultipoleExpansion.html Multipole expansion]

<math display="block">Q_{ij} = \int\, \rho(\mathbf{r})\left(3r_i r_j - \left\|\mathbf{r}\right\|^2\delta_{ij}\right)\, d^3\mathbf{r}</math>

[[de:Quadrupol]]

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.

If each charge is the source of a "<math>1/r</math> potential" field, like the [[electric field|electric]] or [[gravitational field]], the contribution to the field's [[potential]] from the quadrupole moment is:

<math display="block">V_\text{q}(\mathbf{R}) = \frac{k}{|\mathbf{R}|^3} \sum_{i,j} \frac{1}{2} Q_{ij}\, \hat{R}_i \hat{R}_j\ ,</math>

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, <math>\hat{R}_i</math> for <math>i=x,y,z</math> are the Cartesian components of the unit vector pointing from the origin to the field point. Here, <math>k</math> is a constant that depends on the type of field, and the units being used.

==Examples {{Anchor|electric quadrupole}}==

[[File:QuadrupoleContour.svg|right|thumb|200px|Contour plot of the [[equipotential surface]]s of an electric quadrupole field]]

=== 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 [[Electric dipole moment|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.<ref>{{Cite web |date=2016-12-20 |title=3.8: Quadrupole Moment |url=https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Electricity_and_Magnetism_(Tatum)/03%3A_Dipole_and_Quadrupole_Moments/3.08%3A_Quadrupole_Moment |access-date=2024-11-16 |website=Physics LibreTexts |language=en}}</ref> 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]] of an electric charge quadrupole is given by<ref>{{Cite book| author= Jackson, John David| date= 1975| title= Classical Electrodynamics| publisher= [[John Wiley & Sons]]| isbn= 0-471-43132-X| url-access= registration| url= https://archive.org/details/classicalelectro00jack_0}}</ref>

<math display="block">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\ ,</math>

where <math>\varepsilon_0</math> is the [[electric permittivity]], and <math>Q_{ij}</math> follows the definition above.

Alternatively, other sources<ref>{{Cite book| author= Griffiths, David J.| date= 2013| title= Introduction to Electrodynamics, 4th ed.| publisher= Pearson| page= 153,165}}</ref> include the factor of one half in the <math>Q_{ij}</math> tensor itself, such that:

<math display="block">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},</math>

and

<math display="block">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\ ,</math>

which makes more explicit the connection to [[Legendre polynomials]] which result from the multipole expansion, namely here <math display="inline">P_2(x) = \frac{3}{2}x^2 - \frac{1}{2}.</math>

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

<math display="block">Q \equiv \frac{1}{e} \int r^2 (\cos^2(\theta) - 1) \rho(\mathbf{r})\,d^3 \mathbf{r}</math>

where <math> \mathbf{r}</math> is the position within the nucleus and <math>\rho</math> gives the charge density at <math> \mathbf{r}</math>.<ref>{{Cite book |last=Amsler |first=Claude |title=Nuclear and particle physics |date=2015 |publisher=IOP Publishing |isbn=978-0-7503-1140-3 |edition=Version: 20150501 |series=IOP expanding physics |___location=Bristol, UK}}</ref>

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).<ref>{{Cite book |title=Mass spectrometry: an applied approach |date=2019 |publisher=Wiley |isbn=978-1-119-37733-7 |editor-last=Smoluch |editor-first=Marek |edition=Second |series=Wiley series on mass spectrometry |___location=Hoboken, NJ |editor-last2=Grasso |editor-first2=Giuseppe |editor-last3=Suder |editor-first3=Piotr |editor-last4=Silberring |editor-first4=Jerzy}}</ref>

=== Magnetic quadrupole ===

{{More citations needed|section|date=December 2024}}[[File:VFPt quadrupole coils 1.svg|left|thumb|Coils producing a quadrupole field]]

[[File:Magnetic quadrupole moment.svg|right|thumb|Schematic [[quadrupole 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. [[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]]. 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.<ref>{{Cite book |last1=Dattoli |first1=Giuseppe |url=https://iopscience.iop.org/book/mono/978-0-7503-1239-4 |title=Charged Beam Dynamics, Particle Accelerators and Free Electron Lasers |last2=Doria |first2=Andrea |last3=Sabia |first3=Elio |last4=Artioli |first4=Marcello |date=2017-09-01 |publisher=IOP Publishing |isbn=978-0-7503-1239-4 |language=en}}</ref> 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.

