Electric field gradient

The electric field gradient is a physical measurand that describes the change in the field at the location of an atomic nucleus . It is measured in nuclear solid-state physics to characterize materials via the hyperfine interaction and is essential for understanding the local structure. It can be calculated theoretically with the density functional theory .

definition

The energy of the electrical hyperfine interaction between the charge distribution of the nucleus and the extranuclear static electric field can be expanded to multipoles. The monopole term only causes an energy shift and the dipole term disappears, so that the first relevant expansion term is the quadrupole term:

{\ displaystyle E_ {Q} = \ sum _ {ij} Q_ {ij} V_ {ij}}

ij = 1; 2; 3

This can be written as the product of the quadrupole moment and the electric field gradient . Both tensors are of second order. Higher orders have too little effect to be measured with PAC . ${\ displaystyle Q_ {ij}}$ ${\ displaystyle V_ {ij}}$

The electric field gradient is the second derivative of the electric potential at the core: ${\ displaystyle \ Phi ({\ vec {r}})}$

{\ displaystyle V_ {ij} = {\ frac {\ partial ^ {2} \ Phi ({\ vec {r}})} {\ partial x_ {i} \ partial x_ {j}}} = {\ begin { pmatrix} V_ {xx} & 0 & 0 \\ 0 & V_ {yy} & 0 \\ 0 & 0 & V_ {zz} \\\ end {pmatrix}}}

${\ displaystyle V_ {ij}}$ is diagonalized so that:

{\ displaystyle | V_ {zz} | \ geq | V_ {yy} | \ geq | V_ {xx} |}

The matrix is free of traces in the main axis system ( Laplace equation ):

{\ displaystyle V_ {xx} + V_ {yy} + V_ {zz} = 0}

Usually the electric field gradient is defined with the largest share and : ${\ displaystyle V_ {zz}}$ ${\ displaystyle \ eta}$

{\ displaystyle \ eta = {\ frac {V_ {yy} -V_ {xx}} {V_ {zz}}}}

,

{\ displaystyle 0 \ leq \ eta \ leq 1}

Measurement methods

The electric field gradient can be measured directly using methods from nuclear solid state physics: Mössbauer , PAC, NMR and muon spin spectroscopy .

Individual evidence

↑ Günter Schatz, Alois Weidinger, Manfred Deicher: Nuclear solid-state physics: Nuclear physical measurement methods and their applications. 4th edition. Vieweg + Teubner Verlag, 2010, ISBN 978-3-8351-0228-6 .

[1] Günter Schatz, Alois Weidinger, Manfred Deicher: Nuclear solid-state physics: Nuclear physical measurement methods and their applications. 4th edition. Vieweg + Teubner Verlag, 2010, ISBN 978-3-8351-0228-6 .