Fitting Czjzek distributions with ssNake

Quadrupolar NMR spectra of glasses and other disordered solids often cannot be described by a single set of quadrupolar parameters. Instead, the local environment varies from one nucleus to another, producing distributions of the quadrupolar coupling constant, $C_Q$, and the asymmetry parameter, $\eta_Q$. In this tutorial, I will show how such spectra can be fitted in ssNake using the Czjzek model.

The tutorial starts from an already processed spectrum. If your data still require Fourier transformation, phasing, or other basic processing, first see my tutorial on NMR data processing with ssNake.

What does a Czjzek fit describe?

A conventional quadrupolar simulation assigns one $C_Q$ and one $\eta_Q$ value to each site. This is often suitable for a well-defined crystallographic environment but is usually too restrictive for a disordered material.

The standard Czjzek model instead describes a statistical distribution of electric-field-gradient tensors around an environment that would be symmetric before disorder is introduced. Its principal adjustable parameter is $\sigma$, which controls the width of the distribution. A larger $\sigma$ generally produces a broader distribution of $C_Q$ values.

The extended Czjzek model adds a non-zero, locally ordered contribution described by $C_Q^0$ and $\eta_Q^0$. It is therefore useful when the underlying site is already asymmetric and structural disorder produces an additional distribution around that environment.

This distinction matters: the standard and extended models are not merely two different mathematical lineshapes. They represent different assumptions about the local structure, and the more complicated model should only be used when it is justified by the data and by chemical knowledge of the sample.

Load and prepare the spectrum

Open the dataset Czjzek_example using File --> Open. Since this example has already been processed in TopSpin, only a few preparatory steps are required.

First, apply a baseline correction using Tools --> Baseline Correction. Exclude the signal-containing regions from the baseline fit and check that the correction does not remove broad spectral intensity. This is particularly important for disordered quadrupolar spectra, because a slowly varying part of the true lineshape can easily be mistaken for baseline curvature.

Next, normalize the spectrum using Matrix --> Normalize. Normalization is not required by the Czjzek model, but it keeps the component integrals at manageable values and makes it easier to compare different trial fits. Finally, set the horizontal axis to ppm in the Axis tab if it is not already displayed in ppm.

Before fitting, note the experimental conditions that determine the simulated lineshape:

• the observed nucleus and its spin quantum number;
• the magnetic field or Larmor frequency;
• whether the experiment was static or recorded under MAS;
• the MAS frequency;
• whether satellite transitions and spinning sidebands must be included.

Most of this information is normally read from the dataset, but it is worth checking before generating a library. An incorrect field or MAS frequency may still produce a visually plausible fit with physically incorrect parameters.

Open the Czjzek fitting panel

Choose Fitting --> Czjzek. The fitting panel appears below the spectrum. As in the other ssNake fitting routines, the left side contains the simulation, fitting, preference, exclusion, and export controls. The right side contains the parameters of the current fit component.

Remember that a checked box fixes the corresponding parameter, whereas an unchecked box allows it to vary during fitting. This is easy to overlook and is one of the most common reasons for an apparently unresponsive fit.

The most important component parameters are:

Generate the quadrupolar library

Czjzek fitting requires an additional step that is not needed for a simple Lorentzian or Gaussian fit. ssNake first calculates a library of quadrupolar powder patterns over a grid of $C_Q$ and $\eta_Q$ values. During simulation and fitting, these patterns are combined with weights defined by the selected Czjzek distribution. Pre-calculating the library makes the iterative fit much faster.

Click Library in the fitting panel. In the library window, check the experimental settings and define the $C_Q$–$\eta_Q$ grid. For an MAS spectrum, choose the finite-MAS option, enter the experimental spinning frequency, and include enough spinning sidebands to cover all sidebands visible in the experimental spectral window. A static spectrum or a spectrum containing only the central transition requires different settings.

There is a trade-off when choosing the grid. A broad and finely spaced grid is more flexible and accurate, but requires more time and memory. A grid that is too narrow truncates the distribution and can bias the fitted value of $\sigma$. As a useful rule of thumb, the maximum $C_Q$ should reach approximately $4\sigma$ for the largest distribution width that the fit is likely to explore. This is a starting criterion rather than a substitute for inspecting the distribution.

Click Show to display the distribution weights on the current $C_Q$–$\eta_Q$ grid. The intensity should decay well before it reaches the upper $C_Q$ boundary. If the contours are cut off at the edge, increase the maximum $C_Q$ and generate the library again.

Once the experimental settings and grid are satisfactory, click Generate. Library generation may take some time. When it is complete, close the library window and return to the fitting panel.