A changing magnetic quadrupole moment produces [[electromagnetic 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:

<math display="block">V_\text{q}(\mathbf{R}) = -\frac{G}{2|\mathbf{R}|^3} \sum_{i,j} Q_{ij}\, \hat{R}_i \hat{R}_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>{1}/{|\mathbf{R}|^3}</math> term falls quickly.

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| url = http://authors.library.caltech.edu/11159/1/THOrmp80a.pdf }}</ref> 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.<ref>{{Cite book |last=Kennefick |first=Daniel |title=Traveling at the Speed of Thought: Einstein and the Quest for Gravitational Waves |date=2016 |publisher=Princeton University Press |isbn=978-1-4008-8274-8 |___location=Princeton, NJ}}</ref> The magnitude of the gravitational radiation, or gravitational wave, can be described by [[Quadrupole formula|Einstein's quadrupole formula]].<ref>{{Cite journal |last=Chruściel |first=Piotr T. |date=2019 |title=Elements of General Relativity |url=https://link.springer.com/book/10.1007/978-3-030-28416-9 |journal=Compact Textbooks in Mathematics |language=en |doi=10.1007/978-3-030-28416-9 |isbn=978-3-030-28415-2 |issn=2296-4568|url-access=subscription }}</ref>

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

<math display="block">Q_{ij} = M\left(3x_i x_j - |\mathbf{x}|^2 \delta_{ij}\right) </math>

where '''M''' is the mass of each point, <math>x_i</math> are components of the (unit) position vector of one of the points, and <math>\displaystyle \delta_{ij}</math> is the Kronecker delta.<ref>{{Cite book |last1=Miller |first1=M. Coleman |title=Gravitational waves in physics and astrophysics: an artisan's guide |last2=Yunes |first2=Nicolás |date=2021 |publisher=IOP Publishing |isbn=978-0-7503-3051-0 |series=AAS-IOP astronomy. [2021 collection] |___location=Bristol [England] (Temple Circus, Temple Way, Bristol BS1 6HG, UK)}}</ref> 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 [[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.<ref>{{Cite book |last1=Miller |first1=M. Coleman |title=Gravitational waves in physics and astrophysics: an artisan's guide |last2=Yunes |first2=Nicolás |date=2021 |publisher=IOP Publishing |isbn=978-0-7503-3051-0 |series=AAS-IOP astronomy. [2021 collection] |___location=Bristol [England] (Temple Circus, Temple Way, Bristol BS1 6HG, UK)}}</ref>

==Generalization: higher multipoles==

An extreme generalization ("point [[octupole|octopole]]") would be: Eight alternating point charges at the eight corners of a [[parallelepiped]], e.g., of a cube with edge length ''a''. The "octopole moment" of this arrangement would correspond, in the "octopole limit" <math display="inline">\lim_{a\to 0} {a^3 \cdot Q} \to \text{const. } </math> to a nonzero diagonal tensor of order three. Still higher multipoles, e.g. of order <math>2^{\ell}</math>, would be obtained by dipolar (quadrupolar, octopolar, ...) arrangements of point dipoles (quadrupoles, octopoles, ...), not point monopoles, of lower order, e.g., <math>2^{\ell-1}</math>.

==See also==

{{div col|colwidth=15em}}

* [[Multipole expansion]]

* [[Solid harmonics]]

* [[Axial multipole moments]]

* [[Cylindrical multipole moments]]

* [[Spherical multipole moments]]

* [[Laplace expansion (potential)|Laplace expansion]]

* [[Legendre polynomials]]

* [[Quadrupole ion trap]]

* [[Quadrupole mass analyzer]]

* [[Multipolar exchange interaction]]

* [[Star quad cable]]

* [[Magnetic lens]]

* [[Quadrupole formula]]

{{div col end}}

==References==

{{Reflist}}

==External links==

* [http://scienceworld.wolfram.com/physics/MultipoleExpansion.html Multipole expansion]

[[Category:PhysicsElectromagnetism]]

[[Category:Gravity]]

[[Category:Moment (physics)]]