kwadrupol

Kwadrupol, in science often translated as quadrupole, refers to a type of multipole moment describing a charge, mass, or current distribution for which the total monopole and dipole moments vanish, but higher-order moments remain. The concept is used across disciplines, including electrostatics, nuclear physics, and gravitation, to describe static properties and radiative phenomena.

Mathematically, the quadrupole moment is represented by a rank-2 tensor. For discrete charges, Q_ij = ∑ q_a (3 x_a,i x_a,j − r_a^2 δ_ij); for continuous distributions, Q_ij = ∫ ρ(x) (3 x_i x_j − r^2 δ_ij) d^3x. The tensor is traceless and depends on the chosen coordinate system, changing under rotations in a way that reflects the system’s geometry.

In electrostatics, a system with zero net charge and zero dipole moment can still interact with spatial variations of an external electric potential through its quadrupole moment. The interaction energy is U = −(1/6) Q_ij ∂^2 V/∂x_i ∂x_j, evaluated at the system’s location, highlighting how field gradients probe non-spherical charge distributions.

Nuclear and molecular contexts use the electric quadrupole moment to characterize non-spherical charge distributions, influencing spectroscopy through the interaction with the electric field gradient at the nucleus. Nuclear quadrupole moments affect NMR, NQR, and Mössbauer spectra and provide insight into molecular structure.

In gravitation, time-varying quadrupole moments are the leading source of gravitational radiation in general relativity. The quadrupole formula shows that radiated power depends on the third time derivative of the mass distribution’s quadrupole moment; binary star systems are notable examples.