Electric field gradient

Mathematically, the electric field gradient (EFG) is the hessian matrix (the matrix of the second derivatives) of the electrical potential V: :$V\_\{ij\}\; =\; frac\{partial^2\; V\}\{partial\; x\_i\; partial\; x\_j\}$It is an important structural property of a crystalline solid, where it is defined at the location of a nucleus. 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. The individual components V_ij form a symmetrical and traceless tensor. The principal tensor components are usually denoted V_zz, V_yy and V_xx in order of decreasing modulus. Given the tensor's traceless character,

:$qquad\; V\_\{zz\}\; +\; V\_\{yy\}\; +\; V\_\{xx\}\; =\; 0$

holds, which allows for a description of the EFG using only two parameters, V_zz and the "asymmetry" η

:$qquad\; eta\; =\; frac\{V\_\{xx\}\; -\; V\_\{yy\{V\_\{zz$

Any quadrupolar moment of the nucleus interacts with the inhomogeneous field surrounding it, thus the electric field gradient can be measured using several spectroscopic methods, such as nuclear magnetic resonance (NMR), electron paramagnetic resonance (EPR, ESR), nuclear quadrupole resonance (NQR), moessbauer or perturbed angular correlation (PAC), provided the nucleus in question has a quadrupolar moment.