Perturbed angular correlation: Difference between revisions

Revision as of 16:23, 24 November 2019

Revision as of 16:23, 24 November 2019

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</math>

</blockquote>

The publications mostly list <math>\nu_Q</math>. <math>e</math> as [[elementary charge]] and <math>h</math> as [[Planck constant]] are well known or well defined.

The [[quadrupole moment|nuclear quadrupole moment]] <math>Q</math> is often determined only very inaccurately (often only with 2-3 digits).

Because <math>\nu_Q</math> can be determined much more accurately than <math>Q</math>, it is not useful to specify only <math>V_{zz}</math> because of the error propagation.

In addition, <math>\nu_Q</math> is independent of spin! This means that measurements of two different isotopes of the same element can be compared, such as ^199mHg(5/2−), ^197mHg(5/2−) and ^201mHg(9/2−). Further, <math>\nu_Q</math> can be used as finger print method.

For the energy difference then follows:

:<math>\Delta E_Q=\hbar\omega_Q\cdot 3|m^2-m'^2|

</math>

If <math>\eta=0</math>, then:

:<math>\omega_n=n\cdot\omega_0

</math>

with:

:<math>\omega_0=\textrm{min}\left(\frac{\Delta E_Q}{\hbar}\right)

</math>

For integer spins applies:

:<math>\omega_0=3\cdot\omega_Q</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; und &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <math>n=|M^2-M'^2|</math>

For half integer spins applies:

:<math>\omega_0=6\cdot\omega_Q</math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; und &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <math>n=\frac{1}{2}|M^2-M'^2|</math>

The perturbation factor is given by:

:<math>G_{k_1k_2}^{NN}=\sum_ns_{nN}^{k_1k_2}\cos{(n\omega_Q^0t)}

</math>

With the factor for the probabilities of the observed frequencies:

:<math>s_{nN}^{k_1k_2}=\sqrt{(2k_1+1)(2k_2+1)}\cdot\sum_{M,M'}\begin{pmatrix}

I&I&k_1\\

M'&-M&N\\

\end{pmatrix}\begin{pmatrix}

I&I&k_2\\

M'&-M&N\\

\end{pmatrix}

</math>

As far as the magnetic dipole interaction is concerned, the electrical quadrupole interaction also induces a precision of the angular correlation in time and this modulates the quadrupole interaction frequency. This frequency is an overlap of the different transition frequencies <math>\omega_n</math>. The relative amplitudes of the various components depend on the orientation of the electric field gradient relative to the detectors (symmetry axis) and the asymmetry parameter <math>\eta</math>. For a probe with different probe nuclei, one needs a parameter that allows a direct comparison: Therefore, the quadrupole coupling constant <math>\nu_Q</math> independent of the nuclear spin <math>\vec{I}</math> is introduced.

=== General theory ===