Analytical models representing X-ray form factors of ions

Analytical representations of X-ray form factors for ions are examined based on the inverse Mott–Bethe formula. Twelve sources of form-factor data spanning the period 1961–2023 and representing various ranges and grids in sinθ/λ and different precisions are analysed. Generally, 96.1% of all form factors are exactly reproduced by the analytical models.


Introduction
In a previous paper (Thorkildsen, 2023), hereafter denoted GT-I, the inverse Mott-Bethe formula was successfully applied to model X-ray form-factor data for neutral atoms.

Formulas
The basic expression used to model form-factor data for ions is with c n ¼ fc 1 ; . . .; c n g and expðÀ This model is denoted MB[nG + �]. a 0 is the Bohr radius and s ¼ sin �=�.n, giving the number of Gaussians in the model, is treated as a variable.Z 0 is interpreted as the number of electrons and Z is the atomic number of the charged atom in question.�Z = Z À Z 0 is thus the net ionic charge.The traditional model for form factors, referred to as S[nG + c], is Here n has been treated as a constant.n = 2, . . ., 6 have been reported in the literature.

Method
As in GT-I, the fitting procedure here is performed using the Mathematica function NonlinearModelFit (Wolfram Research, 2023) (ii) Expand: Form-factor data sets for ions normally exhibit a greater span in the number of Gaussians, which appears in the final analytical models, than was found in the work on neutral atoms.Thus the Expand part of the analysis, i.e.MB[nG + �] !MB[(n + 1)G + �], which aims to increase in steps the number of parameters in the model by two, giving a better fit to the original data, has been slightly altered: are appended to the parameters obtained using n Gaussians in the previous step of the refinements (the first step is the Search process).Together they represent the new sets of initial values.Subsequently, refinements are conducted for all (remaining) ions in the set.If the refinement for some ions fails, Expand is repeated, first with d (0) = 2.5 A ˚2 and then, if necessary, with d (0) = 10.0A ˚2.The ratios among the d (0) values, ð1; 1 2 ; 2Þ, are usually kept fixed, but the actual values have been the subject of some trial and error.If no new model MB[(n + 1)G + �] is obtained for a given ion, the model MB[nG + �] is regarded as the final representation.
(iii) General comments: In cases where the ratio hj�f ðnÞ 0 ji s À hj�f ðnþ1Þ 0 ji s hj�f ðnÞ 0 ji s < 0:02; model MB[nG + �] is used as the final one.An improvement of less than 2% in the mean absolute error does not warrant an additional Gaussian in the model (this also manifests itself in increasing parameter uncertainties).The constraint for the final set of d values is updated: Otherwise, conditions to be satisfied by the parameters are as in GT-I.Relative parameter uncertainties are always assessed as part of the final verification of the models.In a very few cases this may result in choosing models from a previous step, having one less Gaussian, as the definitive ones.
OFFVi is used for properties common to OFFV1i and OFFV2i.The data set for neutral atoms provided by Volkov and described in GT-I is here denoted by OFFV2.
Complete lists of the species incorporated in the data sets are given in the supporting information.The actual number of species is summarized in Table 1.
The analytical setup for each data set is comprised of model functions MB[nG + �] of equation ( 1).The number of Gaussians involved in the final models is listed in Table 2. n spans the interval n 2 [2, 18].Precisions, number of sampling grid points and number of form factors in the various sets are included in Table 3.A precision of 1 � 10 À 5 is assessed as a convenient practical limit and 1 � 10 À 7 as the lower limit for retaining numerical accuracy throughout the analysis (this only affects MCi and OFFV2i).Sampling grids are summarized below.

