Approximate resolution convolution function for fitting a dispersion gap measured on a triple-axis spectrometer

TL;DR

The paper tackles the problem of accurately determining dispersion gaps from TAS data, where instrumental resolution creates a high-energy tail that biases simple fits. It introduces an analytic resolution-convoluted gap function that assumes a pancake-like $Q$-resolution with two broad directions and one narrow direction and a parabolic dispersion near the gap, applicable to both linear and quadratic gaps under moderate resolution. The method is validated on MnF$_2$ via simulated McStas TAS data and real CAMEA TAS-like data, showing significantly improved gap estimates and robust convergence compared with Gaussian or Gaussian+Voigt tail models. The approach provides a practical, easily implementable tool for TAS data analysis with potential extensions to double gaps and continuum scattering studies when the resolution is well characterized.

Abstract

We present an analytic convoluted-gap function, eq. 11 in the manuscript, for fitting dispersion gaps measured on triple-axis spectrometers (TAS). At the gap, the instrumental resolution skews the signal, producing a high-energy tail that complicates fitting. Our function assumes an instrumental $Q$-resolution with two equal wide and one narrow direction (typical of focused TAS instruments), and a parabolic dispersion at the gap, which is exact for quadratic and accurate for linear dispersions if the resolution is moderate. We demonstrate, that our function outperforms previous methods of fitting a gap, by giving a better fit and more accurate gap determination, seen in figure 4. Here, the anti-ferromagnetically gapped material; MnF$_2$ is simulated in a double-focusing TAS instrument. We also tested our function on experimental data on MnF$_2$ from a TAS-like instrument, where we reproduce the gap size from previous accurate experimentally determined measurements. The function is simple to implement, converges reliably, and we recommend its use for future gap fitting on TAS data.

Figures (13)

• Figure 1: (Left) The linear dispersion in eq. \ref{['eq:AFM_dispersion']}, energy transfer as a function of Q (black line). The instrumental resolution at the gapped Q-position is drawn as coloured ellipsis at varying energy transfers. (Right) The intensity in a constant Q-cut at the gapped position, illustrating how the high-energy tail above the spin gap appears from the instrumental resolution.
• Figure 2: The 2-dimentional part of the $Q$-resolution ellipsoid for a triple-axis-spectrometer is indicated in red. The scattering vector $\textbf{Q}=\textbf{k}_f -\textbf{k}_i$, with the size of the analyzer, determines the 3D $Q$-resolution, which can be split into 3 components; two $dQ$ principal axis in the scattering plane; one narrow and one wide, and $dQ_{vert}$, which is the resolution out of the scattering plane (wide, and not drawn). The instrument also has an energy resolution, which needs to be taken into account.
• Figure 3: The intrinsic instrumental energy resolution at a specific Q is shown in orange and the projected effective resolution in green. For a broad dispersion, one should use the projected resolution (green), while for a narrow dispersion, the energy resolution at a specific Q is more correct. For our approximation, we do not see the correlations between Q and E as in the real case (grey ellipsis), so we assume an ellipsis without a tilt and with the energy resolution depending on the steepness of the dispersion. The black ellipsis indicates the case of a narrow dispersion (orange).
• Figure 4: a) The simulated data is plotted in black errorbars and is normalised to unity. The fitted peak positions are plotted at zero in coloured squares. The simulated MnF$_2$ data with a Gaussian (yellow line), a combined Gaussian, Voigt function (green line) and with our approximate convoluted-gap function (red line). The pink line shows a fit to the convoluted gap function with a fixed $\sigma_E$. The fitted gap positions are the coloured squares at zero and the black line shows the nominal gap size at (100), $\Delta_T=1.062$ meV. The residuals (data minus fit) are plotted below. The same colour coding is used in b) and c), showing the obtained gap values and $\chi^2$ values as a function of slit settings, respectively.
• Figure 5: Fitted $Q$-resolution and energy resolution from our approximate convoluted gap function eq. \ref{['eq:analytical']}. The red line is where only $a$ is fixed in the fit, where in pink is both $a$ and $\sigma_E$ fixed. The triangles, in both figures, indicate the respective calculated resolutions from Table \ref{['tab:resolution_McStas']}. The black crosses in a) are the three calculated $Q$-resolutions averaged. All resolutions are FWHM.