Quadrupole

Not to be confused with two-port network, which is sometimes called quadripole.

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 of a more complex structure reflecting various orders of complexity.

1 Mathematical definition
2 Electric quadrupole
3 Generalization: Higher multipoles
4 Magnetic quadrupole
5 Gravitational quadrupole
6 See also
7 References
8 External links

Mathematical definition

The zero-trace quadrupole moment tensor of a system of charges (or masses, for example) is defined as

$Q_{ij}=\sum_l q_l(3(x_i)_l (x_j)_l-r_l^2\delta_{ij})\ ,$

for a discrete system with individual charges $q_l$ , or

$Q_{ij}=\int\, \rho(x)(3x_i x_j-r^2\delta_{ij})\, d^3x\ ,$

for a continuous system with charge density $\rho(x)$ . The indices $i,j$ run over the Cartesian coordinates $x,y,z$ .

The quadrupole moment tensor has 9 components, but because of the rotational symmetry and zero-trace property, only 5 of these are independent. As with any multipole moment, if a lower-order moment (monopole or dipole 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 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 " $1/r$ " field, like the electric or gravitational field, the contribution to the field's potential from the quadrupole moment is:

$V_q(\mathbf{R})=\frac{k}{|\mathbf{R}|^3} \sum_{i,j} Q_{ij}\, n_i n_j\ ,$

where R is a vector with origin in the system of charges and n is the unit vector in the direction of R. Here, $k$ is a constant that depends on the type of field, and the units being used. The factors $n_i, n_j$ 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 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 of an electric charge quadrupole is given by ^[1]

$V_q(\mathbf{R})=\frac{1}{4\pi \epsilon_0} \frac{1}{2} \frac{1}{|\mathbf{R}|^3} \sum_{i,j} Q_{ij}\, n_i n_j\ ,$

where $\epsilon_0$ 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, e.g. of a cube with edge length a. The "octupole moment" of this arrangement would correspond, in the "octupole limit" $\lim_{a\to 0;\,a^3\cdot Q\to\rm{const.}}$ , to a nonzero diagonal tensor of order three. Still higher multipoles, e.g. of order 2^l, would be obtained by dipolar (quadrupolar, octupolar, ...) arrangements of point dipoles (quadrupoles, octupoles, ...), not point monopoles, of lower order, e.g. 2^l-1.

Magnetic quadrupole

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:

$V_q(\mathbf{R})=G \frac{1}{2} \frac{1}{|\mathbf{R}|^3} \sum_{i,j} Q_{ij}\, n_i n_j\ .$

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 $\frac{1}{|\mathbf{R}|^3}$ 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 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.^[2]

The simplest and most important example of a radiating system is a pair of black holes 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

$Q_{ij}=M(3x_i x_j-\delta_{ij})\ ,$

where M is the mass of each hole, and $x_i$ 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.

References

^ Jackson, John David (1975). Classical Electrodynamics. John Wiley & Sons. ISBN 047143132X.
^ Thorne, Kip S. (April 1980). "Multipole Expansions of Gravitational Radiation". Reviews of Modern Physics 52 (2): 299–339. Bibcode 1980RvMP...52..299T. doi:10.1103/RevModPhys.52.299.

External links

Multipole expansion