Quadrupole

A quadrupole arises from the arrangement shown standing next to two oppositely - equivalent dipoles with arbitrary distance vector, typically called.

In general, any charge or current distribution, provided that it does not possess certain symmetries, a multipole are assigned to second order. To this end, the real potential is approximated by a Taylor expansion. This results in the multipole expansion, inter alia, a quadrupole moment.

Due to the field perpendicular to the axial direction, each array of four alternately polarized electrodes is usually reduced as the " quadrupole " means, even if it does not generate a pure quadrupole field. In AC operation only particles of a certain mass are transmitted through this arrangement, which is why the arrangement is used in mass spectrometers. Another application of an electric quadrupole, the high- frequency - quadrupole accelerator.

Electric quadrupole

The electrical quadrupole is comprised of two positive and two negative charges of equal strength, which form two opposite - the same dipoles. So are the four charges in an alternating arrangement at the corners of a parallelogram (usually even a square ). Mathematically precise, the definition of a designated " quadrupole limit " limit process in which the area of ​​the parallelogram converges to zero, while at the same diverges the charge strength at the corners of the Parallellogramms charges, in such a way that the product remains constant, about where the constant should be positive.

The potential is obtained as the superposition ( superposition ) of Dipolpotentiale? D with corresponding weighting factors:

In the transition to the last equation, the Taylor expansion has been used and the order terms have been neglected. From the multipole expansion we obtain the Quadrupolmomenttensor Q with

Or

For continuous charge distributions. Thus, the potential can be represented alternatively as

Where the Einstein summation convention has been used.

Magnetic quadrupole

The magnetic quadrupole consists of two oppositely directed magnetic dipoles in the distance.

Applications:

• Quadrupole magnet: focusing magnetic particles in particle accelerators and beam lines
• Selective separation in mass spectrometry systems
• Together with nuclear magnetic resonance spectroscopy: Statements about the local geometry of the atomic nucleus in solids.

A spherical magnetic quadrupole field can be generated, for example, with a Maxwell coil.

Gravitational waves

In contrast to electromagnetism gravity has only positive charges ( masses). Therefore, the definition of a gravitational quadrupole as above is not possible on two dipoles. Nevertheless, mass distributions have a quadrupole moment. The lowest order of gravitational waves is a quadrupole, which corresponds in shape to the propagation of the electromagnetic quadrupole radiation.

Higher multipoles

Analog can be treated higher multipoles, so-called octupoles for example by alternating point charges on the eight corners of a parallelepiped, for example, a cube of edge length a, with the " octupole Limes" ( or, more generally approximates a single 2l - pole by superposition of two shifted 2 ( l-1 ) poles of opposite sign ).

Literature

• Horst Stöcker: Paperback physics. 4th edition, published by Harry German, Frankfurt am Main, 2000, ISBN 3-8171-1628-4