### From Wikipedia, the free encyclopedia

In atomic,
molecular, and solid-state
physics, the **electric field gradient**
(**EFG**) measures the rate of change of the electric field at
an atomic
nucleus generated by the electronic 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 resonance
(NMR), electron paramagnetic
resonance (EPR, ESR), nuclear quadrupole
resonance (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 density in the immediate
vicinity of a nucleus. This is because the EFG operator
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 iterations,
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 field 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
equation,
∇^{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 modulus. 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

## References

- Kaufmann, Elton N; Reiner J.
Vianden (1979). "The electric field gradient in noncubic metals".
*Reviews of Modern Physics* **51** (1):
161–214. doi:10.1103/RevModPhys.51.161.