Simulate before fitting

It is tempting to click Fit immediately, but first obtaining a reasonable manual simulation makes the optimization faster and more reliable. Set an approximate Pos, Integral, and $\sigma$, and then click Sim.

Adjust one parameter at a time and simulate again:

• use Pos mainly to align the calculated and experimental spectra;
• use $\sigma$ to adjust the extent and character of the quadrupolar distribution;
• use Integral to match the overall intensity;
• add only as much Lorentzian or Gaussian broadening as is needed to reproduce broadening not already described by the quadrupolar model.

The goal at this stage is not a perfect match. It is to place the optimizer in a physically sensible region of parameter space.

If you substantially increase $\sigma$, return to the Library window and click Show again. The library that was suitable for the starting value may no longer cover the distribution. If necessary, widen the $C_Q$ range and regenerate it before fitting.

Fit with the standard Czjzek model

Decide which parameters should be optimized and uncheck their boxes. In a first round, it is usually sensible to vary only Pos, Integral, and $\sigma$, while keeping less important broadening parameters fixed. Click Fit and inspect the fitted spectrum and residual.

If the fit is stable, the broadening terms can be released in a later round. This staged approach reduces correlations between $\sigma$, Gaussian broadening, and Lorentzian broadening. Allowing all parameters to vary from the beginning may improve the numerical residual while making the physical interpretation less reliable.

Do not judge the result from the RMSD alone. Look for systematic deviations in the central-transition edges, shoulders, and spinning sidebands. A residual with a clear lineshape indicates that the model is missing something, even when its numerical value appears small.

When should the extended Czjzek model be used?

If the standard Czjzek distribution cannot reproduce the characteristic edges or sideband shapes, and there is a structural reason to expect an already asymmetric underlying environment, change Type from the standard to the extended Czjzek model. This activates $C_Q^0$ and $\eta_Q^0$.

Choose physically motivated starting values. Parameters obtained for a related crystalline phase, an ordered analogue, or a less-disordered sample are often more useful than arbitrary guesses. Start by fixing $\eta_Q^0$ if there is not enough information to determine it independently, and release $C_Q^0$ only after a satisfactory manual simulation has been obtained.

Because the extended distribution adds parameters and is computationally more demanding, click Sim after each substantial change. A calculation with $\eta_Q^0 \ne 0$ can be particularly slow.

After changing from the standard to the extended model, $\sigma$ may need to be reduced considerably: part of the width previously assigned to disorder may now be described by the non-zero ordered quadrupolar interaction. Check the distribution in the Library window once more, then click Fit.

Check that the result is meaningful

Before reporting the fitted parameters, I recommend the following checks:

1. Inspect the residual. It should resemble noise rather than a missing spectral component.
2. Recheck the library limits. The fitted distribution must not be truncated at the maximum $C_Q$ value.
3. Repeat the fit from different starting values. Convergence to similar parameters provides more confidence that the result is not a local minimum.
4. Test whether all broadening terms are necessary. Strong correlations between $\sigma$, Gaussian width, and Lorentzian width can make the solution non-unique.
5. Compare with chemical expectations. Values of Pos, $C_Q^0$, $\eta_Q^0$, and $\sigma$ should remain physically plausible.
6. Prefer the simpler model when it is sufficient. An improved residual does not by itself justify the additional parameters of an extended Czjzek distribution or a second site.

The fitted integrals should also be interpreted carefully. They are directly quantitative only if the experimental excitation and detection efficiencies are equivalent for the components being compared. This is not automatically guaranteed for broad quadrupolar spectra.

Export the fit

Use Export/Import to save the fitted parameters. To export the experimental spectrum, individual components, total fit, and residual together, choose Curves to Workspace. In the newly created workspace, set the axis to ppm and use File --> Export --> ASCII or CSV.

For a publication-ready image, use File --> Export --> Figure. I normally show the experimental spectrum, total fit, individual components, and residual, while stating in the caption whether the standard or extended Czjzek model was used.

Summary

The essential steps are:

1. baseline-correct and normalize the processed spectrum;
2. open Fitting --> Czjzek;
3. generate a quadrupolar library using the correct field and MAS settings;
4. verify that the $C_Q$–$\eta_Q$ grid contains the complete distribution;
5. obtain a reasonable manual simulation before fitting;
6. fit the standard Czjzek model first;
7. use the extended model only when the spectrum and structural context justify it;
8. validate the residual, library limits, parameter stability, and physical plausibility before exporting the result.

Further examples are available in the official ssNake tutorial collection. The underlying fitting approach is described in the ssNake software paper.