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Tutorial on properties at the nuclei

Observables near the atomic nuclei.

The purpose of this tutorial is to show how to compute several observables of interest in Mössbauer, NMR, and NQR spectroscopy, namely:

• the electric field gradient,
• the isomer shift

This tutorial should take about 1 hour.

Electric field gradient

Various spectroscopies, including nuclear magnetic resonance and nuclear quadrupole resonance (NMR and NQR), as well as Mössbauer spectroscopy, show spectral features arising from the electric field gradient at the nuclear sites. Note that the electric field gradient (EFG) considered here arises from the distribution of charge within the solid, not due to any external electric fields.

The way that the EFG is observed in spectroscopic experiments is through its coupling to the nuclear electric quadrupole moment. The physics of this coupling is described in various texts, for example [Slichter1978]. Abinit computes the field gradient at each site, and then reports the gradient and its coupling based on input values of the nuclear quadrupole moments.

The electric field and its gradient at each nuclear site arises from the distribution of charge, both electronic and ionic, in the solid. The gradient especially is quite sensitive to the details of the distribution at short range, and so it is necessary to use the PAW formalism to compute the gradient accurately. For charge density $n({\mathbf r})$, the potential $V$ is given by

$$V({\mathbf r})=\int \frac{n({\mathbf r'})}{ |\mathbf{r}-\mathbf{r'}| } d{\mathbf r'}$$

and the electric field gradient is

$$V_{ij} = -\frac{\partial^2}{\partial x_i\partial x_j}V({\mathbf r}).$$

The gradient is computed at each nuclear site, for each source of charge arising from the PAW decomposition (see the tutorial PAW1 ). This is done in the code as follows:

• Valence space described by planewaves: expression for gradient is Fourier-transformed at each nuclear site.
• Ion cores: gradient is computed by an Ewald sum method
• On-site PAW contributions: moments of densities are integrated in real space around each atom, weighted by the gradient operator

The code reports each contribution separately if requested.

The electric field gradient computation is performed at the end of a ground-state calculation, and takes almost no additional time. The tutorial file is for stishovite, a polymorph of SiO$_2$. In addition to typical ground state variables, only two additional variables are added:

prtefg  2
quadmom 0.0 -0.02558

View tests/tutorial/Input/tnuc_1.in

The first variable instructs Abinit to compute and print the electric field gradient, and the second gives the quadrupole moments of the nuclei, in barns, one for each type of atom. A standard source for quadrupole moments is [Pyykko2008]. Here we are considering silicon and oxygen, and in particular Si-29, which has zero quadrupole moment, and O-17, the only stable isotope of oxygen with a non-zero quadrupole moment.

After running the file tnuc_1.in through Abinit, you can find the following near the end of the output file:

 Electric Field Gradient Calculation

 Atom   1, typat   1: Cq =      0.000000 MHz     eta =      0.000000

      efg eigval :     -0.165960
-         eigvec :     -0.000001    -0.000001    -1.000000
      efg eigval :     -0.042510
-         eigvec :      0.707107    -0.707107     0.000000
      efg eigval :      0.208470
-         eigvec :      0.707107     0.707107    -0.000002

      total efg :      0.082980     0.125490    -0.000000
      total efg :      0.125490     0.082980    -0.000000
      total efg :     -0.000000    -0.000000    -0.165960

This fragment gives the gradient at the first atom, which was silicon. Note that the gradient is not zero, but the coupling is—that’s because the quadrupole moment of Si-29 is zero, so although there’s a gradient there’s nothing in the nucleus for it to couple to.

