New Empirical Equation for the Atomic Form Factor Function in the Momentum Transfer Range, q = 0–50 Å−1 for the Elements in the Range 1≤ Z ≤30

Abstract

The importance of Atomic Form Factors (f) is well-known to the scientific community. Tabulated values for f are mostly used in calculating cross-sections and Monte Carlo sampling for the coherent scattering of photons. The uses of these values are subjected to different approximations and interpolation techniques because the available data points for f in the literature for specified momentum-transfer-grids are very limited. In order to make it easier to accurately use the tabulated data, a mathematical expression for f functions would be a great achievement. Therefore, the current study was designed to suggest an empirical expression for the f functions. In the results, an empirical equation for Hubbell's tabulated data for f is created in the momentum transfer range, q = 0–50 Å−1 for the elements in the range 1≤ Z ≤30. The number of applied parameters was seven. The fitting to f showed that the maximum deviation was within 3%, 4% and 5% for the element having, Z = 1–11, Z = 12–22 and Z = 23–30, respectively, while the average deviations were within 0.3–2.25% for all elements (i.e., Z = 1–30). The values generated by the analytical equation were used in the Monte Carlo code instead of Hubbell’s tabulated values. The statistical noise in the Probability Distribution Functions of coherently scattered photons was efficiently removed. Furthermore, it also reduced the dependence on different interpolation techniques and approximations, and on the use of large tabulated data for f with the specified elements.