Results
The parameters of the final models for all data sets are presented in the supporting information.
The quality of the analytical modelling is evaluated in three different ways.(i) When the original data have a common precision, statistical measures are calculated (Table 4).In all cases the differences between the original data points and the model calculations are as expected.The rounding of formfactor values to the actual data precision may be regarded as a stochastic process described by a uniform statistical distribution.(ii) Form factors are calculated at the actual s grids based on the refined models and rounded to the same precision as the original data.The differences in the last significant digit are then compared.The results are presented in Table 5.We see that 96.1% of all modelled form factors exactly reproduce the underlying data.(iii) The distributions of errors {�f 0 = f 0 (data) À f 0 (model)} [presented as histograms in Fig. 1 for four different data compilations, together with the corresponding graphical presentations of �f 0 (s) for the same cases as shown in Fig. 2] also verify that the accuracy of the modelling is determined by the precision (and inherent rounding) of the original data.
An interesting feature is revealed in Fig. 3. Generally, for a given atomic number fewer Gaussians are needed in the modelling when �Z = Z À Z 0 becomes more positive, i.e. for cations with an increasing net charge.
In Fig. 4 the parameters c n and d n for n = 1, . . ., 6 are depicted for ions and neutral atoms based on OFFV2i and OFFV2 data, both rounded to a precision of 1 � 10 À 5 .The ions are grouped according to their atomic number and, in the case of multiple occurrences, lines spanning the parameter values are used for plot markers.One readily observes the resemblance between this pair of figures.The parameters are organized according to increasing values of d, i.e. d n < d n+1 , and the values presented have the largest impact on the high-s value form factors, for which only small differences are expected between the neutral atoms and their associated ions.

Figure 2
Examples of the variation of hj�f 0 ðs; Z 0 ; ZÞji s for the cases shown in Fig. 1.
tudes typically increase and additional Gaussians, which appear in the models when �Z decreases, involve large d values and thus only influence form factors when evaluated for small s values.

Discussion
A few points are worth highlighting.
ITiiii: The compilation by Ibers (1962), which is documented in great detail, is built of contributions from several other authors.The presentation is, however, associated with a specific s grid, not always comprising the grids in the original reports.This is, among others, the case for the Watson & Freeman (1961) form-factor data (denoted SX-67 by Ibers).Slightly different parameter values are obtained, e.g. for the ions of nickel, based on the Ibers presentation compared with the one by Watson & Freeman (1961).In another case, ion S 2À (denoted AX-46) form-factor data are rounded from the original source (Tomiie & Stam, 1958) to fit the chosen s grid.An analysis of the original data set resulted in a slightly better fit than found in the ITiiii analysis.Generally, interpolated data sets give rise to larger residuals following the model refinements.The present fits to the inverse Mott-Bethe formula for C val , Zr 4+ and Hg 2+ are, for some reason, of poorer quality than the fits for the other ions.
ITCi: The form-factor data of Maslen et al. (1992) are, with the exception of those for H À , a copy of those first presented by Cromer & Waber (1974).This set is an original calculation not published elsewhere [i.e.not linked with the form factors of Cromer & Waber (1964)].Identical parameter sets based on the traditional model of equation ( 3), S[4G + c], are provided in both these editions of International Tables (despite the change in the data for H À ).It has further become evident that the published parameters for Ru 4+ and Bi 5+ are in error, leading to e.g.absolute deviations of, respectively, 3.0 and 16.2 for s = 2.0 A ˚À 1 .Excluding these ions from a statistical analysis based on the traditional model adopted in International Tables leads to hj�f 0 ðs; Z 0 ; ZÞji s;Z 0 ;Z = 2.78 � 10 À 3 and hj�f 0 ðs; Z 0 ; ZÞji r:m:s:js;Z 0 ;Z = 5.17 � 10 À 3 , one order of magnitude larger than the values obtained in the present MB modelling.In this analysis the form factors of Tl 3+ exhibit the most prominent deviations, �f 0 2 [0.002-0.003],occurring for s 2 [0.01, 0.06] A ˚À 1 .Waasmaier & Kirfel (1995) analysed the data of Maslen et al. (1992) in model S[5G + c].They extended the data to s max = 6.00A ˚À 1 by using data for neutral atoms for s > 2.00 A ˚À 1 (or s > 1.50 A ˚À 1 ), 'because scattering from valence electrons can be neglected' (Waasmaier & Kirfel, 1995).By applying their 11-parameter models for the restricted ranges actually published, one observes statistical measures a factor of two worse than found using nine-parameter models (Cromer & Waber, 1974).Altogether, it seems that a general

Table 5
Absolute deviations from the original form-factor values using the model calculations amount to 0 (no deviation), 1, or 2 and 3 in the last significant figure of the original data.
The incidences for all compilations are given as percentages.For ITiiii, species 10, 78 and 82 (C val , Zr 4+ and Hg 2+ , respectively) are omitted from the calculation.See also Section 6.

