Nuclear Electric Quadrupoles

Any distribution of electric charge can be written as the sum of simpler charge distributions. This is known as the superposition principle. In many cases it is useful to use "multipole moment" distributions. The lowest multiple moments are often the most important and include:
Moment: MonopoleDipoleQuadrupole
Example
distribution: 		spherical charge positive and negative charge near each other two positive and two negative charges arranged alternately around the corners of a square

Since Nuclei have a state of definite parity, a nuclear charge distribution cannot have an electric dipole contribution. Likewise they will not have a magnetic monopole or quadrupole contribution. (Magnetic contributions come from moving or spinning charges).

When a nucleus is in its equilibrium position in a crystal, the net force on it is zero (by definition). Hence, the Monopole Electric moment (which is the charge of the nucleus, giving the force on it due to neighboring charges) will not lead to energy level splittings. On the other hand, deviations from spherical symmetry for both the nucleus and of the electric charges in the environment surrounding the nucleus can provide a torque on the nucleus and hence energy levels which depend on orientation. That is known as an electric quadrupole coupling.

Not all nuclei have an electric quadrupole moment. Due to symmetry considerations, for example, all nuclei which have a nuclear spin quantum number of 0 or 1/2 cannot have such a moment. Within the periodic table such spin quantum numbers are not all that common, but included are the common isotopes of Hydrogen, Oxygen, and Carbon. The most common isotope of Nitrogen, on the other hand, has a nuclear spin of 1 and a non-zero nuclear electric quadrupole moment.

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