Atom 2 is an oxygen atom, and its entry in the output is:

 Atom   2, typat   2: Cq =      6.603688 MHz     eta =      0.140953

      efg eigval :     -1.098710
-         eigvec :     -0.707107     0.707107     0.000000
      efg eigval :      0.471922
-         eigvec :     -0.000270    -0.000270     1.000000
      efg eigval :      0.626789
-         eigvec :      0.707107     0.707107     0.000382

      total efg :     -0.235961     0.862750     0.000042
      total efg :      0.862750    -0.235961     0.000042
      total efg :      0.000042     0.000042     0.471922


      efg_el :     -0.044260    -0.065290     0.000042
      efg_el :     -0.065290    -0.044260     0.000042
      efg_el :      0.000042     0.000042     0.088520

      efg_ion :     -0.017255     0.306132    -0.000000
      efg_ion :      0.306132    -0.017255    -0.000000
      efg_ion :     -0.000000    -0.000000     0.034509

      efg_paw :     -0.174446     0.621908     0.000000
      efg_paw :      0.621908    -0.174446     0.000000
      efg_paw :      0.000000     0.000000     0.348892

Now we see the electric field gradient coupling, in frequency units, along with the asymmetry of the coupling tensor, and, finally, the three contributions to the total. Note that the valence part, efg_el, is quite small, while the ionic part and the on-site PAW part are larger. In fact, the PAW part is largest – this is why these calculations give very poor results with norm-conserving pseudopotentials, and need the full accuracy of PAW. Experimentally, the nuclear quadrupole coupling for O-17 in stishovite is reported as $6.5\pm 0.1$ MHz, with asymmetry $0.125\pm 0.05$ [Xianyuxue1994].

Fermi contact interaction

The Fermi contact interaction arises from overlap of the electronic wavefunctions with the atomic nucleus, and is an observable for example in Mössbauer spectroscopy [Greenwood1971]. In Mössbauer spectra, the isomer shift $\delta$ is expressed in velocity units as

$$\begin{equation} \label{eq:mossbauershift} \delta = \frac{c}{E_\gamma}\frac{2\pi Z e^2}{3}(|\Psi(R)_A|^2-|\Psi(R)_S|^2)\Delta\langle r^2\rangle , \end{equation}$$

where $\Psi(R)$ is the electronic wavefunction at nuclear site $R$, for the absorber (A) and source (S); $c$ is the speed of light, $E_\gamma$ is the nuclear transition energy, and $Z$ the atomic number; and $\Delta\langle r^2\rangle$ the change in the nuclear size squared. All these quantities are assumed known in the Mössbauer spectrum of interest, except $|\Psi(R)|^2$, the Fermi contact term.

Abinit computes the Fermi contact term in the PAW formalism by using as observable $\delta(R)$, that is, the Dirac delta function at the nuclear site [Zwanziger2009]. Like the electric field gradient computation, the Fermi contact calculation is performed at the end of a ground- state calculation, and takes almost no time. There is a tutorial file for SnO$_2$, which, like stishovite studied above, has the rutile structure. In addition to typical ground state variables, only one additional variable is needed:

prtfc  1

View tests/tutorial/Input/tnuc_2.in

After running this file, inspect the output and look for the phrase “Fermi-contact Term Calculation”. There you’ll find the FC output for each atom; in this case, the Sn atoms, typat 1, yield a contact term of 72.2969 atomic units (charge per volume $e/a^3_0$).

To interpret Mössbauer spectra you need really both a source and an absorber; in the tutorial we provide also a file for $\alpha$-Sn (grey tin, which is non-metallic).

View tests/tutorial/Input/tnuc_3.in

If you run this file, you should find a contact term of 102.3008.

To check your results, you can use experimental data for the isomer shift $\delta$ for known compounds to compute $\Delta\langle r^2\rangle$ in Eq.\ref{eq:mossbauershift} (see [Zwanziger2009]). Using our results above together with standard tin Mössbauer parameters of $E_\gamma = 23.875$ keV and an experimental shift of 2.2 mm/sec for $\alpha$-Sn relative to SnO$_2$, we find $\Delta\langle r^2\rangle = 5.74\times 10^{-3}\mathrm{fm}^2$, in decent agreement with other calculations of 6–7$\times 10^{-3}\mathrm{fm}^2$ [Svane1987], [Svane1997].