In

atomicAtomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. It is primarily concerned with the arrangement of electrons around the nucleus and...

,

molecularMolecular physics is the study of the physical properties of molecules, the chemical bonds between atoms as well as the molecular dynamics. Its most important experimental techniques are the various types of spectroscopy...

, and

solid-state physicsSolid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the large-scale properties of solid materials result from...

, the

**electric field gradient** (

**EFG**) measures the rate of change of the

electric fieldIn physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

at an

atomic nucleusThe nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...

generated by the

electronThe electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

ic charge distribution and the other nuclei. The EFG couples with the nuclear electric quadrupole moment of quadrupolar nuclei (those with spin quantum number greater than one-half) to generate an effect which can be measured using several spectroscopic methods, such as

nuclear magnetic resonanceNuclear magnetic resonance is a physical phenomenon in which magnetic nuclei in a magnetic field absorb and re-emit electromagnetic radiation...

(NMR),

electron paramagnetic resonanceElectron paramagnetic resonance or electron spin resonance spectroscopyis a technique for studying chemical species that have one or more unpaired electrons, such as organic and inorganic free radicals or inorganic complexes possessing a transition metal ion...

(EPR, ESR),

nuclear quadrupole resonanceNuclear quadrupole resonance spectroscopy or NQR is a chemical analysis technique related to nuclear magnetic resonance .-Principle:In NMR, nuclei with spin ≥ 1/2 have a magnetic dipole moment so that their energies are split by a magnetic field, allowing resonance absorption of energy related to...

(NQR), Mössbauer spectroscopy or perturbed angular correlation (PAC). The EFG is non-zero only if the charges surrounding the nucleus violate cubic symmetry and therefore generate an inhomogeneous electric field at the position of the nucleus.

EFGs are highly sensitive to the

electronic densityIn quantum mechanics, and in particular quantum chemistry, the electronic density is a measure of the probability of an electron occupying an infinitesimal element of space surrounding any given point. It is a scalar quantity depending upon three spatial variables and is typically denoted as either...

in the immediate vicinity of a nucleus. This is because the EFG

operatorIn physics, an operator is a function acting on the space of physical states. As a resultof its application on a physical state, another physical state is obtained, very often along withsome extra relevant information....

scales as

*r*^{−3}, where

*r* is the distance from a nucleus. This sensitivity has been used to study effects on charge distribution resulting from substitution, weak interactions, and charge transfer.

## Definition

A given charge distribution of electrons and nuclei,

*ρ*(

**r**), generates an electrostatic potential

*V*(

**r**). The derivative of this potential is the negative of the

electric fieldIn physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

generated. The first derivatives of the field, or the second derivatives of the potential, is the electric field gradient. The nine components of the EFG are thus defined as the second spatial derivatives of the electrostatic potential, evaluated at the position of a nucleus:

For each nucleus, the components

*V*_{ij} are combined as a symmetric 3 × 3 matrix. Under the assumption that the charge distribution generating the electrostatic potential is external to the nucleus, the matrix is traceless, for in that situation

Laplace's equationIn mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as:where ∆ = ∇² is the Laplace operator and \varphi is a scalar function...

, ∇

^{2}*V*(

**r**) = 0, holds. Relaxing this assumption, a more general form of the EFG tensor which retains the symmetry and traceless character is

where ∇

^{2}*V*(

**r**) is evaluated at a given nucleus.

As

*V* (and

*φ*) is symmetric it can be diagonalized. The principal tensor components are usually denoted

*V*_{zz},

*V*_{yy} and

*V*_{xx} in order of decreasing

modulusModulus may refer to:* Modulus a genus of small sea snails* Modulus , a formal product of places of a number field* The absolute value of a real or complex number...

. Given the traceless character, only two of the principal components are independent. Typically these are described by

*V*_{zz} and the

**asymmetry parameter**,

*η*, defined as