Table 4
Statistical properties for compilations having a fixed precision.
MCi: The analysis reveals oscillations in �f 0 (s) for approximately s � 5 A ˚À 1 .These are most prominent for the valence states C val and Si val and all anions.Oscillations are also observed for most of the cations (occurring for s � 1-2 A ˚À 1 ), but in these cases the amplitudes are smaller by at least one order of magnitude.The oscillations disappear when the original data are rounded to a precision of 1 � 10 À 4 .This is depicted for Cl À in Fig. 5 with the corresponding OFFV2i analysis as a reference.
General: Fig. 6 shows the differences in form-factor values of various ions of oxygen and oxygen itself, e.g.f 0 (s|O 2+ ) À f 0 (s|O), for s � 2.0 A ˚À 1 .The data are the sets provided by Volkov (2023) rounded to a precision of 1 � 10 À 5 .The differences observed are roughly one to three orders of magnitude larger than the data precision.Thus, substitution of neutral-atom form-factor data when high-s value data are lacking for associated ions [as in Waasmaier & Kirfel (1995)] should be avoided.Fig. 7 shows the results of a detailed analysis for the ion O 2+ .

Concluding remarks
The modelling of form-factor data of neutral atoms accounted for in GT-I is also appropriate for ions.It gives improved analytical models compared with the traditional ones existing in the literature.The new models are easily implemented and can be applied in all cases where e.g.scattering factors are to be calculated.They are generally very accurate and flexible in such a way that original form-factor calculations, with  different physical features incorporated (Schmidt & Weiss, 1979), are consistently reproduced.

Figure 3
Figure 3 Examples of the variation of j�f 0 ðs; Z 0 ; ZÞj max with the number of Gaussians in the analytical model, based on the original OFFV2i data.(a) Ions of oxygen, including the neutral atom.(b) Fastest and slowest development.For the neutral oxygen atom, data from Olukayode et al. (2023a) are used.

Figure 4 (
Figure 4 (a) and (c) Parameters d n and c n associated with OFFV2i; n = 1, . . ., 6.(b) and (d) Parameters d n and c n associated with neutral atoms included for comparison.In the last case, the parameters emerge from modelling of the extended data set provided by Volkov (cf.GT-I), rounded to a precision of 1 � 10 À 5 .

Figure 6
Figure 6Deviation in form factors between selected ions of oxygen and neutral oxygen for s � 2.0 A ˚À 1 .The figure is based on the inverse Mott-Bethe modelling of the extended data sets byVolkov (2023).
. All observations are associated with unit weights.The analysis leading to the final values of the parameters of the model, {�, c 1 , . .., c n , d 1 , . .., d n }, is a slightly changed version of the one reported in GT-I.This has affected primarily the Search and Expand modules.Repair has become obsolete.(i)Search: The Search module represents the initial part of the procedure and is usually performed only once involving a small number of Gaussians.The random-number generator RandomReal returns initial values for the d parameters (in units of A ˚2), here shown for the default case of three Gaussians: d ðiÞ 1 ¼ RandomReal ½f0:025; 0:250g�; d ðiÞ 2 ¼ RandomReal ½f0:25; 2:50g�; d ðiÞ 3 ¼ RandomReal ½f2:50; 10:00g�: The value 1.0 A ˚is associated with � (i) and c ðiÞ k ; k ¼ 1; . . .; 3. Refinements are then conducted to obtain parameter sets for model MB[3G + �] for all ions in the data set.

Table 2
Number of species with a parameter set involving nG Gaussians.

Table 3
Basic information related to the compilations.

Table 6
Parameters associated with oxygen and associated ions.� and c i are in A ˚, d i in A ˚2. Actual data sets are OFFV2 and OFFV2i.Final models are MB[13G + �] for O 2+ and O 1+ , MB[14G + �] for O and MB[16G + �] for O 1À .