quadrupolemoment

Quadrupole moment is a quantity used in physics to describe the second-order spatial distribution of charge or current and thus the deviation of a system from spherical symmetry. It arises in the multipole expansion of the electrostatic or magnetostatic potential at distances far from the source, where the monopole (total charge) and dipole terms have already been accounted for. The electric quadrupole moment is commonly represented by a second-rank tensor that encodes how the charge distribution extends along different axes.

A common definition for the electric quadrupole moment tensor is Q_ij = ∫ ρ(r) (x_i x_j - (1/3) δ_ij r^2) d^3r, where ρ(r) is the charge density, x_i are Cartesian coordinates, and δ_ij is the Kronecker delta. This form yields a traceless tensor with Q_ii = 0. An alternative convention is Q_ij = ∫ ρ(r) (3 x_i x_j - r^2 δ_ij) d^3r, which is proportional to the same physical content but differs by an overall constant factor. The far-field potential from a localized charge distribution behaves as V(r) ≈ (1/4π ε0) (1/r^3) ∑_{i,j} Q_ij n_i n_j, with n the unit vector r̂. The eigenvalues and eigenvectors of Q identify the principal axes and the degree of asphericity of the distribution.

In practice, quadrupole moments influence molecular spectra, energy level splittings, and transition rates, especially when the system is non-spherical. Magnetic quadrupole moments, defined analogously from current distributions, describe second-order magnetic effects and appear in detailed descriptions of atomic, molecular, and nuclear magnetism.