Perturbed angular correlation: Difference between revisions

Revision as of 16:07, 24 November 2019

Revision as of 16:11, 24 November 2019

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:<math>G_{k_1k_2}^{NN}=e^{\left({-iN\omega_Lt}\right)}

</math>

=== Static electric quadrupole interaction ===

The energy of the hyperfine electrical interaction between the charge distribution of the core and the extranuclear static electric field can be extended 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:

:<math>E_Q=\sum_{ij}Q_{ij}V_{ij}

</math> &nbsp;&nbsp;&nbsp;&nbsp;ij=1;2;3

This can be written as a product of the [[quadrupole moment]] <math>Q_{ij}</math> and the electric field gradient <math>V_{ij}</math>. Both [tensor]s are of second order. Higher orders have too small effect to be measured with PAC.

The electric field gradient is the second derivative of the electric potential <math>\Phi(\vec{r})</math> at the core:

:<math>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}

</math>

<math>V_{ij}</math> becomes diagonalized, that:

:<math>|V_{zz}|\ge|V_{yy}|\ge|V_{xx}|

</math>

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

:<math>V_{xx}+V_{yy}+V_{zz}=0

</math>

Typically, the electric field gradient is defined with the largest proportion <math>V_{zz}</ math> and <math>\eta</math>:

:<math>\eta=\frac{V_{yy}-V_{xx}}{V_{zz}}

</math>,&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<math>0\le\eta\le 1</math>

In cubic crystals, the axis parameters of the unit cell x, y, z are of the same length. Therefore:

:<math>V_{zz}=V_{yy}=V_{xx}</math> and <math>\eta=0</math>

In axisymmetric systems is <math>\eta=0</math> math>.

For axially symmetric electric field gradients, the energy of the substates has the values:

:<math>E_Q=\frac{eQV_{zz}}{4I(2I-1)}\cdot (3m^2-I(I+1))

</math>

The energy difference between two substates, <math>M</math> and <math>M'</math>, is given by:

:<math>\Delta E_Q=E_m-E_{m'}=\frac{eQV_{zz}}{4I(2I-1)}\cdot 3|M^2-M'^2|

</math>

The quadrupole frequency <math>\omega_Q</math> is introduced.

The formulas in the colored frames are important for the evaluation:

<blockquote width="80%;" style="background: #F4FFFF; border: 2px solid #999999; border-right-width: 2px;">

:<math>\omega_Q=\frac{eQV_{zz}}{4I(2I-1)\hbar}=\frac{2\pi eQV_{zz}}{4I(2I-1)h}=\frac{2\pi\nu_Q}{4I(2I-1)}

</math>

</blockquote>

<blockquote width="80%;" style="background: #FFF4FF; border: 2px solid #999999; border-right-width: 2px;">

:<math>\nu_Q=\frac{eQ}{h}V_{zz}=\frac{4I(2I-1)\omega_Q}{2\pi}

</math>

</blockquote>

=== General theory ===